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Journal of Function Spaces and Applications
Volume 2013 (2013), Article ID 813502, 9 pages
Density in Spaces of Interpolation by Hankel Translates of a Basis Function
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Spain
Received 3 May 2013; Accepted 24 June 2013
Academic Editor: Józef Banaś
Copyright © 2013 Cristian Arteaga and Isabel Marrero. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The function spaces 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 . 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 which ensure that the Zemanian spaces ℬμ and ℋμ are dense in . These conditions are shown to be satisfied by the weights giving rise to direct seminorms of the so-called type II.
The Hankel integral transformation is usually defined by where , , and denotes the Bessel function of the first kind and order .
1.1. The Distributional Hankel Transformation
Aiming to define the Hankel transformation in spaces of distributions, Zemanian  introduced the space of all those smooth, complex-valued functions such that Here, and in the sequel, . When topologized by the family of norms , becomes a Fréchet space where is an automorphism provided that . Then the generalized Hankel transformation , defined by transposition on the dual of , is an automorphism of when this latter space is endowed with either its weak* or its strong topology.
Zemanian  also constructed the space as follows. For every , consists of all those such that whenever . This space is endowed with the topology generated by the family of seminorms , given by In this way becomes a Fréchet space. Moreover, if , then and the topology of coincides with that inherited from . This allows to consider the inductive limit . As usual, the dual spaces of and are respectively denoted by and . Since is a dense subspace of , can be regarded as a subspace of .
1.2. The Delsarte Kernel and the Hankel Convolution
Hirschman , Cholewinski , and Haimo  developed a convolution theory on Lebesgue spaces for a variant of the Hankel transformation closely connected to (1). For , straightforward manipulations of the results in [3–5] allow us to define a convolution for as follows. Whenever the integrals involved exist, the Hankel convolution of the functions and is defined as the function where the Hankel translate of is given by Here is the so-called Delsarte kernel. Recall that being the area of the triangle with sides of length if such a triangle exists, and zero otherwise [3, p. 308]. We note that only when . Furthermore, with .
For , denote by the space of Lebesgue measurable functions whose th power is absolutely integrable on with respect to the weight , normed with By we will represent the space of Lebesgue measurable functions such that is essentially bounded, normed with If and , then (5) and (4) exist as continuous functions on . If , then (5) and (4) exist as functions in , 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 , only for . In a series of papers, Betancor and the second named author investigated systematically the generalized #-convolution in wider spaces of distributions, allowing . In this context (cf. ), the Hankel translation was shown to be a continuous operator from into itself. Thus, the Hankel convolution of and can be defined through [7, Definition 3.1]. The formulas respectively extending (11) and (12), hold in the sense of equality in (cf. [7, Proposition 3.5]).
1.3. 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 so-called native spaces, was constructed by several authors upon seminal work of Micchelli  and Madych and Nelson [9–11]. Later, Light and Wayne  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 , or (ii) identify them with functions on the positive real half-axis. For instance, Schaback and Wu  devised a general theory which allows to write multivariate Fourier transforms or convolutions of radial functions as very simple univariate operations. Motivated by , in  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 and (a.e. ) is an integrable radial function, then its -dimensional Fourier transform is also radial and reduces to a 1-dimensional Hankel transform of order [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 above-mentioned variant of the Hankel convolution of order can be seen to coincide with the usual convolution on (cf. [16, Example 3.2]). Thus, for the Hankel convolution structure provides a strict generalization of the Fourier one.
Denote by the class of all those Lebesgue measurable functions such that The following spaces were introduced in .
Definition 1. Let be a continuous function, let be the Bessel differential operator, and let where is the identity operator, is the operator iterated times, and stands for the class of all measurable functions satisfying A seminorm (norm if ) is defined on by setting
In , for 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 is the set of interpolation points; is a complex function defined on (the so-called basis function), connected with through the distributional identity , are Müntz monomials; denotes the Hankel translation operator of order ; and 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  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 [18–20].
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 , in  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 . 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 and are dense in .
1.4. 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 and in 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 , 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 for the set of nonnegative integers and for the function giving the kernel of the Hankel transformation . The following classes of functions will be occasionally used: , formed by the continuous functions on , and , 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].
2. Seminorms of Type II
Denote by 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 is regular, generated by a continuous function on , such that and the limit exists, (ii), (iii) as , where , and (iv) a.e. .
If condition in part (iii) is replaced with the stronger one , then we call (23) a strong type II seminorm.
Example 3 (strong type II seminorm). Set . The change of variables leads to so that . Moreover, , , and [25, Equation 8.6(10)]. Thus, satisfies the strong condition (iii), hence the weak one (with ), as well as the remaining conditions (i), (ii), and (iv) in Definition 2. For , 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 . Consequently, .
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.
Lemma 4. Let . Then (i), (ii)the limit exists, (iii), and (iv), where .
The space consists of all those such that Endowed with the topology generated by the family of seminorms , becomes a Fréchet space [23, Proposition 4.3]. As usual, its dual space will be denoted by . The inclusions being dense, we have [23, Proposition 4.4].
Theorem 5. Let and assume as , for some . Then (i), (ii), (iii)the limit exists, and (iv)some is such that
Proof. From the condition as , it follows that
for some . Define , . Clearly,
As , we have , and hence defines a distribution in .
We claim that . First we observe that while , so that, from (33), because . Denoting by the least positive integer greater than , we may then write which proves our claim. In particular, , and from (34) we conclude that .
Since , [23, Propositions 4.5 and 4.6] yield , for which And since , Lemma 4 entails that . Therefore, for some , A combination of (37) and (38) establishes (32).
Applying [7, Lemma 3.2] to , we find that and the limit exists. Moreover, as , from Lemma 4 we obtain that and the limit exists as well. This shows that and the limit exists. The proof is thus complete.
Proposition 7. If the weight gives rise to a type II seminorm as in Definition 2, then and as for some .
3. Density Results
In this section we prove two density results for weights satisfying the conditions in the thesis of Proposition 7; namely, the Zemanian spaces and are dense in . 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 and hence were shown to be dense in .
Proposition 8. If and there exists such that as , then .
Proof. For , 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 such that . Fix . Then, The first integral on the right-hand side of this identity is finite because : On the other hand, provided that .
Lemma 9. Let satisfy and , where . Define . If is continuous at , then If, moreover, is uniformly continuous, then
Proof. Note that
Put . Since is continuous at , given , there exists such that implies . In view of (4), (5), and (8), we are allowed to write From (7) we find that forces ; in this case, we have . Along with (48) and (8), this leads us to On the other hand, in view of (49) and (8), there exists for which At this point, (46) follows from (50), (51), and (52). Moreover, if is uniformly continuous on , then and hence in the previous argument do not depend on , so that (47) holds.
Lemma 10. Assume as for some . Let satisfy and . Define . Given and , there exists such that
Proof. Fix and . Since , some is such that
As in the proof of Proposition 7, for the second integral on the right-hand side of (55) we arrive at
In order to estimate the first integral on the right-hand side of (55), fix and apply the self-reciprocity of on , along with (42) and (14), to get For any fixed and suitable constants , , we may write with Since and , we find that the right-hand side of (59) is bounded by times an function of . It then follows from (57) that Consequently,
A change of variables leads to The sequence of integrals on the right-hand side of (62) converges to as and is therefore bounded. This follows from Lebesgue's monotone convergence theorem applied to Indeed, the sequence is nonincreasing and , because and even polynomials are multipliers of [24, Lemma 5.3.1]. Thus, from (61), Plugging (56) and (64) into (55) finally yields
The hypothesis that as furnishes for which . Combining this estimate with (65), we obtain provided that is chosen so that . This entails uniformly in and completes the proof.
Theorem 11. Let , and let satisfy as for some . The space is a subspace of . Furthermore, is a dense subspace of , in the following sense: given and , there exists such that .
Proof. From Proposition 8, . As , is a subspace of .
Take such that , , and . For each , define . Let and . From [14, Theorem 2.23], the set is dense in ; therefore, we may assume that . Set . Then , and Lemma 10, applied to the weight , gives such that From (68) and Minkowski's inequality, Note that because . Moreover, since , we have that and is uniformly continuous. Apply then Lemma 9 to obtain satisfying Hence, for that , Finally, define through where the exchange formula (15) has been used. Since , so does [25, Equation 8.1(2)]. And since , we have . Thus , with as required.
Corollary 12. Let , and let satisfy as for some . The space is a dense subspace of , in the following sense: given and , there exists such that .
Proof. It suffices to recall that and apply Theorem 11.
Cristian Arteaga is supported by a 2011 CajaCanarias Research Grant for Postgraduates. Both authors are partially supported by MICINN-FEDER Grant MTM2011-28781 (Spain).
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