WITH RESPECT TO MEASURES RELATED PROBLEMSdownloads.hindawi.com/journals/ijmms/1993/860245.pdfWego back to consider orthogonal polynomials w.r.t, varying measures. Thefollowing result,

Internat. J. Math. & Math. Scl.VOL. 16 NO. 2 (1993) 255-258

255

REMARKS ON ORTHOGONAL POLYNOMIALS WITH RESPECT TOVARYING MEASURES AND RELATED PROBLEMS

XlN LI

Department of MathematicsUniversity of Central Florida

Orlando, FL 32816

(Received March 26, 1992 and in revised form May 4, 1992)

ABSTRACT. We point out the relation between the orthogonal polynomials with respect to

(w.r.t.) varying measures and the so-called orthogonal rationals on the unit circle in the complex

plane. This observation enables us to combine different techniques in the study of these

polynomials and rationals. As an example, we present a simple and short proof for a known result

on the weak-star convergence of orthogonal polynomials w.r.t, varying measures. Some related

problems are also considered.

KEY WORDS AND PHRASES. Orthogonal polynomial and rational, weak-star convergence.

1991 AMS SUBJECT CLASSIFICATION CODES. Primary 42A05, Secondary 33C45, 41A28.

1. INTRODUCTION.Let a be a finite positive measure on the unit circle r: {zECI zl 1}, and let

w,(z) zn+.. be a sequence of polynomials whose zeros {z,,,}’= all lie in the closed unit disk

zl _< 1. Assume

r ,,,(z) 12 < ’then, for each n, we can consider the orthonormal polynomials ,,,m(z)=a,.,z"+---(a,,,>O)w.r.t, d/I w,,(z) 12, i.e., polynomials satisfying

and

] z-J,n,m{z) =0, j=0,1 m,r w,(z)

f i,,.,(z)ld,

2 r iw,(z)l_X.The sequences {,,,(z)}= 0 are called orthogonal polynomials w.r.t, varying measures der/Iw,(z)

(n 1,2 ). They appear in the study of simultaneous Pad approximation and related problems.

The following weak-star convergence result plays an important role in the study of convergence of

{,,,()}= -

256 X. LI

THEOREM 1. (Lopez, [8])limn._.,oo n= 1( z,, l) o, then

in the weak-star topology.

If the zeros of w.(z) satisfy the condition

One nice thing about this result is that it holds without any condition on the finite positive

measure da. We will present a simple and short proof of Theorem 1. But before we get to the

proof, we remark that the so-called orthogonal rationals (with prescribed poles) are also of current

interest, especially in problems related to electrical engineering. (See, e.g., [2, 3, 4, 5]) These

applications were more concerned with the algebraic properties of orthogonal rationals. It turns out

that orthogonal rationals are just special case of orthogonal polynomials w.r.t, varying measures. It

is the main purpose of this note to point out this relation. To see this relation, let us recall the

definition of orthogonal rationals with respect to

Let {zi},= be points in zl _< 1, and denote O,(z): l-I’= 10- z,).DEFINITION 2. (cf. [2]) For n 1,2, the rationals P,(z) with P,(z) a,z" + (a,, > O)

satisfying

(z).for r(z) tQn{z), q 9n- 1}, and -/ F Q-oPn(z) ’-(z)do. =0

rdo.=l,

are called orthogonal rationals w.r.t.

Now, from Definition 2, it can be seen that with Q,(z) treated as w,(z), the corresponding

.,,(z) is P,(z). So orthogonal rationals are indeed special case of orthogonal polynomials w.r.t.

varying measures.

We go back to consider orthogonal polynomials w.r.t, varying measures. The following result,

which is essentially proved in the orthogonal rational setting (cf. e.g., [2]), will help us to simplify

the proof of Theorem 1 in [8].LEMMA 3. If r and s belong to %(w,): {p(z)/w.(z): p ,,}, then

/+/-2 rr(z)7 .,.()1 dz rr(Z)S(z)do."

PROOF. By [6, formula (1.20)], we have

r I,,,,(z)l ldzl r w,(z)

for all p, q 9n. So,

/ p(z) (q(z) Wn(Z) 12 2j./ p(z) (q(z) do’,2-7 r K,o.(z)/I I lagr \w.(z)J

which implies the lemma.

REMARKS. (i) Taking r in Lemma 3 gives us

do..r ,,,(z) dzl r

(1)

ORTHOGONAL POLYNOMIALS WITH RESPECT TO VARYING MEASURES 257

(ii) Lemma 3 is equivalent to the following:

/ z-"t’2"(z) w"(z) 2J/ z-"t’2"(z)do’,Idzl r Iw,(z)[

(2)

for all P2. E 92,,.We also need the following result about weighted approximation with varying weights in

Loo-norm.LEMMA 4. The space

T.: iw.(z)12"P2. E2,,

is dense in c(r) in the L-norm if and only if

lira (1-

PROOF. This is a simple generalization of some known results (see, e.g., [1, p. 244] and [2,Theorem 8..2]). We only mention the following fact (cf. [1, p. 243])

inf II z,.It) ( P2n + l’I ,n,, # o(Z Zn, ,)(Z 1/Z--,;) z.,, #o

PROOF OF THEOREM 1. Now the proof of Theorem 1 follows from (1), (2), and Lemma 4

by using a standard argument (cf. e.g., [9, p. 248]).

Recently, in the study of frequency analysis (see, e.g., [7]), the following orthogonalpolynomials w.r.t, varying measures appear naturally, i.e., polynomials

+ (3,.,, > 0) satisfying

l___f z-,g,n.(z) lwn(z)O2da=O,j=O, mlF

and

/ . ,.(z)w.(z)l 2da 1.2"j F

For convenience, let us assume that the zeros of w,(z) all lie outside the unit circle r, i.e., z,.il > 1.

The asymptotics of ,,.,,(z)w,,(z) are wanted in application ([7]). Here we will not discuss this

problem but instead point out the relation of these polynomials {,.,,} with the following ones

{,,.,,}: Assume An. > 0(1 < <_ n,n >_ 1) are given, then define polynomials ,.,,(z)+ (7,.,, > 0) satisfying

r i=1

for j 0,1 m + n 1, and

(z) 12da + A,, ,. ,,(z,. i) 12 .F i=1

The polynomials {#,.,,} 0 are the orthonormal polynomials w.r.t, measure da/(2,)

+ ’= 1A,.z, having finitely many mass points off the unit circle r (where $, denotes the point.,unit measure with support at z). We have

258 X. LI

THEOREM 5. For each n fixed, there hold

lim ,,,,,(z) ,,,,,,(z)w,,(z),

l<i<n

locally uniformly in C, where m 0,1,2,

PROOF. We use a normal family argument. For n and m chosen, since f rl,,,(z) lZda _< 2,

then the set : {n,m:An,,>O, <_i <_n} as a subset of finite dimensional normed space is a

uniformly bounded set. Thus as a set of analytic functions (polynomials) is a normal family over

C. Let f(z)E 9,+,, be a limit of a subsequence O’c O, then since [’__IA,,, J,,,,,(z,,,)J <_ 1, we

have

lim

’,, ,(z,, ,) 0f(z,,,)

6,,,,, e

for i= 1,2 n. So 1’(z)=wn(z)g(z with some g(z)eg,. Now we just need to use the extremal

property of (monic) orthogonal polynomials to assert that g(z)= Cn,,Jz). This completes the proofof Theorem 5.

REFERENCES

1. ACHIESER, N.I., Theory of Approximation, Frederick Ungar Publ. Co., New York, 1956.

2. BULTHEEL, A., GONZALEZ-VERA, P., HENDRIKSEN, E., & NJASTAD, O., A Szegtltheory for rational functions, manuscript, 1991.

3. DEWILDE, P., & DYM, H., Lossless inverse scattering, digital filters, and estimation theory,IEEE Trans. Inform. Theory, Vol. IT-30 (1984), 644-663.

4. DEWILDE, P., & DYM, H., Schur recursions, error formulas, and convergence of rationalestimators for stationary stochastic sequences, IEEE Tran. Inform. Theory, Vol. IT-27(198I), 446-461

5. DJRBASHIAN, M.M., A survey on the theory of orthogonal systems and some openproblems, Orthogonal Polynomials: Theory and Applications, (P. Nevai ed.), KluwerAcad. Publ., Boston, 1990, 135-146.

6. GERONIMUS, YA., Orthogonal Polynomials. Consultants Bureau, New York, 1961.

7. JONES, W.B., NJASTAD, O., & SAFF, E.B., Szeg6 polynomials associated with Wiener-Levinson filters, J. Comp. Appl. Math. 32 (1990), 387-406.

8. LOPEZ LAGOMASINO, G., Asymptotics of polynomials orthogonal with respect to varyingmeasures, Constr. Approx. 5 (1989), 199-219.

9. MT, A., NEVAI, P. & TOTIK, V., Strong and weak convergence of orthogonalpolynomials, Amer. J. Math. 109 (1987), 239-282.

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WITH RESPECT TO MEASURES RELATED PROBLEMSdownloads.hindawi.com/journals/ijmms/1993/860245.pdfWego back to consider orthogonal polynomials w.r.t, varying measures. Thefollowing result,