Schoenberg Representations and Gramian Matrices of Matérn Functions
aa r X i v : . [ m a t h . C A ] F e b Schoenberg Representations and GramianMatrices of Mat´ern Functions
Yong-Kum Cho Dohie Kim Kyungwon ParkHera YunSeptember 3, 2018
Abstract.
We represent Mat´ern functions in terms of Schoenberg’sintegrals which ensure the positive definiteness and prove the systemsof translates of Mat´ern functions form Riesz sequences in L ( R n ) orSobolev spaces. Our approach is based on a new class of integral trans-forms that generalize Fourier transforms for radial functions. We alsoconsider inverse multi-quadrics and obtain similar results. Keywords.
Bessel function, Fourier transform, Gramian matrix, Hankel-Schoenberg transform, inverse multi-quadrics, Mat´ern function, positive def-inite, Riesz sequence, Schoenberg matrix, Sobolev space.2010 Mathematics Subject Classification: 33C10, 41A05, 42B10, 60E10. Introduction
In many areas of Mathematics, the functions of type M α ( z ) = K α ( z ) z α ( α ∈ R , z >
0) (1.1)arise frequently, referred to as the Mat´ern functions, where K α ( z ) stands forthe modified Bessel function of the second kind of order α .Intimately connected is the family of functions of type φ β ( r ) = (1 + r ) − β ( β > , r ≥
0) (1.2)whose radial extensions to the Euclidean spaces are referred to as the inversemulti-quadrics in the theory of interpolations or spatial statistics.In a fixed Euclidean space, both class of functions, if radially extendedwith suitably rearranged α, β, provide essential ingredients of Sobolev spaces.In their pioneering work [3], N. Aronszajn and K. T. Smith introduced theSobolev space H α ( R n ) , α > , as the space of Bessel potentials, that is,the convolutions ( G α/ ∗ u )( x ) , u ∈ L ( R n ) , where G α/ denotes the radialextension of a special kind of Mat´ern functions defined as follows. Definition 1.1.
For a positive integer n and α > ,G α ( z ) = 12 α − n π n Γ( α ) K α − n ( z ) z α − n ( z > . (1.3)For its radial extension to the Euclidean space R n , we write G α ( x ) = G α ( | x | ) , | x | = √ x · x ( x ∈ R n ) . A characteristic feature of the kernel G α is the Fourier transform c G α ( ξ ) = Z R n e − iξ · x G α ( x ) d x = (cid:0) | ξ | (cid:1) − α , which, together with the intrinsic properties of K α − n/ , enables the authorsto obtain a comprehensive list of functional properties. Let us state only afew of their list which are relevant to the present work (see also [4]).2a) The Sobolev space H α ( R n ) is identified with H α ( R n ) = (cid:26) u ∈ L ( R n ) : Z R n (cid:0) | ξ | (cid:1) α | b u ( ξ ) | dξ < ∞ (cid:27) . In particular, G β ∈ H α ( R n ) if and only if β > (2 α + n ) / . (b) ( G α ∗ G β ) ( x ) = G α + β ( x ) for α > , β > . (c) In the case α > n/ , G α is positive definite on R n . The symmetrickernel G α ( x − y ) is in fact a reproducing kernel for the Hilbert space H α ( R n ) under the inner product (cid:0) u, v (cid:1) H α ( R n ) = (2 π ) − n Z R n b u ( ξ ) b v ( ξ ) (1 + | ξ | ) α dξ. Our primary purpose in the present work is to obtain a set of invariantsfor both classes of functions, that is, those properties valid in any Euclideanspace, related with the positive definiteness and Fourier transforms.We recall that a univariate function φ defined on the interval [0 , ∞ ) issaid to be positive semi-definite on R n if it satisfies N X j =1 N X k =1 φ ( | x j − x k | ) α j α k ≥ α , · · · , α N ∈ C and distinct points x , · · · , x N ∈ R n , where N is arbitrary. If equality in (1.4) holds only if α = · · · = α N = 0 , then it is said to be positive definite on R n .A univariate function which is positive semi-definite or positive definiteon every R n takes the following specific form: Criterion I (I. J. Schoenberg [13]).
A continuous function φ on [0 , ∞ ) is positive semi-definite on every R n if and only if φ ( r ) = Z ∞ e − r t dν ( t ) (1.5) for a finite positive Borel measure ν on [0 , ∞ ) . Moreover, if ν is notconcentrated at zero, then φ is positive definite on every R n . φ β ( r ) = 1Γ( β ) Z ∞ e − r t e − t t β − dt ( β > , (1.6)it is well known that each φ β is positive definite on every R n (see e.g. [16]).Our preliminary observation is the following. Theorem 1.1.
For α > , we have − α Γ( α ) K α ( z ) z α = Z ∞ e − z t f α ( t ) dt ( z ≥ , where f α denotes the probability density defined by f α ( t ) = 12 α Γ( α ) exp (cid:18) − t (cid:19) t − α − . As a consequence, M α is positive definite on every R n . In order to find direct relationships between the functions M α and φ β ,without recourse to their Euclidean extensions, we shall introduce a newclass of integral transforms that incorporates Fourier transforms for radialmeasures and Hankel transforms in certain sense. Definition 1.2.
For λ > − , let J λ denote the Bessel function of the firstkind of order λ and define Ω λ : R → R byΩ λ ( t ) = Γ( λ + 1) (cid:18) t (cid:19) − λ J λ ( t )= Γ( λ + 1) ∞ X k =0 ( − k k ! Γ( λ + k + 1) (cid:18) t (cid:19) k . In the special case λ = ( n − / , with n a positive integer, Ω λ ariseson consideration of the Fourier transforms for radial functions on R n . To bespecific, if F is integrable with F ( x ) = f ( | x | ) for some univariate function f on [0 , ∞ ), then it is well known (see e.g. [14]) that b F ( ξ ) = (2 π ) n/ | ξ | − n − Z ∞ J n − ( | ξ | t ) f ( t ) t n/ dt = 2 π n/ Γ( n/ Z ∞ Ω n − ( | ξ | t ) f ( t ) t n − dt. (1.7)4ore extensively, I. J. Schoenberg noticed that the Fourier transform ofany radial measure on R n is also representable in the above form and set upthe following characterization (see also H. Wendland [16]). Criterion II (I. J. Schoenberg [12], [13]).
A continuous function φ on [0 , ∞ ) is positive semi-definite on R n if and only if φ ( r ) = Z ∞ Ω n − ( rt ) dν ( t ) (1.8) for a finite positive Borel measure ν on [0 , ∞ ) . Moreover, in the casewhen dν ( t ) = f ( t ) t n − dt with continuous f , φ is positive definite on R n if and only if φ is nonnegative and non-vanishing. Our generalization of Schoenberg’s integrals or Fourier transforms forradial measures takes the following form.
Definition 1.3.
The Hankel-Schoenberg transform of order λ > − ν on [0 , ∞ ) is defined by φ ( r ) = Z ∞ Ω λ ( rt ) dν ( t ) (0 ≤ r < ∞ ) . For those Borel measures on [0 , ∞ ) which are absolutely continuous withrespect to Lebesgue measure, it is simple to express the Hankel-Schoenbergtransforms in terms of the classical Hankel transforms for which analoguesof the Fourier inversion theorem and Parseval’s relations are available.Our evaluations will be of the form (cid:0) r (cid:1) − α − λ − = c ( α, λ ) Z ∞ Ω λ ( rt ) (cid:2) K α ( t ) t α (cid:3) t λ +1 dt (1.9)for α + λ +1 > c ( α, λ ). By inversions andorder-changing transforms, we shall obtain a number of representation formu-las for the Mat´ern functions M α in terms of φ β ’s and vice versa, which suitsto Schoenberg’s criterion and makes it possible to find the Fourier transformsof their radial extensions to any Euclidean space.In accordance with the notation of [6], we introduce5 efinition 1.4. For a univariate function φ on [0 , ∞ ) and a set of distinctpoints X = { x j } j ∈ N ⊂ R n , the Schoenberg matrix is defined to be S X ( φ ) = h φ (cid:0) | x j − x k | (cid:1)i j, k ∈ N . (1.10)The notion of Schoenberg matrix comes up instantly with an attempt toconstruct an interpolating functional that matches the values of any functionat each point of X . To state briefly, if S X ( φ ) defines a bounded invertibleoperator on the space ℓ ( N ), then it is possible to construct a Lagrange-typeradial basis sequence (cid:8) u ∗ j (cid:9) j ∈ N by setting u ∗ j ( x ) = ∞ X k =1 c j,k φ ( | x − x k | ) , j = 1 , , · · · , and solving the infinite system u ∗ j ( x k ) = δ j,k , which has a unique solution c j = ( c j, , c j, , · · · ) ∈ ℓ ( N ) for each j . The functional A X ( f )( x ) = ∞ X j =1 f ( x j ) u ∗ j ( x ) , definable on any class of functions, obviously interpolates f at X. The Schoenberg matrices arise under various guises in other fields ofMathematics. In Functional Analysis, for example, it is common that S X ( φ )coincides with the Gramian matrix of a sequence obtained by translatinganother function ψ by X in an appropriate Hilbert space H , that is, S X ( φ ) = h(cid:0) ψ ( | · − x j | ) , ψ ( | · − x k | ) (cid:1) H i j,k ∈ N . In such a circumstance, { ψ ( | x − x j | ) } j ∈ N is a Riesz sequence in H if andonly if S X ( φ ) defines a bounded invertible operator on ℓ ( N ).Our secondary purpose is to study the Schoenberg or Gramian matricesassociated with the Mat´ern functions M α as well as the functions φ β withour focuses on their boundedness and invertibility on ℓ ( N ).Our approaches are substantially based on the recent developments [6],[9] of L. Golinskii et al. in which a list of criteria for the boundedness6nd invertibility are established from several perspectives. To illustrate, theauthors devoted considerable portions of their work in studying the L -basedGramian matrices associated to the Mat´ern functions M α and obtained theirboundedness and invertibility on ℓ ( N ) in the range − n/ < α ≤ . As we shall present below, we shall improve their results by extendingthe range to α > − n/ (cid:8) M α ( | x − x j | ) (cid:9) j ∈ N , where ( x j ) is anarbitrary set of distinct points of R n , is a Riesz sequence in L ( R n ) or theSobolev space of certain specified order.In the same manner, the system of type (cid:8) φ β ( | x − x j | ) (cid:9) j ∈ N will be shownto be a Riesz sequence in the Hilbert space of functions on R n for which φ β ( | x − y | ) is a reproducing kernel. K α In this section we collect some of the basic properties of K α relevant tothe present work, most of which can be found in [1], [5] and [15].For α ∈ C , the modified Bessel function K α is defined by K α ( z ) = π (cid:20) I − α ( z ) − I α ( z )sin ( απ ) (cid:21) , where (2.1) I α ( z ) = ∞ X k =0 k ! Γ ( k + α + 1) (cid:16) z (cid:17) k + α . (2.2)In the case when α happens to be an integer, α = n, this formula should beinterpreted as K n ( z ) = lim α → n K α ( z ) . The Bessel functions I α , K α form afundamental system of solutions to the differential equation z d udz + z dudz − ( z + α ) u = 0 . (2.3)Hereafter, we shall be concerned only with α ∈ R and z > . (K1) By definition, it is evident K − α ( z ) = K α ( z ) . For each integer n , a7eries expansion formula for K n ( z ) is also available. In particular, K ( z ) = − log( z/ I ( z ) + ∞ X k =0 ψ ( k + 1)( k !) (cid:16) z (cid:17) k , (2.4)where ψ denotes the digamma function so that ψ (1) = − γ, ψ ( k + 1) = − γ + k X j =1 j , with γ being the Euler-Mascheroni constant.(K2) For α > − / z > , Schl¨afli’s integrals state K α ( z ) = √ π Γ( α + 1 / (cid:16) z (cid:17) α Z ∞ e − zt (cid:0) t − (cid:1) α − dt = r π e − z z α Γ( α + 1 / Z ∞ e − zt (cid:20) t (cid:18) t (cid:19)(cid:21) α − dt (2.5)in which the latter follows from the former by suitable substitutions.Another form of Schl¨afli’s integral reads K α ( z ) = 12 Z ∞−∞ e − z cosh t − αt dt, (2.6)which holds for any real α and z > . As a consequence, K α ( z ) ispositive on the interval (0 , ∞ ).(K3) From the differential equation (2.3), it follows plainly ddz (cid:2) K α ( z ) z α (cid:3) = − K α − ( z ) z α . By (K2), hence, the Mat´ern function M α ( z ) = K α ( z ) z α is positive andstrictly decreasing on the interval (0 , ∞ ).(K4) Of great significance is the asymptotic behavior of K α for α ≥ . z → , the series expansions (2.2) and (2.4) yield K α ( z ) ∼ ( α − Γ( α ) z − α for α > , − log z for α = 0 . (ii) As z → ∞ , a version of Hankel’s asymptotic formula states K α ( z ) = r π z e − z (cid:20) α − z + O (cid:18) z (cid:19)(cid:21) . (K5) In the special case α = n +1 / n an integer, it is simple to express K α , and hence the Mat´ern function M α , in closed forms on evaluationof Schl¨afli’s integral (2.5). To state M α explicitly, M n + ( z ) = r π e − z z n n X k =0 ( n + k )! k !( n − k )! (2 z ) − k ,M − n − ( z ) = r π e − z z − n − n X k =0 ( n + k )! k !( n − k )! (2 z ) − k , (2.7)where n is a nonnegative integer. A list of positive orders reads M ( z ) = r π e − z ,M ( z ) = r π z ) e − z ,M ( z ) = r π (cid:0) z + z (cid:1) e − z (2.8) To be more precise, (2.2) shows K α ( z ) = 2 α − Γ( α ) z − α (cid:2) O ( z α ∗ ) (cid:3) , where α ∗ = min(2 α,
2) and (2.4) shows K ( z ) = − log z + log 2 − γ + (cid:2) − log( z/ (cid:3) O (cid:0) z (cid:1) . M − ( z ) = r π e − z z ,M − ( z ) = r π (cid:18) z + 1 z (cid:19) e − z ,M − ( z ) = r π (cid:18) z + 3 z + 3 z (cid:19) e − z . (2.9) The purpose of this section is to establish basic properties of the Hankel-Schoenberg transforms which will be used subsequently.To begin with, we list the following properties on the kernels Ω λ whichare deducible from those on the Bessel functions J λ ([5], [15]).(J1) Each Ω λ is of class C ∞ ( R ), even and uniformly bounded by 1 = Ω λ (0) . A theorem of Bessel-Lommel states that it is an oscillatory functionwith an infinity of positive simple zeros. A modification of Hankel’sasymptotic formula for J λ shows that as t → ∞ , Ω λ ( t ) = Γ( λ + 1) √ π (cid:18) t (cid:19) − λ − / (cid:20) cos (cid:18) t − (2 λ + 1) π (cid:19) + O (cid:0) t − (cid:1)(cid:21) . (J2) For λ > − / , Poisson’s integral readsΩ λ ( t ) = 2 B ( λ + 1 / , / Z cos( ts ) (1 − s ) λ − dt, where B stands for the Euler beta function defined by B ( a, b ) = Z t a − (1 − t ) b − dt ( a > , b > . (J3) By Liouville’s theorem, Ω λ is expressible in finite terms by algebraicand trigonometric functions if and only if 2 λ is an odd integer. Indeed,10he Lommel-type recurrence formulaΩ λ ( t ) − Ω λ − ( t ) = t λ ( λ + 1) Ω λ +2 ( t ) ( λ > − n +1 / for any integer n together withΩ − ( t ) = cos t, Ω ( t ) = sin tt . The Hankel transforms of a function f refer to the integrals Z ∞ J λ ( rt ) f ( t ) tdt ( λ ∈ C ) . It follows by definition that the Hankel-Schoenberg transforms can be writtenin terms of the Hankel transforms whenever ν admits an integrable density f ,that is, dν ( t ) = f ( t ) dt. The Hankel-Watson inversion theorem ([15]) statesthat if λ ≥ − / f ( t ) √ t is integrable on [0 , ∞ ), then Z ∞ J λ ( rt ) (cid:20)Z ∞ J λ ( ru ) f ( u ) udu (cid:21) rdr = f ( t + 0) + f ( t − t > f is of bounded variation in a neighborhood of t .An obvious modification yields the following inversion formula which mayserve as an alternative of the Fourier inversion theorem for radial functions. Theorem 3.1. (Inversion)
For λ ≥ − / , assume that Z ∞ | f ( t ) | t − λ − / dt < ∞ . (3.1) Then the following holds for every t > at which f is continuous: φ ( r ) = Z ∞ Ω λ ( rt ) f ( t ) dt implies f ( t ) = t λ +1 λ [Γ( λ + 1)] Z ∞ Ω λ ( rt ) φ ( r ) r λ +1 dr. A version of Parseval’s theorem is deducible from its equivalent for theHankel transforms in a trivial manner.11 heorem 3.2. (Parseval’s relation)
For λ > − , let φ j ( r ) = Z ∞ Ω λ ( rt ) f j ( t ) t λ +1 dt, j = 1 , . If both integrals are absolutely convergent, then Z ∞ f ( t ) f ( t ) t λ +1 dt = 14 λ [Γ( λ + 1)] Z ∞ φ ( r ) φ ( r ) r λ +1 dr. Lemma 3.1.
For λ > ρ > − and r ≥ , Ω λ ( r ) = 2 B ( ρ + 1 , λ − ρ ) Z ∞ Ω ρ ( rt )(1 − t ) λ − ρ − t ρ +1 dt. Proof. If ν denotes the probability measure dν ( t ) = 2 B ( ρ + 1 , λ − ρ ) (1 − t ) λ − ρ − t ρ +1 dt, then it has finite moments of all orders with Z ∞ t k dν ( t ) = Γ( k + ρ + 1)Γ( ρ + 1) · Γ( λ + 1)Γ( k + λ + 1) , k = 0 , , · · · . It follows from integrating termwise, readily justified, that Z ∞ Ω ρ ( rt ) dν ( t ) = Γ( ρ + 1) ∞ X k =0 ( − k k ! Γ( k + ρ + 1) (cid:16) r (cid:17) k Z ∞ t k dν ( t )= Γ( λ + 1) ∞ X k =0 ( − k k ! Γ( k + λ + 1) (cid:16) r (cid:17) k = Ω λ ( r ) . The Hankel-Schoenberg transforms may be regarded as a generalizationof the radial Fourier transforms or Schoenberg’s integrals due to the followingorder-changing interrelations. As usual, we write x + = max ( x,
0) for x ∈ R . heorem 3.3. Let λ > n − with n a positive integer. For any finite positiveBorel measure ν on [0 , ∞ ) which is not concentrated at zero, its Hankel-Schoenberg transform of order λ can be represented as Z ∞ Ω λ ( rt ) dν ( t ) = Z ∞ Ω n − ( rt ) W λ ( ν )( t ) t n − dt, where W λ ( ν )( t ) = 2 B (cid:0) n , λ + 1 − n (cid:1) Z ∞ (cid:18) − t s (cid:19) λ − n + s − n dν ( s ) . Moreover, dµ ( t ) = W λ ( ν )( t ) t n − dt defines a finite positive Borel measure on [0 , ∞ ) with the total mass µ ([0 , ∞ )) = ν ([0 , ∞ )) . Proof.
As a special case of Lemma 3.1, the choice ρ = n − givesΩ λ ( r ) = 2 B (cid:0) n , λ + 1 − n (cid:1) Z ∞ Ω n − ( rs )(1 − s ) λ − n + s n − ds, (3.2)whence the result follows by interchanging the order of integrations.Since Z ∞ (cid:18) − t s (cid:19) λ − n + t n − dt = s n Z (1 − u ) λ − n u n − du for each s > , it is straightforward to find µ ([0 , ∞ )) = Z ∞ W λ ( ν )( t ) t n − dt = 2 B (cid:0) n , λ + 1 − n (cid:1) Z ∞ Z ∞ (cid:18) − t s (cid:19) λ − n + s − n dν ( s ) t n − dt = 2 B (cid:0) n , λ + 1 − n (cid:1) Z ∞ Z ∞ (cid:18) − t s (cid:19) λ − n + t n − dts − n dν ( s )= ν ([0 , ∞ )) . Remark . A positive Borel measure ν on [0 , ∞ ) is concentrated at zero ifit is a constant multiple of Dirac mass at zero, that is, ν = c δ with c > . For such a Borel measure ν , its Hankel-Schoenberg transform is simply Z ∞ Ω λ ( rt ) dν ( t ) = c Ω λ (0) = c. Schoenberg representations
Our aim in this section is to set up Schoenberg’s representations forMat´ern functions which ensure their positive definiteness.
Lemma 4.1.
For α ∈ R and z > , we have K α ( z ) z α = 2 − α − Z ∞ exp (cid:18) − z t − t (cid:19) t − α − dt. (4.1) Proof.
For any real α and z > , if we make substitution ze − t = 2 s in thesecond form of Schl¨afli’s integral (2.6), then K α ( z ) = 12 Z ∞−∞ exp ( − z cosh t − αt ) dt = 2 α − z − α Z ∞ exp (cid:18) − s − z s (cid:19) s α − ds from which (4.1) follows on making another substitution s = 1 / t. In the case α > , it follows from the asymptotic behavior of K α nearzero, as stated in (K4), that the Mat´ern function M α is well defined as acontinuous function on [0 , ∞ ) with the limiting value M α (0) = 2 α − Γ( α ) . For this reason, it will be convenient to consider the following types of Mat´ernfunctions which are frequently used in many fields (see e.g. [7]).
Definition 4.1.
For α > , put M α ( z ) = 2 − α Γ( α ) K α ( z ) z α ( z ≥ . (4.2)We recall that a function φ is said to be completely continuous on [0 , ∞ )if it is continuous on [0 , ∞ ) and satisfies the condition ( − m φ ( m ) ( z ) ≥ m and z > Theorem 4.1. (Theorem 1.1)
For α > , we have M α ( z ) = Z ∞ e − z t f α ( t ) dt ( z ≥ , here f α is the continuous probability density on [0 , ∞ ) defined by f α ( t ) = α Γ( α ) exp (cid:18) − t (cid:19) t − α − for t > , for t = 0 . As a consequence, M α is continuous and positive definite on every R n . Inaddition, the function M α ( √ z ) is also positive definite on every R n andcompletely continuous on [0 , ∞ ) .Proof. As it is elementary to verify that f α is a continuous probability densityon [0 , ∞ ), the statements on M α ( z ) are immediate consequences of Lemma4.1 and Schoenberg’s Criterion I on the positive definiteness.In the special case α = 1 / , we have e − z = Z ∞ e − z t f / ( t ) dt ( z ≥ , (4.3)whence it is straightforward to deduce the integral representations M α (cid:0) √ z (cid:1) = Z ∞ e − zt f α ( t ) dt (4.4)= Z ∞ e − z u g α ( u ) du, (4.5)where g α stands for the function defined by g α (0) = 0 and g α ( u ) = u − / α +1 √ π Γ( α ) Z ∞ exp (cid:18) − t − t u (cid:19) t − α dt for u > . As readily verified, g α is a continuous probability density on[0 , ∞ ) and hence it follows from (4.5) and Schoenberg’s Criterion I that thefunction M α ( √ z ) is positive definite on every R n .That it is completely continuous on [0 , ∞ ) is a consequence of (4.4). It may be proved either by differentiating under the integral sign or by applyingthe well-known theorem of Bernstein-Hausdorff-Widder which states that a function f iscompletely continuous on [0 , ∞ ) if and only if f ( r ) = Z ∞ e − rt dµ ( t ) ( r ≥ µ on [0 , ∞ ) (see e.g. [16]).
15e are now concerned with the second form of Schoenberg’s integrals. Forthe sake of computational facilitation as well as inversion, it is advantageousto consider the Hankel-Schoenberg transforms.As it is conventional, we shall use the notation of Pochhammer and Barnesfor the generalized hypergeometric functions p F q ( a , · · · , a p ; b , · · · , b q ; x ) = ∞ X k =0 ( a ) k · · · ( a p ) k k ! ( b ) k · · · ( b q ) k x k in which the symbol ( a ) k for a non-zero real number a stands for( a ) k = ( a ( a + 1) · · · ( a + k −
1) for k ≥ , k = 1 . The following is easily obtainable from Scl¨afli’s integrals. As it is known,however, we shall omit the proof (see [1], [5], [15]).
Lemma 4.2.
For α ∈ R and β > | α | , we have Z ∞ K α ( t ) t β − dt = 2 β − Γ (cid:18) β + α (cid:19) Γ (cid:18) β − α (cid:19) . (4.6) Lemma 4.3.
Let α ∈ R and β > | α | . For the probability measure dν ( t ) = 12 β − Γ (cid:0) β + α (cid:1) Γ (cid:0) β − α (cid:1) K α ( t ) t β − dt, the Hankel-Schoenberg transform of order λ > − is given by Z ∞ Ω λ ( rt ) dν ( t ) = F (cid:18) β − α , β + α λ + 1; − r (cid:19) . (4.7) Proof.
A simple modification of (4.6) yields Z ∞ t k dν ( t ) = 2 k (cid:18) β + α (cid:19) k (cid:18) β − α (cid:19) k , k = 0 , , , · · · . Integrating termwise, we deduce Z ∞ Ω λ ( rt ) dν ( t ) = ∞ X k =0 ( − k k ! ( λ + 1) k (cid:16) r (cid:17) k Z ∞ t k dt = ∞ X k =0 (cid:0) β + α (cid:1) k (cid:0) β − α (cid:1) k k ! ( λ + 1) k (cid:0) − r (cid:1) k , which is equivalent to the stated formula (4.7).16y obvious cancellation effects, the generalized hypergeometric function(4.7) reduces to the binomial series expansion in the case β = α + 2( λ + 1)or β = − α + 2( λ + 1) . To be precise, we have the following general resultswhich include Schoenberg’s representations for Mat´ern functions.
Theorem 4.2.
Let λ > − and α + λ + 1 > . For each r ≥ , we have (1 + r ) − α − λ − = 12 α +2 λ Γ( λ + 1)Γ( α + λ + 1) × Z ∞ Ω λ ( rt ) (cid:2) K α ( t ) t α (cid:3) t λ +1 dt. (4.8) Moreover, if α + λ + 3 / > in addition, then for each z > ,K α ( z ) z α = 2 α Γ( α + λ + 1)Γ( λ + 1) Z ∞ Ω λ ( zt )(1 + t ) − α − λ − t λ +1 dt. (4.9) Proof.
Formula (4.8) follows from the special case β = α + 2 λ + 2 of (4.7),Lemma 4.3, and Newton’s binomial theorem ∞ X k =0 ( α + λ + 1) k k ! ( − r ) k = (1 + r ) − α − λ − . As the function f ( t ) = K α ( t ) t α +2 λ +1 is continuous on (0 , ∞ ) and Z ∞ | f ( t ) | t − λ − / dt = Z ∞ K α ( t ) t α + λ +1 / dt = 2 α + λ − / Γ (cid:18) α + 2 λ + 34 (cid:19) Γ (cid:18) λ + 34 (cid:19) < ∞ by Lemma 4.2, applicable due to the condition 2 α + λ +3 / > , (4.9) followsfrom inverting (4.8) in accordance with Theorem 3.1.Choosing α, λ suitably or regarding them as variable parameters, onemay exploit these formulas from several perspectives. If we are concernedwith the Fourier transforms in a fixed Euclidean space R n , for example, thefirst formula may be applied to yield the following.17a) For α > , if we recall (1.3) G α ( z ) = 12 α − n π n Γ( α ) K α − n ( z ) z α − n , the special case λ = ( n − / r ) − α = 2 π n/ Γ ( n/ Z ∞ Ω n − ( rt ) G α ( t ) t n − dt so that we obtain the Fourier transform formula c G α ( ξ ) = (1 + | ξ | ) − α . (4.10)(b) As α varies over α > , (4.10) expresses the inverse multi-quadrics ofany positive order in terms of the Fourier transforms of G α ( x ). Onthe contrary, Hankel-Schoenberg transform formula (4.8) enables us toobtain such Fourier representations by varying λ with a fixed α .To be specific, let us fix α > − n/ F α,λ ( z ) = 12 α +2 λ π n Γ (cid:0) λ + 1 − n (cid:1) Γ( α + λ + 1) × Z ∞ z ( s − z ) λ − n (cid:2) K α ( s ) s α (cid:3) sds (4.11)for λ > ( n − / . By Theorem 3.3, we may put (4.8) in the form(1 + r ) − α − λ − = 2 π n/ Γ ( n/ Z ∞ Ω n − ( rt ) F α,λ ( t ) t n − dt. If we write F α,λ ( x ) = F α,λ ( | x | ) , x ∈ R n , then d F α,λ ( ξ ) = (1 + | ξ | ) − α − λ − . (4.12)As λ varies in the range λ > ( n − / , this Fourier transform formularepresents the inverse multi-quadrics of order greater than α + n/ . A noteworthy feature of Mat´ern functions is the following invariancewhich follows immediately from (4.9) by reformulation.18 orollary 4.1. If α > , then for any λ > − , M α ( z ) = Z ∞ Ω λ ( zt ) dν α,λ ( t ) ( z ≥ , (4.13) where ν α,λ denotes the probability measure on [0 , ∞ ) defined by dν α,λ ( t ) = 2 B ( α, λ + 1) (1 + t ) − α − λ − t λ +1 dt. Remark . In view of Schoenberg’s Criterion II, this integral formula with λ = ( n − / n = 1 gives M α ( z ) = 2 B ( α, / Z ∞ cos( zt ) dt (1 + t ) α +1 / , the formula obtained by Basset, Malmst´en and Poisson (see [15]). ℓ ( N ) In this section we shall investigate whether Schoenberg matrices of Mat´ernfunctions or inverse multi-quadrics, in a fixed Euclidean space R n , give riseto bounded invertible operators on the Hilbert space ℓ ( N ).As it is common in the theory of scattered data approximations, we shalldeal with arbitrary sets of type X = { x j ∈ R n : j ∈ N } satisfying δ ( X ) = inf j = k | x j − x k | > , dim [span( X )] = d (5.1)for some 1 ≤ d ≤ n. Our analysis will be based on the following.
Proposition 5.1. ([6])
Let f be a nonnegative function defined on [0 , ∞ ) . (i) Suppose f is monotone decreasing, f (0) = 1 and f ( t ) t d − is integrableon [0 , ∞ ) . Then the Schoenberg matrix S X ( f ) defines a bounded self-adjoint operator on ℓ ( N ) with k S X ( f ) k ≤ d (5 d − δ ( X )] d Z ∞ f ( t ) t d − dt. oreover, if X satisfies the additional separation assumption δ ( X ) > (cid:20) d (5 d − Z ∞ f ( t ) t d − dt (cid:21) /d , then S X ( f ) defines a bounded invertible operator on ℓ ( N ) . (ii) Suppose n ≥ and f admits an integral representation f ( r ) = Z ∞ e − r t dν ( t ) ( r ≥ for a finite positive Borel measure ν such that it is equivalent to Lebesguemeasure on [0 , ∞ ) and satisfies the moment condition Z ∞ t − d/ dν ( t ) < ∞ . Then S X ( f ) defines a bounded invertible operator on ℓ ( N ) .Remark . A positive Borel measure ν on [0 , ∞ ) is equivalent to Lebesguemeasure | · | if both are absolutely continuous with respect to each other. Bythe Radon-Nikodym theorem, a necessary and sufficient condition for ν tobe equivalent to Lebesgue measure is that dν ( t ) = p ( t ) dt for a nonnegativedensity p such that supp( p ) = [0 , ∞ ) and Z I p ( t ) dt = 0 ⇐⇒ | I | = 0for any Borel set I ⊂ [0 , ∞ ) . As the operator norm bound and the invertibility condition of part (i)are slightly different from the original ones presented in [6], we shall give areview of their proof for part (i) in the appendix.Now that Schoenberg’s representations are available for Mat´ern functionsof type (4.2), it is a simple matter to prove the following.
Theorem 5.1.
Let X be an arbitrary set of points of R n satisfying (5.1) .For α > , consider the Schoenberg matrix of M α , S X ( M α ) = h M α ( x j − x k ) i j, k ∈ N . S X ( M α ) defines a bounded self-adjoint operator on ℓ ( N ) with k S X ( M α ) k ≤ d d − (5 d − (cid:0) α + d (cid:1) Γ (cid:0) d (cid:1) [ δ ( X )] d Γ( α ) . (ii) For n ≥ , S X ( M α ) defines a bounded invertible operator on ℓ ( N ) .In the case n = d = 1 , if X satisfies the additional assumption δ ( X ) > (cid:0) α + (cid:1) Γ (cid:0) (cid:1) Γ( α ) , then it defines a bounded invertible operator on ℓ ( N ) .Proof. An application of Lemma 4.2 gives Z ∞ M α ( t ) t d − dt = 2 − α Γ( α ) Z ∞ K α ( t ) t α + d − dt = 2 d − Γ (cid:0) α + d (cid:1) Γ (cid:0) d (cid:1) Γ( α ) . Since M α (0) = 1 and M α is strictly decreasing on the interval [0 , ∞ )as it is noted in (K3), the criterion in the first part of Proposition 5.1 isapplicable and part (i) follows with the stated operator norm bound.Concerning part (ii), we invoke Corollary 4.1 to represent M α ( z ) = Z ∞ e − z t f α ( t ) dt ( z ≥ f α stands for the probability density f α ( t ) = α Γ( α ) exp (cid:18) − t (cid:19) t − α − for t > , t = 0 . Since the measure determined by f α ( t ) dt is obviously equivalent to Lebesguemeasure on [0 , ∞ ) and it is elementary to compute Z ∞ t − d/ f α ( t ) dt = 2 d Γ (cid:0) α + d (cid:1) Γ( α ) < ∞ , the criterion in the second part of Proposition 5.1 implies the invertibilityof S X ( M α ) in the case n ≥ . The last statement on the invertibility when n = d = 1 follows by the first criterion of Proposition 5.1.21 heorem 5.2. For β > n/ , put φ β ( r ) = (1 + r ) − β ( r ≥ . (5.2) Let X be an arbitrary set of points of R n satisfying (5.1) and S X ( φ β ) = h φ β ( x j − x k ) i j, k ∈ N . (i) S X ( φ β ) defines a bounded self-adjoint operator on ℓ ( N ) with k S X ( φ β ) k ≤ d (5 d − B (cid:0) β − d , d (cid:1) δ ( X )] d . (ii) For n ≥ , S X ( φ β ) defines a bounded invertible operator on ℓ ( N ) . Inthe case n = d = 1 , if X satisfies the additional assumption δ ( X ) > B (cid:18) β − , (cid:19) , then it defines a bounded invertible operator on ℓ ( N ) .Proof. By using the aforementioned integral representation φ β ( r ) = 1Γ( β ) Z ∞ e − r t e − t t β − dt ( r ≥ , the proof follows along the same scheme as above. Remark . In connection with the problem of interpolating functions at anarbitrary set of distinct points X , it is an immediate consequence of Theorems5.1, 5.2 that M α , φ β , with α > , β > n/ , could be used in constructingLagrange-type radial basis sequences (cid:8) u ∗ j (cid:9) j ∈ N , by the same process pointedout in the introduction, and the interpolating functional A X ( f )( x ) = ∞ X j =1 f ( x j ) u ∗ j ( x ) . Gramian matrices and Riesz sequences
Now that Schoenberg matrices of Mat´ern functions are shown to inducebounded and invertible operators on ℓ ( N ), it is natural to ask if they generateRiesz sequences or bases in appropriate Hilbert spaces.We recall that a system { f j } j ∈ N of vectors in a Hilbert space H is saidto be a Riesz sequence if its moment space is equal to ℓ ( N ), that is, n m f = (cid:8) ( f, f j ) H (cid:9) j ∈ N : f ∈ H o = ℓ ( N ) . If { f j } j ∈ N is complete in addition, it is called a Riesz basis (see [17]). Aclassical theorem of Bari states a necessary and sufficient condition for thesystem { f j } j ∈ N to be a Riesz sequence is that the Gramian matrixGram (cid:16) { f j } j ∈ N ; H (cid:17) = (cid:2) ( f j , f k ) H (cid:3) j, k ∈ N (6.1)defines a bounded and invertible operators on ℓ ( N ).As for the sequences constructed from translating Mat´ern functions bydistinct points, their Gramian matrices in L ( R n ) or Sobolev spaces turn outto be easily identifiable in terms of Schoenberg matrices.In order not to entangle with parameters, it is convenient to work withthe Bessel potential kernels of (1.3) G α ( x ) = 12 α − n π n Γ( α ) K α − n ( | x | ) | x | α − n . L ( R n ) space Concerning the square integrability, we have the following.
Lemma 6.1.
For λ > − , if α + λ + 1 > , then Z ∞ (cid:2) K α ( t ) t α (cid:3) t λ +1 dt = √ π Γ( α + λ + 1)Γ(2 α + λ + 1)Γ( λ + 1)4 Γ (cid:0) α + λ + (cid:1) . In particular, if α + n/ > with n a positive integer, then Z ∞ (cid:2) K α ( t ) t α (cid:3) t n − dt = √ π Γ (cid:0) α + n (cid:1) Γ (cid:0) α + n (cid:1) Γ (cid:0) n (cid:1) (cid:0) α + n +12 (cid:1) . roof. An application of Parseval’s relation, Theorem 3.2, for the Hankel-Schoenberg transforms to formula (4.8) of Theorem 4.2 gives Z ∞ (cid:2) K α ( t ) t α (cid:3) t λ +1 dt = (cid:2) α + λ Γ( α + λ + 1) (cid:3) Z ∞ r λ +1 dr (1 + r ) α +2 λ +2 . By making substitution u = 1 / (1 + r ) , we compute Z ∞ r λ +1 dr (1 + r ) α +2 λ +2 = 12 Z u α + λ (1 − u ) λ du = 12 B (2 α + λ + 1 , λ + 1)and the stated formula follows on simplifying constants by using Legendre’sduplication formula for the Gamma function. The second stated formulacorresponds to a special case of the first one with λ = n/ − . Theorem 6.1. If α > n/ , then for any x , y ∈ R n , (cid:0) G α ( · − x ) , G α ( · − y ) (cid:1) L ( R n ) = G α ( x − y ) . (6.2) As a consequence, for any sequence of distinct points ( x j ) j ∈ N ⊂ R n , theGramian matrix of the system (cid:8) G α ( x − x j ) (cid:9) j ∈ N ⊂ L ( R n ) coincides withthe Schoenberg matrix of G α , that is, Gram (cid:16)(cid:8) G α ( x − x j ) (cid:9) j ∈ N ; L ( R n ) (cid:17) = h G α ( x j − x k ) i j, k ∈ N . Proof.
By Lemma 6.1, G α ∈ L ( R n ) . Due to radial symmetry, (cid:0) G α ( · − x ) , G α ( · − y ) (cid:1) L ( R n ) = Z R n G α ( u − x ) G α ( u − y ) d u = Z R n G α ( x − y − w ) G α ( w ) d w = ( G α ∗ G α ) ( x − y ) . On the Fourier transform side, formula (4.10) gives \ G α ∗ G α ( ξ ) = (1 + | ξ | ) − α = d G α ( ξ ) , whence G α ∗ G α = G α and the result follows.24 emark . This result extends the work of L. Golinskii et al. [6] in whichthe authors dealt only with the range n/ < α ≤ n/ . (a) In the case when both α − n/ α − n/ L inner products explicitly with the aid of (K5).As illustrations in R , we take α = 1 , Z R e −| u − x |−| u − y | | u − x | | u − y | d u = 2 π e −| x − y | , Z R e −| u − x |−| u − y | d u = π e −| x − y | (cid:18) | x − y | + | x − y | (cid:19) for which the first formula is of considerable interest in the spectralanalysis for the Schr¨odinger equations (see [9]).(b) To reformulate (6.2) in a more direct fashion, put F α ( x ) = 12 α + n − π n Γ (cid:0) α + n (cid:1) K α ( | x | ) | x | α . (6.3)As an alternative of (6.2), if α > − n/ , then (cid:0) F α ( · − x ) , F α ( · − y ) (cid:1) L ( R n ) = F α + n ( x − y ) . (6.4)As it is shown in Theorem 5.1 that the Schoenberg matrices of G α ( z ) = Γ(2 α − n/ π ) n/ Γ(2 α ) M α − n/ ( z ) ( z > ℓ ( N ) as long as α > n/ , weobtain the following from Bari’s theorem and Theorem 6.1. Corollary 6.1.
Let α > n/ and X = { x j ∈ R n : j ∈ N } be arbitrary with δ ( X ) = inf j = k | x j − x k | > . (i) If n ≥ , then (cid:8) G α ( x − x j ) (cid:9) j ∈ N forms a Riesz sequence in L ( R n ) . (ii) In the case n = 1 , if X is separated with δ ( X ) > α )Γ(1 / α − / , then (cid:8) G α ( x − x j ) (cid:9) j ∈ N forms a Riesz sequence in L ( R ) . .2 Results on Sobolev spaces An important feature of the Sobolev space H α ( R n ) with α > n/ G α ( x − y ) so that itmay be viewed as the space of functions of type f ( x ) = ∞ X j =1 a j G α ( x − x j ) , where ( a j ) ∈ ℓ ( N ) and ( x j ) ⊂ R n are arbitrary (see [2]). Thus it is reason-able to expect that the system { G α ( x − x j ) } j ∈ N ⊂ H α ( R n ) may serve as aRiesz sequence or a Riesz basis in its closed linear span once the translationpoints ( x j ) were scattered all over some planes of R n .As a matter of fact, the reproducing property implies (cid:0) G α ( · − x ) , G α ( · − y ) (cid:1) H α ( R n ) = G α ( x − y ) (6.5)for all x , y ∈ R n and our foregoing analysis yields Theorem 6.2.
Let α > n/ and X = { x j ∈ R n : j ∈ N } be arbitrary with δ ( X ) = inf j = k | x j − x k | > . (i) If n ≥ , then (cid:8) G α ( x − x j ) (cid:9) j ∈ N forms a Riesz sequence in H α ( R n ) . (ii) In the case n = 1 , if X is separated with δ ( X ) > α )Γ(1 / α − / , then (cid:8) G α ( x − x j ) (cid:9) j ∈ N forms a Riesz sequence in H α ( R ) . Regarding the problem of determining if the sequences of translates byinverse multi-quadrics give rise to Riesz sequences, we introduce a class offunction spaces defined in terms of Fourier transforms as follows.
Definition 6.1.
For α > , K α ( R n ) = ( f ∈ C ( R n ) ∩ L ( R n ) : Z R n (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) dξK α ( | ξ | ) | ξ | α < ∞ ) .
26 theorem of R. Schaback [10] and H. Wendland ([16], Theorem 10.27)states if Φ ∈ C ( R n ) ∩ L ( R n ) , real-valued and positive definite, the Hilbertspace of functions on R n with the reproducing kernel Φ( x − y ) coincides with H ( R n ) = ( f ∈ C ( R n ) ∩ L ( R n ) : Z R d (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) dξ b Φ( ξ ) < ∞ ) for which the inner product is defined by (cid:0) f, g (cid:1) H ( R n ) = (2 π ) − n Z R n b f ( ξ ) b g ( ξ ) dξ b Φ( ξ ) . As a consequence, it is simple to find that the space K α ( R n ) arises as areproducing kernel Hilbert space with an appropriate multi-quadrics as itsreproducing kernel. To be precise, we have the following results. Theorem 6.3.
For β > n/ , consider the inverse multi-quadrics φ β ( x ) = (1 + | x | ) − β . (i) The Hilbert space of functions on R n with the reproducing kernel φ β coincides with K β − n/ ( R n ) for which the inner product is defined by (cid:0) f, g (cid:1) K β − n/ ( R n ) = (2 π ) − n Z R n b f ( ξ ) b g ( ξ ) dξG β ( ξ ) . (6.6)(ii) Let X = { x j ∈ R n : j ∈ N } be arbitrary with δ ( X ) = inf j = k | x j − x k | > . Then the system (cid:8) φ β ( x − x j ) (cid:9) j ∈ N forms a Riesz sequence in K β − n/ ( R n ) for any n ≥ and for n = 1 under the additional assumption δ ( X ) > B ( β − / , / . Proof.
Obviously, φ β is continuous, integrable and positive definite. By anapplication of the Hankel-Schoenberg transform formula for φ β as stated in27orollary 4.1, we have c φ β ( ξ ) = (2 π ) n G β ( ξ ) and hence part (i) follows bythe aforementioned theorem of Schaback and Wendland.By the reproducing property, the Gramian matrix is given byGram (cid:16)(cid:8) φ β ( x − x j ) (cid:9) j ∈ N ; K β − n/ ( R n ) (cid:17) = h φ β ( x j − x k ) i j, k ∈ N . and part (ii) follows immediately from Theorem 5.2. Remark . As the Mat´ern functions of positive order are bounded smoothfunctions with exponential decays, it is evident K α ( R n ) ⊂ H ∞ ( R n ) for any α > . In the special case β = ( n + 1) / , we note K / ( R n ) = (cid:26) f ∈ C ( R n ) ∩ L ( R n ) : Z R n e | ξ | (cid:12)(cid:12) b f ( ξ ) (cid:12)(cid:12) dξ < ∞ (cid:27) , which is the reproducing kernel Hilbert space with the Poisson kernel φ n +12 ( x ) = (1 + | x | ) − n +12 . ℓ ( N ) -Boundedness For the sake of completeness, we reproduce the proof of L. Golinskii etal. [6] for part (i) of Proposition 5.1 which states
Suppose that f is a nonnegative monotone decreasing function on [0 , ∞ ) such that f (0) = 1 and the function f ( t ) t d − is integrable on [0 , ∞ ) . For any X ⊂ R n satisfying the condition (5.1) , the Schoenberg matrix S X ( f ) defines a bounded self-adjoint operator on ℓ ( N ) with k S X ( f ) k ≤ d (5 d − δ ( X )] d Z ∞ f ( t ) t d − dt. (7.1)28 roof. Let us write δ = δ ( X ) and assume span( X ) ≃ R d for simplicity.We fix j and estimate the infinite sum A j ≡ ∞ X k =1 f ( | x k − x j | ) = 1 + ∞ X m =1 X x k ∈ X m f ( | x k − x j | ) , where X m = (cid:8) x k ∈ X : mδ ≤ | x k − x j | < ( m + 1) δ (cid:9) . In terms of the open balls B ( x k , δ/ ⊂ R n , a geometric inspection reveals X m ) ≤ vol (cid:16)n y ∈ R d : (cid:0) m − (cid:1) δ ≤ | y − x j | < (cid:0) m + (cid:1) δ o(cid:17) vol (cid:0) B ( x k , δ/ ∩ R d (cid:1) = (2 m + 3) d − (2 m − d ≤ (5 d − m d − , which implies A j ≤ ∞ X m =1 (5 d − m d − f ( mδ ) . As f is monotone decreasing on [0 , ∞ ), Z ∞ f ( tδ ) t d − dt = ∞ X m =1 Z mm − f ( tδ ) t d − dt ≥ ∞ X m =1 f ( mδ ) (cid:20) m d − ( m − d d (cid:21) ≥ ∞ X m =1 f ( mδ ) m d − d , which yields ∞ X m =1 f ( mδ ) m d − ≤ dδ d Z ∞ f ( t ) t d − dt. Inserting this estimate into the above sum, we are led to A j ≤ d (5 d − δ d Z ∞ f ( t ) t d − dt . Since this estimate is independent of j , the result follows by Schur’s test.29 emark . By Schur’s test, (7.1) implies k I − S X ( f ) k ≤ d (5 d − δ ( X )] d Z ∞ f ( t ) t d − dt and the right side is strictly less than 1 if δ ( X ) > (cid:20) d (5 d − Z ∞ f ( t ) t d − dt (cid:21) /d . For such a set X , S X ( f ) defines a bounded invertible operator on ℓ ( N ). Acknowledgements.
Yong-Kum Cho is supported by National ResearchFoundation of Korea Grant funded by the Korean Government (
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