Compactly supported reproducing kernels for L^2-based Sobolev spaces and Hankel-Schoenberg transforms
aa r X i v : . [ m a t h . C A ] F e b Compactly supported reproducing kernelsfor L -based Sobolev spacesand Hankel-Schoenberg transforms Yong-Kum Cho ∗† February 21, 2017
Abstract.
We exhibit three classes of compactly supported functionswhich provide reproducing kernels for the Sobolev spaces H δ ( R d ) ofarbitrary order δ > d/ . Our method of construction is based on a newclass of oscillatory integral transforms that incorporate radial Fouriertransforms and Hankel transforms.
Keywords.
Askey’s class, Bessel function, Bessel potential kernel, binomialdensity, Fourier transform, generalized hypergeometric function, Hankel-Schoenberg transform, Hilbert space, positive definite, reproducing kernel,Sobolev space, Wendland’s function.2010 Mathematics Subject Classification: 33C10, 41A05, 42B10, 60E10. ∗ Department of Mathematics, College of Natural Sciences, Chung-Ang University, 84Heukseok-Ro, Dongjak-Gu, Seoul 156-756, Korea (e-mail: [email protected]) † This research was supported by National Research Foundation of Korea Grant fundedby the Korean Government ( Introduction
In this paper we shall deal with the problem of constructing compactlysupported radial functions Φ on R d such that the symmetric kernels Φ( x − y )could serve as reproducing kernels for Sobolev spaces under appropriate innerproducts. Due to advantageous aspects in practical applications, the problemhas become an important issue in various fields of Mathematics including thetheory of interpolations, spatial statistics and machine learning.In their pioneering work [3], N. Aronszajn and K. T. Smith introducedthe Sobolev space H δ ( R d ) of order δ > G δ/ ∗ u )( x ) , where u ∈ L ( R d ) and G δ ( x ) = 12 δ − d π d Γ( δ ) K δ − d ( | x | ) | x | δ − d . (1.1)As usual, | x | = √ x · x denotes the Euclidean norm for each x ∈ R d and K δ − d/ stands for the modified Bessel function of order δ − d/ . Often referred to as Mat´ern functions (see e.g. [12]), the Bessel potentialkernels G δ are are integrable with the Fourier transforms c G δ ( ξ ) = Z R d e − iξ · x G δ ( x ) d x = (cid:0) | ξ | (cid:1) − δ . (1.2)As a consequence, the Sobolev space of order δ may be identified with H δ ( R d ) = n u ∈ L ( R d ) : (cid:0) | · | (cid:1) δ/ b u ∈ L ( R d ) o , which becomes a Hilbert space under the inner product (cid:0) u, v (cid:1) H δ ( R d ) = (2 π ) − d Z R d (cid:0) | ξ | (cid:1) δ b u ( ξ ) b v ( ξ ) dξ. In the case δ > d/ , N. Aronszajn and K. T. Smith noticed further that H δ ( R d ) ⊂ C ( R d ) continuously and H δ ( R d ) is a reproducing kernel Hilbertspace with kernel G δ ( x − y ), that is, for every u ∈ H δ ( R d ) , x ∈ R d , (i) G δ ( · − x ) ∈ H δ ( R d ) and(ii) u ( x ) = (cid:0) u, G δ ( · − x ) (cid:1) H δ ( R d ) G δ ).In connection with the problem of our consideration, there is a standardframework on reproducing kernel Hilbert spaces of functions on R d whichresembles the structure of Sobolev spaces and reads as follows. For a givenreal-valued positive definite function Φ ∈ C ( R d ) ∩ L ( R d ) , if we define F Φ ( R d ) = ( u ∈ C ( R d ) ∩ L ( R d ) : Z R d (cid:12)(cid:12)b u ( ξ ) (cid:12)(cid:12) dξ b Φ( ξ ) < ∞ ) , (cid:0) u, v (cid:1) F Φ ( R d ) = (2 π ) − d Z R d b u ( ξ ) b v ( ξ ) dξ b Φ( ξ ) , then F Φ ( R d ) becomes a Hilbert space with a reproducing kernel Φ( x − y )(see [16], [24] and also [2] for more general properties).On account of this framework, we shall focus on constructing compactlysupported radial functions Φ ∈ C ( R d ) ∩ L ( R d ) which are positive definiteand subject to the Fourier transform estimates C (1 + | ξ | ) − δ ≤ b Φ( ξ ) ≤ C (1 + | ξ | ) − δ (1.3)for some δ > d/ C , C . An initiative construction had been started by H. Wendland ([23], [24])who introduced a family of polynomials on [0 , ∞ ) defined by P d,m ( r ) = c m Z r (cid:0) t − r (cid:1) m − t (1 − t )[ d ] + m +1 dt,P d ( r ) = (1 − r )[ d ] +1 , (1.4)for 0 ≤ r ≤ m is a positive integer and c m isa constant, and proved P d,m ( | x − y | ) is a reproducing kernel for the Sobolevspace H d +12 + m ( R d ) and so is P d ( | x − y | ) for H d +12 ( R d ) if d ≥ . In an attempt to cover the missing cases, R. Schaback ([17]) introduceda family of non-polynomial functions defined by R d,m ( r ) = c m Z r (cid:0) t − r (cid:1) m − t (1 − t ) d + m +1 dt (1.5)3or 0 ≤ r ≤ m is a nonnegative integer, andproved R d,m ( | x − y | ) is a reproducing kernel for H d + m +1 ( R d ) if d is even(see S. Hubbert [14] for computational aspects).In order to deal with fractional orders, A. Chernih and S. Hubbert ([7])further generalized Wendland’s functions in the form S d,α ( r ) = c α Z r (cid:0) t − r (cid:1) α − t (1 − t ) d +12 + α dt (1.6)for 0 ≤ r ≤ α > c α is a constant, andproved that S d,α ( | x − y | ) is a reproducing kernel for H d +12 + α ( R d ) . Our primary aim in the present paper is to obtain a family of compactlysupported radial functions which provide reproducing kernels for the Sobolevspaces H δ ( R d ) of any order δ > d/ H δ ( R d ) of order δ = d + 12 , δ > max (cid:18) , d (cid:19) , δ > d , separately. One of these classes include the compactly supported functionsof Wendland, Schaback, Chernih and Hubbert as special instances.4 distinctive feature of our construction is that the Fourier transformis explicit, which enables us to specify the inner product about which thereproducing property holds. As an illustration, it will be shown that thefunction A ( x ) = (1 − | x | ) , x ∈ R , has the Fourier transform c A ( ξ ) = 4 ξ (cid:18) − sin ξξ (cid:19) , which is strictly positive and behaves like the Cauchy-Poisson kernel, and A ( x − y ) is a reproducing kernel for H ( R ) under the inner product (cid:0) u, v (cid:1) A ( R ) = 18 π Z ∞−∞ b u ( ξ ) b v ( ξ ) ξ dξξ − sin ξ . Notation.
We shall use the following notation in what follows. • The Euler beta function will be denoted by B ( a, b ) = Z t a − (1 − t ) b − dt ( a > , b > . • The generalized hypergeometric functions will be denoted by p F q ( a , · · · , a p ; b , · · · , b q ; r ) = ∞ X k =0 ( a ) k · · · ( a p ) k k ! ( b ) k · · · ( b q ) k r k in which ( a ) k = a ( a + 1) · · · ( a + k −
1) if k ≥ a ) = 1 for anyreal number a . • The positive part of x ∈ R will be denoted by x + = max( x, . • We shall write f ( x ) ≈ g ( x ) for x ∈ X for two real-valued functions f, g defined on X to indicate there exist positive constants c , c suchthat c g ( x ) ≤ f ( x ) ≤ c g ( x ) for all x ∈ X. Positive definite functions
We recall that a function Φ on R d is said to be positive semi-definite if N X j =1 N X k =1 Φ ( x j − x k ) z j z k ≥ z , · · · , z N ∈ C and x , · · · , x N ∈ R d . If equality holdsonly when z = · · · = z N = 0 , Φ is said to be positive definite .A well-known theorem of S. Bochner states that a continuous function Φis positive semi-definite if and only if it is the Fourier transform of some finitenonnegative Borel measure µ on R d . If the carrier of µ contains an open set,then Φ = b µ is positive definite. In particular, the Fourier transform of anonnegative function f ∈ L ( R d ) is positive definite if the essential supportof f contains an open set (see [24]) A univariate function φ on [0 , ∞ ) is said to be positive semi-definite orpositive definite on R d if the radial extension x φ ( | x | ) , x ∈ R d , is positivesemi-definite or positive definite in the above sense. To state sufficient ornecessary conditions in terms of Fourier transforms, we shall introduce thefollowing kernels, more extensive than being needed, which will serve as thekernels of Hankel-Schoenberg transforms to be studied later. Definition 2.1.
For λ > − , define Ω λ : R → R byΩ λ ( t ) = Γ( λ + 1) ∞ X k =0 ( − k k ! Γ( λ + k + 1) (cid:18) t (cid:19) k = Γ( λ + 1) ( t/ − λ J λ ( t ) , where J λ denotes the Bessel function of the first kind of order λ . The carrier of a nonnegative Borel measure µ on R d is defined to be R d \ (cid:8) O ⊂ R d : O is open and µ ( O ) = 0 (cid:9) . If µ is absolutely continuous with respect to Lebesgue measure, dµ ( x ) = f ( x ) d x with anonnegative f ∈ L ( R d ) , then the carrier of µ equals to the essential support of f , thecomplement of the largest open subset of R d on which f = 0 almost everywhere.
6n the special case λ = ( d − / , with d ≥ λ ariseson consideration of the Fourier transform of the area measure σ on the unitsphere S d − of the Euclidean space R d in the form1 | S d − | Z S d − e − iξ · x dσ ( x ) = Ω d − ( | ξ | ) . An immediate consequence is that if F is integrable and radial with F ( x ) = f ( | x | ) for some univariate function f on [0 , ∞ ), then its Fouriertransform is easily evaluated as b F ( ξ ) = Z ∞ (cid:18)Z S d − e − itξ · x dσ ( x ) (cid:19) f ( t ) t d − dt = | S d − | Z ∞ Ω d − ( | ξ | t ) f ( t ) t d − dt. Since it is simple to find Ω − / ( t ) = cos t by definition, this formula continuesto hold true for d = 1 if we interpret | S | = 2 . In summary, we have the following which are substantially due to I. J.Schoenberg [19] (see also [10], [11], [20], [24]).
Proposition 2.1.
For f ∈ L (cid:0) [0 , ∞ ) , t d − dt (cid:1) , put φ ( r ) = Z ∞ Ω d − ( rt ) f ( t ) t d − dt ( r ≥ . (i) If F ( x ) = f ( | x | ) , x ∈ R d , then the Fourier transform of F is given by b F ( ξ ) = 2 π d/ Γ( d/ φ ( | ξ | ) . (ii) If f is nonnegative and the essential support of f contains an openinterval, then φ is positive definite on R d .Proof. Part (i) is what we have mentioned as above. Concerning part (ii), ifthe essential support of a nonnegative function f contains an open interval,say, I = ( a, b ) , then the essential support of F ( x ) = f ( | x | ) contains theopen annulus { x ∈ R d : a < | x | < b } and the assertion follows.7 emark . The integral defined in the statement is often called the d -dimensional radial Fourier transform and formally denoted as F d ( f )( r ) = 2 π d/ Γ( d/ Z ∞ Ω d − ( rt ) f ( t ) t d − dt ( r ≥ . (2.1)As the kernel Ω d − will be shown to be uniformly bounded, the integral makessense on the class of finite Borel measures on [0 , ∞ ). Indeed, Schoenberg’soriginal theorem states that a continuous function φ on [0 , ∞ ) is positivesemi-definite on R d if and only if φ ( r ) = Z ∞ Ω d − ( rt ) dν ( t ) ( r ≥ ν on [0 , ∞ ). As it is classical (see [22] for instance), the Hankel transforms of a function f ∈ L (cid:0) [0 , ∞ ) , √ tdt (cid:1) refer to the integrals of type Z ∞ J λ ( rt ) f ( t ) tdt ( λ ≥ − / . As a generalization of both Fourier transforms of radial functions andHankel transforms, we shall consider the following integral transforms.
Definition 3.1.
The Hankel-Schoenberg transform of order λ ≥ − / f on [0 , ∞ ) is defined to be φ ( r ) = Z ∞ Ω λ ( rt ) f ( t ) dt ( r ≥ f . For thismatter, we shall begin with investigating the kernels.8 .1 Kernels Ω λ In many aspects, each Ω λ is similar in nature to the cardinal sine functionsin tt = ∞ Y k =1 (cid:18) − t k π (cid:19) which coincides with the special case λ = 1 / . To be more specific, we listthe following properties of Ω λ ’s which are deducible from the theory of Besselfunctions J λ in a straightforward manner (see [1], [8], [22]).(P1) Each Ω λ is of class C ∞ ( R ), even and uniformly bounded by 1 = Ω λ (0) . The kernels Ω λ satisfy the Bessel-type differential equationsΩ λ ′′ ( t ) + 2 λ + 1 t Ω λ ′ ( t ) + Ω λ ( t ) = 0as well as the Lommel-type recurrence relationsΩ λ ′ ( t ) = − t λ + 1) Ω λ +1 ( t ) , Ω λ ( t ) − Ω λ − ( t ) = t λ ( λ + 1) Ω λ +2 ( t ) . (3.1)(P2) An asymptotic formula due to Hankel states 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) . (P3) Ω λ is oscillatory with an infinity of simple zeros. Arranging the positivezeros of J λ in the ascending order 0 < j λ, < j λ, < · · · , Ω λ can berepresented as the infinite productΩ λ ( t ) = ∞ Y k =1 − t j λ,k ! . (P4) Due to Liouville, Ω λ is expressible in finite terms by algebraic andtrigonometric functions if and only if 2 λ is an odd integer. Indeed,Ω − / ( t ) = cos t , Ω / ( t ) = sin tt (3.2)9nd recurrence formula (3.1) may be used to express Ω n +1 / , with n aninteger, in finite terms by elementary functions. For example,Ω / ( t ) = 3 (cid:18) sin t − t cos tt (cid:19) , Ω / ( t ) = 15 (cid:20) (3 − t ) sin t − t cos tt (cid:21) . (3.3)(P5) For λ > − / , Poisson’s integral readsΩ λ ( t ) = 2 B ( λ + 1 / , / Z cos( ts ) (1 − s ) λ − ds. Owing to the boundedness and asymptotic behavior of Ω λ described asin (P1), (P2), it is evident that the Hankel-Schoenberg transform of order λ is well defined on the class L ([0 , ∞ )) or L (cid:0) [0 , ∞ ) , t − λ − / dt (cid:1) . The Hankel-Watson inversion theorem ([22]) states that if λ ≥ − / f ( s ) √ s is integrable on [0 , ∞ ), then Z ∞ J λ ( rt ) (cid:20)Z ∞ J λ ( rs ) f ( s ) sds (cid:21) rdr = f ( t + 0) + f ( t − t > f is of bounded variation in a neighborhood of t .As it is straightforward to express Hankel-Schoenberg transforms in termsof Hankel transforms, an obvious modification yields the following. Proposition 3.1. (Inversion)
For λ ≥ − / , assume that Z ∞ | f ( t ) | t − λ − / dt < ∞ . (3.4) Then the following holds for every t > at which f is continuous: φ ( r ) = Z ∞ Ω λ ( rt ) f ( t ) dt ( r > implies f ( t ) = t λ +1 λ [Γ( λ + 1)] Z ∞ Ω λ ( rt ) φ ( r ) r λ +1 dr. emark . In the case λ = ( d − / , this formula may be considered asan alternative of the Fourier inversion theorem for radial functions. Usefulto the present circumstance is the inversion of f ∈ L ([0 , ∩ C ((0 , φ ( r ) = Z Ω d − ( rt ) f ( t ) t d − dt ( r > ⇒ f ( t ) = 12 d − [Γ( d/ Z ∞ Ω d − ( rt ) φ ( r ) r d − dr ( t > . (3.5) As the radial Fourier transforms of different dimensions are known to beinterrelated by certain dimension walk transforms, the Hankel-Schoenbergtransforms of different orders turn out to be related with each other.
Lemma 3.1.
For ν > − and α > , β > , we have F (cid:18) β ; α + β, ν + 1; − r (cid:19) = Z ∞ Ω ν ( rt ) f ( t ) dt ( r ≥ , where f is the probability density on [0 , ∞ ) defined by f ( t ) = 2 B ( α, β ) (1 − t ) α − t β − . Proof.
An elementary computation shows Z ∞ t k f ( t ) dt = ( β ) k ( α + β ) k , k = 0 , , , · · · , and integrating termwise yields Z ∞ Ω ν ( rt ) f ( t ) dt = ∞ X k =0 ( − k k ! ( ν + 1) k (cid:16) r (cid:17) k Z ∞ t k f ( t ) dt = ∞ X k =0 ( β ) k k ! ( α + β ) k ( ν + 1) k (cid:18) − r (cid:19) k . Theorem 3.1.
Let d be a positive integer and λ > d/ − . (i) For each r ≥ , we have Ω λ ( r ) = 2 B (cid:0) λ + 1 − d , d (cid:1) Z ∞ Ω d − ( rt )(1 − t ) λ − d + t d − dt. (ii) If f ∈ L ([0 , ∞ )) , then for each r ≥ , Z ∞ Ω λ ( rt ) f ( t ) dt = Z ∞ Ω d − ( rt ) I λ ( f )( t ) t d − dt, where I λ ( f )( t ) = 2 B (cid:0) λ + 1 − d , d (cid:1) Z ∞ t (cid:0) s − t (cid:1) λ − d s − λ f ( s ) ds. Moreover, I λ ( f ) ∈ L (cid:0) [0 , ∞ ) , t d − dt (cid:1) with Z ∞ | I λ ( f )( t ) | t d − dt ≤ Z ∞ | f ( t ) | dt. Proof.
The special choices of ν = d/ − , α = λ + 1 − d/ , β = d/ λ ( r ) = F (cid:18) λ + 1; − r (cid:19) . As for part (ii), we first notice Z ∞ Z ∞ t (cid:0) s − t (cid:1) λ − d s − λ | f ( s ) | ds t d − dt = Z ∞ (cid:20)Z s (cid:0) s − t (cid:1) λ − d t d − dt (cid:21) s − λ | f ( s ) | ds = Z ∞ (cid:20)Z (cid:0) − u (cid:1) λ − d u d − du (cid:21) | f ( s ) | ds = B (cid:0) λ + 1 − d , d (cid:1) Z ∞ | f ( s ) | ds, I λ ( f ) ∈ L (cid:0) [0 , ∞ ) , t d − dt (cid:1) and the last estimate follows. The statedformula is a simple consequence of part (i) on interchanging the order ofintegrations, which is legitimate due to Fubini’s theorem. Remark . If d = 1 , part (i) reduces to Poisson’s integral (P5). The so-called descending-dimension walks of radial Fourier transforms are specialinstances of this theorem. In fact, if we take λ = ( d + k − / , with d, k positive integers, and write I λ = I k for simplicity, then the formula of part(ii) applied to the function f ( t ) t d + k − yields Z ∞ Ω d + k − ( rt ) f ( t ) t d + k − dt = Z ∞ Ω d − ( rt ) I k ( f )( t ) t d − dt,I k ( f )( t ) = 2 B (cid:0) k , d (cid:1) Z ∞ t (cid:0) s − t (cid:1) k − sf ( s ) ds. (3.6)In the notation of (2.1), it reads F d + k ( f )( r ) = π k/ Γ (cid:0) d (cid:1) Γ (cid:0) d + k (cid:1) F d ( I k ( f )) ( r ) , (3.7)which expresses the ( d + k )-dimensional radial Fourier transform of f as d -dimensional radial Fourier transform of I k ( f ). We refer to [17], [18] and [24]for more detailed results on dimension walks. As the first step of our construction, we shall consider all possible binomialdensities and evaluate their Hankel-Schoenberg transforms.
Lemma 4.1.
Let λ > − and α > , β > . For each r ≥ , Z ∞ Ω λ ( rt ) p ( t ) dt = F (cid:18) β , β + 12 ; α + β , α + β + 12 , λ + 1; − r (cid:19) , here p is the probability density on [0 , ∞ ) defined by p ( t ) = 1 B ( α, β ) (1 − t ) α − t β − . Proof.
By applying Legendre’s duplication formula for the gamma functionrepeatedly, it is elementary to compute Z ∞ t k p ( t ) dt = B ( α, k + β ) B ( α, β ) = (cid:0) β (cid:1) k (cid:0) β +12 (cid:1) k (cid:0) α + β (cid:1) k (cid:0) α + β +12 (cid:1) k for k = 0 , , , · · · , and integrating termwise yields the stated result.After reducing the generalized hypergeometric functions of Lemma 4.1to the ones of type F , we shall investigate their asymptotic properties forwhich our analysis will be based on the following lemma which has beenstudied by many authors including R. Askey and H. Pollard [5], J. Steinig[21] and culminated in the present form by J. Fields and M. Ismail [9]. Lemma 4.2.
For ρ > , ν > , put U ( ρ, ν ; x ) = F (cid:18) ν ; ρν, ρν + 12 ; − x (cid:19) ( x ∈ R ) . (i) If ρ = 1 , it is identical to the function Ω ν − / . (ii) If ρ = 1 , then as | x | → ∞ ,U ( ρ, ν ; x ) = Γ(2 ρν )Γ(2 ρν − ν ) | x | − ν h O (cid:0) | x | − (cid:1) i + Γ(2 ρν )2 ν − Γ( ν ) | x | − ν ( ρ − ) (cid:20) cos (cid:16) | x | − ρνπ + νπ (cid:17) + O (cid:0) | x | − (cid:1)(cid:21) . (iii) If either ρ ≥ , ν > or ρ ≥ , ν > , then U ( ρ, ν ; x ) ≈ (1 + | x | ) − ν for x ∈ R . In particular, U ( ρ, ν ; x ) > for every x ∈ R . Askey’s class for H d +12 ( R d ) In the special case δ = ( d + 1) / , the Bessel potential kernel G δ , whichgives a reproducing kernel for H d +12 ( R d ) under the usual inner product,coincides with the exponential of −| x | and its Fourier transform is nothingbut the Cauchy-Poisson kernel (see appendix). To be precise, we have G d +12 ( x ) = 12 d π d − Γ (cid:0) d +12 (cid:1) e −| x | , [ G d +12 ( ξ ) = (1 + | ξ | ) − d +12 . A large class of compactly supported functions, often referred to as Askey’sclass ([4], [13], [24]), turn out to be also available as reproducing kernels undersuitable inner products in this case.
Theorem 5.1.
For a positive integer d , assume that α satisfies α ≥ d +12 if d ≥ and α ≥ if d = 1 . Define Λ d,α ( r ) = F (cid:18) d + 12 ; d + α + 12 , d + α + 22 ; − r (cid:19) ( r ≥ . (i) Λ d,α is positive definite on R d with Λ d,α ( r ) = 1 B ( α + 1 , d ) Z ∞ Ω d − ( rt ) (1 − t ) α + t d − dt ( r ≥ . (ii) 0 < Λ d,α ( r ) ≤ for each r ≥ and Λ d,α ( r ) = Γ( d + α + 1)Γ( α ) r − d − h O (cid:0) r − (cid:1) i + Γ( d + α + 1)2 d − Γ (cid:0) d +12 (cid:1) r − ( d +2 α +1)2 (cid:20) cos (cid:18) r − ( d + 2 α + 1) π (cid:19) + O (cid:0) r − (cid:1)(cid:21) as r → ∞ . Moreover, Λ d,α ( r ) ≈ (1 + r ) − d − for r ≥ . (iii) Λ d,α ∈ C ([0 , ∞ )) ∩ L (cid:0) [0 , ∞ ) , r d − dr (cid:1) and (1 − t ) α + = 2Γ( α + 1)Γ (cid:0) d +12 (cid:1) √ π Γ( α + d + 1)Γ (cid:0) d (cid:1) Z ∞ Ω d − ( rt )Λ d,α ( r ) r d − dr ( t ≥ . As a consequence, the function t (1 − t ) α + is positive definite on R d . roof. The integral representation of part (i) corresponds to the special caseof Lemma 4.1 with λ = d − , β = d for which we replace α by α + 1. Thepositive definiteness of Λ d,α is an immediate consequence of Proposition 2.1.As to part (ii), while the uniform bound | Λ d,α ( r ) | ≤ d,α ( r ) = U (cid:18) d + α + 1 d + 1 , d + 12 ; r (cid:19) . As to part (iii), the property Λ d,α ∈ C ([0 , ∞ )) ∩ L (cid:0) [0 , ∞ ) , r d − dr (cid:1) isobvious. For t > , the stated integral representation follows by invertingthe formula of part (i) in accordance with Proposition 3.1, particularly with(3.5). By continuity, it continues to hold true for t = 0 . Finally, the positivedefiniteness follows again from Proposition 2.1.On consideration of radial extensions, we obtain the following in which γ d,α = 2 d π d − Γ( α + 1)Γ (cid:0) d +12 (cid:1) Γ( α + d + 1) . Corollary 5.1.
For α ≥ d +12 if d ≥ and α ≥ if d = 1 , put A α ( x ) = (1 − | x | ) α + ( x ∈ R d ) . Then each A α is continuous and positive definite with c A α ( ξ ) = γ d,α Λ d,α ( | ξ | ) ( ξ ∈ R d ) . As a consequence, A α ( x − y ) is a reproducing kernel for the Sobolev space H d +12 ( R d ) with respect to the inner product defined by (cid:0) u, v (cid:1) A α ( R d ) = (2 π ) − d · γ d,α Z R d b u ( ξ ) b v ( ξ ) dξ Λ d,α ( | ξ | ) . emark . If we put Λ d,α ( x ) = Λ d,α ( | x | ) , x ∈ R d , for simplicity, then theinversion formula of part (iii) in Theorem 5.1 shows d Λ d,α ( ξ ) = π d +12 Γ( α + d + 1)Γ( α + 1)Γ (cid:0) d +12 (cid:1) A α ( ξ ) ( ξ ∈ R d ) . Thus Λ d,α is an example of band-limited functions, the class of L functionswhose Fourier transforms are compactly supported.In the odd dimensional case, Λ d,α is expressible in terms of algebraic andtrigonometric functions if α happens to be an integer. As illustrations, wepresent the following examples:(a) In the case d = 1 , the formula of part (i) in Theorem 5.1 reduces toΛ ,α ( r ) = ( α + 1) Z cos( xt )(1 − t ) α dt ( α ≥ . With the choice of minimal α = 2 and α = 3 , we haveΛ , ( r ) = 6 r (cid:18) − sin rr (cid:19) , Λ , ( r ) = 12 r ( − (cid:20) sin( r/ r/ (cid:21) ) in which each formula must be understood as the limiting value at r = 0 .(b) In the case d = 3 , the formula of part (i) in Theorem 5.1 reduces toΛ ,α ( r ) = ( α + 3)( α + 2)( α + 1)2 r Z sin( rt )(1 − t ) α tdt ( α ≥ . With the choice of minimal α = 2 , we haveΛ , ( r ) = 60 r (cid:18) r − rr (cid:19) with the same interpretation at r = 0 as above.17 Compactly supported reproducing kernelsfor H δ ( R d ) with δ > max (1 , d/ Due to an obvious cancellation effect, the generalized hypergeometricfunction of Lemma 4.1 in the special case β = 2 λ + 1 reduces to F (cid:18) λ + 12 ; α + 2 λ + 12 , α + 2 λ + 22 ; − r (cid:19) = 1 B ( α, λ + 1) Z ∞ Ω λ ( rt )(1 − t ) α − t λ dt. (6.1)Expressing in the form of U -function defined in Lemma 4.2, it is simpleto find that this function is strictly positive if λ > / , α ≥ λ + 1 / . Thechoice of minimal value α = λ + 1 / F (cid:18) λ + 12 ; 32 (cid:18) λ + 12 (cid:19) , (cid:18) λ + 12 (cid:19) + 12 ; − r (cid:19) = 1 B (cid:0) λ + , λ + 1 (cid:1) Z ∞ Ω λ ( rt )(1 − t ) λ − + t λ dt. (6.2)Rearranging parameters λ + 1 / δ and representing the last Hankel-Schoenberg transforms in terms of radial Fourier transforms, that is, thoseintegrals with kernels Ω d − , we are led to the following class of functions. Definition 6.1.
For a positive integer d and δ > d/ , defineΦ d,δ ( t ) = 1 B (2 δ − d, δ ) Z t ( s − t ) δ − d +12 (1 − s ) δ − ds for 0 ≤ t ≤ Lemma 6.1.
For a positive integer d and δ > d/ , the integral in thedefinition of Φ d,δ converges and the following properties hold: (i) Φ d,δ is continuous, strictly decreasing on [0 , and ≤ Φ d,δ ≤ . (ii) Φ d,δ ( t ) ≈ (1 − t ) δ − d +12 on [0 , . If δ = d +12 , then Φ d,δ ( t ) = (1 − t ) d +12 + . (iv) If δ > d +12 , then for ≤ t ≤ , Φ d,δ ( t ) = 1 B (2 δ − d − , δ + 1) Z t ( s − t ) δ − d +32 s (1 − s ) δ ds. Proof.
For δ ≥ d +12 , as the function Φ d,δ is dominated by1 B (2 δ − d, δ ) Z s δ − d − (1 − s ) δ − ds = 1 , the convergence of the defining integral is obvious. Under the transformation s θ + (1 − θ ) t, ≤ θ ≤ , we may writeΦ d,δ ( t ) = (1 − t ) δ − d +12 V ( t ) , where V ( t ) = 1 B (2 δ − d, δ ) Z θ δ − d +12 (1 − θ ) δ − (cid:2) t + θ (1 − t ) (cid:3) δ − d +12 dθ . In the case d < δ < d +12 , if we observe2 δ − d +12 ≤ (cid:2) t + θ (1 − t ) (cid:3) δ − d +12 ≤ θ δ − d +12 for 0 ≤ t ≤ θ > , it is simple to infer that V ( t )converges uniformly on [0 ,
1] with 0 ≤ V ( t ) ≤ d,δ is welldefined. Bounding V ( t ) in this way, we also deduce part (ii) plainly.As the convergence is ensured, part (i) can be verified easily. Part (iii) istrivial and part (iv) is a simple consequence of integrating by parts. Remark . Noteworthy are the following special instances of part (iv).(a) In the case δ = d/ k + 1 / , k ∈ N , Φ d,δ coincides with Wendland’sfunction P d,k , defined in (1.4), in the odd dimensions.(b) In the case δ = d/ m +1 , with m a nonnegative integer, Φ d,δ coincideswith Schaback’s function R d,m , defined in (1.5), in every dimension.Likewise, if δ = ( d + 1) / α, α > , Φ d,δ coincides with the function S d,α of Chernih and Hubbert, defined in (1.6), in every dimension.19n the statement below, we shall denote ω d,δ = 2 − d Γ (3 δ ) Γ (cid:0) δ − d (cid:1) Γ ( δ ) Γ (3 δ − d ) Γ (cid:0) d (cid:1) . (6.3) Theorem 6.1.
For a positive integer d and δ > max (1 , d/ , define W δ ( r ) = F (cid:18) δ ; 3 δ , δ + 12 ; − r (cid:19) ( r ≥ . (i) W δ is positive definite on R d with W δ ( r ) = 1 B ( δ, δ ) Z ∞ Ω δ − ( rt )(1 − t ) δ − t δ − dt = ω d,δ Z ∞ Ω d − ( rt ) Φ d,δ ( t ) t d − dt. (ii) 0 < W δ ( r ) ≤ for each r ≥ and as r → ∞ ,W δ ( r ) = Γ (3 δ )Γ ( δ ) r − δ (cid:20) r − δπ )2 δ − (cid:21) + O (cid:0) r − δ − (cid:1) . Moreover, W δ ( r ) ≈ (1 + r ) − δ for r ≥ . (iii) W δ ∈ C ([0 , ∞ )) ∩ L (cid:0) [0 , ∞ ) , r d − dr (cid:1) and Φ d,δ ( t ) = 12 d − [Γ( d/ ω d,δ Z ∞ Ω d − ( rt ) W δ ( r ) r d − dr ( t ≥ . As a consequence, Φ d,δ is positive definite on R d .Proof. If we set λ = δ − / W δ ( r ) = 1 B ( δ, δ ) Z ∞ Ω δ − ( rt )(1 − t ) δ − t δ − dt = C ( d, δ ) Z ∞ Ω d − ( rt ) Φ d,δ ( t ) t d − dt, C ( d, δ ) = B (2 δ − d, δ ) B (cid:0) δ + − d , d (cid:1) B ( δ, δ ) . Simplifying with the aid of Legendre’s duplication formula for the gammafunction, it is elementary to see C ( d, δ ) = ω d,δ . The positive definiteness of W δ is an immediate consequence of Proposition 2.1 and part (i) is proved.In view of the identification W δ ( r ) = U (cid:18) , δ ; r (cid:19) , part (ii) is a consequence of Lemma 4.2 and Lemma 6.1.As to part (iii), that W δ ∈ C ([0 , ∞ )) ∩ L (cid:0) [0 , ∞ ) , r d − dr (cid:1) is obvious. For t > , the stated representation follows by inverting the formula of part (i)in accordance with (3.5). By continuity, it continues to hold true for t = 0 . Finally, the positive definiteness follows again from Proposition 2.1.As an immediate corollary, we obtain what we aim to accomplish. Tosimplify notation, we shall write ζ d,δ = 2 π d/ Γ( d/ · ω d,δ . (6.4) Corollary 6.1.
For δ > max (1 , d/ , let Φ d,δ ( x ) = Φ d,δ ( | x | ) , x ∈ R d . Then Φ d,δ is continuous and positive definite with d Φ d,δ ( ξ ) = ζ d,δ W δ ( | ξ | ) ( ξ ∈ R d ) . As a consequence, Φ d,δ ( x − y ) is a reproducing kernel for the Sobolev space H δ ( R d ) with respect to the inner product defined by (cid:0) u, v (cid:1) Φ d,δ ( R d ) = (2 π ) − d · ζ d,δ Z R d b u ( ξ ) b v ( ξ ) dξW δ ( | ξ | ) . Remark . In the special case δ = ( d + 1) / α, α > , A. Chernih andS. Hubbert also obtained the Fourier transform W δ (Theorem 2.1, [7]), butthe authors did not give the integral representation formula nor the inversionformula as stated in the first equation of part (ii), part (iii) of Theorem 6.1,respectively. We supplement a few computational aspects as follows.21a) In some special instances, it is possible to evaluate Φ d,δ in closed formsas the following list shows.Φ d,δ ( r ) on the interval [0 , δ = d +12 d ≥ − r ) d +12 d = 1 (1 − r ) (1 + 3 r ) δ = 2 d = 2 (1 + 2 r ) √ − r − r log (cid:16) √ − r r (cid:17) d = 3 (1 − r ) d = 1 (1 − r ) (1 − r + 8 r ) δ = 3 d = 2 h (4 − r − r ) √ − r + 15 r (6 + r ) log (cid:16) √ − r r (cid:17) i d = 3 (1 − r ) (1 + 4 r )(b) In the case when δ is an integer, one may use the representation formulaof part (i), Theorem 6.1, to express W δ in a closed form involvingalgebraic and trigonometric functions. To illustrate, let us take W ( r ) = F (cid:18)
2; 3 ,
72 ; − r (cid:19) ( r ≥ . We evaluate W ( r ) = 20 Z ∞ Ω / ( rt )(1 − t ) + t dt = 60 r (cid:26)Z sin( rt )(1 − t ) dt − r Z cos( rt )(1 − t ) tdt (cid:27) = 120 r (cid:16) r (cid:17) −
180 sin rr . We should point out this closed form is consistent with the asymptoticformula stated in part (ii) of Theorem 6.1 which reads W ( r ) = 120 r (cid:16) r (cid:17) + O (cid:0) r − (cid:1) . A smoother family of compactly supportedreproducing kernels
Due to the restriction δ > max (1 , d/ , there are missing cases in thepreceding results, namely, the cases 1 / < δ ≤ H δ ( R ) . Although the particular instance δ = 1 is covered inCorollary 5.1, the case 1 / < δ < H δ ( R d ) of any order δ > d/ λ + 1 / > , α ≥ λ + 1 . Choosingthe minimal α = 2 λ + 1 and setting λ + 1 / δ, it reduces to F (cid:18) δ ; 2 δ, δ + 12 ; − r (cid:19) = 1 B (2 δ, δ ) Z ∞ Ω δ − ( rt )(1 − t ) δ − t δ − dt, (7.1)which is strictly positive for any δ > . For δ > d/ , an application of order-walk transformation yields Z ∞ Ω δ − ( rt )(1 − t ) δ − t δ − dt = Z ∞ Ω d − ( rt ) I δ ( t ) t d − dt (7.2)in which I δ stands for the function supported in [0 ,
1] and defined by I δ ( t ) = 2 B (cid:0) δ + − d , d (cid:1) Z t ( s − t ) δ − d +12 (1 − s ) δ − ds for 0 ≤ t ≤ . Normalizing the constant, we introduce
Definition 7.1.
For a positive integer d and δ > d/ , defineΨ d,δ ( t ) = 1 B (2 δ − d, δ ) Z t ( s − t ) δ − d +12 (1 − s ) δ − ds for 0 ≤ t ≤ d,δ , we deduce its basic properties in thesame way as stated and proved in Lemma 6.1. Lemma 7.1.
For a positive integer d and δ > d/ , the integral in thedefinition of Ψ d,δ converges and the following properties hold: (i) Ψ d,δ is continuous, strictly decreasing on [0 , and ≤ Ψ d,δ ≤ . (ii) Ψ d,δ ( t ) ≈ (1 − t ) δ − d +12 on [0 , . (iii) If δ = d +12 , then Ψ d,δ ( t ) = (1 − t ) d +1+ . (iv) If δ > d +12 , then for ≤ t ≤ , Ψ d,δ ( t ) = 1 B (2 δ − d − , δ + 1) Z t ( s − t ) δ − d +32 s (1 − s ) δ ds. Combining (7.1), (7.2) in terms of Ψ d,δ , we obtain the following analog ofTheorem 6.1 without difficulty in which we write τ d,δ = 2 − d Γ (4 δ ) Γ (cid:0) δ − d (cid:1) Γ ( δ ) Γ (4 δ − d ) Γ (cid:0) d (cid:1) . (7.3) Theorem 7.1.
For a positive integer d and δ > d/ , define Q δ ( r ) = F (cid:18) δ ; 2 δ, δ + 12 ; − r (cid:19) ( r ≥ . (i) Q δ is positive definite on R d with Q δ ( r ) = 1 B (2 δ, δ ) Z ∞ Ω δ − ( rt )(1 − t ) δ − t δ − dt = τ d,δ Z ∞ Ω d − ( rt ) Ψ d,δ ( t ) t d − dt. (ii) 0 < Q δ ( r ) ≤ for each r ≥ and as r → ∞ ,Q δ ( r ) = Γ (4 δ )Γ (2 δ ) r − δ + O (cid:0) r − min (2 δ +2 , δ ) (cid:1) . Moreover, Q δ ( r ) ≈ (1 + r ) − δ for r ≥ . Q δ ∈ C ([0 , ∞ )) ∩ L (cid:0) [0 , ∞ ) , r d − dr (cid:1) and Ψ d,δ ( t ) = 12 d − [Γ( d/ τ d,δ Z ∞ Ω d − ( rt ) Q δ ( r ) r d − dr ( t ≥ . As a consequence, Ψ d,δ is positive definite on R d . As an immediate corollary, we obtain the following in which η d,δ = 2 π d/ Γ( d/ · τ d,δ . (7.4) Corollary 7.1.
For δ > d/ , let Ψ d,δ ( x ) = Ψ d,δ ( | x | ) , x ∈ R d . Then Ψ d,δ iscontinuous and positive definite with d Ψ d,δ ( ξ ) = η d,δ Q δ ( | ξ | ) ( ξ ∈ R d ) . As a consequence, Ψ d,δ ( x − y ) is a reproducing kernel for the Sobolev space H δ ( R d ) with respect to the inner product defined by (cid:0) u, v (cid:1) Ψ d,δ ( R d ) = (2 π ) − d · η d,δ Z R d b u ( ξ ) b v ( ξ ) dξQ δ ( | ξ | ) . Remark . In view of parts (ii), (iii) of Lemma 6.1, it is evident that Ψ d,δ is much smoother than Φ d,δ is, if both parameters d, δ are fixed. A possibledisadvantage in practical applications, however, is that Ψ d,δ involves higheralgebraic powers than Φ d,δ does.(a) As illustrations, we have the following evaluations:Ψ d,δ ( r ) on the interval [0 , d ≥ δ = d +12 (1 − r ) d +1 δ =
32 12 h (2 + 13 r ) √ − r − r (4 + r ) log (cid:16) √ − r r (cid:17) i d = 1 δ = 2 (1 − r ) (1 + 5 r ) δ = 3 (1 − r ) (1 + 8 r + 21 r ) δ = (1 − r ) (1 + 6 r ) d = 2 δ =
72 13 (1 − r ) (3 + 27 r + 80 r ) δ = (1 − r ) (1 + 12 r + 57 r + 112 r ) d = 3 δ = 3 (1 − r ) (1 + 7 r ) δ = 4 (1 − r ) (1 + 10 r + 33 r )25b) As before, one may use the representation formula of part (i), Theorem7.1, to express Q δ in a closed form involving algebraic and trigonometricfunctions in some instances. To illustrate, let us take Q ( r ) = F (cid:18)
2; 4 ,
92 ; − r (cid:19) ( r ≥ . In this special case, we evaluate Q ( r ) = 140 Z ∞ Ω / ( rt )(1 − t ) t dt = 420 r (cid:26)Z sin( rt )(1 − t ) dt − r Z cos( rt ) t (1 − t ) dt (cid:27) = 840 r (cid:0) r − r + 15 sin r − r cos r (cid:1) , which is consistent with the asymptotic formula Q ( r ) = 840 r + O (cid:0) r − (cid:1) . In addition to the Fourier transform formulas, the Bessel potential kernels G δ (or Mat´ern functions) possess a number of important properties and arisein many areas of Mathematics with various disguises. As we are concernedwith constructing possible replacements of Bessel potential kernels in thesubject of reproducing kernels for Sobolev spaces, it may be instructive torecall some of their very basic properties (see [1], [22]).(a) Each G δ is smooth away from the origin and subject to the asymptotic26ehavior, modulo multiplicative constants, described as follows:(i) As | x | → ∞ , G δ ( x ) ∼ e −| x | | x | δ − d +12 . (ii) As | x | → , G δ ( x ) ∼ δ > d/ , − log | x | if δ = d/ , | x | δ − d if δ < d/ . (b) Due to Schl¨afli’s integral representations, 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, (8.1)which is valid for α > − / z > , it is easy to see G d +12 ( x ) = K ( | x | ) p | x | d − π d Γ (cid:0) d +12 (cid:1) = e −| x | d π d − Γ (cid:0) d +12 (cid:1) . (c) More generally, if m is a nonnegative integer, then G m + d +12 ( x ) = e −| x | | x | m m + d π d − Γ (cid:0) m + d +12 (cid:1) m X k =0 ( m + k )! k !( m − k )! (2 | x | ) − k , (8.2)which can be deduced easily from Schl¨afli’s integrals. References [1] M. Abramowitz and I. A. Stegun (editors),
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