Density in Spaces of Interpolation by Hankel Translates of a Basis Function

The function spaces Y m (m ∈ Z + ) arising in the theory of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined through a seminorm which is expressed in terms of the Hankel transform of each function and involves a weight w. At least two special classes of weights allow to write these indirect seminorms in direct form, that is, in terms of the function itself rather than its Hankel transform. In this paper, we give fairly general conditions on w which ensure that the Zemanian spaces B μ and H μ (μ > −1/2) are dense in Y m (m ∈ Z + ). These conditions are shown to be satisfied by the weights giving rise to direct seminorms of the so-called type II.

1.1.The Distributional Hankel Transformation.Aiming to define the Hankel transformation in spaces of distributions, Zemanian [1] introduced the space H  of all those smooth, complex-valued functions  = () ( ∈ ) such that ] , () = max Here, and in the sequel,  =   = /.When topologized by the family of norms {] , } ∈Z + , H  becomes a Fréchet space where ℎ  is an automorphism provided that  ≥ −1/2.Then the generalized Hankel transformation ℎ   , defined by transposition on the dual H   of H  , is an automorphism of H   when this latter space is endowed with either its weak * or its strong topology.
Zemanian [2] also constructed the space B  as follows.For every  > 0, B , consists of all those  ∈ H  such that () = 0 whenever  > .This space is endowed with the topology generated by the family of seminorms { , } ∈Z + , given by  , () = sup ∈]0,[       ( −1 )   −−1/2  ()       ( ∈ B , ,  ∈ Z + ) . ( In this way B , becomes a Fréchet space.Moreover, if 0 <  < , then B , ⊂ B , and the topology of B , coincides with that inherited from B , .This allows to consider the inductive limit B  = ⋃ >0 B , .As usual, the dual spaces of B  and B , are respectively denoted by B   and B  , ( > 0).Since B  is a dense subspace of H  , H   can be regarded as a subspace of B   .
For 1 ≤  < ∞, denote by    the space of Lebesgue measurable functions whose th power is absolutely integrable on  with respect to the weight  +1/2 , normed with By  ∞  we will represent the space of Lebesgue measurable functions  = () ( ∈ ) such that  −−1/2 () ( ∈ ) is essentially bounded, normed with If  ∈  1  and  ∈  ∞  , then (5) and ( 4) exist as continuous functions on .If ,  ∈  1  , then (5) and (4) exist as functions in  1  , and, moreover, the formula and the exchange formula hold.
The study of the Hankel convolution on compactly supported distributions was initiated by de Sousa Pinto [6], only for  = 0.In a series of papers, Betancor and the second named author investigated systematically the generalized #convolution in wider spaces of distributions, allowing  > −1/2.In this context (cf.[7]), the Hankel translation was shown to be a continuous operator from H  into itself.Thus, the Hankel convolution # ∈ H   of  ∈ H   and  ∈ H  can be defined through [7, Definition 3.1].The formulas respectively extending (11) and (12), hold in the sense of equality in H   (cf.[7, Proposition 3.5]).

Interpolation by Hankel Translates of a Basis Function.
In approximation theory, radially symmetric, (conditionally) positive definite functions are used to solve scattered data interpolation problems in Euclidean space.The setting for a variational approach to such interpolation problems, the socalled native spaces, was constructed by several authors upon seminal work of Micchelli [8] and Madych and Nelson [9][10][11].
Later, Light and Wayne [12] ideated an alternative approach in which the distributional theory of the Fourier transformation plays a prominent role.When dealing with interpolation by radial basis functions, one can either (i) keep treating the involved functions as radially symmetric functions on R  ( ∈ N), or (ii) identify them with functions on the positive real half-axis.For instance, Schaback and Wu [13] devised a general theory which allows to write multivariate Fourier transforms or convolutions of radial functions as very simple univariate operations.Motivated by [12], in [14] we benefited from the Hankel transformation and the Hankel convolution in order to provide (ii) with an adequate theoretical support.This new approach generalizes and improves (i) in a sense that is made precise next.
Recall that if  ∈ N and () =  0 (||) (a.e. ∈ R  ) is an integrable radial function, then its -dimensional Fourier transform is also radial and reduces to a 1-dimensional Hankel transform of order /2 − 1 [15, Theorem 3.3]: Actually, since it turns out that on radial univariate -even-functions, the Fourier transformation, which agrees with the Fourier-cosine transformation, coincides (up to a multiplicative constant) with the Hankel transform as well.Similarly, the abovementioned variant of the Hankel convolution of order  = −1/2 can be seen to coincide with the usual convolution on R (cf.[16,Example 3.2]).Thus, for 2 + 2 ∉ N the Hankel convolution structure provides a strict generalization of the Fourier one.
Denote by  1 , the class of all those Lebesgue measurable functions  = () ( ∈ ) such that The following spaces were introduced in [14].
Definition 1.Let  = () > 0 ( ∈ ) be a continuous function, let be the Bessel differential operator, and let where  0  is the identity operator,    ( ∈ N) is the operator   iterated  times, and  2 , stands for the class of all measurable functions  = () ( ∈ ) satisfying A seminorm (norm if  = 0) is defined on   by setting In [14], for  ∈ N and suitable conditions on the weight  related to the values of , the spaces   were shown to consist of continuous functions on .Also, interpolants to  ∈   of the form were obtained, where { 1 , . . .,   } ⊂  is the set of interpolation points; Φ ∈ H   is a complex function defined on  (the so-called basis function), connected with  through the distributional identity , () =  2++1/2 ( ∈ Z + , 0 ≤  ≤  − 1) are Müntz monomials;   ( ∈ ) denotes the Hankel translation operator of order ; and   ,   (,  ∈ Z + , 1 ≤  ≤ , 0 ≤  ≤  − 1) are complex coefficients.When applied to scattered data interpolation the previous scheme leaves a greater variety of manageable kernels at our disposal, which could be useful in handling mathematical models built upon a class of radial basis functions depending on the order  and whose performance is expected to improve by adjusting , as it happens with the family of Matérn kernels in [17,Supplement,p. 6]; the examples and numerical experiments exhibited in [14] seem to support this view.Other potential applications of interpolation by Hankel translates of a basis function are in the field of radial basis function neural networks [18][19][20].
It may be observed that the seminorm in ( 23) is written in terms of the Hankel transform of the function  (an indirect seminorm) rather than  itself (a direct seminorm).The latter is more convenient for the purpose of obtaining error estimates, however.Motivated by [21], in [22] we expressed the indirect seminorm (23) in two equivalent direct forms, which were referred to as seminorms of type I and type II.
Here we want to use type II seminorms to gain a deeper understanding of the spaces   ( ∈ Z + ).We show that, under rather general conditions on the weight , which are satisfied by those weights giving rise to seminorms of type II, the Zemanian spaces B  and H  are dense in   ( ∈ Z + ).

Structure and Notation.
This paper is organized as follows.In Section 2 we recall the definition of a seminorm of type II and introduce the notion of strong type II seminorm.Also, we prove that those weights giving rise to type II seminorms are integrable near zero and exhibit polynomial growth at infinity.With the aid of some preliminary lemmas concerning Hankel approximate identities, the density of B  and H  in   ( ∈ Z + ) is finally proved in Section 3.
Throughout the rest of this paper, the positive real axis will be always denoted by , while  will stand for a real number strictly greater than −1/2, and  will represent a suitable positive constant, depending only on the opportune subscripts (if any), whose value may vary from line to line.Moreover, we shall adhere to the notations Z + = N ∪ {0} for the set of nonnegative integers and J  () =  1/2   () ( ∈ ) for the function giving the kernel of the Hankel transformation ℎ  .The following classes of functions will be occasionally used: C, formed by the continuous functions on , and E, consisting of all those infinitely differentiable functions on .For the operational rules of the Hankel transformation and further properties of the Hankel translation and Hankel convolution that eventually might be required, both in the classical and the generalized senses, the reader is mainly referred to [3-5, 7, 23, 24].

Seminorms of Type II
Denote by  1  , the class of all those measurable functions  = () ( ∈ ) such that Definition 2. A seminorm of the form given in ( 23) is called a type II seminorm provided that where (i) the distribution  ∈ H   is regular, generated by a continuous function on , such that  ∈  ∞  and the limit lim  → 0+  −−1/2 () exists,   8.6 (10)].Thus,  satisfies the strong condition (iii), hence the weak one (with  =  + 1/2), as well as the remaining conditions (i), (ii), and (iv) in Definition 2. For  = 1/2, with the aid of Maple 14, the weight defined by ( 27) is found to be and the expression in parentheses can be seen to be positive for  ̸ = 0. Consequently, () > 0 ( ∈ ).
Theorem 5 later will show that condition (i) in Definition 2 above is somewhat redundant and, at the same time, will shed some light on how to construct weights giving rise to seminorms of type II.The following preliminary result is well known; we include it for the sake of completeness.
Proof.Associate  to  as in Definition 2, and let  = ℎ   .From ( 5) and ( 8) we have or Consequently, there exists  > 0 for which This establishes the proposition.

Density Results
In this section we prove two density results for weights  = () > 0 ( ∈ ) satisfying the conditions in the thesis of Proof.For  = 0, this is [14, Theorem 2.12]; the proof below runs along similar lines, and we include it for completeness.The following operational rule of the Hankel transformation will be used [24,Equation 5.4(5)]: From the hypothesis, there exist ,  > 0 such that () ≤   ( > ).Fix  ∈ H  .Then, The first integral on the right-hand side of this identity is finite because  ∈  1 , : On the other hand, provided that 2 > 4 +  + 3 + 5/2.
Endowed with the topology generated by the family of seminorms { ,, } (,)∈N×Z + , E  becomes a Fréchet space [23, Proposition 4.3].As usual, its dual space will be denoted by E   .The inclusions B  ⊂ H  ⊂ E  being dense, we have E   ⊂ H   ⊂ B   [23, Proposition 4.4].
[14,osition 7; namely, the Zemanian spaces B  and H  are dense in   ( ∈ Z + ).Both of these results will therefore hold true for spaces endowed with type II seminorms.In this way, direct seminorms allow us to establish direct counterparts of[14, Theorem 2.23], where the space { ∈   : ℎ    ∈ B  } and hence { ∈   : ℎ    ∈ H  } were shown to be dense in   ( ∈ Z + ).