MATH 590: Meshfree Methods

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MATH 590: Meshfree Methods

Chapter 7: Conditionally Positive Definite Functions

Greg Fasshauer

Department of Applied Mathematics Illinois Institute of Technology

Fall 2010

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Outline

1 Conditionally Positive Definite Functions Defined

2 CPD Functions and Generalized Fourier Transforms

[email protected] MATH 590 – Chapter 7 2

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Conditionally Positive Definite Functions Defined

Outline

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In this chapter we generalize positive definite functions to conditionally positive definite and strictly conditionally positive definite functions of order m.

These functions provide a natural generalization of RBF interpolation with polynomial reproduction.

Examples of strictly conditionally positive definite (radial) functions are given in the next chapter.

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Definition

A complex-valued continuous function Φ is calledconditionally positive definite of order m on R^sif

N

X

j=1 N

X

k =1

c_jc_kΦ(x_j− xk) ≥0 (1)

for any N pairwise distinct pointsx₁, . . . ,x_N ∈ R^s, and c = [c₁, . . . ,c_N]^T ∈ C^N satisfying

N

X

j=1

c_jp(x_j) =0,

for any complex-valued polynomial p of degree at most m − 1.

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An immediate observation is Lemma

A function that is (strictly) conditionally positive definite of order m on R^s is also (strictly) conditionally positive definite of any higher order. In particular, 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 case m = 0 yields the class of (strictly) positive definite functions, i.e., (strictly) conditionally positive definite functions of order zero are (strictly) positive definite.

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Proof.

The first statement follows immediately from the definition.

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

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Proof.

The first statement follows immediately from the definition.

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

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

for more details) Theorem

A real-valued continuous even function Φ is calledconditionally positive definite of order m on R^s if

N

X

j=1 N

X

k =1

c_jc_kΦ(x_j− x_k) ≥0 (2) for any N pairwise distinct pointsx₁, . . . ,x_N ∈ R^s, and

c = [c₁, . . . ,c_N]^T ∈ R^N satisfying

N

X

j=1

c_jp(x_j) =0,

for any real-valued polynomial p of degree at most m − 1.

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Remark

If the function Φ is strictly conditionally positive definite of order m, then the matrix A with entries A_jk = Φ(x_j − x_k)can be interpreted as being positive definite on the space of vectorsc such that

N

X

j=1

c_jp(x_j) =0, p ∈ Π^s_m−1.

In this senseA is positive definite on the space of vectorsc

“perpendicular” to s-variate polynomials of degree at most m − 1.

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Remark

If the function Φ is strictly conditionally positive definite of order m, then the matrix A with entries A_jk = Φ(x_j − x_k)can be interpreted as being positive definite on the space of vectorsc such that

N

X

j=1

c_jp(x_j) =0, p ∈ Π^s_m−1.

In this senseA is positive definite on the space of vectorsc

“perpendicular” to s-variate polynomials of degree at most m − 1.

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We can now generalize the theorem we had in the previous chapter for constant precision interpolation to the case of general polynomial reproduction:

Theorem

If the real-valued even function Φ is strictly conditionally positive definite of order m on R^sand the pointsx₁, . . . ,x_N form an (m − 1)-unisolvent set, then the system of linear equations

A P

P^T O

c d

=

y 0

(3)

is uniquely solvable.

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We can now generalize the theorem we had in the previous chapter for constant precision interpolation to the case of general polynomial reproduction:

Theorem

If the real-valued even function Φ is strictly conditionally positive definite of order m on R^sand the pointsx₁, . . . ,x_Nform an (m − 1)-unisolvent set, then the system of linear equations

A P

P^T O

c d

=

y 0

(3)

is uniquely solvable.

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Proof

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

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

Weshow that [c, d]^T =0 is the only possible solution. Multiplication of the top block of (3) byc^T yields

c^TAc + c^TPd = 0.

From the bottom block of the system we know P^Tc = 0. This implies c^TP =0^T, and therefore

c^TAc = 0. (4)

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Proof

Weshow that [c, d]^T =0 is the only possible solution. Multiplication of the top block of (3) byc^T yields

c^TAc + c^TPd = 0.

c^TAc = 0. (4)

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Proof

Weshow that [c, d]^T =0 is the only possible solution.

Multiplication of the top block of (3) byc^T yields c^TAc + c^TPd = 0.

c^TAc = 0. (4)

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Proof

c^TAc = 0. (4)

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Proof

c^TAc = 0. (4)

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Sincethe function Φ is strictly conditionally positive definite of order m by assumptionwe know that the above quadratic form of A (with coefficients such that P^Tc = 0) is zero only forc = 0.

Therefore (4) tells us thatc = 0.

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

Ac + Pd = 0 of the homogeneous version of (3).

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CPD Functions and Generalized Fourier Transforms

Outline

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As before,integral characterizationshelp us identify functions that are strictly conditionally positive definite of order m on R^s.

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

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

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

In order to formulate theFourier transform characterizationof strictly conditionally positive definite functions of order m on R^s we require some advanced tools from analysis (see Appendix B).

First we define theSchwartz spaceof rapidly decreasing test functions S = {γ ∈ C^∞(R^s) : lim

kxk→∞x^α(D^βγ)(x) = 0, α, β ∈ N^s0}, where we use the multi-index notation

D^β = ∂^|β|

∂x₁^β¹· · · ∂x_s^β^s, |β| =

s

X

i=1

βi.

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In order to formulate theFourier transform characterizationof strictly conditionally positive definite functions of order m on R^s we require some advanced tools from analysis (see Appendix B).

First we define theSchwartz spaceof rapidly decreasing test functions S = {γ ∈ C^∞(R^s) : lim

kxk→∞x^α(D^βγ)(x) = 0, α, β ∈ N^s0}, where we use the multi-index notation

D^β = ∂^|β|

∂x₁^β¹· · · ∂x_s^β^s, |β| =

s

X

i=1

β_i.

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Some properties of the Schwartz space

S consists of all those functions γ ∈ C^∞(R^s)which, together with all their derivatives, decay faster than any power of 1/kxk.

S contains the space C₀^∞(R^s), the space of all infinitely differentiable functions on R^s with compact support. C₀^∞(R^s)is atrue subspaceof S since, e.g., the function γ(x) = e^−kxk² belongs to S but not to C₀^∞(R^s).

γ ∈ S has aclassical Fourier transformˆγ which is also in S.

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Some properties of the Schwartz space

S contains the space C₀^∞(R^s), the space of all infinitely differentiable functions on R^s with compact support.

C₀^∞(R^s)is atrue subspaceof S since, e.g., the function γ(x) = e^−kxk² belongs to S but not to C₀^∞(R^s).

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Some properties of the Schwartz space

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Some properties of the Schwartz space

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Of particular importance are the followingsubspaces S_m of S S_m = {γ ∈ S : γ(x) = O(kxk^m)for kxk → 0, m ∈ N0}.

Furthermore, the set V ofslowly increasing functionsis given by V = {f ∈ C(R^s) : |f (x)| ≤ |p(x)| for some polynomial p ∈ Π^s}.

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Of particular importance are the followingsubspaces S_m of S S_m = {γ ∈ S : γ(x) = O(kxk^m)for kxk → 0, m ∈ N0}.

Furthermore, the set V ofslowly increasing functionsis given by V = {f ∈ C(R^s) : |f (x)| ≤ |p(x)| for some polynomial p ∈ Π^s}.

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The generalized Fourier transform is now given by Definition

Let f ∈ V be complex-valued. A continuous function ˆf : R^s\ {0} → C is called thegeneralized Fourier transformof f if there exists an integer m ∈ N0such that

Z

R^s

f (x)ˆγ(x)dx = Z

R^s

ˆf (x)γ(x)dx

is satisfied for all γ ∈ S_2m.

The smallest such integer m is called theorderof ˆf.

Remark

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

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The generalized Fourier transform is now given by Definition

Let f ∈ V be complex-valued. A continuous function ˆf : R^s\ {0} → C is called thegeneralized Fourier transformof f if there exists an integer m ∈ N0such that

Z

R^s

f (x)ˆγ(x)dx = Z

R^s

ˆf (x)γ(x)dx

is satisfied for all γ ∈ S_2m.

The smallest such integer m is called theorderof ˆf.

Remark

Various definitions of the generalized Fourier transform exist in the

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Since one can show that thegeneralized Fourier transform of an s-variate polynomial of degree at most 2m is zero, it follows that the inverse generalized Fourier transform is only unique up to addition of such a polynomial.

Theorder of the generalized Fourier transformis nothing but the order of the singularity at the originof the generalized Fourier transform.

Forfunctions in L₁(R^s)the generalized Fourier transform coincides with theclassical Fourier transform.

For functions in L₂(R^s)it coincides with the distributional Fourier transform.

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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 the original 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 generalized Fourier transform ˆΦof order m which is continuous on R^s\ {0}. Then Φis strictly conditionally positive definite of order m if and only if ˆΦis non-negative and non-vanishing.

Remark

This theorem states that strictly conditionally positive definite functions on R^s are characterized by theorder of the singularity of their

generalized Fourier transform at the origin, provided that this generalized Fourier transform is non-negative and non-zero.

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This general approach originated in the manuscript

[Madych and Nelson (1983)]. Many more details can be found in the original 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 generalized Fourier transform ˆΦof order m which is continuous on R^s\ {0}. Then Φis strictly conditionally positive definite of order m if and only if ˆΦis non-negative and non-vanishing.

Remark

This theorem states that strictly conditionally positive definite functions on R^s are characterized by theorder of the singularity of their

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Since integral characterizationssimilar to our earlier theorems of Schoenberg for positive definite radial functionsare so complicated in the conditionally positive definite case we do not pursue the concept of a conditionally positive definite radial functionhere.

Such theorems can be found in [Guo et al. (1993a)].

Examples of radial functions via the Fourier transform approach are given in the next chapter.

In Chapter 9 we will explore the connection between completely and multiply monotone functions and conditionally positive definite radial functions.

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Since integral characterizationssimilar to our earlier theorems of Schoenberg for positive definite radial functionsare so complicated in the conditionally positive definite case we do not pursue the concept of a conditionally positive definite radial functionhere.

Such theorems can be found in [Guo et al. (1993a)].

Examples of radial functions via the Fourier transform approach are given in the next chapter.

In Chapter 9 we will explore the connection between completely and multiply monotone functions and conditionally positive definite radial functions.

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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).

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Appendix References

References II

Guo, K., Hu, S. and Sun, X. (1993a).

Conditionally positive definite functions and Laplace-Stieltjes integrals.

J. Approx. Theory74, pp. 249–265.

Iske, A. (1994).

Charakterisierung bedingt positiv definiter Funktionen für multivariate Interpolationsmethoden 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 multivariate interpolation.