MATH 590: Meshfree Methodsfass/Notes590_Ch7.pdfby assumptionwe know that the above quadratic form of A (with coefﬁcients such that PTc = 0) is zero only for c = 0. Therefore (4)

MATH 590: Meshfree MethodsChapter 7: Conditionally Positive Definite Functions

Greg Fasshauer

Department of Applied MathematicsIllinois Institute of Technology

Fall 2010

[email protected] MATH 590 – Chapter 7 1

http://math.iit.edu/~fass

Outline

1 Conditionally Positive Definite Functions Defined

2 CPD Functions and Generalized Fourier Transforms



Conditionally Positive Definite Functions Defined

Outline






In this chapter we generalize positive definite functions toconditionally positive definite and strictly conditionally positivedefinite functions of order m.These functions provide a natural generalization of RBFinterpolation with polynomial reproduction.Examples of strictly conditionally positive definite (radial) functionsare given in the next chapter.




Definition

A complex-valued continuous function Φ is called conditionally positivedefinite of order m on Rs if

N∑j=1

N∑k=1

cjck Φ(x j − xk ) ≥ 0 (1)

for any N pairwise distinct points x1, . . . ,xN ∈ Rs, andc = [c1, . . . , cN ]T ∈ CN satisfying

N∑j=1

cjp(x j) = 0,

for any complex-valued polynomial p of degree at most m − 1.The function Φ is called strictly conditionally positive definite of order mon Rs if the quadratic form (1) is zero only for c ≡ 0.




An immediate observation is

LemmaA function that is (strictly) conditionally positive definite of order m onRs is also (strictly) conditionally positive definite of any higher order. Inparticular, a (strictly) positive definite function is always (strictly)conditionally positive definite of any order.

Proof.The first statement follows immediately from the definition.The second statement is true since (by convention) the casem = 0 yields the class of (strictly) positive definite functions, i.e.,(strictly) conditionally positive definite functions of order zero are(strictly) positive definite.






Proof.The first statement follows immediately from the definition.

The second statement is true since (by convention) the casem = 0 yields the class of (strictly) positive definite functions, i.e.,(strictly) conditionally positive definite functions of order zero are(strictly) positive definite.






Proof.The first statement follows immediately from the definition.The second statement is true since (by convention) the casem = 0 yields the class of (strictly) positive definite functions, i.e.,(strictly) conditionally positive definite functions of order zero are(strictly) positive definite.




As for positive definite functions we also have (see [Wendland (2005a)]for more details)

TheoremA real-valued continuous even function Φ is called conditionallypositive definite of order m on Rs if

N∑j=1

N∑k=1

cjck Φ(x j − xk ) ≥ 0 (2)

for any N pairwise distinct points x1, . . . ,xN ∈ Rs, andc = [c1, . . . , cN ]T ∈ RN satisfying

N∑j=1

cjp(x j) = 0,

for any real-valued polynomial p of degree at most m − 1.The function Φ is called strictly conditionally positive definite of order mon Rs is zero only for c ≡ 0.




RemarkIf the function Φ is strictly conditionally positive definite of order m,then the matrix A with entries Ajk = Φ(x j − xk ) can be interpreted asbeing positive definite on the space of vectors c such that

N∑j=1

cjp(x j) = 0, p ∈ Πsm−1.

In this sense A is positive definite on the space of vectors c“perpendicular” to s-variate polynomials of degree at most m − 1.




RemarkIf the function Φ is strictly conditionally positive definite of order m,then the matrix A with entries Ajk = Φ(x j − xk ) can be interpreted asbeing positive definite on the space of vectors c such that

N∑j=1

cjp(x j) = 0, p ∈ Πsm−1.

In this sense A is positive definite on the space of vectors c“perpendicular” to s-variate polynomials of degree at most m − 1.




We can now generalize the theorem we had in the previous chapter forconstant precision interpolation to the case of general polynomialreproduction:

Theorem

If the real-valued even function Φ is strictly conditionally positivedefinite of order m on Rs and the points x1, . . . ,xN form an(m − 1)-unisolvent set, then the system of linear equations[

A PPT O

] [cd

]=

[y0

](3)

is uniquely solvable.




We can now generalize the theorem we had in the previous chapter forconstant precision interpolation to the case of general polynomialreproduction:

Theorem

If the real-valued even function Φ is strictly conditionally positivedefinite of order m on Rs and the points x1, . . . ,xN form an(m − 1)-unisolvent set, then the system of linear equations[

A PPT O

] [cd

]=

[y0

](3)

is uniquely solvable.




Proof

The proof is almost identical to the proof of the earlier theorem forconstant reproduction.

Assume [c,d ]T is a solution of the homogeneous linear system, i.e.,with y = 0.

We show that [c,d ]T = 0 is the only possible solution.

Multiplication of the top block of (3) by cT yields

cT Ac + cT Pd = 0.

From the bottom block of the system we know PT c = 0. This impliescT P = 0T , and therefore

cT Ac = 0. (4)




Proof





cT Ac + cT Pd = 0.


cT Ac = 0. (4)




Proof





cT Ac + cT Pd = 0.


cT Ac = 0. (4)




Proof





cT Ac + cT Pd = 0.


cT Ac = 0. (4)




Proof





cT Ac + cT Pd = 0.


cT Ac = 0. (4)




Since the function Φ is strictly conditionally positive definite of order mby assumption we know that the above quadratic form of A (withcoefficients such that PT c = 0) is zero only for c = 0.

Therefore (4) tells us that c = 0.

The unisolvency of the data sites, i.e., the linear independence of thecolumns of P (c.f. one of our earlier remarks), and the fact that c = 0guarantee d = 0 from the top block

Ac + Pd = 0

of the homogeneous version of (3). �







Ac + Pd = 0








Ac + Pd = 0




CPD Functions and Generalized Fourier Transforms

Outline






As before, integral characterizations help us identify functions that arestrictly conditionally positive definite of order m on Rs.

An integral characterization of conditionally positive definite functionsof order m, i.e., a generalization of Bochner’s theorem, can be found inthe paper [Sun (1993b)].

However, since the subject matter is rather complicated, and since itdoes not really help us solve the scattered data interpolation problem,we do not mention any details here.















CPD Functions and Generalized Fourier Transforms The Schwartz Space and the Generalized Fourier Transform

In order to formulate the Fourier transform characterization of strictlyconditionally positive definite functions of order m on Rs we requiresome advanced tools from analysis (see Appendix B).

First we define the Schwartz space of rapidly decreasing test functions

S = {γ ∈ C∞(Rs) : lim‖x‖→∞

xα(Dβγ)(x) = 0, α,β ∈ Ns0},

where we use the multi-index notation

Dβ =∂|β|

∂xβ11 · · · ∂xβs

s, |β| =

s∑i=1

βi .




In order to formulate the Fourier transform characterization of strictlyconditionally positive definite functions of order m on Rs we requiresome advanced tools from analysis (see Appendix B).

First we define the Schwartz space of rapidly decreasing test functions

S = {γ ∈ C∞(Rs) : lim‖x‖→∞

xα(Dβγ)(x) = 0, α,β ∈ Ns0},

where we use the multi-index notation

Dβ =∂|β|

∂xβ11 · · · ∂xβs

s, |β| =

s∑i=1

βi .




Some properties of the Schwartz space

S consists of all those functions γ ∈ C∞(Rs) which, together withall their derivatives, decay faster than any power of 1/‖x‖.

S contains the space C∞0 (Rs), the space of all infinitelydifferentiable functions on Rs with compact support.C∞0 (Rs) is a true subspace of S since, e.g., the functionγ(x) = e−‖x‖

2belongs to S but not to C∞0 (Rs).

γ ∈ S has a classical Fourier transform γ̂ which is also in S.





S consists of all those functions γ ∈ C∞(Rs) which, together withall their derivatives, decay faster than any power of 1/‖x‖.S contains the space C∞0 (Rs), the space of all infinitelydifferentiable functions on Rs with compact support.

C∞0 (Rs) is a true subspace of S since, e.g., the functionγ(x) = e−‖x‖







S consists of all those functions γ ∈ C∞(Rs) which, together withall their derivatives, decay faster than any power of 1/‖x‖.S contains the space C∞0 (Rs), the space of all infinitelydifferentiable functions on Rs with compact support.C∞0 (Rs) is a true subspace of S since, e.g., the functionγ(x) = e−‖x‖







S consists of all those functions γ ∈ C∞(Rs) which, together withall their derivatives, decay faster than any power of 1/‖x‖.S contains the space C∞0 (Rs), the space of all infinitelydifferentiable functions on Rs with compact support.C∞0 (Rs) is a true subspace of S since, e.g., the functionγ(x) = e−‖x‖






Of particular importance are the following subspaces Sm of S

Sm = {γ ∈ S : γ(x) = O(‖x‖m) for ‖x‖ → 0, m ∈ N0}.

Furthermore, the set V of slowly increasing functions is given by

V = {f ∈ C(Rs) : |f (x)| ≤ |p(x)| for some polynomial p ∈ Πs}.




Of particular importance are the following subspaces Sm of S

Sm = {γ ∈ S : γ(x) = O(‖x‖m) for ‖x‖ → 0, m ∈ N0}.

Furthermore, the set V of slowly increasing functions is given by

V = {f ∈ C(Rs) : |f (x)| ≤ |p(x)| for some polynomial p ∈ Πs}.




The generalized Fourier transform is now given by

Definition

Let f ∈ V be complex-valued. A continuous function f̂ : Rs \ {0} → C iscalled the generalized Fourier transform of f if there exists an integerm ∈ N0 such that ∫

Rsf (x)γ̂(x)dx =

∫Rs

f̂ (x)γ(x)dx

is satisfied for all γ ∈ S2m.The smallest such integer m is called the order of f̂ .

RemarkVarious definitions of the generalized Fourier transform exist in theliterature (see, e.g., [Gel’fand and Vilenkin (1964)]).




The generalized Fourier transform is now given by

Definition

Let f ∈ V be complex-valued. A continuous function f̂ : Rs \ {0} → C iscalled the generalized Fourier transform of f if there exists an integerm ∈ N0 such that ∫

Rsf (x)γ̂(x)dx =

∫Rs

f̂ (x)γ(x)dx

is satisfied for all γ ∈ S2m.The smallest such integer m is called the order of f̂ .

RemarkVarious definitions of the generalized Fourier transform exist in theliterature (see, e.g., [Gel’fand and Vilenkin (1964)]).




Since one can show that the generalized Fourier transform of ans-variate polynomial of degree at most 2m is zero, it follows thatthe inverse generalized Fourier transform is only unique up toaddition of such a polynomial.

The order of the generalized Fourier transform is nothing but theorder of the singularity at the origin of the generalized Fouriertransform.For functions in L1(Rs) the generalized Fourier transformcoincides with the classical Fourier transform.For functions in L2(Rs) it coincides with the distributional Fouriertransform.




Since one can show that the generalized Fourier transform of ans-variate polynomial of degree at most 2m is zero, it follows thatthe inverse generalized Fourier transform is only unique up toaddition of such a polynomial.The order of the generalized Fourier transform is nothing but theorder of the singularity at the origin of the generalized Fouriertransform.

For functions in L1(Rs) the generalized Fourier transformcoincides with the classical Fourier transform.For functions in L2(Rs) it coincides with the distributional Fouriertransform.




Since one can show that the generalized Fourier transform of ans-variate polynomial of degree at most 2m is zero, it follows thatthe inverse generalized Fourier transform is only unique up toaddition of such a polynomial.The order of the generalized Fourier transform is nothing but theorder of the singularity at the origin of the generalized Fouriertransform.For functions in L1(Rs) the generalized Fourier transformcoincides with the classical Fourier transform.

For functions in L2(Rs) it coincides with the distributional Fouriertransform.




Since one can show that the generalized Fourier transform of ans-variate polynomial of degree at most 2m is zero, it follows thatthe inverse generalized Fourier transform is only unique up toaddition of such a polynomial.The order of the generalized Fourier transform is nothing but theorder of the singularity at the origin of the generalized Fouriertransform.For functions in L1(Rs) the generalized Fourier transformcoincides with the classical Fourier transform.For functions in L2(Rs) it coincides with the distributional Fouriertransform.



CPD Functions and Generalized Fourier Transforms Fourier Transform Characterization

This general approach originated in the manuscript[Madych and Nelson (1983)]. Many more details can be found in theoriginal literature as well as in [Wendland (2005a)].The following result is due to [Iske (1994)].

Theorem

Suppose the complex-valued function Φ ∈ V possesses a generalizedFourier transform Φ̂ of order m which is continuous on Rs \ {0}. ThenΦ is strictly conditionally positive definite of order m if and only if Φ̂ isnon-negative and non-vanishing.

RemarkThis theorem states that strictly conditionally positive definite functionson Rs are characterized by the order of the singularity of theirgeneralized Fourier transform at the origin, provided that thisgeneralized Fourier transform is non-negative and non-zero.




This general approach originated in the manuscript[Madych and Nelson (1983)]. Many more details can be found in theoriginal literature as well as in [Wendland (2005a)].The following result is due to [Iske (1994)].

Theorem

Suppose the complex-valued function Φ ∈ V possesses a generalizedFourier transform Φ̂ of order m which is continuous on Rs \ {0}. ThenΦ is strictly conditionally positive definite of order m if and only if Φ̂ isnon-negative and non-vanishing.

RemarkThis theorem states that strictly conditionally positive definite functionson Rs are characterized by the order of the singularity of theirgeneralized Fourier transform at the origin, provided that thisgeneralized Fourier transform is non-negative and non-zero.




Since integral characterizations similar to our earlier theorems ofSchoenberg for positive definite radial functions are so complicated inthe conditionally positive definite case we do not pursue the concept ofa conditionally positive definite radial function here.Such theorems can be found in [Guo et al. (1993a)].

Examples of radial functions via the Fourier transform approachare given in the next chapter.In Chapter 9 we will explore the connection between completelyand multiply monotone functions and conditionally positive definiteradial functions.




Since integral characterizations similar to our earlier theorems ofSchoenberg for positive definite radial functions are so complicated inthe conditionally positive definite case we do not pursue the concept ofa conditionally positive definite radial function here.Such theorems can be found in [Guo et al. (1993a)].

Examples of radial functions via the Fourier transform approachare given in the next chapter.In Chapter 9 we will explore the connection between completelyand multiply monotone functions and conditionally positive definiteradial functions.



Appendix References

References I

Buhmann, M. D. (2003).Radial Basis Functions: Theory and Implementations.Cambridge University Press.

Fasshauer, G. E. (2007).Meshfree Approximation Methods with MATLAB.World Scientific Publishers.

Gel’fand, I. M. and Vilenkin, N. Ya. (1964).Generalized Functions Vol. 4.Academic Press (New York).

Iske, A. (2004).Multiresolution Methods in Scattered Data Modelling.Lecture Notes in Computational Science and Engineering 37, Springer Verlag(Berlin).

Wendland, H. (2005a).Scattered Data Approximation.Cambridge University Press (Cambridge).



Appendix References

References II

Guo, K., Hu, S. and Sun, X. (1993a).Conditionally positive definite functions and Laplace-Stieltjes integrals.J. Approx. Theory 74, pp. 249–265.

Iske, A. (1994).Charakterisierung bedingt positiv definiter Funktionen für multivariateInterpolationsmethoden mit radial Basisfunktionen.Ph.D. Dissertation, Universität Göttingen.

Madych, W. R. and Nelson, S. A. (1983).Multivariate interpolation: a variational theory.manuscript.

Sun, X. (1993b).Conditionally positive definite functions and their application to multivariateinterpolation.J. Approx. Theory 74, pp. 159–180.



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MATH 590: Meshfree Methodsfass/Notes590_Ch7.pdfby assumptionwe know that the above quadratic form of A (with coefﬁcients such that PTc = 0) is zero only for c = 0. Therefore (4)