Finite-Dimensional Approximations of Operators in …yigordon/papers/AGKh.pdfthe constructed approximations, the methods of nonstandard analysis are used. Mathematics Subject Classiﬁcations

Acta Applicandae Mathematicae64: 33–73, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

33

Finite-Dimensional Approximations of Operators inthe Hilbert Spaces of Functions on LocallyCompact Abelian Groups

S. ALBEVERIO1, E. I. GORDON2? and A. YU. KHRENNIKOV3

1Rheinische Friedrich-Wilhelms-Universitaet Bonn, Institut fuer Angewandte Mathematik Abtlg.Stochastik, Wegelerstrasse 6, 53115 Bonn, Germany. e-mail: [email protected] Mathematics, University of Illinois, 1409 West Green Street, Urbana, IL 61801 U.S.A.e-mail: [email protected] of Mathematics, Statistics and Computer Science, University of Väjö,S 35 195 Sweden

(Received: 6 October 1998; in final form: 5 June 2000)

Abstract. A new approach to the approximation of operators in the Hilbert space of functionson a locally compact Abelian (LCA) group is developed. This approach is based on sampling thesymbols of such operators. To choose the points for sampling, we use the approximations of LCAgroups by finite groups, which were introduced and investigated by Gordon. In the case of the groupRn, the constructed approximations include the finite-dimensional approximations of the coordinateand linear momentum operators, suggested by Schwinger. The finite-dimensional approximationsof the Schrödinger operator based on Schwinger’s approximations were considered by Digernes,Varadarajan, and Varadhan inRev. Math. Phys.6 (4) (1994), 621–648 where the convergence ofeigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operatorwas proved in the case of a positive potential increasing at infinity. Here this result is extendedto the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. Weconsider the approximations of p-adic Schrödinger operators as an example. For the investigation ofthe constructed approximations, the methods of nonstandard analysis are used.

Mathematics Subject Classifications (2000):26E35, 30G06, 22B99, 35Q40, 81C05.

Key words: finite-dimensional approximations, locally compact Abelian groups, Schrödinger oper-ators, p-adic numbers, nonstandard analysis.

Introduction

In the paper [Sch], an approach to the approximation of infinite-dimensional quan-tum systems by finite-dimensional systems was outlined. This approach can bedescribed in the following way. LetN = 2M+1 ∈ N. Consider anN-dimensionalspaceEN , which consists of the vectorsx = 〈x−M, . . . , x0, . . . , xM〉 and two op-erators in it – the operatorUN of shift modN : (UNx)n = xn+1, where+ is the

? Supported by Russian Foundations for Basic Research, project No. 98-01-00790.

34 S. ALBEVERIO ET AL.

addition modN ; and the operatorVN of multiplication by the character:(VNx)n =exp[2πin/N] · xn.

It is well known that these two operators are unitary equivalent:UN =FNVNF

−1N , where(FN)nk = 1/

√N exp[2πink/N] is the matrix of finite Fourier

transform, and satisfy the following commutative relation

UNVN = exp2πi

NVNUN.

For |m|, |n| 6 M, this relation implies the following:

UmNV

nN = exp

2πimn

NV nNU

mN . (0.1)

SinceUN andVN are unitary operators, they can be represented in the formUN = eiεPN , VN = eiεQN , wherePN andQN are Hermitian operators also inter-laced by the finite Fourier transformPN = FNQNF

−1N , ε > 0. The spectrum of

PN andQN is {kε | k = −M, . . . ,0, . . . ,M}. So, whenN → ∞, ε → 0, thisspectrum becomes dense inR. If we choose

ε =√

2π

N, (0.2)

thenPN → P andQN → Q in the sense of Definition 1 below, whereP andQ are the momentum and coordinate operators inL2(R), respectively. Ifnε → u

andmε → v, thenUN → eiuP , VN → eivQ and the commutation relations (0.2)converge to the Weyl commutation relations

eiuPeivQ = exp(iuv)eivQeiuP .

The operators

X(n,m) = N− 12 exp

(−πimn

N

)V mN U

nN, |n|, |m| 6 M

form an orthonormal basis in the algebra of linear operatorsL(EN) with respectto the scalar product induced by the trace. The idea of approximation of operatorsin L2(R), which is contained implicitly in [Sch], is to approximate an operator Awith the Weyl symbolf (p, q) by operators of the form

1

N

M∑n,m=−M

fN(n,m)X(m, n), (0.3)

wherefN = FN ⊗ FN(fN) andfN(m, n) = f (mε, nε).There are no proofs or even precise formulations concerning this approximation

in [Sch]. The convergence ofPN andQN to P andQ respectively and of the com-mutation relations (0.2) to Weyl commutation relations was rigorously investigatedby the second author [Gor] by means of nonstandard analysis.

FINITE-DIMENSIONAL APPROXIMATIONS OF OPERATORS 35

The finite-dimensional approximations of the Schrödinger operator inL2(Rn)

were considered in [DVV]. The approach to approximation used there is the fol-lowing. The space of functions on a finite mesh with the width equal toε in somebounded domain ofRn is used instead ofL2(Rn). The Laplacian1 is approxi-mated by an appropriate finite differences operator1d with some natural boundaryconditions (e.g. periodic ones) and the potentialV – by its ‘table’Vd . Thus, theSchrödinger operatorH = −1 + V is approximated by the finite-dimensionaloperatorsHd = −1d + Vd . The authors considered only positive potentialV (x),going to infinity, whenx →∞. So the HamiltonianH has a pure discrete spectrumwith finite multiplicities. It was shown in [DVV] that in this case, the eigenvectors(eigenvalues) ofHd converge to the ‘tables’ of eigenfunctions (eigenvalues) of theoriginal Hamiltonian whenε→ 0 and the sizes of the domain increases toRn.

According to Schwinger’s approach, described above, we have to consider themeshes with the widthε and the number of pointsN connected by the equality (0.2)and approximateH by the operatorsHN = P 2

N +V (QN). This approximation wasalso considered in [DVV] and the same result about convergence of eigenvectorsand eigenfunctions was also proved there. In [Var], the question about a similarfinite-dimensional approximation of p-adic and adelic quantum dynamical systemswas mentioned. Using the theory of approximations of locally compact Abelian(LCA) groups by finite groups, which was developed by the second author (a de-tailed account of this theory can be found in the monograph [Gor]), it is possible toconstruct analogous approximations for the operators in the Hilbert space of func-tions on an arbitrary LCA group. The idea of our approximation is the following.LetG be an LCA group,G – its dual group. Consider the Haar measureµ onGand the Haar measureµ on G, dual toµ. This means that the Fourier transformFG: L2(µ)→ L2(µ), defined by the formula

FG(ϕ)(χ) =∫G

ϕ(g)χ(g)dµ(g), (0.4)

preserves the inner product.For ξ ∈ G,h ∈ G, define the unitary operatorX(ξ, h) = VhUξ in the space

L2(G). HereUξ is the operator of shift byξ andVh is the operator of multiplicationby the characterh. Note that, for the case ofG = R, the operatorsX(ξ, h) differfrom those in (0.3) by a constant factor. This difference will be discussed below.Note that for a finite Abelian groupG, the family {X(g, χ) | g ∈ G,χ ∈ G},as before, forms a basis in the algebra of linear operators in the spaceC(G) offunctions onG.

By the theory of approximations of LCA groups mentioned above, one canconstruct a sequence of triples〈Gn, jn, jn〉, whereGn is a finite Abelian group,jn: Gn → G, jn: Gn → G – injective maps, which may not be homomorphisms,but which are close to homomorphisms in some natural sense (see Definition 3).Moreover, the union of images of the groupsGn (Gn) is dense inG (G) and ifjn(gn)→ ξ ∈ G, jn(χn)→ h ∈ G, thenχn(gn)→ h(g).


For any good enough functionf : G × G → C, one can define the linearoperator (may be unbounded)Af : L2(G)→ L2(G) (the operator with the symbolf ) by the formula

Af =∫∫

G×Gf (h, ξ)VhUξ dµ(h)dµ(ξ), (0.5)

wheref = FG ⊗ FG(f ).Now the approximating operatorsA(n)f : C(Gn) → C(Gn) for Af are defined

by the formula

A(n)f =

1

|Gn|∑

g∈Gn,χ∈Gnfn(χ, g)X(g, χ), (0.6)

where

fn = FGn ⊗ FGn(fn) and fn(g, χ) = f (jn(g), jn(χ)).

For the case whereG = R, the dual groupG can be identified withR. ThenGN = {−M, . . . ,0, . . . ,M} is the additive group modN and its dual groupGN isisomorphic toGN . So these groups can be identified too. Now to get an approxi-mation ofR, we have to define the mapsjn, jn: Gn→ R. These maps are definedby the formulajn(k) = jn(k) = kε, wherek ∈ GN andε satisfies (0.2).

Note that, in the case ofG = R, the symbol defined by formula (0.5) is not theWeyl (symmetrical) symbol of the operator – it is the so-calledqp-symbol. So theapproximating operator, which is defined by formula (0.6) with the approximationof the groupR described above, in general is not the same as (0.3). The case ofsymmetric symbols will be discussed at the end of Section 3. Theqp-symbols foroperators in the spacesL2(Qn

p) were introduced by V. S. Vladimirov [VVZ]. Themain considerations of this paper are related to the operators on an LCA groupG

with symbols of the form

f (x, χ) = a(x) + b(χ). (0.7)

It is easy to see that the Weyl symbols of such operators are equal to theirqp-symbols. If in (0.7)a(x) > 0 anda(x)→∞, whenx →∞, then the operatorAfis called an operator of the Schrödinger type.

If the symbol (0.7) of a Schrödinger-type operator is such thatb(χ) is boundedfrom below andb(χ)→∞, whenχ →∞, then this operator has a pure discretespectrum with finite multiplicities. For these operators, we prove here a theorem,analogous to the one proved in [DVV] for the Schrödinger operator itself which wehave mentioned above. In fact, we prove that the eigenvalues of the approximat-ing operatorsA(n)f (0.6) converge to the eigenvalues ofAf and their eigenvectorsconverge to the tables of its eigenfunctions.

It is well known that every LCA group is a direct product ofRn for somen > 0and of a group with a compact open subgroup. Each of these two cases needs to


be considered separately. The case ofRn can be investigated by the methods of thepaper [DVV]. We shall consider it in another paper. Here the case of groups with acompact open subgroup is considered. We consider the fieldQp of p-adic numbersas an example.

One of the difficulties of the investigation of convergence here is that the op-eratorsA(n)f act in different spacesC(Gn) which are also not connected with eachother in any natural way. These spaces are not closely connected withL2(G). Wecan only construct a linear mapTn for eachn from a dense subspaceY ⊆ L2(G)

of continuous functions ontoXn = C(Gn): Tn(f ) = f ◦ jn. Thus,Tn(f ) is thefunctionf sampled in the points ofjn(Gn), i.e. the table off .

As is shown in LCA groups approximation theory, if〈Gn, jn, jn〉 is an approx-imating sequence for an LCA groupG, then there exists a sequence of positivereals1n such that, for every integrable almost everywhere continuous functionf

which decreases fast enough at infinity (e.g. with compact support), the followingequality holds:∫

G

f dµ = limn→∞1n

∑g∈Gn

f (jn(g)). (0.8)

Now if we wish to define a norm‖ ‖n onXn by the formula

‖ϕ‖n =(1n

∑g∈Gn|ϕ(g)|2

)1/2

, (0.9)

then our mapsTn have the following property:

∀f ∈ Y limn→∞‖Tn(f )‖n = ‖f ‖. (0.10)

Here‖f ‖ is the usual norm off in L2(µ).The following definition is a particular case of the definition of discrete conver-

gence, which comes up in [Stu] and is discussed in [Rei], Section 6.3.

DEFINITION 1. (1) LetX, Xn, n ∈ N, be Banach spaces. Suppose that there ex-ist a dense subspaceY ⊆ X and a sequence of surjective linear operatorsTn: Y →Xn which satisfy (0.10). Then we say that the sequence〈Xn, Tn〉 is a discreteapproximation (d.a.) of the spaceX. If Y = X, we say that this discrete approxi-mation is strong.

A sequencexn ∈ Xn converges discretely tof ∈ Y if ‖Tnf − xn‖n → 0, whenn→∞.

(2) Let 〈Xn, Tn〉 be a d.a. ofX, A: X → X – a linear operator (may be un-bounded) andAn: Xn → Xn – a sequence of linear operators. Denote by DAp(A)

the subspace ofY which consists of all suchf ∈ Y thatAf ∈ Y and

limn→∞‖TnAf − AnTnf ‖n = 0 (0.11)

(this means thatAnTnf converges discretely toAf ).


We say that DAp(A) is the domain of approximability (ofA byAn). If DAp(A)is dense inY , then we say that the sequence of operatorsAn converges discretelyto the operatorA. If the discrete approximation〈Xn, Tn〉 is strong and‖TnA −AnTn‖n → 0, whenn→∞, then we say that the discrete convergence is uniform.

LEMMA 1. If a discrete approximation〈Xn, Tn〉 is strong then the sequence‖Tn‖nis uniformly bounded.

A proof of this lemma can be found in [Tre].The case ofY = X is also used in [DVV]. There the spacesXn of functions

on the finite meshes are embedded intoL2(Rn) as spaces of step functions andthe orthonormal projections onto these spaces are used as the linear operatorsTn.Note that, in this case,Tnf differs from the table off , even for a continuousfunctionf . But for smoothf , it is easy to see that the difference betweenTnf andits table converges to 0. The possibility to consider the operatorsAn as operatorswhich act inX (equal toL2(Rn) in this case) makes the investigation of spectraconvergence essentially easier. The key result necessary to prove this convergenceis the so-called uniform compactness [DVV] of operatorsAn which means that⋃n∈NAn(B), whereB is the unit ball inL2(Rn), is relatively compact. This notion

was introduced in [Ans]. In what follows, we use a more general notion of com-pactness of a sequence of operatorsAn: Xn→ Xn which for our particular case isequivalent to the notion of discrete compactness, introduced in [Rei], Section 7.3.

For the case of an arbitrary locally compact Abelian groupG with a com-pact open subgroup, we construct here such embeddings that are based on theapproximations of the LCA groups mentioned above.

In this paper, we use methods of nonstandard analysis for investigating spectraconvergence for our approximations. Using them, we can construct a nonseparableBanach spaceX from the sequence of Banach spacesXn, satisfying Definition 1,and an embeddingt: X → X. This X is the nonstandard hull of a spaceXnwith an infinitely large, in the sense of nonstandard analysis, integern. From thestandard point of view,X is constructed from the ultraproduct of{Xn | n ∈ N}. Ifthe operatorsAn of Definition 1 have uniformly bounded norms, then they definethe operatorA: X → X and the condition of discrete convergence becomesequivalent to commutativity of the following diagram, used in [Gor] and [Wol]:

A

X −→ X

t ↓ ↓ t

X −→ XA

(0.12)

In what follows, we consider only the case of Hilbert spaces. It is easy to see thatif A is compact or bothA andA are self-adjoint, then the commutativity of thediagram (0.12) implies the inclusionσ (A) ⊆ σ (A), whereσ (B) is the spectrumof operatorB. It is proved using nonstandard analysis that the spectrumσ (A)


consists of limit points of the set⋃n∈N σ (An) and is purely discrete [Gor, Wol]

(see also Theorem 7.16 in [Rei]). Note that, asX is nonseparable, soσ (A) canbe uncountable, e.g. equal to some closed interval inR. Now to get the necessaryspectra convergence, it is sufficient to prove the equalityσ (A) = σ (A), whichfollows, for example, from the inclusiont(X) = A(X), which here replaces thecondition of uniform compactness.

We consider here only self-adjoint operators with discrete spectrum and com-pact resolvents but it seems that this approach may be useful in more generalcases. Some general results about convergence of spectra of operatorsAn whichconverge discretely to an operatorA in the general case of Banach spaces, butunder the assumption thatY = X (see Definition 1) were obtained by Räbigerand Wolff [RWo, Wol], who also used the nonstandard approach. For the specialcase of self-adjoint integral operators on compact groups, the same approach wasemployed in [Gor].

We develop the necessary nonstandard techniques in Section 1. The basic con-cepts and results of nonstandard analysis can be found in [AFHL]. In Section 2 wepresent briefly the results about approximations of LCA groups by finite groupsfrom [Gor] and introduce and investigate one discrete approximation of the spaceL2(G), whereG is an LCA group with a compact open subgroup, based on theapproximations of this group by finite groups. In Section 3 we consider the finite-dimensional approximations of operators inL2(G), based on the approximationof their symbols and prove the theorems about spectral convergence for Hilbert–Schmidt operators and for some Schrödinger-type operators.

1. Nonstandard Approach to Discrete Approximations

We fix someω1-saturated universe∗U. Let ∗R be the field of hyperreals in it. Ift ∈ ∗R is infinite (i.e.,∀a ∈ R, |t| > a), then we writet ∼ ∞. If t is notinfinite, then we call it finite or bounded and write|t| � ∞ (if we know thatt ispositive, then we omit| |). If t is infinitesimal (t−1 ∼ ∞), then we writet ≈ 0.Two hyperreals are infinitesimally close ift1 − t2 ≈ 0. Every finite hyperrealt isinfinitesimally close to a unique standard real, which is called the standard part oft , or the shadow oft and is denoted by◦t .

LetX andXn be Hilbert spaces with inner products〈·, ·〉 and〈·, ·〉n respectively.Assume that they satisfy Definition 1, i.e.〈Xn, Tn〉 is a d.a. ofX. Fix any infiniteintegerN ∈ ∗N \N and consider the internal Hilbert spaceXN and its two externalsubspaces

X(b)N = {x ∈ XN | ‖x‖n �∞}

and

X(0)N = {x ∈ XN | ‖x‖n ≈ 0}.

The quotient spaceX#N = X

(b)N /X

(0)N is called the nonstandard hull ofXN . For

x ∈ X(b)N , we denote byx# its image inX#

N . The spaceX#N is equipped with the


natural inner product, which will be denoted by(·, ·): (x#, y#) = ◦〈x, y〉N . It iswell known [AFHL] thatX#

N is a Hilbert space, nonseparable if dimXN ∼ ∞. IfdimXN ∈ N, then dimXN = dimX#

N .In what follows, we denoteX#

N by X. Define an embeddingt: X → X. LetY ⊆ X be the dense subspace satisfying Definition 1. Forf ∈ Y , let t(f ) =TN(f )

#. By (0.10) and the nonstandard definition of a limit,‖f ‖ ≈ ‖TN(f )‖N .This obviously implies the equality‖f ‖ = ‖t(f )‖. So the linear operatort: Y →X is norm-preserving and, thus, it has a unique extension to the wholeX, whichwe also denote byt.

Consider now a sequence of linear operatorsAn: Xn → Xn which convergesdiscretely to a bounded operatorA: X → X (see Definition 1(2)). First, assumethat the operatorsAn are uniformly bounded (i.e.∃C∀n‖An‖n 6 C). Then‖AN‖Nis also bounded and thusAN defines a bounded linear operatorA#

N : X → X

by the formulaA#N(x

#) = AN(x)# ∀x ∈ X(b)

N . In what follows, we denote thisoperator byA. It is easy to show that‖A‖ = ◦‖AN‖N . The condition of discreteconvergence (0.11) implies that∀f ∈ Y, ‖ANTNf − TNAf ‖N ≈ 0. This meansthat∀f ∈ YAt(f ) = tAf . As Y is dense inX and all the operators in the latterequality are bounded, we come to the following proposition:

PROPOSITION 1. If a sequence of linear operatorsAn: Xn → Xn is uniformlybounded, then it converges discretely to a bounded linear operatorA: X→ X ifffor an arbitraryN ∈ ∗N \ N the diagram(0.12) is commutative.

This proposition immediately implies the following one:

PROPOSITION 2. If a sequence of linear operatorsAn: Xn → Xn is uniformlybounded and converges discretely to a bounded linear operatorA: X → X thenits domain of approximability isDAp(A) = {f ∈ Y | Af ∈ Y }. In particular, ifY = X and the equality(0.11) holds on some dense subset ofX, then it holds onthe wholeX.

From now on, we assume that all the spacesXn are finite-dimensional and allthe operatorsAn andA are normal (self-adjoint). ThenA is also normal (resp. self-adjoint). In this case, it is easy to prove ([Gor]; Proposition 1.3.5, see also [Mo])thatσ (A) = {◦λ | λ ∈ σ (AN)} and this spectrum is purely discrete, i.e. it consistsonly of eigenvalues.?

We also assume thatAn converges discretely toA. Obviously for normal opera-tors, the commutativity of diagram (0.12) implies the inclusionσ (A) ⊆ σ (A). So,the eigenfunctions ofA , which correspond to the points of the spectrumσ (A), canbe considered as generalized eigenfunctions ofA. It is interesting to compare thisconstruction with Gel’fand’s triples, but in this paper we consider only the case ofcompact operators or operators with compact resolvents.

Below we use the following notation:B(λ) = Ker(B − λ), for an arbitraryoperatorB.? In more general cases this equality does not hold (see [Wol] for counterexamples).


PROPOSITION 3. If A is compact operator andA(X) ⊆ t(X), then

(1) σ (A) = σ (A);(2) if λ ∈ σ (A) andλ 6= 0 thenA(λ) = t(A(λ)), and sodimA(λ) = dimA(λ).

Proof. We have to show that, under our assumptions,σ (A) ⊇ σ (A) and eacheigenfunctionf /∈ KerA of A is equal tot(x), wherex is an eigenfunction ofA,corresponding to the same eigenvalue. Indeed, letλ ∈ σ (A). We can assume thatλ 6= 0. Otherwiseλ ∈ σ (A), sinceA is compact and self-adjoint. Asσ (A) consistsonly of eigenvalues, there existsf ∈ X such thatAf = λf . But by our conditionAf ∈ t(X), and sinceλ 6= 0, alsof ∈ t(X). So there exists a uniquex ∈ Xsuch thatf = t(x). By the commutativity of diagram (0.12),t(Ax) = A(t(x)) =Af = λf . Obviously, t(λx) = λf and ast is injective, we obtain thatx is aneigenvalue ofA, corresponding toλ. 2PROPOSITION 4. Under the conditions of Proposition3, if A andAN are self-adjoint, then

(1) for eachλ ∈ σ (A), λ 6= 0, the setσλ = {ν ∈ σ (AN) | ν ≈ λ} is finite,? i.e.σλ = {ν1, . . . , νk}, wherek ∈ N;

(2) for eachi 6 m dimAN(νi) = mi is finite and

∑ki=1mi = s = dimA(λ);

(3) (AN(ν1)⊕· · ·⊕AN(νm))# = A(λ). This implies that if〈xi1, . . . , ximi 〉 is an ortho-normal basis inAN

(νi), i = 1, . . . , k, then〈(x11)

#, . . . , (x1m1)#, . . . , (xk1)

#, . . . ,

(xkmk )#〉 is an orthonormal basis inA(λ);

(4) if, under the conditions of items(1)–(3), A(λ) ⊆ DAp(A), then there exists anorthonormal basis〈y1

1, . . . , y1m1, . . . , yk1, . . . , y

kmk, 〉 in A(λ) such thatTNyij ≈

xij i = 1, . . . , k; j = 1, . . . , mi .

Proof. Obviously, if ν ≈ λ andx ∈ AN(ν), thenx# ∈ A(λ). If ν1, . . . , νk ∈ σλand are pairwise different, then anyx1 ∈ AN(ν1), . . . , xk ∈ AN(νk) are pairwise or-thogonal and thusx#

1, . . . , x#k are pairwise orthogonal. This proves thatk 6 s. The

inverse inequality is proved in Proposition 1.3.20 in [Gor]. Assertion (3) followsfrom (2). Assertion (4) follows from the definition of the embeddingt. 2DEFINITION 2. If a sequence of operatorsAn (not necessarily discretely con-verging to some operatorA) satisfies the conditionA(X) ⊆ t(X), then it is calledquasicompact.

A motivation for such a terminology is the following. Assume for a momentthat the discrete approximation〈Xn, Tn〉 is strong (Definition 1). Then the con-dition of Definition 2 means that for any bounded elementx ∈ XN , its image? We say that a set is finite if its cardinality lies inN and hyperfinite if it is internal and its

cardinality lies in∗N. Obviously, every finite set is internal. The same terminology is used for thedimension of an internal linear space.


ANx is infinitesimally close toTNy for some (standard!) elementy ∈ X. Nowrecall the nonstandard criterion of operator compactness: an operator is compactiff the image of any bounded element is nearstandard. This definition is equiv-alent to the definition of discrete compactness of a sequence of operators [Rei],p. 194.

One simple sufficient condition of quasicompactness of a sequenceAn is thefollowing (cf. [RWo]).?

PROPOSITION 5. If a sequenceAn converges discretely to a compact operatorA, this convergence is uniform(see Definition 1)and the d.a.〈Xn, Tn〉 satisfies thefollowing condition

supn

sup‖z‖=1

(inf{‖x‖n | Tnx = z}) <∞, (1.1)

then the sequenceAn is quasicompact.Proof. Let ξ ∈ X, then there exists a bounded elementx ∈ XN such that

ξ = x#. Since all the operatorsTn are surjective, by the transfer principle, thereexistsf ∈ ∗X such thatx = TNf . By condition (1.1), suchf can be chosenbounded. By the condition of uniform convergence‖ANTN − TN ∗A‖N ≈ 0 andthusANx = ANTNf ≈ TN

∗Af . AsA is compact andf bounded, there exists astandard elementh ∈ X such that∗Af ≈ h. By Lemma 1.1,‖TN‖N is finite andsoTN ∗Af ≈ TNh, i.e.ANx ≈ TNh andA(ξ) = t(h). 2

Remark 1.Below we shall use the strong discrete convergence in the situa-tion [DVV] when there exist norm preserving embeddingsin: Xn → X andTn = i−1

n ◦ pn, wherepn: X → in(Xn) is the orthonormal projection. It is easy tosee that, in this case, the condition (1.1) holds.

Unfortunately, the uniform convergence does not occur often enough. So inmany interesting cases, it is not so simple to prove the quasicompactness ofAn.First of all, here it is necessary to find out the conditions for an arbitrary elementx ∈ XN to be infinitesimally close to an element of the formTNy for somestandardy ∈ X. One such criterion, which is useful for our approximations of Schrödingertype operators, will be proved below.

The definition of quasicompactness has a quite natural standard version if weassume that the condition of Definition 1 holds for any infinite integerN . Thenthe ordinary arguments of nonstandard analysis allow us to prove the followingproposition, which holds for arbitrary operatorsAn in arbitrary Banach spacesXNandX.

We denote the unit ball inXN with center in the point 0 byDN and the ball inXN with center in a pointx and radius equal toε byDε

N(x).

? The following proposition holds for the case of arbitrary Banach spacesXn andX.


PROPOSITION 6. If a discrete approximation〈Xn, Tn〉 is strong, then a sequenceof operatorsAn is quasicompact iff

∀ε > 0∃finB ⊆ X∃n0∀N > n0

(AN(DN) ⊆

⋃y∈B

DεN(TNy)

). (1.2)

Proof.Suppose first that (1.2) holds. Fix an arbitrary standardε > 0. Then, bythe transfer principle, for any infinite integerN ∈ ∗N \ N, we have the followinginclusion:

AN(DN) ⊆⋃y∈B

DεN(TNy).

Fix any x ∈ DN . The previous inclusion implies that, for any standardn ∈ N,there exists a standardyn ∈ X such that‖ANx − TNyn‖N < n−1. Since‖TNyn −TNym‖N ≈ ‖yn − ym‖ for all standardn andm, the sequence{yn | n ∈ N} is aCauchy sequence inX and thus it converges to some (standard) elementy ∈ X.Now it is obvious that‖ANx − TNy‖ ≈ 0.

Suppose now that (1.2) is false. Going over to the negations, we get that thefollowing holds:

∃ε > 0∀finB ⊆ X∀n0∃N > n0

((AN(DN) 6⊆

⋃y∈B

DεN(TNy)

). (1.3)

Let a standardε0 > 0 satisfy (1.3). Consider a hyperfinite setB ⊆ ∗X, suchthatX ⊆ B. Then applying the transfer principle to (1.3), we get the existence ofN ∈ ∗N \ N, such that

AN(DN) 6⊆⋃y∈B

DεN(TNy).

Thus there existsx ∈ DN such that the distance fromx to any element of theform TNy for some standardy is greater than the standardε0. This proves that thecondition of Definition 2 is not true for thisN . 2

Remark 2.In this proof we used a nonstandard universe which is Card(X)+-saturated, or satisfies the Robinson’s concurrence principle. It is easy to see that,for separableX, it is enough to use aℵ1-saturated universe.

Let us compare our notion of quasicompactness with the notion of compactnessof the sequenceAn used in [DVV]. In [DVV] only the situation described in Re-mark 1 is considered. In this situation, the operatorAn can be identified with theoperatorA′n = inAnTn, which acts inX. And according to [DVV], the sequenceAnis compact iff

⋃n A′n(D), whereD is the unit ball inX, is relatively compact. It is

easy to see that [DVV]-compactness ofAn implies its quasicompactness. Indeed,


let x ∈ DN for anyN ∈ ∗N \ N. TheniN(x) ∈ D andA′N iNx = iNANx ≈ y forsome standardy ∈ X by the condition of [DVV]-compactness and the well-knownnonstandard compactness criterion. Now the trivial translation of Propositions 3and 4 into standard language gives us the following theorem, which generalizesLemma 1 of Section 1 of [DVV]:

THEOREM 1. LetA be a compact self-adjoint operator in the Hilbert spaceX,〈Xn, Tn〉 – a discrete approximation ofX, andAn: Xn → Xn – a quasicompactsequence of self-adjoint operators which converges discretely toA. Then

(1) σ (A) is equal to the set of nonisolated limit points of⋃n σ (An);

(2) if 0 6= λ ∈ σ (A) andJ is a neighborhood ofλ which contains no other pointsof σ (A), thenλ is a unique nonisolated limit point ofJ ∩⋃n σ (An);

(3) if in the condition of item(2), Mλn =

∑ν∈σ(An)∩J An

(ν), then, for all largeenoughn, dimMλ

n = dimA(λ) = s and there exists a sequence of orthonormalbases〈f n1 , . . . , f ns 〉 in Mλ

n which converges discretely to an orthonormal basis〈f1, . . . , fs〉 in A(λ).

Certainly, all our arguments cannot be used for the case of an unbounded self-adjoint operatorA, because, in this case, ifAn converges discretely toA, thenthe norm ofAN is infinite and there are problems even with the definition ofA#N (see [Kru]). But in this case, we can use slightly modified results about the

strong resolvent convergence (cf., e.g., [RSa], Theorem VIII.19 and also [Fat],Theorem 5.7.6).

If λ /∈ σ (A), then we denote the corresponding resolvent, as usual, byRλ(A) =(A− λI)−1, whereI is the identity operator.

PROPOSITION 7. If A is self-adjoint and the domain of approximabilityDAp(A)of A byAn is an essential domain ofA, then for eachλ /∈ cl(σ (A) ∪⋃n σ (An)),the sequenceRλ(An) converges discretely toRλ(A).

Proof. First of all we shall prove our proposition forλ = ±i, which obviouslysatisfies our assumptions. Let us consider only the case of−i. The considera-tions in the other case are quite similar. By Definition 1, we have to prove theequality (0.11) withR−i(A) andR−i(An) for some dense subsetY ⊆ X. LetY = {(A + i)ϕ | ϕ ∈ DAp(A)}. ThenY is dense inX because DAp(A) is anessential domain ofA. Fix any infinite integerN ∈ ∗N \ N. Then(

(A+ i)−1 − (AN + i)−1)(A+ i)ϕ = (AN + i)−1(AN − A)ϕ.

Sinceϕ ∈ DAp(A), we have(AN − A)ϕ ≈ 0. For arbitraryB ⊆ R andλ ∈ C,denote the distance betweenλ andB by ρ(λ,B). Let S = cl(σ (A) ∪⋃n σ (An))

If λ satisfies the conditions of our proposition, thenρ(λ, S) > 0. Then, by thetransfer principle,∥∥(AN + i)−1

∥∥ = ρ(λ, σ (AN))−1 6 ρ(λ, S)−1� +∞.


Thus,R−i (AN)(A+ i)ϕ ≈ R−i(A)(A + i)ϕ. This proves thatR−i(An) convergesdiscretely toR−i(A).

We shall now prove that if our proposition holds for someλ0, satisfying thenecessary conditions, and if

‖λ− λ0‖ < ρ(λ0, S), (1.4)

then our proposition also holds forλ. Obviously this is enough to prove our propo-sition, because eachλ /∈ S can be connected either withi or with−i by a smoothcurve lying entirely inC \ S, and thus can be approached fromi or−i by finitelymany circles of a radius less thenρ(λ, S).

The functionsRλ(A) andRλ(An) for all n ∈ N are analytic in the circle (1.4),and can be expanded there into the following, uniformly converging series:

Rλ(A) =∞∑m=0

(λ− λ0)mRm+1

λ0(A),

(1.5)

Rλ(An) =∞∑m=0

(λ− λ0)mRm+1

λ0(An).

We have to prove that for any infiniteN ∈ ∗N \ N and for any standardf ∈Y, TNRλ(A)f ≈ Rλ(AN)TNf . Since it holds forλ0, then for any standardk ∈ N,

k∑m=0

(λ− λ0)mTNR

m+1λ0

(A)f ≈k∑

m=0

(λ− λ0)mRm+1

λ0(AN)TNf. (1.6)

Fix an arbitraryε > 0. Then, by equalities (1.5), there exists ann0 such that fork > n0∥∥∥∥∥Rλ(A)f −

k∑m=0

(λ− λ0)mRm+1

λ0(A)f

∥∥∥∥∥ < ε.Denote the function on the left part of the last inequality byh. Sinceh is standard,we have‖TNh‖ ≈ ‖h‖. This implies that∥∥∥∥∥TNRλ(A)f −

k∑m=0

(λ− λ0)mTNR

m+1λ0

(A)f

∥∥∥∥∥N

6 ε. (1.7)

Let us show that the convergence of the series (1.5) is uniform with respect ton. We have

‖Rλ0(An)‖ = maxν{ν | ν ∈ σ (An)} = ρ(λ0, σ (An))

−1 6 ρ(λ0, S)−1.

Soq = |λ− λ0| · ‖Rλ0(An)‖ < 1 and thus∥∥∥∥∥∞∑

m=k+1

(λ− λ0)mRm+1

λ0(An)

∥∥∥∥∥ 6 ρ(λ0, S)−1 · qk+1

1− q → 0, whenk→∞.


Now since‖TNf ‖N is finite (TNf ≈ f ), there existsn1 such that fork > n1∥∥∥∥∥Rλ(An)TNf −k∑

m=0

(λ− λ0)mRm+1

λ0(AN)TNf

∥∥∥∥∥N

< ε. (1.8)

Now taking a standardk > max{n0, n1}, we obtain by (1.6), (1.7) and (1.8) that‖TNRλ(A)f − Rλ(AN)TNf ‖N 6 2ε and, as this holds for an arbitrary standardε > 0, the proposition is proved. 2

The following proposition is a simple corollary of Proposition 7:

PROPOSITION 8. If, under the conditions of Proposition7, the resolventsRλ(A)of a self-adjoint operatorA are compact and for some realλ /∈ S (hereS is thesame as in the proof of Proposition7) the sequenceRλ(A) is quasicompact, thenthe assertions(1)–(3) of Theorem1 holds. 2

If we assume that our d.a.〈Xn, Tn〉 is strong and satisfies (1.1), then by Proposi-tion 5, the quasicompactness of the sequence of resolvents follows from its uniformconvergence. Sufficient conditions for uniform convergence of the resolvents aregiven by the following proposition (cf. [RSa], Theorem 8.25):

PROPOSITION 9.Suppose that in the conditions of Proposition7 the d.a.〈Xn, Tn〉is strong, satisfies(1.1), and for some essential domainD ⊆ DA(A) of A thefollowing condition holds:

limn→∞ sup

ϕ∈D, ‖ϕ‖A=1‖(TnA− AnTn)ϕ‖n = 0,

where‖ϕ‖A = ‖Aϕ‖+‖ϕ‖. Then the convergence ofRλ(An) toRλ(A) is uniform.

This proposition will not be used below, so we omit its proof.

2. Discrete Approximations of the Hilbert Spaces of Functions on LCAGroups

1. First of all we formulate the main results about finite approximations of LCAgroups from [Gor].

DEFINITION 3. LetG be a locally compact group andρ – some left invariantmetric on it. The sequence(Gn, jn) of finite groupsGn and mapsjn: Gn → G iscalled an approximating sequence (a.s.) forG if, for anyε > 0 and for any compactK ⊆ G, there existsN > 0 such, that for alln > N ,

(1) jn(Gn) is anε – network forK?;(2) ∀g1, g2 ∈ j−1

n (K) ρ(jn(g1 ◦n g±12 ), jn(g1) · jn(g2)

±1) < ε, where◦n is themultiplication inGn.

? Here we do not require that anε-network ofK consists of elements ofK .


(3) jn(en) = e, whereen ande are units in the groupsGn andG, respectively.

A locally compact group is calledapproximableif it has an approximatingsequence.

For approximable groups, the Haar integral can be represented as the limit ofRiemann integral sums.

THEOREM 2. LetG be an approximable locally compact group,µ – Haar mea-sure onG,U – a relatively compact neighborhood of the unit inG, K – the familyof all compact subsets ofG. Then, iff : G → C is bounded, almostµ-everywherecontinuous, and satisfies the following condition:

∀ε > 0 ∃K ∈ K∃n0∀n > n0∀B ⊆ j−1n (G \K),

1n ·∑g∈B|f (jn(g))| < ε, where1n = µ(U)

|j−1n (U)| , (2.1)

thenf ∈ L1(µ) and satisfies(0.8).

Obviously condition (2.1) holds for every compact groupG.The1n, defined in (2.1) will be called thenormalizing multiplier(n.m.) of a.p.

(Gn, jn). We can take any sequence1′n, equivalent to1n for an n.m.For a compact groupG we take1n = |Gn|−1 and for discrete groups we take

1n = 1.

NOTATION. For eachp > 1, we denote bySp(G) the space of functionsf , suchthat |f |p satisfies the conditions of Theorem 2.

In this paper, we consider only LCA groups. In this case, the approximatingfinite groups are also supposed to be Abelian. Now we are going to define, forevery a.s.(Gn, jn) of a LCA groupG, a special approximation sequence(Gn, jn)

for its dual groupG.First, we introduce the following notation. Let0 be the base of neighborhoods

of the unit inG, which consists of the neighborhoods of the formU(K, ε), whereK ⊆ G is compact, and

U(K, ε) = {γ ∈ G | ∀g ∈ K |γ (g)− 1| < ε}.For χ ∈ Gn, we say thatχ in U(K, ε) strongly (χ ∈s U(K, ε)) if jn(χ) ∈U(K, ε), andχ is in U(K, ε) weakly (χ ∈w U(K, ε)) if ∀g ∈ j−1

n (K) |χ(g)−1| < ε.DEFINITION 4. Let(Gn, jn) be an a.s. for a LCA groupG and(Gn, jn) – an a.s.for its dual groupG. We call this pair of a.s. mutually dual (the first is dual to thesecond and vice versa) if the following two conditions are fulfilled:

(i) ∀U ∈ 0 ∃U ′ ∈ 0 ∃n0 ∀n > n0 ∀χ ∈ Gn (χ ∈w U ′ ⇒ χ ∈s U).


(ii) Ifgn ∈ Gn, χn ∈ Gn, lim

n→∞ jn(gn) = ξ, limn→∞ jn(χn) = χ,

thenχn(gn)→ χ(ξ) whenn→∞.

Remark 3.It is easy to prove that, for groups which have a compact opensubgroup, condition (i) follows from condition (ii)

THEOREM 3. For every LCA groupG, there exists a pair of dual approximationsfor G andG, respectively.

The dual approximations are necessary for approximating the Fourier transformonL2(G) (see (0.4)).

PROPOSITION 10.If (Gn, jn) and(Gn, jn) is a pair of dual a.s. forG and1n isa normalizing multiplier forµ, then1n = (|Gn|·1n)

−1 is a normalizing multiplierfor the Haar measureµ.

Note that for a finite Abelian groupGn, its dual groupGn is isomorphic toGn,thus|Gn| = |Gn|.

The Fourier transformFn: L2(Gn) → L2(Gn) is defined by the formula

Fn(ϕ)(χ) = 1n ·∑g∈Gn

ϕ(g)χ(g) (2.2)

and the inverse Fourier transformF−1n by

F−1n (ψ)(g) = 1n ·

∑χ∈Gn

ψ(χ)χ(g). (2.2′)

Now we are able to formulate the approximation theorem for the Fourier trans-form.

THEOREM 4. If (Gn, jn) and(Gn, jn) is a pair of dual a.s. forG andf ∈ S2(G),Tnf = f ◦ jn: Gn → C (the table of the function f, sampled in the points ofjn(Gn)), Tnf = f ◦ jn: Gn → C (the table of the functionf , sampled in thepoints ofjn(Gn)), then

limn→∞ 1n ·

∑χ∈Gn|Tnf (χ)− Fn(Tnf )(χ)|2 = 0.

Here and belowf is the Fourier transform off , defined by the formula (0.4).In what follows, we denote the spaceCGn (resp.CGn) equipped with the norm

‖ϕ‖(p)n =(1n

∑g∈Gn|ϕ(g)|p

)1/p (resp.‖ψ‖(p)n =

(1n

∑χ∈jn|ψ(χ)|p

)1/p)byLp(Gn) (respLp(Gn)).


Forp = 2, this space will also be denoted byXn (Xn), and the index (2) willbe omitted in the notation of norms (cf. (0.9)). The spacesL2(G) andL2(G) willbe denoted asX andX, respectively. We denote byY (resp.Y ), the dense subsetof X (resp.X) consisting of functionsf such that|f |2 satisfies the conditions ofTheorem 2 with respect to a.s.(Gn, jn) (resp.(Gn, jn)). The following propositionis an obvious corollary of Theorem 2 and Proposition 10:

PROPOSITION 11. The sequences〈Lp(Gn), Tn〉 and 〈Lp(Gn), Tn〉 are discreteapproximations of the spacesLp(µ) andLp(µ), respectively.

Here and belowTn: Y → Xn and Tn: Y → Xn are the operators defined inTheorem 4. This theorem also shows that the sequenceFn of finite Fourier trans-forms converges discretely to the Fourier transformF in the sense of an obviousgeneralization of Definition 1.

2. Here we consider approximations of groups with a compact open subgroup anddiscrete approximations of Hilbert spaces of functions on such groups. We shallconsiderRn and the operators inL2(Rn) in another paper.

We begin with the case of a discrete groupG. In this case,X = l2(G) andDefinition 3 can be reformulated in the following way [VeGo]:

PROPOSITION 12.If G is a discrete group and(Gn, jn) is an a.s. in the sense ofDefinition3, then

(1) lim−→ jn(Gn) = G as a set;

(2) ∀a, b ∈ G ∃n0 ∈ N ∀n > n0 ∀g, h ∈ Gn(jn(g) = a, jn(h) = b⇒→ jn(g ◦n h±1) = a · b±1);

(3) jn(en) = e. 2As in this case

∫Gf dµ =∑ξ∈G f (ξ),

∀f ∈ l1(G)∀ε > 0∃finK ⊆ G∑ξ∈G\K

|f (ξ)| < ε. (2.3)

The condition (1) of Proposition 12 implies that

∀finK ⊆ G∃n0 ∈ N∀n > n0jn(Gn) ⊇ K. (2.4)

For a discrete groupG, a setK ⊆ G is compact iff it is finite and the normalizingmultiplier1n is equal to 1. Equations (2.3) and (2.4) show that in this case condi-tion (2.1) holds for any integrable functionf onG. This implies thatY = X and sothe d.a.〈Xn, Tn〉 is strong. It is also easy to construct a norm preserving embeddingin: Xn → X which is right inverse toTn and thus condition (1.1) is also true forthe d.a.〈Xn, Tn〉 (see Remark 1). This embedding is defined by the formula

in(ϕ)(ξ) ={ϕ(g), ξ = jn(g),0, ξ /∈ jn(Gn).

(2.5)


Now we shall consider the general case of an LCA groupGwith a compact opensubgroupK. LetL = G/K, thenL is a discrete group andL ⊆ G (L = {p ∈ G |p|K = 1}). Let µ be the Haar measure onG such thatµ(K) = 1. Then the dualHaar measureµ on G is such thatµ(L) = 1. The discrete groupK is isomorphicto G/L. Let {al | l ∈ L} be a complete system of distinct representatives of cosetsofG/K, and{ph | h ∈ K} – a complete system of distinct representatives of cosetsof G/L. Note that, fork ∈ K, ph(k) = h(k). If a ∈ G andp ∈ G, then there existuniquel ∈ L, h ∈ K, k ∈ K, s ∈ L such thata = al + k, p = ph + s and

p(a) = ph(al) · s(l) · h(k). (2.6)

Let ϕ ∈ L2(G) andl ∈ L. Denote byϕl a function inL2(K) such thatϕl(k) =ϕ(al + k). Then obviously‖ϕ‖2 = ∑

l∈L ‖ϕl‖2. Using the Fourier transform onthe compact groupK, we obtainϕl(k) = ∑h∈K clhh(k). The mapϕ ↔ 〈clh | l ∈L, h ∈ K〉 is an isomorphismι between the Hilbert spacesL2(G) and

l2(L× K) ={〈clh | l ∈ L, h ∈ K〉 |

∑l∈L,h∈K

|clh|2 < +∞}

which depends on the chosen system of distinct representatives of cosets ofG/K

{al | l ∈ L}.Similarly for any functionψ ∈ L2(G) and h ∈ K, we define the function

ψh ∈ L2(L) such thatψh(s) = ψ(ph + s). Thenψh(s) = ∑l∈L dlhs(l) and

again the mapψ ↔ 〈dlh | l ∈ L, h ∈ K〉 is an isomorphismι between theHilbert spacesL2(L) andl2(L×K)which depends on the chosen system of distinctrepresentatives of cosets ofG/L {ph | h ∈ K}. We have

ψ(ph + s) =∑l∈L

dhls(l). (2.7)

Let nowϕ ∈ L2(G). Consider the Fourier transformϕ(p) = ∫Gϕ(g)p(g)dµ(g).

By (2.6)

ϕ(ph + s) =∑l∈L

∫K

ϕl(k)ph(al) · s(l) · h(k)dµ(k).

As∫Kϕl(k)h(k)dµ(k) = clh ands(l) = s(−l), we have

ϕ(ph + s) =∑l∈L

ph(a−l)c−lhs(l).

Comparing this formula with (2.7), we see that the Fourier transformF : L2(G)→L2(G) is equivalent to the unitary operator inl2(L × K) which is defined by thefollowing matrix:

f (h, l, h′, l′) = ph′(l′) · δh,h′δ−l,l′ . (2.8)


Denote byDfin the subspace of functionsϕ ∈ L2(G) such thatϕ and ϕ havecompact support, and letDfin be the subspace ofL2(G) with the same property.As the matrix (2.8) has only one element unequal to zero in each line and in eachcolumn, we obviously haveF (Dfin) = Dfin.

PROPOSITION 13.Letϕ ∈ L2(G),

ι(f ) = 〈clh | l ∈ L, h ∈ K〉 and ι(f ) = 〈dlh | l ∈ L, h ∈ K〉.Thenϕ ∈ Dfin iff there exist finite subsetsA ∈ L andB ∈ K such that for〈l, h〉 /∈A× B, clh = 0. In this case there exist also finite subsetsR ∈ L andS ∈ K suchthat, for 〈l, h〉 /∈ R × S, dlh = 0.

This proposition follows immediately from the fact that the matrix (2.8) hasonly one element unequal to zero in each line and in each column. 2

Let now〈Gn, jn〉, 〈Gn, jn〉 be a pair of dual a.s. forG.

PROPOSITION 14.There existsn0 ∈ N such that for alln > n0 the setsj−1n (K) =

Kn and j−1n (L) = Ln are subgroups ofGn and Gn, respectively, the sequences

〈Kn, jn|Kn〉, 〈Ln, jn|Ln〉 are approximating sequences forK and L, respectively,andLn is the dual group forLn = Gn/Kn.

In what follows we assume thatKn andLn are subgroups ofGn andGn respec-tively, for all n ∈ N.

DEFINITION 5. We say that a.s.〈Ln, j ′n〉 of a discrete groupL is compatiblewith an a.s.〈Gn, jn〉 if for any finite subsetB ⊆ L there existsn0 ∈ N such that,for all n > n0 and for allλ ∈ Ln if j ′n(λ) = l ∈ B thenj−1

n (l) = λ.

PROPOSITION 15. Let 〈Ln, j ′n〉 be a dual a.s. for the approximating sequence〈Ln, jn|Ln〉 of L. Then it is compatible with the a.s.〈Gn, jn〉. Similarly an a.s.〈Kn, j ′n〉 dual to the a.s.〈Kn, jn|Kn〉 is compatible with the as〈Gn, jn〉 (recallthat Kn = Gn/Ln).

PROPOSITION 16.Let {al | l ∈ L} be a complete system of distinct representa-tives of cosets ofG/K. Then for an arbitraryε > 0 and a finite subsetB ⊆ L, forall large enoughn ∈ N there exists a complete system{αλ | λ ∈ Ln} of distinctrepresentatives of cosets ofGn/Kn such that for allλ ∈ j ′−1

n (B) ρ(aj ′n(λ), αλ) < ε(cf. Definition 3). Here〈Ln, j ′n〉 is an a.s. forL compatible with〈Gn, jn〉.

The last three propositions will be proved in Section4 with the help of nonstan-dard analysis.

EXAMPLE. LetG be the additive group of the fieldQp of p-adic numbers. Choosetwo sequences of integersr, s →∞ and letn = r+s. LetGn be the additive groupof the ringZ/pnZ = {0,1, . . . , pn−1} (this representation of the ring is important


for us). Definejn: Gn→ Qp by the formulajn(k) = k/pr . It is easy to prove that〈Gn, jn〉 is an a.s. forG [Gor]. The dual groupG = Qp is isomorphic toQp: everycharacter ofQp is of the formχξ (η) = exp(2πi{ξη}), ξ ∈ Qp, the mapξ → χξ isa topological isomorphism. So we can identifyQp andQp and consider a dual a.s.as some other a.s. ofQp. We also identifyGn with Gn. Now a dual a.s.〈Gn, jn〉 isdefined by the formulajn(m) = m/ps [Gor].

We take for a compact open subgroupK of G the additive group of the ringZp of p-adic integers. Then we haveKn = j−1

n (K) = prGn = {k · pr | k =0,1, . . . , ps − 1}.

To define the quotient groupL = G/K denote byQ(p) the additive group ofrational numbers of the formm/pl, l > 0. Then obviouslyL is isomorphic toQ(p)/Z. The quotient groupLn = Gn/Kn is isomorphic toZ/prZ = {0,1, . . . ,pr −1}. Definej ′n: Ln→ L by the formulaj ′n(t) = t/pr . It is easy to verify that〈Ln, j ′n〉 is an a.s. forL and that it is compatible with an a.s.〈Gn, jn〉.

The set{k/pl | k < pl, (k, p) = 1} is a complete system of distinct represen-tatives of cosets ofG/K and the set{0,1, . . . , pr − 1} is the complete system ofdistinct representatives of cosets ofLn = Gn/Kn which satisfies Proposition 16.We omit the simple proofs of these assertions.

According to our identificationsL = Zp = K andK = L. So

Ln = {0,1, . . . , pr − 1} ∼= {k · ps | k = 0,1, . . . , pr − 1},Kn = {0,1, . . . , ps − 1}

and if j ′n: Kn → L is defined by the formulaj ′n(u) = u/ps, then〈Kn, j ′n〉 is ana.s. of the quotient groupG/L = K = Lwhich is compatible with an a.s.〈Gn, jn〉.The set{0,1, . . . , ps − 1} is the complete system of distinct representatives ofcosets ofKn = Gn/Ln which satisfies Proposition 16 for this a.s.

Now it is easy to construct the matrix (2.8) for the Fourier transform in this case.Let 0p = {〈m, l〉 | (m, p) = 1,0 6 m < pl}. Then obviously we can considerl2(L× K) asl2(02

p) and the matrix (2.8) has the following form:

f (〈m, l〉, 〈u, v〉, 〈m′, l′〉, 〈u′, v′〉)(2.9)

= exp

(−2πim′u′

pl′+v′

)δ〈m,l〉,〈m′,l′〉δ〈pv−u,v〉,〈u′,v′〉

(we used the fact that, inQ(p), −(u/pv) = (pv − u)/pv).Similar arguments can be used for the finite Fourier transformFn: L2(Gn) →

L2(Gn). We identifyL2(Gn) with l2(Ln × Kn) by the formula

ϕ(l + kpr) =ps−1∑h=0

clh exp2πikh

ps,

ϕ ∈ L2(Gn), 06 l < pr − 1, 06 k < ps − 1,


andL2(Gn) with l2(Ln × Kn) by the formula

ψ(v + tps) =pr−1∑l=0

dlv exp2πitl

pr,

ψ ∈ L2(Gn), 06 v < ps − 1, 06 t < pr − 1.

Now

Fn(ϕ) = ϕ(v + tps) = 1

ps

pn−1∑u=0

ϕ(u)exp

(−2πiu(v + tps)

pn

)

= 1

ps

pr−1∑l=0

ps−1∑k=0

ps−1∑h=0

clh exp2πikh

psexp

(−2πi(l + kpr)(v + tps)

pn

)

=pr−1∑l=0

clv exp

(−2πilv

pn

)exp

(−2πilt

pr

).

The matrix (2.8) for this case is given by

f (l, v, l′, v′) = exp

(−2πil′v′

pn

)δpr−l,l′δvv′. (2.10)

Note that the well-known Cooley–Tukey algorithm for fast Fourier transform isbased exactly on the same computations [AuT].

Similar considerations can be used for each groupQa for an arbitrary sequenceof positive integersa = {an | n ∈ Z} (for the definition of this group see [HeR],where it is denoted by�a). A pair of dual a.s. forQa was described in the Intro-duction of the book [Gor]. It is well known (see for example [Gur]) that any totallydisconnected LCA groupG is isomorphic toQa for an appropriatea.

3. Usually the d.a.〈Xn, Tn〉 is not strong but it can be changed a little to becomea strong one. We are going to construct a strong d.a.〈Xn, Sn〉 of the spaceX,which satisfies the condition (1.1) and such, that for anyf from some dense subsetof Y, ‖Tnf − Snf ‖n → 0. Obviously in this case the d.a.Sn defines the sameembeddingt: X→ X as the d.a.Tn.

One strong d.a. satisfying the necessary conditions for the case ofRn was con-structed in [DVV]. So we have only to do the job for the groups with a compactopen subgroup. As it was noticed in the beginning of Section2 if G is a discretegroup, then the d.a.〈Xn, Tn〉 is strong and satisfies (1.1).

Now we consider the case of a compact groupG. Recall that in this case anormalizing multiplier is given by1n = |Gn|−1. The d.a.〈Xn, Tn〉 is not strong inthis case. The dense subspaceY ⊆ X consists of bounded almost everywherecontinuous functions, and it is easy to see thatTn can not be extended to thewholeX.


We defineSn: X→ Xn by the following formula:

Sn(f )(g) =∑χ∈Gn

f (jn(χ))χ(g). (2.11)

PROPOSITION 17.The sequence〈Xn, Sn〉 is a strong d.a. ofX such that for allf ∈ Y , ‖Tnf − Snf ‖n → 0 and it satisfies condition(1.1).

Proof. As {χ(g) | χ ∈ Gn} is an orthonormal basis inXn so we have‖Sn(f )‖2 = ∑

χ∈Gn |f (jn(χ))|2. The groupGn is discrete and thus the d.a.〈Xn, Tn〉 of X is strong by the previous considerations. So

limn→∞

∑χ∈Gn|f (jn(χ))|2 = ‖f ‖2 = ‖f ‖2,

and‖Sn(f )‖ → ‖f ‖.If f ∈ Y , then by Theorem 4 (which obviously also holds for inverse Fourier

transforms)‖Tn(f ) − F−1n Tnf ‖ → 0 whenn → ∞. Formula (2.2′) shows that

F−1n Tnf = Sn(f ).

To prove that the d.a.〈Xn, Sn〉 satisfies (1.1), we construct an embeddingin:Xn→ X by the formula

in(ϕ)(ξ) =∑χ∈Gn

Fn(ϕ)(χ)jn(χ)(ξ).

Thenin(Xn) = {∑χ∈Gn cχ jn(χ)}, and it is easy to verify that for anyf ∈ in(Xn)in(Sn(f )) = f . This shows that ifpn: X→ in(Xn) is the orthonormal projection,thenSn = i−1

n ◦ pn, and this proves the condition (1.1) (cf. Remark 1). 2Let nowG be an LCA group with a compact open subgroupK, L = G/K,

〈Gn, jn〉, 〈Gn, jn〉 – a pair of dual a.s. forG, Kn satisfying Proposition 14,Ln =Gn/Kn, 〈Ln, j ′n〉 – an a.s. forL, compatible with〈Gn, jn〉.

Fix a complete system{al | l ∈ L} of distinct representatives of cosets ofL.Passing to subsequences, if necessary, we can assume by Proposition 16 that thereexists a complete system{αλ | λ ∈ Ln} of distinct representatives of cosets ofLnfor eachn ∈ N such that

∀l ∈ L, limn→∞ jn(αj ′−1

n (l)) = al. (2.12)

Denote byS ′n the operator fromL2(K) ontoL2(Kn) defined by the formula (2.11).Define the operatorSn: X→ Xn by the formula

Snϕ(αλ + ξ) = S ′nϕj ′n(λ)(ξ ), ξ ∈ Kn, λ ∈ Ln. (2.13)

Hereϕl is defined as in the previous section:ϕl(k) = ϕ(al + k).PROPOSITION 18. The sequence〈Xn, Sn〉 is a strong d.a. ofX which satisfiescondition(1.1) and, for allϕ ∈ S2(G), ‖Tnϕ − Snϕ‖n→ 0, whenn→∞.


Proof.The considerations of the previous section show that the mapϕ ↔ 〈ϕl |l ∈ L〉 is an isomorphism between the Hilbert spacesX and

∏l∈L X(l), where each

X(l) is inL2(K). Similarly, the mapψ ↔ 〈ψλ | λ ∈ Ln〉, whereψλ(ξ) = ψ(αλ +ξ), is an isomorphism between the Hilbert spacesXn and

∏l∈L X

(λ)n , where each

X(λ)n isL2(Kn). Identifying our Hilbert spaces by this isomorphisms we see that

Sn(〈ϕl | l ∈ L〉) = 〈S ′nϕjn(λ) | λ ∈ Ln〉.As S ′n satisfies Proposition 17, this proves immediately the first part of our propo-sition. The remaining part of it will be proved in Section4 with the help of non-standard analysis. 2

In a similar way we can define the mapSn: L2(G) → L2(Gn) which satis-fies Proposition 18. Let{πν | ν ∈ Kn} be the complete system of representa-tives of cosets ofGn/Ln, which satisfies Proposition 16 for the a.s.〈Gn, jn〉 andS ′n : L2(L) → L2(Ln) – the operator which satisfies Proposition 17 for the a.s.〈Ln, jn|Ln〉 of L. Then

Snψ(πν + η) = S ′nψj ′n(ν)(η), ν ∈ Kn, η ∈ Ln.Let us return to the example of Section2. Then for an almost everywhere

continuous functionϕ ∈ L2(Qp)

(Tnϕ)(j + kpr) = ϕ(jn(j + kpr)) = ϕ(j

pr+ k

)= ϕ〈l,m〉(k) =

∑〈u,v〉∈0p

c〈l,m〉〈u,v〉 exp2πiku

pv. (2.14)

Here〈l, m〉 ∈ 0p andl/pm = j/pr .To calculateSnϕ we remark that in our caseL = Zp, soj ′n: Kn → L is just a

dual approximation tojn|Kn. Now using formula (2.11) forS ′n, we get

Snϕ(j + kpr)= S ′nϕj ′n(j)(jn(kpr)) = S ′nϕ〈l,m〉(k)=∑w∈Kn

c〈l,m〉j ′n(w) exp2πiwk

ps=∑v6s

∑〈u,v〉∈0p

c〈l,m〉〈u,v〉 exp2πiku

pv. (2.15)

If ϕ ∈ Dfin (cf. Proposition 13) andr, s are such that form > r, v > s c〈l,m〉〈u,v〉= 0, n = r + s, then formulae (2.14) and (2.15) show thatSnϕ = Tnϕ.

Comparing (2.9) and (2.10) with (2.15), we see that for an arbitraryf ∈ L2(Qp),Sn(f ) = Fn(Snf ), and iff ∈ Dfin, thenTn(f ) = Fn(Tnf ). AsDfin is dense inXwe have proved Theorem 4 for this case.

Remark.Theorem 4, together with the construction of strong d.a. and Proposi-tion 18, imply Theorem 6.1 of [DHV] about the approximation of LCA groups by


finite ones in the sense of Weyl systems for the case of groups with compact opensubgroups. To derive Theorem 6.1 of [DHV] from Theorem 4 for the case ofRn, itis necessary to use the strong d.a. ofL2(Rn), introduced in [DVV].

4. Now we shall reformulate all introduced definitions and results about the LCAgroupG in nonstandard language. We denote by ns(∗G), the set of all nearstandardelements of∗G and by ns(∗G), the set of all nearstandard elements of∗G. Thefollowing proposition can be easily proved with the help of the routine procedureof translating nonstandard propositions into standard ones.

PROPOSITION 19. (i)A sequence〈Gn, jn〉 is an a.s. forG iff for anyN ∈ ∗N\Nthe following conditions are fulfilled.

(1) ∀stξ ∈ G∃g ∈ GN jN(g) ≈ ξ ;(2) ∀g1, g2 ∈ j−1

N (ns(∗G)) jN(g1 ± g2) ≈ jN(g1)± jN(g2);(3) jN(eGN ) = eG.

(ii) An a.s.〈Gn, jN 〉 for G is dual to an a.s.〈Gn, jn〉 ofG iff for anyN ∈ ∗N\Nthe following conditions are fulfilled.

(1) If χ ∈ GN is such that,∀g ∈ j−1n (ns(∗G)), χ(g) ≈ 1, thenjN (χ) ≈ 0.

(2) If jN(g) ∈ ns(∗G) and jN (χ) ∈ ns(∗G), thenjN (χ)(jN(g)) ≈ χ(g).(iii) If 〈Gn, jn〉 is an a.s. forG, then a functionf : G → C satisfies the

condition (2.1) of Theorem2 iff for anyN ∈ ∗N \ N and for any internalB ⊆j−1N (∗G \ ns(∗G))

1N ·∑g∈B|∗f (jN(g))| ≈ 0. (2.16)

If f is bounded, almost everywhere continuous and satisfies(2.16), then∫G

f dµ = ◦(1N

∑g∈GN

f (jN(g))

).

(iv) In the conditions of Theorem4 the following relation holds for everyN ∈∗N \ N

1N ·∑χ∈GN|TN ∗f (χ)− FN(TN∗f )(χ)|2 ≈ 0.

(v) A sequence〈Gn, jn〉 is an a.s. for a discrete groupG iff for anyN ∈ ∗N \N, j−1

N (G) is an external subgroup ofGN and jN |j−1N (G): j−1

N (G) → G is anisomorphism(we remark that in the case of discrete groups one hasG = ns(∗G)).

(vi) If G has a compact open subgroupK then for anyN ∈ ∗N \ N j−1N (∗K) is

an internal subgroup ofGN .(vii) Under the conditions of Definition5 an a.s.〈Ln, j ′n〉 of a discrete group

L is compatible with an a.s.〈Gn, jn〉 iff for anyN ∈ ∗N \ N and for any standardl ∈ L ∀λ ∈ LN (j ′N(λ) = l ⇔ j−1

N (l) = λ). 2


Proof of Proposition 14.Fix N ∈ ∗N \N. We have to show thatKN = j−1N (∗K)

is a subgroup ofGN . Let jN(a) ∈ ∗K, jN(b) ∈ ∗K. ThenjN(a ± b) ≈ jN(a) ±jN(b) ∈ ∗K and thusjN(a ± b) ∈ ∗K sinceK is open and compact simulta-neously. This proves thatKN is a subgroup. The fact that〈KN, jN |KN 〉 satisfiesconditions (1) and (2) of Proposition 19 is obvious. This proves that〈Kn, jn|Kn〉 isan a.s. ofK. The proof for a dual a.s. is similar. We have to show only thatLN isthe dual group ofLN = GN/KN . This means that∀κ ∈ GN

jN(κ) ∈ ∗L ⇐⇒ κ|KN ≡ 1.

If jN (κ) ∈ ∗L then jN (κ)|∗K ≡ 1. Then forg ∈ KN 1 = jN (κ)(jN(g)) ≈ κ(g)(note thatκ andg are nearstandard sinceL andK are compact). Soκ|KN ≈ 1 and,by Lemma 2.2.20 of [Gor],κ|KN ≡ 1.

Let now κ|KN ≡ 1. Then for allg ∈ GN such thatjN(g) ≈ 0, κ(g) = 1.This implies (cf. the proof of Theorem 2.4.16 in [Gor]) thatjN (κ) ∈ ns(∗G).Let k ∈ ∗K. Then there existsg ∈ KN such thatjN(g) ≈ k. Then jN (κ)(k) ≈jN (κ)(jN(g)) ≈ κ(g) = 1, i.e., jN(κ)|∗K ≈ 1; and sinceK is discrete,jN(κ)|∗K ≡ 1. 2

Proof of Proposition 15.To prove Proposition 15 we use the equivalent de-finition of a compatible a.s. which is contained in Proposition 19(vii). Letλ ∈LN and j ′N(λ) = l ∈ L. By Proposition 19(ii) this means that for anyκ ∈LN, jN(κ)(l) ≈ κ(λ). We have to prove thatj−1

N (l) = λ. We remark that thereis only oneλ′ ∈ GN such thatjN(λ′) = l. Indeed, ifjN(λ′) = jN(λ′′) = l then fora ∈ λ′, b ∈ λ′′ jN(a − b) ≈ jN(a) − jN(b) ∈ ∗K. SinceK is compact and openthis means thatjN(a − b) ∈ K, thusa − b ∈ KN andλ′ = λ′′. Now we see that∀κ ∈ LN , κ(λ) ≈ κ(λ′), and by the proof of Theorem 2.4.16 in [Gor], this impliesthe equalityλ′ = λ. 2

Proof of Proposition 16.It is easy to see that the nonstandard version of Propo-sition 16 is the following.

Let {al | l ∈ L} be a complete system of distinct representatives of cosetsof G/K. Then for everyN ∈ ∗N \ N there exists complete system{αλ | λ ∈LN} of distinct representatives of cosets ofGN/KN such that for allλ ∈ j ′−1

N (L),jN(αλ) ≈ al, wherej ′N(λ) = l.

Let R = j−1N∗{al | l ∈ L} ⊆ GN . Define an internal equivalence relation∼

on R such thatg ∼ h ⇔ g − h ∈ KN , and letR′ = R/∼. Define an internalsubsetS ⊆ R′ such thatS = {r ∈ R′ | |r| = 1}, S ′ = ⋃

S. Then the previousconsiderations show that for every standardl ∈ L, j−1

N (al) ∈ S ′. Let S ′′ = {λ ∈LN | ∃g ∈ S ′(g ∈ λ)} andT be a system of distinct representatives of cosets ofL \ S ′′. Then obviouslyS ′ ∩ T = ∅ andS ′ ∪ T is a complete system of distinctrepresentatives of cosets ofLN . 2

Proof of the Second Part of Proposition 18.Let at first ϕ be a continuousfunction with a compact support. Then there exists a standard finite setA ⊆ L


such that∀k ∈ K, ϕ(al + k) = 0, whenl /∈ A. Fix anyN ∈ ∗N \ N. We haveto prove that‖TNϕ − SNϕ‖ ≈ 0. Let {αλ | λ ∈ LN } be a complete system ofdistinct representatives of cosets ofLN , which satisfies the nonstandard version ofProposition 16 (cf. the proof of this proposition). Then forg ∈ KN ,

TNϕ(αλ + g) = ϕ ◦ jN(αλ + g) 6= 0

iff j ′N(λ) ∈ A. Let j ′N(λ) = l ∈ A. Then

TNϕ(αλ + g) = ϕ(jN(αλ + g)) ≈ ϕ(jN(αλ)+ jN(g)) ≈ ϕ(al + jM(g))= TNϕl(g).

Here we used the fact thatjN(αλ) ≈ al , Proposition 19(i)(2), and the fact that acontinuous function with compact support is uniformly continuous, and so evenfor nonstandardα andβ if α ≈ β, thenϕ(α) ≈ ϕ(β). Now by (2.13)SNϕ(αλ +g) = S ′Nϕl(g). If j ′N(λ) = l ∈ A then by Proposition 17TNϕl ≈ S ′nϕl, andif j ′N(λ) = l /∈ A thenϕl = 0 and thusS ′Nϕl = 0. As the cardinality ofA isstandardly finite,‖TNϕ − SNϕ‖N ≈ 0.

Let now ϕ be an arbitrary function inS2(G). Fix an arbitrary standardε >0. Then there exists a continuous functionψ with a compact support such that‖ϕ − ψ‖ < ε. Then

‖TN(ϕ)− TN(ψ)‖N = ‖TN(ϕ − ψ)‖N ≈ ‖ϕ − ψ‖by the definition of a discrete approximation. By the same reason‖SN(ϕ)− SN(ψ)‖N ≈ ‖ϕ − ψ‖ and thus

‖TN(ϕ)− TN(ψ)‖N + ‖SN(ϕ)− SN(ψ)‖N < 2ε.

Now

‖TNϕ − SNϕ‖N 6 ‖TN(ϕ)− TN(ψ)‖N + ‖TN(ψ)− SN(ψ)‖N ++ ‖SN(ϕ)− SN(ψ)‖N < 5ε

since ‖TN(ψ) − SN(ψ)‖N ≈ 0 and as it holds for an arbitrary standardε,‖TNϕ − SNϕ‖N ≈ 0. 2

ForN ∈ ∗N \N we define the spaceX#N = X and the operatort: X→ X as in

the beginning of Section 1. HereXN = L2(GN) andX = L2(G) (cf. the notationsbefore Proposition 11). Proposition 3 shows that it is important to be able to answerthe following question for an arbitraryϕ ∈ X(b)

N (cf. the beginning of Section 1):doesϕ# ∈ tX?

DEFINITION 6. We say that an elementϕ ∈ X(b)N is nearstandard if there exists

f ∈ X such thatϕ# = t(f ).


We shall use the following notations:

H(GN) = GN \ j−1N (ns(∗G)), H(GN) = GN \ j−1

N (ns(∗G)),

H(LN) = LN \ j ′−1N (L), H(KN) = KN \ j ′−1

N (K).

THEOREM 5. An elementϕ ∈ X(b)N is nearstandard iff the following two condi-

tions are fulfilled:

(1) for any internalB ⊆ H(GN),1N

∑g∈B |ϕ(g)|2 ≈ 0;

(2) for any internalC ⊆ H(GN), 1N

∑χ∈C |FN(ϕ)(χ)|2 ≈ 0.

Proof. Necessity. Let ϕ = t(f ). AsDfin is dense inX (cf. Proposition 13) wecan assume that for each standardε > 0 there existsψ ∈ Dfin such that

1N

∑g∈GN|ϕ(g)− ψ(jN(g))|2 < ε.

As the Fourier transform is a norm preserving operator, and by Theorem 4,ψ◦jN ≈FN(ψ ◦ jN), so

1N

∑χ∈GN|FN(ϕ)(χ)− ψ(jN(χ))|2 < ε.

Certainly the same inequalities hold if we restrict summations to any internal sub-sets ofGN andGN respectively. Now asψ andψ both have compact support, forany internalB ⊆ H(GN) andC ⊆ H(GN) ψ ◦ jN |B = 0 andψ ◦ jN |C = 0(as suppψ is compact∗ suppψ ⊆ ns(∗G); the same forψ). Now our inequalitiesshow that

1N

∑g∈B|ϕ(g)|2 < ε, 1N

∑χ∈C|FN(ϕ)(χ)|2 < ε.

And as this holds for an arbitrary standardε > 0, necessity is proved.Sufficiency. Let ϕ satisfy the conditions of the theorem. Fix the complete sys-

tems{αλ | λ ∈ LN } and {πν | ν ∈ KN } of distinct representatives of cosets ofthe groupsLN andGN/LN , respectively, which satisfy the nonstandard version ofProposition 16 (cf. the proof of Propositions 15 and 16). Forλ ∈ LN, k ∈ KN

ϕ(αλ + k) =∑ν∈KN

σλ,νν(k).

Fix any internalP ∈ H(LN). ThenB = P + KN ⊆ H(GN). Note that in ourcase the normalizing multiplier1N = |KN |−1 (we takeK as a relatively compactneighborhood of zero inG, cf. Theorem 2) and1N = |LN |−1 by Proposition 10.Using the fact that{ν(k) | ν ∈ KN } is an orthonormal basis inL2(KN), we have


by the first condition of our theorem

0 ≈ |KN |−1∑g∈B|ϕ(g)|2 = |KN |−1

∑λ∈P

∑k∈KN|ϕ(αλ + k)|2

(2.18)= |KN |−1

∑λ∈P

∑k∈KN|∑ν∈KN

σλνν(k)|2 =∑λ∈P

∑ν∈KN|σλν|2.

Similarly for ν ∈ KN, γ ∈ LNFN(ϕ)(πν + γ ) =

∑λ∈LN

σλ,νγ (λ),

and for any internal

Q ⊆ H(KN),∑ν∈Q

∑λ∈LN|σλ,ν|2 ≈ 0.

We can repeat the calculations which brought us to formula (2.8) for the FouriertransformFN . Then we get thatσλ,ν = πν(αλ)σ−λν and thus|σλ,ν| = |σ−λν|. So forany internalQ ⊆ H(KN)∑

ν∈Q

∑λ∈LN|σλ,ν|2 ≈ 0. (2.19)

As our groupG is separable, the groupsL andK are countable. Thus there existincreasing sequences of finite subsetsA′m ⊆ L andB ′m ⊆ K such that

L =⋃m∈N

A′m, K =⋃m∈N

B ′m.

Let

Am = j ′−1N (A

′m), Bm = j ′−1

N (B′m).

Define a sequenceϕm ∈ XN, m ∈ N by the formula

ϕm(αλ + k) =∑ν∈ Bm

σλνν(k), λ ∈ Am,0, λ /∈ Am.

We are going to show that inX

ϕ# = limm→∞ ϕ

#m. (2.20)

It is enough to show that for any standardε > 0 there exists a standardm0 suchthat for allm > m0, ‖ϕ − ϕm‖ < ε. But similarly to the considerations forL2(G),it is easy to see that the mapϕ ↔ σλ,ν | λ ∈ LN, ν ∈ KN 〉 is an isomorphism


between the Hilbert spacesL2(GN) andl2(LN × KN). Thus

‖ϕ − ϕm‖2 =∑

〈λ,ν〉∈LN×KN\Am×Bm|σλν|2.

It is easy to see that for anyM ∈ ∗N \ N,

L ⊆ A′M ⊆ ∗L, K ⊆ B ′M ⊆ ∗K.Now

P = LN \ AM ⊆ H(LN) and Q = KN \ BM ⊆ H(KN).Let S ⊆ LN × KN \AM × BM . Then∑

〈λ,ν〉∈S|σλν|2 6

∑〈λ,ν〉∈P×KN

|σλν|2+∑

〈λ,ν〉∈LN×Q|σλν|2

and both sums on the right side of this inequality are infinitesimal by (2.18)and (2.19). Thus,‖ϕ−ϕM‖2 ≈ 0. Define an internal setC = {m ∈ ∗N | ‖ϕ−ϕm‖ <ε}. We have just shown thatC contains allM ∈ ∗N\N. Thus there exists a standardm0 such thatC contains allm > m0. This proves (2.20).

As‖ϕ‖ is finite,∑

λ∈LN |σλν|2 is finite and thus eachσλν is finite. Forl ∈ L, h ∈K let clh = ◦σj ′−1

N (l)j ′−1N (k)

and definefm ∈ Dfin, m ∈ N, by the formula

fm(al + ξ) =

∑h∈ B ′m

clhh(ξ), l ∈ A′m,

0, l /∈ A′m.Now, by Propositions 16 and 18,

SN(fm)(αλ + k) =∑ν∈ Bm

cjN (λ)j ′N(ν)ν(k), λ ∈ Am,0, λ /∈ Am,

andSN(fm) ≈ TN(fm). Thus,SN(fm)# = t(fm). On the other hand,

‖SN(fm)− ϕm‖2 =∑λ∈ Am

∑ν∈ Bm|σλν − cjN (λ)j ′N(ν)|2.

This quantity is infinitesimal becauseσλν ≈ cjN (λ)j ′N (ν) and the cardinality ofAm×Bm is standardly finite. Now by (2.20)ϕ# = limm→∞ t(fm) and soϕ# ∈ t(X). 23. Approximations of Pseudodifferential Operators on LCA Groups with

Compact Open Subgroups

Let G be an LCA group,f : G × G → C – a measurable function. In the In-troduction, we defined a pseudodifferential operatorAf with the symbolf by


formula (0.5) and an approximating sequence of finite-dimensional operatorsA(n)f

by formula (0.6). Simple computations, using the formula∫G

χ(ξ)dµ(χ) = δ(ξ), where∫G

ϕ(ξ)δ(ξ)dµ(ξ) = ϕ(0),

which is well known from the theory of distributions on LCA groups, show that foranyψ ∈ L2(G),Afψ can be calculated by the formula

Afψ(x) =∫G

f (x, χ)ψ(χ)χ(x)dµ(χ). (3.1)

Surely there is no problem in giving rigorous justifications for these computa-tions, but it is not necessary, as we can assume formula (3.1) for the definitionof a pseudodifferential operator with the symbolf (this is in analogy with aqp-symbol).

Similar computations give us an analog formula forA(n)f : if ϕ ∈ Xn andx ∈ Gn,

then

A(n)f ϕ(x) = 1n

∑χ∈Gn

f (jn(x), jn(χ))Fn(ϕ)(χ)χ(x). (3.2)

We consider only the case of a groupG with a compact open subgroupK in thisparagraph. As beforeL = G/K, and we fix a complete system of representatives ofcosets{al | l ∈ L} and a complete system of representatives of cosets{ph | h ∈ K}.1. First of all we shall consider Hilbert–Schmidt (HS) operators. The followingproposition is well known in the classical theory of pseudodifferential operators(see, for example, [BSh]) and almost obvious in our case also.

PROPOSITION 20.An operatorAf is an HS operator ifff ∈ L2(G× G). In thiscase

‖Af ‖ 6∫ ∫

G×G|f (x, χ)|2 dµ(x)dµ(χ). (3.3)

Proof. The proof follows immediately from the fact that the kernel ofAf –K(x, y) = (FGf )(x, x − y) (hereFG is the Fourier transform over the secondvariable) and thus‖K(x, y)‖L2(G

2) = ‖f (x, χ)‖L2(G×G). 2Obviously〈jn, jn〉: Gn×Gn→ G×G is an approximating sequence forG×G.

Let us denote byS(2)n : L2(Gn × Gn) → L2(G × G) the map defined in Proposi-tion 18 for this a.s. Thus〈L2(Gn× Gn), S

(2)n 〉 is a strong discrete approximation of

L2(G× G) and we can define one more approximating operator forAf :

B(n)f ϕ(x) = 1n

∑χ∈Gn

(S(2)n f )(x, χ)Fn(ϕ)(χ)χ(x). (3.4)


THEOREM 6. If Af is a HS operator, then

(1) the sequenceB(n)f , defined by formula(3.4), is uniformly bounded:∃n0∀n >n0, ‖Bn‖ 6 ‖f (x, χ)‖L2(G×G);

(2) the sequenceB(n)f converges discretely toAf with respect to the strong discreteapproximation〈Xn, Sn〉 from Proposition18 and this convergence is uniform(Definition1).

(3) if f ∈ S2(G× G) (see the notation after Theorem2) then‖An−Bn‖ → 0, and thus statements(1) and(2) of this theorem hold also forthe sequenceA(n)f .

Using this theorem, Proposition 5, Theorem 1 and Proposition 18, we get thefollowing

COROLLARY 1. If Af is a self-adjoint HS operator, then

(1) the spectrumσ (Af ) is equal to the set of nonisolated limit points of⋃n σ (B

(n)f );

(2) if 0 6= λ ∈ σ (Af ) andJ is a neighborhood ofλ which contains no other pointsof σ (Af ), thenλ is the unique nonisolated limit point ofJ ∩⋃n σ (B

(n)f );

The sequencesA(n)f andB(n)f may not be self-adjoint even ifAf is self-adjoint,so the remaining part of Theorem 1 for this case will be formulated in the followingway (see also3 of this section).

COROLLARY 2. If, under the conditions of Corollary1, Af is self-adjoint andthe sequence of self-adjoint operatorsC(n) : Xn → Xn is such that‖B(n)f −C(n)‖n→ 0, whenn→∞, then

(1) if Mλn =

∑ν∈σ(C(n))∩J C

(n)(ν) (see item(2) of the previous corollary), then forall large enoughn, dimMλ

n = dimAf(λ) = s and there exists a sequence

of orthonormal bases〈f n1 , . . . , f ns 〉 in Mλn which converges discretely to an

orthonormal basis〈f1, . . . , fs〉 in Af (λ);(2) if under the conditions of item(1), f1, . . . , fs ∈ S2(G), then the sequence

of orthonormal bases〈f n1 , . . . , f ns 〉 converges discretely to〈f1, . . . , fs〉 withrespect to the d.a.〈Xn, Tn〉;

Proof of Theorem 6.It is convenient to use nonstandard analysis in this proof.Fix N ∈ ∗N \ N. Obviously, for anyn ∈ ∗N, ‖Bn‖ 6 ‖S(2)n f ‖n. We used here thefacts that‖Fn‖ = 1 and|χ(x)| = 1 also. By the definition of discrete approxima-tion, ‖S(2)N f ‖N ≈ ‖f ‖. If f ∈ S2(G× G), then‖S(2)N f − TNf ‖N ≈ 0. This provesstatements (1) and (3) of the theorem.

To prove statement (2) we use the isomorphisms betweenL2(G) andl2(L× K)andL2(G) andl2(L × K) described in3 of Section 2, and obtain the formula forAf considered as an operator froml2(L × K) to itself. As before we denote this


isomorphisms byι: L2(G)→ l2(L× K) andι : L2(G)→ l2(L× K) respectively.Then (see (2.7)) forϕ ∈ L2(G)

ϕ(ph + s) =∑l∈L( ι ϕ )(l, h)s(l),

Af ϕ(al + k) =∑h∈K

(ιAf ϕ)(l, h)h(k).

Similarly,

f (al + k, ph + s) =∑h′∈K

∑l′∈L( ιf )(l, h, l′, h′) ¯s(l′)h′(k),

whereιf = (ι⊗ ι )f .Now simple computations, using (2.6) and (3.1), show that

(ιAf ϕ)(l, h) =∑h′,l′( ιf )(l, h′, l + l′, h− h′)ph′(al)( ι ϕ )(l′, h′). (3.5)

Fix a complete system of representatives of cosets{αλ | λ ∈ LN } and a completesystem of representatives of cosets{πν | ν ∈ KN } which satisfies the nonstandardversion of Proposition 16 (cf. the proof of this proposition). ObviouslyK × L is acompact open subgroup ofG×G,L×K = G×G/K×L, {〈al, ph〉 | l ∈ L, h ∈ K}is a complete system of representatives of cosets and{〈αλ, πν〉 | λ ∈ LN, ν ∈ KN }is a complete system of representatives of cosets ofLN × KN = GN × GN/KN ×LN , which satisfies the nonstandard version of Proposition 16. It is easy to see that

S(2)N f (αλ + ξ, πν + η)=∑λ′∈LN

∑ν ′∈KN

( ιf )(j ′N(λ), j ′N(ν), j′N(λ

′)j ′N(ν′))η(λ′)ν′(ξ),

where〈LN, j ′N〉 is an a.s. forL, compatible with the a.s.〈GN, jN 〉 (see Defini-tion 5) and〈KN, j ′N〉 is an a.s. forK , dual to the a.s.〈KN, jN |KN 〉 for K.

As above we have the isomorphismsιN : L2(GN) → l2(LN × KN) and ιN :L2(GN)→ l2(LN × KN), defined by the formulae

ϕ(αλ + ξ) =∑ν∈KN

(ιNϕ)(λ, ν)ν(ξ); ψ(πν + η) =∑λ∈LN

(ιNψ)(λ, ν)η(λ).

In a similar way as for (3.5), we obtain a formula forB(N)f :

(ιB(N)f ϕ)(λ, ν)

=∑ν ′,λ′( ιf )(j ′N(λ), j ′N(ν

′), j ′N(λ+ λ′), j ′N(ν − ν′))×

× πν ′(αλ)( ιNFN(ϕ))(λ′, ν′). (3.6)


We have to prove that, for anyψ ∈ L2(G),

‖ιNB(N)f SNψ − ιNSNAfψ‖ ≈ 0. (3.7)

Let us first do some necessary calculations. It immediately follows from the defin-itions that

(ιNSNψ)(λ, ν) = (ιψ)(j ′N(λ), j ′N(ν)). (3.8)

By (2.8)

ιψ (j ′N(λ), j ′N(ν)) = ιψ(−j ′N(la), j ′N(ν))pj ′N(ν)(a−j ′N (λ)). (3.9)

Similar computations for finite groups and the formula (3.8) show that

( ιNFN(SNψ))(λ, ν) = (ιψ)(j ′N(−λ), j ′N(ν))πν(α−λ). (3.10)

By (3.5) and (3.8)

(ιNSNAfψ)(λ, ν)

=∑l′,h′( ιf )(j ′N(λ), h

′, j ′N(λ)+ l′, j ′N(ν)− h′)ph′(aj ′N(λ))ιψ(l′, h′).(3.11)

By (3.6) and (3.10)

(ιB(N)f SNψ)(λ, ν)

=∑ν ′,λ′( ιf )(j ′N(λ), j ′N(ν

′), j ′N(λ+ λ′), j ′N(ν − ν′))×

× πν ′(αλ)(ιψ)(j ′N(−λ′), j ′N(ν′))πν ′(α−λ′). (3.12)

Let us suppose thatf ∈ D(2)fin . This means that there exist (standard!) finite

subsetsA ⊆ L andB ⊆ K such that( ιf )(l, h, l′, h′) = 0 if 〈l, h, l′, h′〉 /∈ (A ×B)2. The spaceD(2)

fin is obviously dense inL2(G × G). Choose the standard finitesetsA1, B1 such that(A − A) ∪ A ⊆ A1 ⊆ L and(B − B) ∪ B ⊆ B1 ⊆ K LetC = j ′−1

N (A1), D = j ′−1N (B1). Then (3.11) shows that(ιNSNAfψ)(λ, ν) = 0,

when〈λ, ν〉 /∈ C ×D, and we can restrict the variablesl′, h′ to the setsA1 andB1,respectively. Since the finite setsA1 andB1 are standard, then by Proposition 12,for α, α′ ∈ C andβ, β ′ ∈ D, we havej ′N(α ± α′) = j ′N(α) ± j ′N(α′) andj ′N(β ± β ′) = j ′N(β)± j ′N(β ′).

These remarks and the equality (3.9) show that (3.11) can be rewritten in theform

(ιNSNAfψ)(λ, ν)

=∑λ′∈C

∑ν ′∈D

( ιf )(j ′N(λ), j ′N(ν′), j ′N(λ+ λ′), j ′N(ν − ν′))×

× (ιψ)(−j ′N(λ), j ′N(ν))pj ′N(ν ′)(aj ′N(λ))pj ′N (ν ′)(a−j ′N(λ′)). (3.13)


For the same reasons we can assume that the variablesλ′ and ν′ in the sumon the right side of equality (3.12) range over the setsC andD, respectively,and (ιB(N)f SNψ)(λ, ν) = 0, when〈λ, ν〉 /∈ C × D. Now comparing (3.12) and(3.13), we see that the term of the sums of these equalities differ by the coefficientsπν ′(αλ)πν ′(α−λ′) in (3.12) andpj ′N(ν ′)(aj ′N(λ))pj ′N (ν ′)(a−j ′N(λ′)) in (3.13). But thesecoefficients are infinitesimaly close by the nonstandard version of Proposition 16.Thus the left-hand sides of the equalities (3.12) and (3.13) are infinitesimally closesince the sums on the right-hand parts have only finitely many nonzero terms. Therelation (3.7) holds now for the same reason. Thus statement (2) of our theoremholds forf ∈ D(2)

fin . The general case of our theorem follows immediately from the

LEMMA 2. If the statement(2) of Theorem6 holds for allAf with f in somedense subsetY ⊆ L2(G× G), then it holds for allf ∈ L2(G× G).

Proof. We have to prove that‖SNAf − B(N)f SN‖ ≈ 0. Fix any standardε > 0and chooseψ ∈ Y such that‖f − ψ‖ < ε. Then

‖SNAf − B(N)f SN‖6 ‖SNAf − SNAψ‖ + ‖SNAψ − B(N)ψ SN‖ + ‖B(N)ψ SN − B(N)f SN‖.

Now ‖SNAψ − B(N)ψ SN‖ ≈ 0 by the conditions of the lemma. By Lemma 1‖SN‖is finite, and thus

‖SNAf − SNAψ‖ 6 ◦‖SN‖ · ‖Af−ψ‖ 6 ◦‖SN‖ · ‖f − ψ‖ 6 ◦‖SN‖ε,‖B(N)ψ SN − B(N)f SN‖ 6 ‖B(N)ψ−f ‖◦‖SN‖ 6 ◦‖SN‖ · ‖f − ψ‖ 6 ◦‖SN‖ε,

and asε is an arbitrary standard positive real, this proves the lemma. 22. In this section we consider the Schrödinger type operators (0.7). Up to the end ofthis section we suppose thata(x) andb(χ) are real, almost everywhere continuous,locally bounded,a(x)→∞, whenx →∞, andb(χ)→∞, whenχ →∞. OurgroupG and its a.s. satisfy all the assumptions of the previous section.

It is easy to see that the formula (3.1) in this case is equivalent to the followingformula

Afψ(x) = a(x) · ψ(x)+ b ∗ ψ(x), (3.14)

where∗-is the convolution inL1(G) and b is the inverse Fourier transform ofb,which is considered here as a distribution onG.

Similarly, the discrete approximation (3.2) satisfies in this case the followingformula

A(n)f ϕ(x) = a(jn(x)) · ϕ(x)+ F (−1)

n (b ◦ jn) ∗ ϕ(x). (3.15)

By analogy with [VVZ], where it was proved forQp, it can be proved thatfor an arbitrary LCA groupG with a compact open subgroup a Schrödinger type


operatorAf has under our assumptions a discrete spectrum, which consists of realeigenvalues

α1 6 λ2 6 · · · 6 λk . . .→∞; k→∞,and the multiplicity of each eigenvalue is finite.

It is easy to see that in this case the operatorsA(n)f are self-adjoint.

THEOREM 7. If Af is a Schrödinger type operator witha and b satisfying theabove assumptions and the domain of approximability ofAf by the sequenceA(n)f(3.2) is an essential domain ofAf , then

(1) the spectrumσ (Af ) is equal to the set of nonisolated limit points of⋃n σ (A

(n)f );

(2) if J is a neighborhood ofλ which contains no other points ofσ (Af ), thenλ isthe unique nonisolated limit point ofJ ∩⋃n σ (A

(n)f );

(3) if under the condition of item(2) Mλn =

∑ν∈σ(A(n)f )∩J A

(n)f

(ν), then for all large

enoughn, dimMλn = dimAf

(λ) = s and there exists a sequence of orthonor-mal bases〈f n1 , . . . , f ns 〉 in Mλ

n which converges discretely to an orthonormalbasis〈f1, . . . , fs〉 in Af

(λ) with respect to the d.a.〈Xn, Sn〉 (and with respectto the d.a.〈Xn, Tn〉 if the eigenfunctions ofAf belong toS(G)).

Proof. Without loss of generality we may assume thata and b are positive.Then the operatorsAf andA(n)f are positive,−1 /∈ cl(σ (Af ) ∪⋃n σ (A

(n)f ) and by

Proposition 8 it is enough to show that the sequenceR−1(A(n)f ) is quasicompact.

Thus we have to prove that forN ∈ ∗N \ N and for an arbitraryψ ∈ Xn, if‖(A(N)f + I )ψ‖N is finite thenψ satisfies the conditions of Theorem 5. Note that

since‖R−1(A(N)f )‖ 6 1 we have that‖ψ‖N is finite and thus((A(N)f + I )ψ,ψ) is

finite. But((A

(N)f + I )ψ,ψ

) = (a · ψ,ψ)+ (F (−1)N (b) ∗ ψ,ψ) + (ψ,ψ)

and (F(−1)N (b) ∗ ψ,ψ) = (b · FN(ψ), FN(ψ)).

Thus,(a ·ψ,ψ) and(b ·FN(ψ), FN(ψ)) are both finite. Suppose now that the firstcondition of Theorem 5 is not fulfilled. Then there exists an internalB ⊆ H(GN)

and a standardc > 0 such that1N

∑x∈B |ψ(x)|2 > c. Sincea(x) → ∞ when

x →∞ there existsL ∼ ∞ such that∀x ∈ B, a(x) > L. Now

(a · ψ,ψ) > 1N

∑x∈B

a(x)|ψ(x)|2 > Lc.

This contradicts the condition of finiteness of(a ·ψ,ψ). The considerations for thesecond condition of Theorem 5 are similar. 2


Now we are going to describe one class of Schrödinger type operators whichsatisfy the conditions of the previous theorem. We shall use the isomorphismsι

andι, defined in the proof of Theorem 6. In spite of the fact thata /∈ L2(G) andb /∈ L2(G), we shall denote byιa andι b the functions onl2(L× K) which satisfythe following equalities:

a(al + k) =∑h∈K

(ιa)(l, h)h(k);

b(ph + s) =∑l∈L( ι b)(l, h)s(l).

PROPOSITION 21. If Af is a Schrödinger type operator and in addition to theabove assumptionsa andb satisfy the following conditions:

∀l ∈ L, S(l) = {h ∈ K | (ιa)(l, h) 6= 0} is finite;∀h ∈ K, T (h) = {l ∈ L | ( ι b)(l, h) 6= 0} is finite;

thenDfin is an essential domain ofAf which is a domain of approximability ofAfbyA(n)f .

Proof. Recall thatDfin is a space of functionsϕ ∈ L2(G) such that bothϕ andϕ have compact supports. This means that there exist finite subsetsA[ϕ] ⊆ L andB[ϕ] ⊆ K such that(ιϕ)(l, h) = 0 when〈l, h〉 /∈ A[ϕ] × B[ϕ]. Now simplecomputations, using formula (2.8), show that

(ιAf ϕ)(l, h)

=∑

h′∈B[ϕ](ιa)(l, h′ − h) · (ιϕ)(l, h′)+

+∑l′∈A[ϕ]

(ι b)(l − l′, h) · (ιϕ)(l′, h) · ph(a−l ) · ph(al′). (3.16)

This formula shows thatAf ϕ ∈ Dfin and

A[Af ϕ] ⊆ A[ϕ] ∪ (A[ϕ] +⋃

h∈B[ϕ]T (h)) = A′[ϕ];

B[Af ϕ] ⊆ B[ϕ] ∪ (B[ϕ] −⋃l∈A[ϕ]

S(l)) = B ′[ϕ].

For an arbitraryϕ ∈ D(Af ) ιAf ϕ satisfies (3.16) withA[ϕ] = L, B[ϕ] = K .For arbitrary finite setsA ⊆ L, B ⊆ K , denote byP(A,B) the orthonormalprojector ofL2(G) onto the space of functionsϕ such thatA[ϕ] ⊆ A, B[ϕ] ⊆ B.Now by (3.16) it is easy to see thatP(A,B)Af ϕ = AfP (A′, B ′)ϕ, where

A′ = A ∪(A−

⋃h∈B

T (h)

); B ′ = B ∪

(B +

⋃l∈AS(l)

).


For an arbitraryε > 0 we can find finiteA ⊆ L, B ⊆ K such that‖P(A,B)ϕ −ϕ‖ < ε and ‖P(A,B)Af ϕ − Af ϕ‖ < ε. SinceA ⊆ A′, B ⊆ B ′ we have‖P(A′, B ′)ϕ − ϕ‖ < ε. So, ifψ = P(A′, B ′)ϕ, thenψ ∈ Dfin and‖ϕ − ψ‖ <ε, ‖Af ϕ − Afψ‖ < ε. This proves thatDfin is an essential domain forAf .

Now we have to show that, forN ∈ ∗N \ N and for any standardψ ∈ Dfin,SN(Afψ) ≈ A

(N)f SNψ . Here it is convenient to rewrite (3.14) and (3.15) in the

form

Afψ(x) = a(x) · ψ(x)+ F −1(b · ψ)(x);A(N)f ϕ(x) = a(jN (x)) · ϕ(x)+ F−1

N (b · FNϕ)(x).Obviously

suppψ ⊆⋃l∈A[ψ]

al +K = C; suppψ ⊆⋃

h∈B[ψ]ph + L = D.

Formulae (3.8) and (3.9) show that

suppSNψ ⊆⋃{αλ +KN | λ ∈ j ′−1

N (A[ψ])} = CN;(3.17)

suppFNSNψ ⊆⋃{πν + LN | ν ∈ j ′−1

N (B[ψ])} = DN.

But

SNAfψ = SN(a · ψ)+ SNF −1(b · ψ)and

A(N)f SNψ = a ◦ jN · SNψ + F−1

N (b ◦ jN · FNSNψ).Sinceψ has a compact support,a · ψ belongs to the subspaceY of Proposition 17and thusSN(a · ψ) ≈ TN(a · ψ) = a ◦ jN · ψ ◦ jN . We have

‖a ◦ jN · SNψ − a ◦ jN · ψ ◦ jN‖2N= 1N

∑x∈CN|a(jN(x))(SNψ(x)− ψ(jN(x))|2 ≈ 0,

sincea is bounded onC. SoSN(a ·ψ) ≈ a ◦ jN · SNψ , and it remains to be provedthatSNF −1(b · ψ) ≈ F−1

N (b ◦ jN · FNSNψ).SinceAfψ ∈ Dfin ⊆ Y and, as it was just proved,a · ψ ∈ Y , F −1(b · ψ) ∈ Y ,

and thusSNF −1(b · ψ) ≈ TNF −1(b · ψ) = F −1(b · ψ) ◦ jN . SinceSNψ ≈ TNψandFN is bounded, we have

FNSNψ ≈ FNTNψ ≈ ψ ◦ jN .Obviously suppψ ◦ jN ⊆ DN . Now using (3.17) and the fact thatb is bounded onD, we obtain thatb◦ jN ·FNSNψ ≈ b◦ jN ·ψ ◦ jN , and thusF−1

N (b◦ jN ·FNSNψ) ≈F−1N (b ◦ jN · ψ ◦ jN ) ≈ F −1(b · ψ) ◦ jN . 2


EXAMPLE. We return to the example of2 of Section 2. So in this case,G = Qp

and alsoG = Qp up to isomorphism. In [VVZ] Schrödinger type operators withsymbols of the form

f (x, χ) = a(|x|p)+ b(|χ |p)were considered. Ifa(|x|p) → ∞ whenx → ∞ andb(|χ |p) → ∞ whenχ →∞, then such operators satisfy the conditions of the last proposition, because thefunctionsa andb are constant on the cosetsQp/Zp, and thus the cardinalities ofthe setsS(l) andT (h) of the proposition are equal to 1. Using the pair of dualapproximating sequences forQp, described in the example in2 of Section 2, thereis no problem in writing the approximating operatorsA(n)f defined by (3.15) for thiscase: forn = r + s

A(n)f ϕ(k) = a(pr |k|p)+

1

pn

n−1∑j,m=0

b(ps|m|p)ϕ(k − j)exp2πijm

n.

3. The symbols, considered in the previous sections, are generalizations of the well-knownqp-symbols to the case of arbitrary LCA groups. The connection betweenthe qp-symbols of the operatorsA andA∗ is not very simple, and thus the con-ditions whichf has to satisfy forAf to be self-adjoint are rather complicated,and they do not imply the self-adjointness ofA(n)f or B(n)f for self-adjointAf . Inthe theory of pseudodifferential operators inL2(Rn) the symmetrical, or Weyl,symbols are also considered. An operatorWf with Weyl symbolf is self-adjointiff f is real [BSh].

The notion of Weyl symbol can be generalized to the case of a LCA groupG ifdivision by two is defined inG. This means that∀x ∈ G, ∃y ∈ G, y + y = x and∀y ∈ G, y + y = 0⇒ y = 0. We shall denote thisy by 1

2x and suppose that theoperationx 7→ 1

2x is continuous inG. Note that ifG admits division by two, thenG also admits it:12χ(x) = χ(1

2x).The operatorWf with the Weyl symbolf : G × G → C is defined by the

formula (cf. (0.5)):

Wf =∫ ∫

G×Gf (y, γ )UyVγ γ (

12y) dµ(γ )dµ(y), (3.18)

where

f (y, γ ) =∫ ∫

G×Gf (x, χ)χ(γ )γ (x)dµ(χ)dµ(x).

It is easy to see that this operator is symmetric ifff is real; and iff is of the forma(x)+ b(χ), thenAf = Wf .

It is easy to compute the kernelKf (x, y) of the operatorWf . Let us denote byf (2)(x, y) the inverse Fourier transform off (x, χ) with respect to the variableχ :

f (2)(x, y) =∫G

f (x, χ)χ(y)dµ(χ).


Then

Kf (x, y) = f (2)(x + y

2, y − x

).

If the groupsGn of our a.s.〈Gn, jn〉 also admit the division by two we saythat the division by two inG is approximable by〈Gn, jn〉 if (cf. Definition 3 andTheorem 2)

∀ε > 0,∀K ∈ K, ∃N > 0,∀n > N,∀g ∈ j (−1)N (K),

ρ

(jn

(g

2

), 1

2jn(x)

)< ε.

It is easy to see that ifp is an odd prime number, then the division by two inQp isapproximable by the a.s. described in the example in Section 2.

If the division by two is approximable then we define the sequence of approxi-mating operatorsW(n)

f in the following way. Define a functionfn: Gn × Gn → Cby the formula:

fn(g, κ) = f (jn(g), jn(κ)),and letfn be the finite Fourier transform offn:

fn(h, χ) = 1

|Gn|∑g,κ

fn(g, κ)χ(g)κ(h).

Then

W(n)f =

1

|Gn|∑h,χ

fn(h, χ)UhVχχ(12h),

where(Uhϕ)(g) = ϕ(g + h), (Vχϕ)(g) = χ(g) · ϕ(g).Denote byf (2)n the inverse Fourier transform offn with respect to the second

variable:

f (2)n (g, s) = 1n

∑κ

fn(g, κ)κ(s).

Then

(W(n)f ϕ)(s) = 1n

∑g

f (2)n

(s + g

2, g − s

)ϕ(g).

Similarly to Theorem 6 we can prove the following

PROPOSITION 22. If f ∈ S2(G × G), then the sequenceW(n)f converges dis-

cretely toWf with respect to the strong discrete approximation〈Xn, Sn〉 fromProposition18 and this convergence is uniform.


Corollaries 1 and 2 of Theorem 6 can be applied to this case. Moreover, Corol-lary 2 holds just for the sequenceW(n)

f , because these operators are self-adjoint (thefunctionfn is real!).

Acknowledgement

The authors are very grateful to Professor M. Wolff (Tübingen) and to ProfessorI. Schereshevskii (Nizhnii Novgorod) for helpful discussions.

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Finite-Dimensional Approximations of Operators in …yigordon/papers/AGKh.pdfthe constructed approximations, the methods of nonstandard analysis are used. Mathematics Subject Classiﬁcations