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

Authors: Cristian Arteaga; Isabel Marrero
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

The function spacesYm(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 ℬμ and ℋμ  (μ>−1/2) are dense in Ym(m∈ℤ+). These conditions are shown to be satisfied by the weights giving rise to direct seminorms of the so-called type II.


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## Body

## 1. Introduction

The Hankel integral transformation is usually defined by(1)(hμφ)(x)=∫0∞‍φ(t)𝒥μ(xt)dt(x∈I),
where I=]0,∞[, 𝒥μ(z)=z1/2Jμ(z)(z∈I), and Jμ denotes the Bessel function of the first kind and order μ∈ℝ.


### 1.1. The Distributional Hankel Transformation

Aiming to define the Hankel transformation in spaces of distributions, Zemanian [1] introduced the space ℋμ of all those smooth, complex-valued functions φ=φ(x)(x∈I) such that
(2)νμ,r(φ)=max0≤k≤rsupx∈I|(1+x2)r(x-1D)kx-μ-1/2φ(x)|<∞(r∈ℤ+).
Here, and in the sequel, D=Dx=d/dx. When topologized by the family of norms {νμ,r}r∈ℤ+, ℋμ becomes a Fréchet space where hμ is an automorphism provided that μ≥-1/2. Then the generalized Hankel transformation hμ′, defined by transposition on the dual ℋμ′ of ℋμ, is an automorphism of ℋμ′ when this latter space is endowed with either its weak* or its strong topology.Zemanian [2] also constructed the space ℬμ as follows. For every a>0, ℬμ,a consists of all those φ∈ℋμ such that φ(x)=0 whenever x>a. This space is endowed with the topology generated by the family of seminorms {γμ,r}r∈ℤ+, given by
(3)γμ,r(φ)=supx∈]0,a[|(x-1D)rx-μ-1/2φ(x)|(φ∈ℬμ,a,r∈ℤ+).
In this way ℬμ,a becomes a Fréchet space. Moreover, if 0<a<b, then ℬμ,a⊂ℬμ,b and the topology of ℬμ,a coincides with that inherited from ℬμ,b. This allows to consider the inductive limit ℬμ=⋃a>0‍ℬμ,a. As usual, the dual spaces of ℬμ and ℬμ,a are respectively denoted by ℬμ′ and ℬμ,a′(a>0). Since ℬμ is a dense subspace of ℋμ, ℋμ′ can be regarded as a subspace of ℬμ′.


### 1.2. The Delsarte Kernel and the Hankel Convolution

Hirschman [3], Cholewinski [4], and Haimo [5] developed a convolution theory on Lebesgue spaces for a variant of the Hankel transformation closely connected to (1). For μ>-1/2, straightforward manipulations of the results in [3–5] allow us to define a convolution for hμ as follows. Whenever the integrals involved exist, the Hankel convolution of the functions φ=φ(x) and ϕ=ϕ(x)(x∈I) is defined as the function
(4)(φ#ϕ)(x)=∫0∞‍φ(y)(τxϕ)(y)dy(x∈I),
where the Hankel translate τxϕ of ϕ is given by
(5)(τxϕ)(y)=∫0∞‍ϕ(z)Dμ(x,y,z)dz(x,y∈I).
Here
(6)Dμ(x,y,z)=∫0∞‍t-μ-1/2𝒥μ(xt)𝒥μ(yt)𝒥μ(zt)dt(x,y,z∈I)
is the so-called Delsarte kernel. Recall that
(7)Dμ(x,y,z)=2μ-1Γ(μ+1/2)π1/2(xyz)1/2-μΔ(x,y,z)2μ-1(x,y,z∈I),Δ(x,y,z) being the area of the triangle with sides of length x,y,z if such a triangle exists, and zero otherwise [3, p.  308]. We note that Dμ(x,y,z)≠0 only when |x-y|<z<x+y(x,y,z∈I). Furthermore,
(8)∫0∞‍Dμ(x,y,z)zμ+1/2dz=cμ-1(xy)μ+1/2(x,y∈I),
with cμ=2μΓ(μ+1).For1≤p<∞, denote by Lμp the space of Lebesgue measurable functions whose pth power is absolutely integrable on I with respect to the weight tμ+1/2, normed with
(9)∥u∥μ,p=(∫0∞‍|u(t)|ptμ+1/2dt)1/p(u∈Lμp).
By Lμ∞ we will represent the space of Lebesgue measurable functions u=u(t)(t∈I) such that t-μ-1/2u(t)(t∈I) is essentially bounded, normed with
(10)∥u∥μ,∞=esssupt∈I|t-μ-1/2u(t)|(u∈Lμ∞).
If φ∈Lμ1 and ϕ∈Lμ∞, then (5) and (4) exist as continuous functions on I. If φ,ϕ∈Lμ1, then (5) and (4) exist as functions in Lμ1, and, moreover, the formula
(11)hμ(τyϕ)(x)=x-μ-1/2𝒥μ(xy)(hμϕ)(x)(x,y∈I)
and the exchange formula
(12)hμ(φ#ϕ)(x)=x-μ-1/2(hμφ)(x)(hμϕ)(x)(x∈I)
hold.The study of the Hankel convolution on compactly supported distributions was initiated by de Sousa Pinto [6], only for μ=0. In a series of papers, Betancor and the second named author investigated systematically the generalized #-convolution in wider spaces of distributions, allowing μ>-1/2. In this context (cf. [7]), the Hankel translation was shown to be a continuous operator from ℋμ into itself. Thus, the Hankel convolution f#φ∈ℋμ′ of f∈ℋμ′ and φ∈ℋμ can be defined through
(13)(f#φ)(x)=〈f,τxφ〉(x∈I)
[7, Definition  3.1]. The formulas
(14)hμ′(τyf)(x)=x-μ-1/2𝒥μ(xy)(hμ′f)(x)(y∈I),(15)hμ′(f#φ)(x)=x-μ-1/2(hμφ)(x)(hμ′f)(x),
respectively extending (11) and (12), hold in the sense of equality in ℋμ′ (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 [8] and Madych and Nelson [9–11]. 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 onℝd(d∈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 ifd∈ℕ and f(x)=f0(|x|) (a.e. x∈ℝd) is an integrable radial function, then its d-dimensional Fourier transform
(16)f^(ξ)=∫ℝd‍f(x)e-2πix·ξdx(ξ∈ℝd)
is also radial and reduces to a 1-dimensional Hankel transform of order d/2-1 [15, Theorem  3.3]:
(17)f^(ξ)=F0(|ξ|)=2π|ξ|-(d/2-1)×∫0∞f0(s)Jd/2-1(2π|ξ|s)sd/2dsJd/2-1(2π|ξ|s)sd/2dsk(ξ∈ℝd).
Actually, since
(18)J-1/2(z)=(2πz)1/2cosz(z∈I),
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 μ=-1/2 can be seen to coincide with the usual convolution on ℝ (cf. [16, Example  3.2]). Thus, for 2μ+2∉ℕ the Hankel convolution structure provides a strict generalization of the Fourier one.Denote byLμ,l1 the class of all those Lebesgue measurable functions u=u(t)(t∈I) such that
(19)∫0a‍|u(t)|tμ+1/2dt<∞(a>0).
The following spaces were introduced in [14].Definition 1.
Letw=w(t)>0(t∈I) be a continuous function, let
(20)Sμ=Sμ,t=t-μ-1/2Dtt2μ+1Dtt-μ-1/2
be the Bessel differential operator, and let
(21)Ym={f∈ℋμ′:hμ′Sμmf∈Lμ,l1∩Lμ,w2}(m∈ℤ+),
where Sμ0 is the identity operator, Sμm(m∈ℕ) is the operator Sμ iterated m times, and Lμ,w2 stands for the class of all measurable functions u=u(t)(t∈I) satisfying
(22)∥u∥μ,w=(∫0∞‍|u(t)|2w(t)tμ+1/2dt)1/2<∞.
A seminorm (norm if m=0) is defined on Ym by setting
(23)|f|m=(∫0∞‍|(hμ′Sμmf)(t)|2w(t)tμ+1/2dt)1/2(f∈Ym).In [14], for m∈ℕ and suitable conditions on the weight w related to the values of m, the spaces Ym were shown to consist of continuous functions on I. Also, interpolants to f∈Ym of the form
(24)(Uf)(x)=∑i=1n‍αi(τaiΦ)(x)+∑j=0m-1‍βjpμ,j(x)(x∈I)
were obtained, where {a1,…,an}⊂I is the set of interpolation points; Φ∈ℋμ′ is a complex function defined on I (the so-called basis function), connected with w through the distributional identity
(25)t4m(hμ′Φ)(t)=1w(t);pμ,j(x)=x2j+μ+1/2(j∈ℤ+, 0≤j≤m-1) are Müntz monomials; τz(z∈I) denotes the Hankel translation operator of order μ; and αi,βj(i,j∈ℤ+,1≤i≤n,0≤j≤m-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 orderμ and whose performance is expected to improve by adjusting μ, 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 [18–20].It may be observed that the seminorm in (23) is written in terms of the Hankel transform of the function f (an indirect seminorm) rather than f 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 Ym(m∈ℤ+). We show that, under rather general conditions on the weight w, which are satisfied by those weights giving rise to seminorms of type II, the Zemanian spaces ℬμ and ℋμ are dense in Ym(m∈ℤ+).


### 1.4.  Structure and Notation

This paper is organized as follows. In Section2 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 ℬμ and ℋμ in Ym(m∈ℤ+) is finally proved in Section 3.Throughout the rest of this paper, the positive real axis will be always denoted byI, while μ will stand for a real number strictly greater than -1/2, and C 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 𝒥μ(z)=z1/2Jμ(z)(z∈I) for the function giving the kernel of the Hankel transformation hμ. The following classes of functions will be occasionally used: 𝒞, formed by the continuous functions on I, and ℰ, consisting of all those infinitely differentiable functions on I. 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].


## 1.1. The Distributional Hankel Transformation

Aiming to define the Hankel transformation in spaces of distributions, Zemanian [1] introduced the space ℋμ of all those smooth, complex-valued functions φ=φ(x)(x∈I) such that
(2)νμ,r(φ)=max0≤k≤rsupx∈I|(1+x2)r(x-1D)kx-μ-1/2φ(x)|<∞(r∈ℤ+).
Here, and in the sequel, D=Dx=d/dx. When topologized by the family of norms {νμ,r}r∈ℤ+, ℋμ becomes a Fréchet space where hμ is an automorphism provided that μ≥-1/2. Then the generalized Hankel transformation hμ′, defined by transposition on the dual ℋμ′ of ℋμ, is an automorphism of ℋμ′ when this latter space is endowed with either its weak* or its strong topology.Zemanian [2] also constructed the space ℬμ as follows. For every a>0, ℬμ,a consists of all those φ∈ℋμ such that φ(x)=0 whenever x>a. This space is endowed with the topology generated by the family of seminorms {γμ,r}r∈ℤ+, given by
(3)γμ,r(φ)=supx∈]0,a[|(x-1D)rx-μ-1/2φ(x)|(φ∈ℬμ,a,r∈ℤ+).
In this way ℬμ,a becomes a Fréchet space. Moreover, if 0<a<b, then ℬμ,a⊂ℬμ,b and the topology of ℬμ,a coincides with that inherited from ℬμ,b. This allows to consider the inductive limit ℬμ=⋃a>0‍ℬμ,a. As usual, the dual spaces of ℬμ and ℬμ,a are respectively denoted by ℬμ′ and ℬμ,a′(a>0). Since ℬμ is a dense subspace of ℋμ, ℋμ′ can be regarded as a subspace of ℬμ′.


## 1.2. The Delsarte Kernel and the Hankel Convolution

Hirschman [3], Cholewinski [4], and Haimo [5] developed a convolution theory on Lebesgue spaces for a variant of the Hankel transformation closely connected to (1). For μ>-1/2, straightforward manipulations of the results in [3–5] allow us to define a convolution for hμ as follows. Whenever the integrals involved exist, the Hankel convolution of the functions φ=φ(x) and ϕ=ϕ(x)(x∈I) is defined as the function
(4)(φ#ϕ)(x)=∫0∞‍φ(y)(τxϕ)(y)dy(x∈I),
where the Hankel translate τxϕ of ϕ is given by
(5)(τxϕ)(y)=∫0∞‍ϕ(z)Dμ(x,y,z)dz(x,y∈I).
Here
(6)Dμ(x,y,z)=∫0∞‍t-μ-1/2𝒥μ(xt)𝒥μ(yt)𝒥μ(zt)dt(x,y,z∈I)
is the so-called Delsarte kernel. Recall that
(7)Dμ(x,y,z)=2μ-1Γ(μ+1/2)π1/2(xyz)1/2-μΔ(x,y,z)2μ-1(x,y,z∈I),Δ(x,y,z) being the area of the triangle with sides of length x,y,z if such a triangle exists, and zero otherwise [3, p.  308]. We note that Dμ(x,y,z)≠0 only when |x-y|<z<x+y(x,y,z∈I). Furthermore,
(8)∫0∞‍Dμ(x,y,z)zμ+1/2dz=cμ-1(xy)μ+1/2(x,y∈I),
with cμ=2μΓ(μ+1).For1≤p<∞, denote by Lμp the space of Lebesgue measurable functions whose pth power is absolutely integrable on I with respect to the weight tμ+1/2, normed with
(9)∥u∥μ,p=(∫0∞‍|u(t)|ptμ+1/2dt)1/p(u∈Lμp).
By Lμ∞ we will represent the space of Lebesgue measurable functions u=u(t)(t∈I) such that t-μ-1/2u(t)(t∈I) is essentially bounded, normed with
(10)∥u∥μ,∞=esssupt∈I|t-μ-1/2u(t)|(u∈Lμ∞).
If φ∈Lμ1 and ϕ∈Lμ∞, then (5) and (4) exist as continuous functions on I. If φ,ϕ∈Lμ1, then (5) and (4) exist as functions in Lμ1, and, moreover, the formula
(11)hμ(τyϕ)(x)=x-μ-1/2𝒥μ(xy)(hμϕ)(x)(x,y∈I)
and the exchange formula
(12)hμ(φ#ϕ)(x)=x-μ-1/2(hμφ)(x)(hμϕ)(x)(x∈I)
hold.The study of the Hankel convolution on compactly supported distributions was initiated by de Sousa Pinto [6], only for μ=0. In a series of papers, Betancor and the second named author investigated systematically the generalized #-convolution in wider spaces of distributions, allowing μ>-1/2. In this context (cf. [7]), the Hankel translation was shown to be a continuous operator from ℋμ into itself. Thus, the Hankel convolution f#φ∈ℋμ′ of f∈ℋμ′ and φ∈ℋμ can be defined through
(13)(f#φ)(x)=〈f,τxφ〉(x∈I)
[7, Definition  3.1]. The formulas
(14)hμ′(τyf)(x)=x-μ-1/2𝒥μ(xy)(hμ′f)(x)(y∈I),(15)hμ′(f#φ)(x)=x-μ-1/2(hμφ)(x)(hμ′f)(x),
respectively extending (11) and (12), hold in the sense of equality in ℋμ′ (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 [8] and Madych and Nelson [9–11]. 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 onℝd(d∈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 ifd∈ℕ and f(x)=f0(|x|) (a.e. x∈ℝd) is an integrable radial function, then its d-dimensional Fourier transform
(16)f^(ξ)=∫ℝd‍f(x)e-2πix·ξdx(ξ∈ℝd)
is also radial and reduces to a 1-dimensional Hankel transform of order d/2-1 [15, Theorem  3.3]:
(17)f^(ξ)=F0(|ξ|)=2π|ξ|-(d/2-1)×∫0∞f0(s)Jd/2-1(2π|ξ|s)sd/2dsJd/2-1(2π|ξ|s)sd/2dsk(ξ∈ℝd).
Actually, since
(18)J-1/2(z)=(2πz)1/2cosz(z∈I),
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 μ=-1/2 can be seen to coincide with the usual convolution on ℝ (cf. [16, Example  3.2]). Thus, for 2μ+2∉ℕ the Hankel convolution structure provides a strict generalization of the Fourier one.Denote byLμ,l1 the class of all those Lebesgue measurable functions u=u(t)(t∈I) such that
(19)∫0a‍|u(t)|tμ+1/2dt<∞(a>0).
The following spaces were introduced in [14].Definition 1.
Letw=w(t)>0(t∈I) be a continuous function, let
(20)Sμ=Sμ,t=t-μ-1/2Dtt2μ+1Dtt-μ-1/2
be the Bessel differential operator, and let
(21)Ym={f∈ℋμ′:hμ′Sμmf∈Lμ,l1∩Lμ,w2}(m∈ℤ+),
where Sμ0 is the identity operator, Sμm(m∈ℕ) is the operator Sμ iterated m times, and Lμ,w2 stands for the class of all measurable functions u=u(t)(t∈I) satisfying
(22)∥u∥μ,w=(∫0∞‍|u(t)|2w(t)tμ+1/2dt)1/2<∞.
A seminorm (norm if m=0) is defined on Ym by setting
(23)|f|m=(∫0∞‍|(hμ′Sμmf)(t)|2w(t)tμ+1/2dt)1/2(f∈Ym).In [14], for m∈ℕ and suitable conditions on the weight w related to the values of m, the spaces Ym were shown to consist of continuous functions on I. Also, interpolants to f∈Ym of the form
(24)(Uf)(x)=∑i=1n‍αi(τaiΦ)(x)+∑j=0m-1‍βjpμ,j(x)(x∈I)
were obtained, where {a1,…,an}⊂I is the set of interpolation points; Φ∈ℋμ′ is a complex function defined on I (the so-called basis function), connected with w through the distributional identity
(25)t4m(hμ′Φ)(t)=1w(t);pμ,j(x)=x2j+μ+1/2(j∈ℤ+, 0≤j≤m-1) are Müntz monomials; τz(z∈I) denotes the Hankel translation operator of order μ; and αi,βj(i,j∈ℤ+,1≤i≤n,0≤j≤m-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 orderμ and whose performance is expected to improve by adjusting μ, 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 [18–20].It may be observed that the seminorm in (23) is written in terms of the Hankel transform of the function f (an indirect seminorm) rather than f 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 Ym(m∈ℤ+). We show that, under rather general conditions on the weight w, which are satisfied by those weights giving rise to seminorms of type II, the Zemanian spaces ℬμ and ℋμ are dense in Ym(m∈ℤ+).


## 1.4.  Structure and Notation

This paper is organized as follows. In Section2 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 ℬμ and ℋμ in Ym(m∈ℤ+) is finally proved in Section 3.Throughout the rest of this paper, the positive real axis will be always denoted byI, while μ will stand for a real number strictly greater than -1/2, and C 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 𝒥μ(z)=z1/2Jμ(z)(z∈I) for the function giving the kernel of the Hankel transformation hμ. The following classes of functions will be occasionally used: 𝒞, formed by the continuous functions on I, and ℰ, consisting of all those infinitely differentiable functions on I. 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 byLμ,c1 the class of all those measurable functions u=u(t)(t∈I) such that
(26)∫a∞‍|u(t)|tμ+1/2dt<∞(a>0).Definition 2.
A seminorm of the form given in (23) is called a type II seminorm provided that
(27)w(t)=2cμ-1t-2μ-1W(t)-t-3μ-3/2(τtW)(t)-cμ-1t-μ-1/2limz→0+z-μ-1/2W(z)>0(t∈I),
where (i)
the distributionW∈ℋμ′ is regular, generated by a continuous function on I, such that W∈Lμ∞ and the limit limt→0+t-μ-1/2W(t) exists,(ii)
hμ′W∈Lμ,c1,(iii)
(hμ′W)(x)=𝒪(xγ) as x→0+, where γ+μ+7/2>0, and(iv)
(hμ′W)(x)≤0 a.e. x∈I.
If conditionγ+μ+7/2>0 in part (iii) is replaced with the stronger one γ+μ+3/2>0, then we call (23) a strong type II seminorm.Example 3 (strong type II seminorm).
SetW(t)=-tμ+1/2e-t2/2(t∈I). The change of variables t2/2=x leads to
(28)∫0∞‍|W(t)|tμ+1/2dt=∫0∞‍e-t2/2t2μ+1dt=2μ∫0∞‍e-xxμdx=2μΓ(μ+1)=cμ,
so that W∈Lμ1⊂ℋμ′. Moreover, limt→0+t-μ-1/2W(t)=-1, limt→∞t-μ-1/2W(t)=0, and (hμ′W)(x)=W(x)(x∈I) [25, Equation  8.6(10)]. Thus, W satisfies the strong condition (iii), hence the weak one (with γ=μ+1/2), as well as the remaining conditions (i), (ii), and (iv) in Definition 2. For μ=1/2, with the aid of Maple 14, the weight defined by (27) is found to be
(29)w(t)=(2π)-1/2t-3(1-e-2t2+2t2-4t2e-t2/2)(t∈I),
and the expression in parentheses can be seen to be positive for t≠0. Consequently, w(t)>0(t∈I).Theorem5 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.
Letf∈Lμ1. Then (i)
hμf∈𝒞,(ii)
the limitlimx→0+x-μ-1/2(hμf)(x) exists,(iii)
limx→∞x-μ-1/2(hμf)(x)=0, and(iv)
∥hμf∥μ,∞≤cμ-1∥f∥μ,1, where cμ=2μΓ(μ+1).Proof.
It suffices to observe that(30)x-μ-1/2(hμf)(x)=∫0∞‍f(t)(xt)-μJμ(xt)tμ+1/2dt(x∈I),
where f∈Lμ1 and the function z-μJμ(z)(z∈I) is continuous, with supz∈I|z-μJμ(z)|≤cμ-1 [26, Equation  9.1.62], limz→0+z-μJμ(z)=cμ-1 [26, Equation  9.1.7], and limz→∞z-μJμ(z)=0 [26, Equation  9.2.1].The spaceℰμ consists of all those u∈ℰ such that
(31)κμ,l,r(u)=supz∈]0,l[|(z-1D)rz-μ-1/2u(z)|<∞(l∈ℕ,r∈ℤ+).
Endowed with the topology generated by the family of seminorms {κμ,l,r}(l,r)∈ℕ×ℤ+, ℰμ becomes a Fréchet space [23, Proposition  4.3]. As usual, its dual space will be denoted by ℰμ′. The inclusions ℬμ⊂ℋμ⊂ℰμ being dense, we have ℰμ′⊂ℋμ′⊂ℬμ′ [23, Proposition  4.4].Theorem 5.
Letf∈Lμ,c1 and assume f(t)=𝒪(tγ) as t→0+, for some γ+μ+3/2>0. Then (i)
f∈ℋμ′,(ii)
hμ′f∈𝒞,(iii)
the limitlimx→0+x-μ-1/2(hμ′f)(x) exists, and(iv)
somer∈ℕ is such that
(32)(hμ′f)(x)=𝒪(x2r+μ+1/2)asx⟶∞.Proof.
From the conditionf(t)=𝒪(tγ) as t→0+, it follows that
(33)|f(t)|≤Ctγ(0<t<a)
for some a,C>0. Define g(t)=0(0<t<a), g(t)=f(t)(t≥a). Clearly,
(34)f(t)=g(t)+(f-g)(t)(t∈I).
As f∈Lμ,c1, we have g∈Lμ1, and hence g defines a distribution in ℋμ′.
We claim thatf-g∈ℰμ′. First we observe that (f-g)(t)=f(t)(0<t<a) while (f-g)(t)=0(t≥a), so that, from (33),
(35)B=∫0∞‍|(f-g)(t)|tμ+1/2dt≤C∫0a‍tγ+μ+1/2dt<∞,
because γ+μ+3/2>0. Denoting by l the least positive integer greater than a, we may then write
(36)|〈f-g,ϕ〉|=|∫0∞‍(f-g)(t)ϕ(t)dt|≤Bκμ,l,0(ϕ)(ϕ∈ℰμ),
which proves our claim. In particular, f-g∈ℋμ′, and from (34) we conclude that f∈ℋμ′.
Sincef-g∈ℰμ′, [23, Propositions  4.5 and 4.6] yield b,C>0, r∈ℕ for which
(37)|x-μ-1/2hμ′(f-g)(x)|≤Cx2r(x>b).
And since g∈Lμ1, Lemma 4 entails that limx→∞x-μ-1/2(hμ′g)(x)=0. Therefore, for some c>0,
(38)|x-μ-1/2(hμ′g)(x)|≤Cx2r(x>c).
A combination of (37) and (38) establishes (32).
Applying [7, Lemma  3.2] to f-g∈ℰμ′⊂ℰ, we find that hμ′(f-g)∈𝒞 and the limit limx→0+x-μ-1/2hμ′(f-g)(x) exists. Moreover, as g∈Lμ1, from Lemma 4 we obtain that hμ′g∈𝒞 and the limit limx→0+x-μ-1/2(hμ′g)(x) exists as well. This shows that hμ′f∈𝒞 and the limit limx→0+x-μ-1/2(hμ′f)(x) exists. The proof is thus complete.Corollary 6.
Assumef∈Lμ,c1 and f(t)=𝒪(tγ) as t→0+ for some γ+μ+3/2>0, so that, by Theorem 5, f∈ℋμ′. If f(t)≤0 a.e. t∈I and W=hμ′f∈Lμ∞, then W satisfies conditions (i) to (iv) of Definition 2.Proof.
From Theorem5 we find that W∈ℋμ′∩𝒞 and the limit limt→0+t-μ-1/2W(t) exists. This yields condition (i) in Definition 2. Since f=hμ′W, conditions (ii) to (iv) are trivially satisfied.Proposition 7.
If the weightw gives rise to a type II seminorm as in Definition 2, then w∈Lμ,l1 and w(t)=𝒪(tγ) as t→∞ for some γ∈ℝ.Proof.
AssociateW to w as in Definition 2, and let f=hμ′W. From (5) and (8) we have
(39)|(τtW)(t)|=|∫0∞‍W(z)Dμ(t,t,z)dz|≤∫0∞‍|z-μ-1/2W(z)|Dμ(t,t,z)zμ+1/2dz≤∥W∥μ,∞∫0∞‍Dμ(t,t,z)zμ+1/2dz=cμ-1∥W∥μ,∞t2μ+1(t∈I),
or
(40)|t-2μ-1(τtW)(t)|≤cμ-1∥W∥μ,∞(t∈I).
Consequently, there exists C>0 for which
(41)w(t)=cμ-1t-μ-1/2|limz→0+z-μ-1/22t-μ-1/2W(t)-cμt-2μ-1(τtW)(t)=cμ-1t-1/2-limz→0+z-μ-1/2W(z)|≤cμ-1t-μ-1/2(2∥W∥μ,∞+∥W∥μ,∞+∥W∥μ,∞)=Ct-μ-1/2(t∈I).
This establishes the proposition.


## 3. Density Results

In this section we prove two density results for weightsw=w(x)>0(x∈I) satisfying the conditions in the thesis of Proposition 7; namely, the Zemanian spaces ℬμ and ℋμ are dense in Ym(m∈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 {u∈Ym:hμ′u∈ℬμ} and hence {u∈Ym:hμ′u∈ℋμ} were shown to be dense in Ym(m∈ℤ+).Proposition 8.
Ifw∈Lμ,l1 and there exists γ∈ℝ such that w(x)=𝒪(xγ) as x→∞, then ℋμ⊂Ym(m∈ℤ+).Proof.
Form=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)]:
(42)hμ(Sμφ)(t)=-t2(hμφ)(t)(t∈I,φ∈ℋμ).
From the hypothesis, there exista,C>0 such that w(x)≤Cxγ(x>a). Fix φ∈ℋμ. Then,
(43)∫0∞‍|(hμSμmφ)(t)|2w(t)tμ+1/2dt={∫0a‍+∫a∞‍}|(hμSμmφ)(t)|2w(t)tμ+1/2dt.
The first integral on the right-hand side of this identity is finite because w∈Lμ,l1:
(44)∫0a‍|(hμSμmφ)(t)|2w(t)tμ+1/2dt=∫0a‍t4m|(hμφ)(t)|2w(t)tμ+1/2dt=∫0a‍t4m+2μ+1|t-μ-1/2(hμφ)(t)|2w(t)tμ+1/2dt≤a4m+2μ+1supz∈I|z-μ-1/2(hμφ)(z)|2×∫0a‍w(t)tμ+1/2dt<∞.
On the other hand,
(45)∫a∞‍|(hμSμmφ)(t)|2w(t)tμ+1/2dt=∫a∞‍t4m|(hμφ)(t)|2w(t)tμ+1/2dt=∫a∞‍t4m-2n+2μ+1|tn-μ-1/2(hμφ)(t)|2×w(t)tμ+1/2dt≤Csupz∈I|zn-μ-1/2(hμφ)(z)|2×∫a∞‍t4m-2n+γ+3μ+3/2dt<∞,
provided that 2n>4m+γ+3μ+5/2.Lemma 9.
Letψ∈Lμ1 satisfy ψ(x)≥0(x∈I) and ∫0∞‍ψ(x)xμ+1/2dx=cμ, where cμ=2μΓ(μ+1). Define ψk(x)=kμ+3/2ψ(kx)(x∈I,k∈ℕ). If ϕ∈Lμ∞ is continuous at x0∈I, then
(46)limk→∞(ϕ#ψk)(x0)=ϕ(x0).
If, moreover, x-μ-1/2ϕ(x)(x∈I) is uniformly continuous, then
(47)limk→∞∥ϕ#ψk-ϕ∥μ,∞=0.Proof.
Note that(48)∥ψk∥μ,1=cμ(k∈ℕ),(49)limk→∞∫δ∞‍ψk(x)xμ+1/2dx=0(δ>0).
Putφ(x)=x-μ-1/2ϕ(x)(x∈I). Since φ is continuous at x0, given ɛ>0, there exists δ=δ(x0,ɛ)>0 such that |x-x0|<δ implies |φ(x)-φ(x0)|<ɛ. In view of (4), (5), and (8), we are allowed to write
(50)(ϕ#ψk)(x0)-ϕ(x0)=∫0∞∫0∞[φ(x)-φ(x0)]×ψk(y)Dμ(x0,x,y)xμ+1/2dxdy={∫0δ‍+∫δ∞‍}∫0∞‍[φ(x)-φ(x0)]×ψk(y)Dμ(x0,x,y)xμ+1/2dxdy.
From (7) we find that Dμ(x0,x,y)≠0(0<y<δ) forces |x-x0|<δ; in this case, we have |φ(x)-φ(x0)|<ɛ. Along with (48) and (8), this leads us to
(51)|∫0δ‍∫0∞‍[φ(x)-φ(x0)]ψk(y)Dμ(x0,x,y)xμ+1/2dxdy|≤ɛ∫0δ‍∫0∞‍ψk(y)Dμ(x0,x,y)xμ+1/2dxdy≤ɛcμ-1∫0δ‍ψk(y)(x0y)μ+1/2dy≤ɛx0μ+1/2.
On the other hand, in view of (49) and (8), there exists k0∈ℕ for which
(52)|∫δ∞‍∫0∞‍[φ(x)-φ(x0)]ψk(y)Dμ(x0,x,y)xμ+1/2dxdy|≤2∥ϕ∥μ,∞∫δ∞‍∫0∞‍ψk(y)Dμ(x0,x,y)xμ+1/2dxdy=2cμ-1∥ϕ∥μ,∞∫δ∞‍ψk(y)(x0y)μ+1/2dy<ɛx0μ+1/2(k∈ℕ,k≥k0).
At this point, (46) follows from (50), (51), and (52). Moreover, if φ is uniformly continuous on I, then δ and hence k0 in the previous argument do not depend on x0, so that (47) holds.Lemma 10.
Assumew(x)=𝒪(xγ) as x→∞ for some γ∈ℝ. Let ψ∈ℋμ satisfy ψ(t)≥0(t∈I) and ∫0∞‍ψ(t)tμ+1/2dt=cμ. Define ψk(t)=kμ+3/2ψ(kt)(t∈I,k∈ℕ). Given f∈ℋμ and ɛ>0, there exists a>0 such that
(53)∫a∞‍|(f#ψk)(t)|2w(t)tμ+1/2dt<ɛ(k∈ℕ).Proof.
Fixr,k∈ℕ and x∈I. Since ψ∈Lμ1, some b>0 is such that
(54)∫b∞‍ψ(t)tμ+1/2dt≤1(1+x2)r.
Clearly,
(55)|(f#ψk)(x)|=|∫0∞‍(τxf)(t)ψk(t)dt|=|{∫0b‍+∫b∞‍}(τxf)(t)ψk(t)dt|≤|∫0b‍(τxf)(t)ψk(t)dt|+|∫b∞‍(τxf)(t)ψk(t)dt|.
As in the proof of Proposition7, for the second integral on the right-hand side of (55) we arrive at
(56)|∫b∞‍(τxf)(t)ψk(t)dt|≤cμ-1∥f∥μ,∞xμ+1/2∫b∞‍ψk(t)tμ+1/2dt=Cxμ+1/2∫kb∞‍ψ(t)tμ+1/2dt≤Cxμ+1/2∫b∞‍ψ(t)tμ+1/2dt≤Cxμ+1/2(1+x2)r.
In order to estimate the first integral on the right-hand side of (55), fix t∈]0,b[ and apply the self-reciprocity of hμ on ℋμ, along with (42) and (14), to get
(57)|t-μ-1/2(τxf)(t)|=xμ+1/2(1+x2)r|x-μ-1/2hμ[(1-Sμ,z)r×(zt)-μJμ(zt)(hμf)(z)(1-Sμ,z)r](x)x-μ-1/2hμ[(1-Sμ,z)r|≤cμ-1xμ+1/2(1+x2)r∫0∞‍|(1-Sμ,z)r(zt)-μ×Jμ(zt)(hμf)(z)(1-Sμ,z)r(zt)-μ|zμ+1/2dz.
For any fixed z∈I and suitable constants an,i(n,i∈ℕ, 0≤i≤n≤r), we may write
(58)(1-Sμ,z)r(zt)-μJμ(zt)(hμf)(z)=∑n=0r‍(-1)n(rn)Sμ,zn(zt)-μJμ(zt)(hμf)(z)=zμ+1/2∑n=0r‍(-1)n(rn)z-μ-1/2Sμ,zn(zt)-μJμ(zt)(hμf)(z)=zμ+1/2∑n=0r‍(-1)n(rn)×∑i=0n‍an,iz2i(z-1Dz)n+i(zt)-μJμ(zt)z-μ-1/2(hμf)(z),
with
(59)z2i+μ+1/2(z-1Dz)n+i(zt)-μJμ(zt)z-μ-1/2(hμf)(z)=∑j=0n+i‍(n+ij)[(z-1Dz)j(zt)-μJμ(zt)]×[z2i+μ+1/2(z-1Dz)n+i-jz-μ-1/2(hμf)(z)]=∑j=0n+i‍(n+ij)[(-t2)j(zt)-μ-jJμ+j(zt)]×[z2i+μ+1/2(z-1Dz)n+i-jz-μ-1/2(hμf)(z)].
Since supy∈I|y-νJν(y)|≤cν-1(ν≥-1/2) and hμf∈ℋμ, we find that the right-hand side of (59) is bounded by (1+t2)2r times an Lμ1 function of z∈I. It then follows from (57) that
(60)|t-μ-1/2(τxf)(t)|≤Cxμ+1/2(1+x2)r(1+t2)2r(t∈]0,b[).
Consequently,
(61)|∫0b‍(τxf)(t)ψk(t)dt|≤∫0b‍|t-μ-1/2(τxf)(t)|ψk(t)tμ+1/2dt≤Cxμ+1/2(1+x2)r∫0b‍(1+t2)2rψk(t)tμ+1/2dt.
A change of variables leads to(62)∫0b‍(1+t2)2rψk(t)tμ+1/2dt=∫0kb‍(1+t2k2)2rψ(t)tμ+1/2dt≤∫0∞‍(1+t2k2)2rψ(t)tμ+1/2dt.
The sequence of integrals on the right-hand side of (62) converges to cμ as k→∞ and is therefore bounded. This follows from Lebesgue's monotone convergence theorem applied to
(63)φn(t)=(1+t2n2)2rψ(t)(t∈I,n∈ℕ).
Indeed, the sequence {φn}n∈ℕ is nonincreasing and φ1(t)=(1+t2)2rψ(t)∈ℋμ⊂Lμ1, because ψ∈ℋμ and even polynomials are multipliers of ℋμ [24, Lemma  5.3.1]. Thus, from (61),
(64)|∫0b‍(τxf)(t)ψk(t)dt|≤Cxμ+1/2(1+x2)r.
Plugging (56) and (64) into (55) finally yields
(65)|(f#ψk)(x)|2≤Cx2μ+1(1+x2)2r(x∈I).
The hypothesis thatw(x)=𝒪(xγ) as x→∞ furnishes s,C>0 for which w(x)≤Cxγ(x>s). Combining this estimate with (65), we obtain
(66)∫a∞‍|(f#ψk)(x)|2w(x)xμ+1/2dx≤C∫a∞‍xγ+3μ+3/2(1+x2)2rdx<∞(a>s),
provided that r∈ℕ is chosen so that 4r>γ+3μ+5/2. This entails
(67)lima→∞∫a∞‍|(f#ψk)(x)|2w(x)xμ+1/2dx=0
uniformly in k∈ℕ and completes the proof.Theorem 11.
Letm∈ℤ+, and let w∈Lμ,l1 satisfy w(x)=𝒪(xγ) as x→∞ for some γ∈ℝ. The space ℬμ is a subspace of Ym. Furthermore, ℬμ is a dense subspace of Ym, in the following sense: given f∈Ym and ɛ>0, there exists u∈ℬμ such that |f-u|m<ɛ.Proof.
From Proposition8, ℋμ⊂Ym. As ℬμ⊂ℋμ, ℬμ is a subspace of Ym.
Takeψ∈ℋμ such that ψ(t)≥0(t∈I), ∫0∞‍ψ(t)tμ+1/2dt=cμ, and hμψ∈ℬμ. For each k∈ℕ, define ψk(t)=kμ+3/2ψ(kt)(t∈I). Let f∈Ym and ɛ>0. From [14, Theorem  2.23], the set {u∈Ym:hμ′u∈ℬμ} is dense in Ym; therefore, we may assume that hμ′f∈ℬμ. Set gk(t)=(hμ′f#ψk)(t)(t∈I,k∈ℕ). Then gk∈ℋμ(k∈ℕ), and Lemma 10, applied to the weight v(t)=t4mw(t)(t∈I), gives a>0 such that
(68)∫a∞‍|(hμ′f)(t)|2w(t)t4m+μ+1/2dt=0,∫a∞‍|gk(t)|2w(t)t4m+μ+1/2dt<ɛ2(k∈ℕ).
From (68) and Minkowski's inequality,
(69)∫0∞‍|(hμ′f-gk)(t)|2w(t)t4m+μ+1/2dt=∫0a‍|t-μ-1/2(hμ′f-gk)(t)|2w(t)t4m+3μ+3/2dt+∫a∞‍|(hμ′f-gk)(t)|2w(t)t4m+μ+1/2dt≤∥hμ′f-gk∥μ,∞2∫0a‍w(t)t4m+3μ+3/2dt+[{∫a∞‍|gk(t)|2w(t)t4m+μ+12dt}12{∫a∞‍|(hμ′f)(t)|2w(t)t4m+μ+1/2dt}1/2+{∫a∞‍|gk(t)|2w(t)t4m+μ+1/2dt}1/2]2<∥hμ′f-gk∥μ,∞2∫0a‍w(t)t4m+3μ+3/2dt+ɛ2.
Note that
(70)∫0a‍w(t)t4m+3μ+3/2dt<∞,
because w∈Lμ,l1. Moreover, since hμ′f∈ℬμ, we have that hμ′f∈Lμ∞ and t-μ-1/2(hμ′f)(t)(t∈I) is uniformly continuous. Apply then Lemma 9 to obtain k∈ℕ satisfying
(71)∥hμ′f-gk∥μ,∞2<ɛ2∫0a‍w(t)t4m+3μ+3/2dt.
Hence, for that k,
(72)∫0∞‍|(hμ′f-gk)(t)|2w(t)t4m+μ+1/2dt<ɛ.
Finally, define u∈ℋμ′ through
(73)u(t)=(hμ′gk)(t)=hμ′(hμ′f#ψk)(t)=t-μ-1/2f(t)(hμψk)(t)(t∈I),
where the exchange formula (15) has been used. Since hμψ∈ℬμ, so does (hμψk)(t)=kμ+1/2(hμψ)(t/k)(t∈I) [25, Equation  8.1(2)]. And since hμ′f∈ℬμ⊂ℋμ, we have f∈ℋμ⊂ℰμ. Thus u∈ℬμ, with
(74)|f-u|m2=∫0∞‍|hμ′Sμm(f-u)(t)|2w(t)tμ+1/2dt=∫0∞‍|(hμ′f-gk)(t)|2w(t)t4m+μ+1/2dt<ɛ,
as required.Corollary 12.
Letm∈ℤ+, and let w∈Lμ,l1 satisfy w(x)=𝒪(xγ) as x→∞ for some γ∈ℝ. The space ℋμ is a dense subspace of Ym, in the following sense: given f∈Ym and ɛ>0, there exists u∈ℋμ such that |f-u|m<ɛ.Proof.
It suffices to recall thatℬμ⊂ℋμ and apply Theorem 11.



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