MATH 590: Meshfree Methods

MATH 590: Meshfree Methods Chapter 9: Conditionally Positive Definite Radial Functions Greg Fasshauer Department of Applied Mathematics Illinois Institute of Technology Fall 2010 fasshauer@iit.edu MATH 590 Chapter 9 1

Outline 1 CPD Radial Functions and Completely Monotone Functions 2 CPD Radial Functions and Multiply Monotone Functions 3 Special Properties of CPD Functions of Order One fasshauer@iit.edu MATH 590 Chapter 9 2

As for strictly positive definite radial functions, we will be able to connect strictly conditionally positive definite radial functions to completely monotone and multiply monotone functions. This connection will provide a criterion for checking conditional positive definiteness of radial functions which is easier to use than the generalized Fourier transform characterization in the previous chapters. As before, if we want to rely on complete monotonity then we are limited to radial functions. fasshauer@iit.edu MATH 590 Chapter 9 3

CPD Radial Functions and Completely Monotone Functions Outline 1 CPD Radial Functions and Completely Monotone Functions 2 CPD Radial Functions and Multiply Monotone Functions 3 Special Properties of CPD Functions of Order One fasshauer@iit.edu MATH 590 Chapter 9 4

CPD Radial Functions and Completely Monotone Functions We now focus on conditionally positive definite functions that are radial on R s for all s. fasshauer@iit.edu MATH 590 Chapter 9 5

CPD Radial Functions and Completely Monotone Functions We now focus on conditionally positive definite functions that are radial on R s for all s. The paper [Guo et al. (1993a)] contains an integral characterization for such functions but is too technical to be included here. fasshauer@iit.edu MATH 590 Chapter 9 5

CPD Radial Functions and Completely Monotone Functions We now focus on conditionally positive definite functions that are radial on R s for all s. The paper [Guo et al. (1993a)] contains an integral characterization for such functions but is too technical to be included here. Another important result in [Guo et al. (1993a)] is a characterization of conditionally positive definite radial functions on R s for all s in terms of (derivatives of) completely monotone functions. Theorem Let ϕ C[0, ) C (0, ). Then the function Φ = ϕ( 2 ) is conditionally positive definite of order m and radial on R s for all s if and only if ( 1) m ϕ (m) is completely monotone on (0, ). fasshauer@iit.edu MATH 590 Chapter 9 5

CPD Radial Functions and Completely Monotone Functions Proof. For m = 0 this is Schoenberg s characterization of positive definite radial functions on R s for all s in terms of completely monotone functions (see p. 12 of the slides of Chapter 5). fasshauer@iit.edu MATH 590 Chapter 9 6

CPD Radial Functions and Completely Monotone Functions Proof. For m = 0 this is Schoenberg s characterization of positive definite radial functions on R s for all s in terms of completely monotone functions (see p. 12 of the slides of Chapter 5). The fact that complete monotonicity implies conditional positive definiteness was proved in [Micchelli (1986)]. fasshauer@iit.edu MATH 590 Chapter 9 6

CPD Radial Functions and Completely Monotone Functions Proof. For m = 0 this is Schoenberg s characterization of positive definite radial functions on R s for all s in terms of completely monotone functions (see p. 12 of the slides of Chapter 5). The fact that complete monotonicity implies conditional positive definiteness was proved in [Micchelli (1986)]. Micchelli also conjectured that the converse holds and gave a simple proof for this in the case m = 1. fasshauer@iit.edu MATH 590 Chapter 9 6

CPD Radial Functions and Completely Monotone Functions Proof. For m = 0 this is Schoenberg s characterization of positive definite radial functions on R s for all s in terms of completely monotone functions (see p. 12 of the slides of Chapter 5). The fact that complete monotonicity implies conditional positive definiteness was proved in [Micchelli (1986)]. Micchelli also conjectured that the converse holds and gave a simple proof for this in the case m = 1. The remaining part of the theorem is shown in [Guo et al. (1993a)]. fasshauer@iit.edu MATH 590 Chapter 9 6

CPD Radial Functions and Completely Monotone Functions In order to get strict conditional positive definiteness we need to generalize the earlier theorem due to Schoenberg, i.e., the fact that ϕ not be constant. This leads to (see [Wendland (2005a)]) fasshauer@iit.edu MATH 590 Chapter 9 7

CPD Radial Functions and Completely Monotone Functions In order to get strict conditional positive definiteness we need to generalize the earlier theorem due to Schoenberg, i.e., the fact that ϕ not be constant. This leads to (see [Wendland (2005a)]) Theorem If ϕ is as in the previous theorem and not a polynomial of degree at most m, then Φ is strictly conditionally positive definite of order m and radial on R s for all s. fasshauer@iit.edu MATH 590 Chapter 9 7

CPD Radial Functions and Completely Monotone Functions We can now more easily verify the conditional positive definiteness of the functions listed in the previous chapter. fasshauer@iit.edu MATH 590 Chapter 9 8

CPD Radial Functions and Completely Monotone Functions We can now more easily verify the conditional positive definiteness of the functions listed in the previous chapter. Example The functions ϕ(r) = ( 1) β (1 + r) β, 0 < β / N fasshauer@iit.edu MATH 590 Chapter 9 8

CPD Radial Functions and Completely Monotone Functions We can now more easily verify the conditional positive definiteness of the functions listed in the previous chapter. Example The functions ϕ(r) = ( 1) β (1 + r) β, 0 < β / N give rise to ϕ (l) (r) = ( 1) β β(β 1) (β l + 1)(1 + r) β l fasshauer@iit.edu MATH 590 Chapter 9 8

CPD Radial Functions and Completely Monotone Functions We can now more easily verify the conditional positive definiteness of the functions listed in the previous chapter. Example The functions ϕ(r) = ( 1) β (1 + r) β, 0 < β / N give rise to ϕ (l) (r) = ( 1) β β(β 1) (β l + 1)(1 + r) β l so that ( 1) β ϕ ( β ) (r) = β(β 1) (β β + 1)(1 + r) β β is completely monotone. fasshauer@iit.edu MATH 590 Chapter 9 8

CPD Radial Functions and Completely Monotone Functions We can now more easily verify the conditional positive definiteness of the functions listed in the previous chapter. Example The functions ϕ(r) = ( 1) β (1 + r) β, 0 < β / N give rise to so that ϕ (l) (r) = ( 1) β β(β 1) (β l + 1)(1 + r) β l ( 1) β ϕ ( β ) (r) = β(β 1) (β β + 1)(1 + r) β β is completely monotone. Moreover, m = β is the smallest possible m such that ( 1) m ϕ (m) is completely monotone. fasshauer@iit.edu MATH 590 Chapter 9 8

CPD Radial Functions and Completely Monotone Functions Example (cont.) Since β / N we know that ϕ is not a polynomial, and therefore the generalized multiquadrics Φ( x ) = ( 1) β (1 + x 2 ) β, β > 0, β / N, are strictly conditionally positive definite of order m β and radial on R s for all values of s. fasshauer@iit.edu MATH 590 Chapter 9 9

CPD Radial Functions and Completely Monotone Functions Example The functions ϕ(r) = ( 1) β/2 r β/2, 0 < β / 2N, fasshauer@iit.edu MATH 590 Chapter 9 10

CPD Radial Functions and Completely Monotone Functions Example The functions give rise to ϕ(r) = ( 1) β/2 r β/2, 0 < β / 2N, ϕ (l) (r) = ( 1) β/2 β 2 ( ) ( ) β β 2 1 2 l + 1 r β/2 l fasshauer@iit.edu MATH 590 Chapter 9 10

CPD Radial Functions and Completely Monotone Functions Example The functions give rise to so that ϕ(r) = ( 1) β/2 r β/2, 0 < β / 2N, ϕ (l) (r) = ( 1) β/2 β 2 ( 1) β/2 ϕ ( β/2 ) (r) = β 2 is completely monotone ( ) ( ) β β 2 1 2 l + 1 r β/2 l ( ) ( β β 2 1 2 ) β + 1 r β/2 β/2 2 fasshauer@iit.edu MATH 590 Chapter 9 10

CPD Radial Functions and Completely Monotone Functions Example The functions give rise to so that ϕ(r) = ( 1) β/2 r β/2, 0 < β / 2N, ϕ (l) (r) = ( 1) β/2 β 2 ( 1) β/2 ϕ ( β/2 ) (r) = β 2 ( ) ( ) β β 2 1 2 l + 1 r β/2 l ( ) ( β β 2 1 2 ) β + 1 r β/2 β/2 2 is completely monotone and m = β/2 is the smallest possible m such that ( 1) m ϕ (m) is completely monotone. fasshauer@iit.edu MATH 590 Chapter 9 10

CPD Radial Functions and Completely Monotone Functions Example (cont.) Since β is not an even integer ϕ is not a polynomial, and therefore, the radial powers Φ( x ) = ( 1) β/2 x β, β > 0, β / 2N, are strictly conditionally positive definite of order m β/2 and radial on R s for all s. fasshauer@iit.edu MATH 590 Chapter 9 11

CPD Radial Functions and Completely Monotone Functions Example The thin plate splines Φ( x ) = ( 1) β+1 x 2β log x, β N, are strictly conditionally positive definite of order m β + 1 and radial on R s for all s. fasshauer@iit.edu MATH 590 Chapter 9 12

CPD Radial Functions and Completely Monotone Functions Example The thin plate splines Φ( x ) = ( 1) β+1 x 2β log x, β N, are strictly conditionally positive definite of order m β + 1 and radial on R s for all s. To see this we observe that 2Φ( x ) = ( 1) β+1 x 2β log( x 2 ). fasshauer@iit.edu MATH 590 Chapter 9 12

CPD Radial Functions and Completely Monotone Functions Example The thin plate splines Φ( x ) = ( 1) β+1 x 2β log x, β N, are strictly conditionally positive definite of order m β + 1 and radial on R s for all s. To see this we observe that Therefore, we let 2Φ( x ) = ( 1) β+1 x 2β log( x 2 ). ϕ(r) = ( 1) β+1 r β log r, β N, which is obviously not a polynomial. fasshauer@iit.edu MATH 590 Chapter 9 12

CPD Radial Functions and Completely Monotone Functions Example (cont.) Differentiating ϕ(r) = ( 1) β+1 r β log r, β N, we get ϕ (l) (r) = ( 1) β+1 β(β 1) (β l+1)r β l log r +p l (r), 1 l β, with p l a polynomial of degree β l. fasshauer@iit.edu MATH 590 Chapter 9 13

CPD Radial Functions and Completely Monotone Functions Example (cont.) Differentiating we get ϕ(r) = ( 1) β+1 r β log r, β N, ϕ (l) (r) = ( 1) β+1 β(β 1) (β l+1)r β l log r +p l (r), 1 l β, with p l a polynomial of degree β l. Therefore, taking l = β we have ϕ (β) (r) = ( 1) β+1 β! log r + C and ϕ (β+1) β+1 β! (r) = ( 1) r, which is completely monotone on (0, ). fasshauer@iit.edu MATH 590 Chapter 9 13

CPD Radial Functions and Multiply Monotone Functions Outline 1 CPD Radial Functions and Completely Monotone Functions 2 CPD Radial Functions and Multiply Monotone Functions 3 Special Properties of CPD Functions of Order One fasshauer@iit.edu MATH 590 Chapter 9 14

CPD Radial Functions and Multiply Monotone Functions We state a theorem originally proved in [Micchelli (1986)], but in a slightly stronger version due to [Buhmann (1993a)] which relates conditionally positive definite radial functions of order m on R s (for some fixed value of s) to multiply monotone functions. Buhmann s version of the theorem ensures strict conditional positive definiteness. fasshauer@iit.edu MATH 590 Chapter 9 15

CPD Radial Functions and Multiply Monotone Functions We state a theorem originally proved in [Micchelli (1986)], but in a slightly stronger version due to [Buhmann (1993a)] which relates conditionally positive definite radial functions of order m on R s (for some fixed value of s) to multiply monotone functions. Buhmann s version of the theorem ensures strict conditional positive definiteness. Theorem Let k = s/2 m + 2 be a positive integer, and suppose ϕ C m 1 [0, ) is not a polynomial of degree at most m. If ( 1) m ϕ (m) is k-times monotone on (0, ) but not constant, then ϕ is strictly conditionally positive definite of order m and radial on R s for any s such that s/2 k + m 2. fasshauer@iit.edu MATH 590 Chapter 9 15

CPD Radial Functions and Multiply Monotone Functions Just as we showed earlier that compactly supported radial functions cannot be strictly positive definite on R s for all s, it is important to note that there are no truly conditionally positive definite functions with compact support. fasshauer@iit.edu MATH 590 Chapter 9 16

CPD Radial Functions and Multiply Monotone Functions Just as we showed earlier that compactly supported radial functions cannot be strictly positive definite on R s for all s, it is important to note that there are no truly conditionally positive definite functions with compact support. More precisely (see [Wendland (2005a)]), Theorem Assume that the complex-valued function Φ C(R s ) has compact support. If Φ is strictly conditionally positive definite of (minimal) order m, then m is necessarily zero, i.e., Φ is already strictly positive definite. fasshauer@iit.edu MATH 590 Chapter 9 16

CPD Radial Functions and Multiply Monotone Functions Proof. The hypotheses on Φ ensure that it is integrable, and therefore it possesses a classical Fourier transform ˆΦ which is continuous. fasshauer@iit.edu MATH 590 Chapter 9 17

CPD Radial Functions and Multiply Monotone Functions Proof. The hypotheses on Φ ensure that it is integrable, and therefore it possesses a classical Fourier transform ˆΦ which is continuous. For integrable functions the generalized Fourier transform coincides with the classical Fourier transform. fasshauer@iit.edu MATH 590 Chapter 9 17

CPD Radial Functions and Multiply Monotone Functions Proof. The hypotheses on Φ ensure that it is integrable, and therefore it possesses a classical Fourier transform ˆΦ which is continuous. For integrable functions the generalized Fourier transform coincides with the classical Fourier transform. Iske s theorem from Chapter 7 ensures that ˆΦ is non-negative on R s \ {0} and not identically equal to zero. fasshauer@iit.edu MATH 590 Chapter 9 17

CPD Radial Functions and Multiply Monotone Functions Proof. The hypotheses on Φ ensure that it is integrable, and therefore it possesses a classical Fourier transform ˆΦ which is continuous. For integrable functions the generalized Fourier transform coincides with the classical Fourier transform. Iske s theorem from Chapter 7 ensures that ˆΦ is non-negative on R s \ {0} and not identically equal to zero. By continuity we also get ˆΦ(0) 0, and Wendland s theorem from Chapter 3 shows that Φ is strictly positive definite. fasshauer@iit.edu MATH 590 Chapter 9 17

CPD Radial Functions and Multiply Monotone Functions Our previous theorem relating conditionally positive definite and multiply monotone functions together with the theorem on compactly supported functions implies that if we perform m-fold anti-differentiation on a non-constant k-times monotone function, then we obtain a function that is strictly positive definite and radial on R s for s/2 k + m 2. fasshauer@iit.edu MATH 590 Chapter 9 18

CPD Radial Functions and Multiply Monotone Functions Our previous theorem relating conditionally positive definite and multiply monotone functions together with the theorem on compactly supported functions implies that if we perform m-fold anti-differentiation on a non-constant k-times monotone function, then we obtain a function that is strictly positive definite and radial on R s for s/2 k + m 2. Example The truncated power function ϕ k (r) = (1 r) k + is k-times monotone (see Chapter 5). fasshauer@iit.edu MATH 590 Chapter 9 18

CPD Radial Functions and Multiply Monotone Functions Our previous theorem relating conditionally positive definite and multiply monotone functions together with the theorem on compactly supported functions implies that if we perform m-fold anti-differentiation on a non-constant k-times monotone function, then we obtain a function that is strictly positive definite and radial on R s for s/2 k + m 2. Example The truncated power function ϕ k (r) = (1 r) k + is k-times monotone (see Chapter 5). To avoid the integration constant we compute the anti-derivative via the integral operator I of Chapter 5, i.e., Iϕ k (r) = r ϕ k (t)dt fasshauer@iit.edu MATH 590 Chapter 9 18

CPD Radial Functions and Multiply Monotone Functions Our previous theorem relating conditionally positive definite and multiply monotone functions together with the theorem on compactly supported functions implies that if we perform m-fold anti-differentiation on a non-constant k-times monotone function, then we obtain a function that is strictly positive definite and radial on R s for s/2 k + m 2. Example The truncated power function ϕ k (r) = (1 r) k + is k-times monotone (see Chapter 5). To avoid the integration constant we compute the anti-derivative via the integral operator I of Chapter 5, i.e., Iϕ k (r) = r ϕ k (t)dt = r (1 t) k +dt fasshauer@iit.edu MATH 590 Chapter 9 18

CPD Radial Functions and Multiply Monotone Functions Our previous theorem relating conditionally positive definite and multiply monotone functions together with the theorem on compactly supported functions implies that if we perform m-fold anti-differentiation on a non-constant k-times monotone function, then we obtain a function that is strictly positive definite and radial on R s for s/2 k + m 2. Example The truncated power function ϕ k (r) = (1 r) k + is k-times monotone (see Chapter 5). To avoid the integration constant we compute the anti-derivative via the integral operator I of Chapter 5, i.e., Iϕ k (r) = r ϕ k (t)dt = r (1 t) k +dt = 1 (1 r)k+1 + k + 1. fasshauer@iit.edu MATH 590 Chapter 9 18

CPD Radial Functions and Multiply Monotone Functions Example (cont.) If we apply m-fold anti-differentiation we get I m ϕ k (r) = I I m 1 ϕ k (r) fasshauer@iit.edu MATH 590 Chapter 9 19

CPD Radial Functions and Multiply Monotone Functions Example (cont.) If we apply m-fold anti-differentiation we get I m ϕ k (r) = I I m 1 ϕ k (r) = ( 1) m (1 r)k+m + (k + 1)(k + 2) (k + m). fasshauer@iit.edu MATH 590 Chapter 9 19

CPD Radial Functions and Multiply Monotone Functions Example (cont.) If we apply m-fold anti-differentiation we get I m ϕ k (r) = I I m 1 ϕ k (r) = Therefore, by the function ( 1) m (1 r)k+m + (k + 1)(k + 2) (k + m). theorem relating conditionally positive definite and multiply monotone functions, the ϕ(r) = (1 r) k+m + is strictly conditionally positive definite of order m and radial on R s for s/2 k + m 2, fasshauer@iit.edu MATH 590 Chapter 9 19

CPD Radial Functions and Multiply Monotone Functions Example (cont.) If we apply m-fold anti-differentiation we get I m ϕ k (r) = I I m 1 ϕ k (r) = Therefore, by the function ( 1) m (1 r)k+m + (k + 1)(k + 2) (k + m). theorem relating conditionally positive definite and multiply monotone functions, the ϕ(r) = (1 r) k+m + is strictly conditionally positive definite of order m and radial on R s for s/2 k + m 2, and by the theorem on compactly supported functions it is even strictly positive definite and radial on R s. fasshauer@iit.edu MATH 590 Chapter 9 19

CPD Radial Functions and Multiply Monotone Functions Example (cont.) If we apply m-fold anti-differentiation we get I m ϕ k (r) = I I m 1 ϕ k (r) = Therefore, by the function ( 1) m (1 r)k+m + (k + 1)(k + 2) (k + m). theorem relating conditionally positive definite and multiply monotone functions, the ϕ(r) = (1 r) k+m + is strictly conditionally positive definite of order m and radial on R s for s/2 k + m 2, and by the theorem on compactly supported functions it is even strictly positive definite and radial on R s. Remark This was also observed in Chapter 5 where we saw that ϕ is even strictly positive definite and radial on R s for s/2 k + m 1. fasshauer@iit.edu MATH 590 Chapter 9 19

CPD Radial Functions and Multiply Monotone Functions Remark We see that we can construct strictly positive definite compactly supported radial functions by anti-differentiating the truncated power function. This is essentially the approach taken by Wendland to construct his popular compactly supported radial basis functions. We provide more details of his construction in Chapter 11. fasshauer@iit.edu MATH 590 Chapter 9 20

Outline Special Properties of CPD Functions of Order One 1 CPD Radial Functions and Completely Monotone Functions 2 CPD Radial Functions and Multiply Monotone Functions 3 Special Properties of CPD Functions of Order One fasshauer@iit.edu MATH 590 Chapter 9 21

Special Properties of CPD Functions of Order One Since an N N matrix that is conditionally positive definite of order one is positive definite on a subspace of dimension N 1 it has the interesting property that at least N 1 of its eigenvalues are positive (see the proof of the theorem on the next slide). fasshauer@iit.edu MATH 590 Chapter 9 22

Special Properties of CPD Functions of Order One Since an N N matrix that is conditionally positive definite of order one is positive definite on a subspace of dimension N 1 it has the interesting property that at least N 1 of its eigenvalues are positive (see the proof of the theorem on the next slide). This follows immediately from the Courant-Fischer theorem of linear algebra (see e.g., [Meyer (2000), p. 550]): Theorem (Courant-Fischer) Let A be a real symmetric N N matrix with eigenvalues λ 1 λ 2 λ N, then λ k = max min x T Ax dimv=k x V x =1 and λ k = min max x T Ax. dimv=n k+1 x V x =1 fasshauer@iit.edu MATH 590 Chapter 9 22

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. Proof. Let λ 1 λ 2 λ N denote the eigenvalues of A. fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. Proof. Let λ 1 λ 2 λ N denote the eigenvalues of A. From the Courant-Fischer theorem we get λ N 1 = max min x T Ax dimv=n 1 x V x =1 fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. Proof. Let λ 1 λ 2 λ N denote the eigenvalues of A. From the Courant-Fischer theorem we get λ N 1 = max min x T Ax dimv=n 1 x V x =1 min c T Ac c: c k =0 c =1 fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. Proof. Let λ 1 λ 2 λ N denote the eigenvalues of A. From the Courant-Fischer theorem we get λ N 1 = max min x T Ax dimv=n 1 x V x =1 so that A has at least N 1 positive eigenvalues. min c T Ac > 0, c: c k =0 c =1 fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One With an additional assumption on A we can make an even stronger statement. Theorem An N N matrix A which is conditionally positive definite of order one and has a non-positive trace possesses one negative and N 1 positive eigenvalues. Proof. Let λ 1 λ 2 λ N denote the eigenvalues of A. From the Courant-Fischer theorem we get λ N 1 = max min x T Ax dimv=n 1 x V x =1 min c T Ac > 0, c: c k =0 c =1 so that A has at least N 1 positive eigenvalues. But since tr(a) = N k=1 λ k 0, A also must have at least one negative eigenvalue. fasshauer@iit.edu MATH 590 Chapter 9 23

Special Properties of CPD Functions of Order One Remark The additional hypothesis of the previous theorem is satisfied for the interpolation matrix resulting from (the negative) of RBFs such as Hardy s multiquadric or the linear radial function ϕ(r) = r since its diagonal elements correspond to the value of the basic function at the origin. fasshauer@iit.edu MATH 590 Chapter 9 24

Special Properties of CPD Functions of Order One Remark The additional hypothesis of the previous theorem is satisfied for the interpolation matrix resulting from (the negative) of RBFs such as Hardy s multiquadric or the linear radial function ϕ(r) = r since its diagonal elements correspond to the value of the basic function at the origin. We will now use this theorem to conclude that we can use radial functions that are strictly conditionally positive definite of order one (such as the multiquadric, 0 < β < 1, and the norm basic function) without appending the constant term to solve the scattered data interpolation problem. fasshauer@iit.edu MATH 590 Chapter 9 24

Special Properties of CPD Functions of Order One Remark The additional hypothesis of the previous theorem is satisfied for the interpolation matrix resulting from (the negative) of RBFs such as Hardy s multiquadric or the linear radial function ϕ(r) = r since its diagonal elements correspond to the value of the basic function at the origin. We will now use this theorem to conclude that we can use radial functions that are strictly conditionally positive definite of order one (such as the multiquadric, 0 < β < 1, and the norm basic function) without appending the constant term to solve the scattered data interpolation problem. This was first proved by [Micchelli (1986)] and motivated by Hardy s earlier work with multiquadrics and Franke s conjecture that the matrix A is non-singular in this case (see [Franke (1982a)]). fasshauer@iit.edu MATH 590 Chapter 9 24

Special Properties of CPD Functions of Order One Theorem (Interpolation) Suppose Φ is strictly conditionally positive definite of order one and that Φ(0) 0. Then for any distinct points x 1,..., x N R s the matrix A with entries A jk = Φ(x j x k ) has N 1 positive and one negative eigenvalue, and is therefore non-singular. fasshauer@iit.edu MATH 590 Chapter 9 25

Special Properties of CPD Functions of Order One Theorem (Interpolation) Suppose Φ is strictly conditionally positive definite of order one and that Φ(0) 0. Then for any distinct points x 1,..., x N R s the matrix A with entries A jk = Φ(x j x k ) has N 1 positive and one negative eigenvalue, and is therefore non-singular. Proof. Clearly, the matrix A is conditionally positive definite of order one. fasshauer@iit.edu MATH 590 Chapter 9 25

Special Properties of CPD Functions of Order One Theorem (Interpolation) Suppose Φ is strictly conditionally positive definite of order one and that Φ(0) 0. Then for any distinct points x 1,..., x N R s the matrix A with entries A jk = Φ(x j x k ) has N 1 positive and one negative eigenvalue, and is therefore non-singular. Proof. Clearly, the matrix A is conditionally positive definite of order one. Moreover, the trace of A is given by tr(a) = NΦ(0) 0. fasshauer@iit.edu MATH 590 Chapter 9 25

Special Properties of CPD Functions of Order One Theorem (Interpolation) Suppose Φ is strictly conditionally positive definite of order one and that Φ(0) 0. Then for any distinct points x 1,..., x N R s the matrix A with entries A jk = Φ(x j x k ) has N 1 positive and one negative eigenvalue, and is therefore non-singular. Proof. Clearly, the matrix A is conditionally positive definite of order one. Moreover, the trace of A is given by tr(a) = NΦ(0) 0. Therefore, Micchelli s theorem applies and the statement follows. fasshauer@iit.edu MATH 590 Chapter 9 25

Special Properties of CPD Functions of Order One As mentioned above, Micchelli s theorem covers the generalized multiquadrics Φ(x) = (1 + x 2 ) β with 0 < β < 1 (which includes the Hardy multiquadric) and the radial powers Φ(x) = x β for 0 < β < 2 (including the Euclidean distance function). fasshauer@iit.edu MATH 590 Chapter 9 26

Special Properties of CPD Functions of Order One As mentioned above, Micchelli s theorem covers the generalized multiquadrics Φ(x) = (1 + x 2 ) β with 0 < β < 1 (which includes the Hardy multiquadric) and the radial powers Φ(x) = x β for 0 < β < 2 (including the Euclidean distance function). Another special property of a conditionally positive definite function of order one is Lemma If C is an arbitrary real constant and the real even function Φ is (strictly) conditionally positive definite of order one, then Φ + C is also (strictly) conditionally positive definite of order one. fasshauer@iit.edu MATH 590 Chapter 9 26

Special Properties of CPD Functions of Order One Proof. Simply consider N N c j c k [Φ(x j x k ) + C] = j=1 k=1 N N N N c j c k Φ(x j x k ) + c j c k C. j=1 k=1 j=1 k=1 fasshauer@iit.edu MATH 590 Chapter 9 27

Special Properties of CPD Functions of Order One Proof. Simply consider N N c j c k [Φ(x j x k ) + C] = j=1 k=1 N N N N c j c k Φ(x j x k ) + c j c k C. j=1 k=1 j=1 k=1 The second term on the right is zero since Φ is conditionally positive definite of order one, i.e., N j=1 c j = 0, and thus the statement follows. fasshauer@iit.edu MATH 590 Chapter 9 27

References I Appendix References 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. Iske, A. (2004). Multiresolution Methods in Scattered Data Modelling. Lecture Notes in Computational Science and Engineering 37, Springer Verlag (Berlin). Meyer, C. D. (2000). Matrix Analysis and Applied Linear Algebra. SIAM (Philadelphia). Wendland, H. (2005a). Scattered Data Approximation. Cambridge University Press (Cambridge). fasshauer@iit.edu MATH 590 Chapter 9 28

References II Appendix References Buhmann, M. D. (1993a). New developments in the theory of radial basis function interpolation. in Multivariate Approximation: From CAGD to Wavelets, Kurt Jetter and Florencio Utreras (eds.), World Scientific Publishing (Singapore), pp. 35 75. Franke, R. (1982a). Scattered data interpolation: tests of some methods. Math. Comp. 48, pp. 181 200. Guo, K., Hu, S. and Sun, X. (1993a). Conditionally positive definite functions and Laplace-Stieltjes integrals. J. Approx. Theory 74, pp. 249 265. Micchelli, C. A. (1986). Interpolation of scattered data: distance matrices and conditionally positive definite functions. Constr. Approx. 2, pp. 11 22. fasshauer@iit.edu MATH 590 Chapter 9 29