# The Role of Nonpolynomiality in Uniform Approximation by RBF Networks of Hankel Translates

Authors: Isabel Marrero
Journal: Journal of Function Spaces
 (2019)

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## Abstract

Givenμ>-1/2 and c∈I=]0,∞[, let the space Cμ,c (respectively, Cμ) consist of all those continuous functions u on ]0,c] (respectively, I) such that the limit limz→0+⁡z-μ-1/2u(z) exists and is finite; Cμ,c is endowed with the uniform norm uμ,∞,c=supz∈[0,c]⁡z-μ-1/2u(z)(u∈Cμ,c). Assume ϕ∈Cμ defines an absolutely regular Hankel-transformable distribution. Then, the linear span of dilates and Hankel translates of ϕ is dense in Cμ,c for all c∈I if, and only if, ϕ∉πμ, where πμ=span{t2n+μ+1/2:n∈Z+}.


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## Body

## 1. Introduction and Motivation

### 1.1. RBFNNs

The radial basis function (RBF) method is nowadays one of the primary tools for interpolating multidimensional scattered data. Its simple form and ability to accurately approximate an underlying function have made the method increasingly popular in several different types of applications, some of which include cartography, medical imaging, the numerical solution of partial differential equations, and neural networks (see, e.g., [1] and references therein).Radial basis function neural networks (RBFNNs) as such were introduced in the 1980s by Broomhead and Lowe [2] and soon applied to problems of supervised learning such as regression, classification, and time series prediction [3, 4]. This type of network falls within the general class of nonlinear, single hidden layer feedforward neural networks. Given d∈N, the family of RBFNNs consists of all those functions v:Rd→R of the form(1)vx=∑i=1mwiKx-ziσi=∑i=1mwigx-ziσi,where (i)
m∈N is the number of kernel nodes in the hidden layer(ii)
(w1,…,wm)∈Rm is the vector of weights from the ith kernel node to the output nodes(iii)
x∈Rd is an input vector(iv)
K is a radially symmetric kernel function of a unit in the hidden layer(v)
zi∈Rd and σi∈R are the centroid and smoothing factor (or width) of the ith kernel node (i∈N,1≤i≤m), respectively(vi)
g:[0,∞[→R is the so-called activation function, which characterizes the kernel shape, often a GaussianThe smoothing factors may be the same in all kernel nodes of a RBFNN or may vary across them. Park and Sandberg [5, 6] proved that under mild conditions on the kernel K (or the activation function g) both classes of RBFNNs (with either the same or varying smoothing factors across nodes) have the universal approximation property, meaning that they are dense in suitable spaces of continuous or integrable functions. Chen and Chen [7] considered RBFNNs with a continuous activation function ϕ in the hidden layer defining a tempered distribution in R and proved that the necessary and sufficient condition for such networks to uniformly approximate every continuous function on compacta is that ϕ is not an even polynomial. Nonpolynomiality is straightforwardly seen to be a necessary condition for these approximations and has been found necessary and sufficient for other types of networks to possess the universal approximation property as well, cf. [8–12]. In this paper we aim to extend the result in [7] to RBFNNs of Hankel translates. The precise meaning of this extension will be clarified in due course.


### 1.2. The Hankel Transformation and the Hankel Translation

LetI=]0,∞[. The Hankel integral transformation is usually defined by (2)hμφx=∫0∞φtJμxtdtx∈I, where Jμ(z)=z1/2Jμ(z)(z∈I) and Jμ denotes the Bessel function of the first kind and order μ∈R.Aiming to obtain a distributional extension ofhμ, Zemanian introduced new spaces of test and generalized functions. The space Hμ [13, 14] consists of all those smooth, complex-valued functions φ=φ(x)(x∈I) such that (3)νμ,pφ=max0≤k≤psupx∈I1+x2px-1Dkx-μ-1/2φx<∞p∈Z+. When topologized by the family of norms {νμ,p}p∈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. For μ∈R and a∈I, Zemanian [15] also introduced the space Bμ,a of all those smooth functions φ=φ(x)(x∈I) such that φ(x)=0(x>a) and(4)δμ,nφ=max0≤k≤nsupx∈Ix-1Dkx-μ-1/2φx<∞n∈Z+.Endowed with the topology generated by the family of seminorms {δμ,k}k∈Z+, Bμ,a becomes a Fréchet space. The strict inductive limit Bμ of the family {Bμ,a}a∈I satisfies Bμ⊂Hμ, with continuous embedding. Since Bμ is dense in Hμ, it turns out that Hμ′ can be regarded as a subspace of Bμ′, the dual of Bμ.The study of the Hankel #-convolution in spaces of generalized functions was initiated by Sousa Pinto [16], only on compactly-supported distributions and for μ=0. In a series of papers [17–19], Betancor and the author investigated systematically the generalized #-convolution in wider spaces of distributions, allowing μ>-1/2. In this context, the Hankel convolution φ#ϕ∈Hμ of φ,ϕ∈Hμ is defined as the function (5)φ#ϕx=∫0∞φyτxϕydyx∈I, where the Hankel translate τxϕ∈Hμ of ϕ∈Hμ is given by (6)τxϕy=∫0∞ϕzDμx,y,zdzx,y∈I. Here, for x,y,z∈I, (7)Dμx,y,z=∫0∞t-μ-1/2JμxtJμytJμztdt=z2-x-y2μ-1/2x+y2-z2μ-1/223μ-1π1/2Γμ+1/2xyzμ-1/2,x-y<z<x+y0,otherwise is the so-called Delsarte kernel. Note that Dμ(x,y,z)≥0, suppDμ(x,y,·)=x-y,x+y, Dμ(x,y,z) is symmetric in x,y,z, and(8)∫0∞Dμx,y,zzμ+1/2dz=cμ-1xyμ+1/2,where cμ=2μΓ(μ+1). Therefore, for any ϕ∈Hμ we have (9)τxϕy=τyϕxx,y∈I.The formula (10)hμτyϕx=x-μ-1/2Jμxyhμϕxx,y∈I and the exchange formula (11)hμφ#ϕx=x-μ-1/2hμφxhμϕxx∈I hold pointwise. The Hankel translation is defined on Hμ′ by transposition. The Hankel convolution f#φ∈Hμ′ of f∈Hμ′ and φ∈Hμ is defined [19, Definition 3.1] by (12)f#φx=f,τxφx∈I. The formulas(13)hμ′τyfx=x-μ-1/2Jμxyhμ′fxy∈Iand(14)hμ′f#φx=x-μ-1/2hμφxhμ′fxhold in the sense of equality in Hμ′ (cf. [19, Proposition 3.5]).The spaceO of all those smooth functions θ=θ(x)(x∈I) with the property that to every k∈Z+ there corresponds nk∈Z+ satisfying(15)supx∈I1+x2-nkx-1Dkθx<∞was characterized as the space of multipliers of Hμ and Hμ′ [20, Theorems 2.3 and 2.9].The generalized Hankel transformationhμ′ establishes an isomorphism between xμ+1/2O and the subspace Oμ,#′ of Hμ′ consisting of the Hankel convolution operators on Hμ and Hμ′, which is a homeomorphism under the natural topologies of xμ+1/2O and Oμ,#′ [19, Propositions 4.2 and 5.2]. The distribution given by(16)δμ,φ=cμlimx→0+x-μ-1/2φxφ∈Hμsatisfies δμ∈Oμ,#′ and f#δμ=δμ#f=f(f∈Hμ′), cf. [19, Proposition 4.7] and [21, Proposition 3].For the operational rules of the Hankel transformation and further properties of the Hankel translation and Hankel convolution that will be required, in particular those involving the Bessel differential operator(17)Sμ=Sμ,x=x-μ-1/2Dxx2μ+1Dxx-μ-1/2=Dx2-4μ2-14x2, the reader is mainly referred to [14, 17, 19]. Here we will highlight the following [14, Equation 5.5(8)]:(18)hμ′Sμfx=-x2hμ′fxf∈Hμ′.Ifd∈N and ϕ(x)=ϕ0(x) (a.e. x∈Rd) is an integrable radial function, then its d-dimensional Fourier transform (19)ϕ^ξ=2π-d/2∫Rdϕxe-ix·ξdxξ∈Rd is also radial and becomes a 1-dimensional Hankel transform of order d/2-1 [22, Theorem IV.3.3]: (20)ϕ^ξ=Φ0ξ=ξ-d-1/2∫0∞ϕ0ssd-1/2Jd/2-1ξsdsξ∈Rd. Actually, since (21)J-1/2z=2π1/2cos⁡zz∈I, it turns out that, on radial univariate—even—functions, the Fourier transformation, which reduces to a Fourier-cosine transformation, coincides with the Hankel transform of order μ=-1/2; similarly, the Hankel translation and Hankel convolution of order μ=-1/2 can be seen to coincide (modulo a multiplicative constant) with the usual translation and convolution on R (cf. [23, Example 3.2]). Thus for 2μ+2∉N the Hankel translation and the Hankel convolution provide strict generalizations of the usual translation and convolution operators, inasmuch as arbitrary orders μ≥-1/2 are allowed.


### 1.3. RBFNNs of Hankel Translates

Motivated by the fact that the Hankel transformation is best adapted to deal with radial functions, Arteaga and the author [24–27] have proved that the Hankel transformation and the Hankel convolution are suitable tools for the description and analysis of a RBF interpolation scheme by functions of the form (22)ux=∑i=1nαiτaiϕx+∑j=0m-1βjpμ,jxx∈I, where μ≥-1/2, ϕ is a complex function defined on I (the so-called basis function), pμ,j(x)=x2j+μ+1/2(j∈Z+,0≤j≤m-1) is a Müntz monomial, τz=τμ,z(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.In analogy to the standard case (1), we set the family S1(ϕ)=Sμ,1(ϕ) of RBFNNs of Hankel translates of order μ>-1/2 to consist of all those functions v:I→R which can be represented as(23)vx=∑i=1mwiτziλσiϕxx∈I,where m∈N is the number of kernel nodes in the hidden layer, for i∈N, 1≤i≤m, wi∈R is the weight from the ith kernel node to the output node, and zi,σi∈I are, respectively, the centroid and the smoothing factor of the ith kernel node. Further, ϕ is a kernel function of a unit in the hidden layer which, in this case, coincides with the activation function and, as above, τz(z∈I) denotes the Hankel translation operator, while (λrϕ)(t)=ϕ(rt)(r,t∈I) is a dilation operator. Note that, for d=1 and μ=-1/2, (23) becomes (1).An investigation on the universal approximation capabilities of a closely related class of RBFNNs defined on the nonnegative real axis has been carried out in several papers by Arteaga and the author [28–30]. It should be remarked that the results in the present paper can be derived neither from [24–27], where only the interpolation problem is addressed, nor from [28–30], where RBFNNs are constructed using the Bessel-Kingman hypergroup translation (or Delsarte translation) instead of the Hankel one, and where the universal approximation property, which is studied mainly in spaces of integrable functions, requires in turn integrability of the basis function.


### 1.4. Objectives

In the sequel we assumeμ>-1/2 and consider the following spaces:(i)
Givenc∈I, Cμ,c will denote the linear space of all those continuous functions u on [0,c] such that the limit(24)limz→0+z-μ-1/2uzexists and is finite. When endowed with the norm(25)uμ,∞,c=supz∈0,cz-μ-1/2uzu∈Cμ,c,Cμ,c becomes a Banach space. In fact, the map (26)u↦z-μ-1/2uzis an isometry fromCμ,c onto C[0,c], the space of all continuous functions on [0,c] with the uniform norm.(ii)
The linear spaceCμ consists of all those continuous functions u on I such that the limit (24) exists and is finite. Endowed with the topology generated by the family of seminorms {ρμ,n}n∈N, where (27)ρμ,nu=supz∈0,nz-μ-1/2uz<∞u∈Cμ,n∈N,Cμ becomes a Fréchet space. Note that sequential convergence in Cμ is equivalent to convergence in Cμ,c for all c∈I.(iii)
The spaceEμ consists of all those smooth functions u on I such that the limits (28)limz→0+z-1Dmz-μ-1/2uzm∈Z+exist and are finite. Endowed with the topology generated by the family of seminorms{κμ,n,m}(n,m)∈N×Z+, where (29)κμ,n,mu=supz∈0,nz-1Dmz-μ-1/2uz<∞u∈Eμ,n∈N,m∈Z+,Eμ becomes a Fréchet space.Our aim here is to find necessary and sufficient conditions on the basis functionϕ for the family of RBFNNs S1(ϕ) to have the universal approximation property. More precisely, the above mentioned result in [7] is extended to the Hankel setting in the following way. Given ϕ∈Cμ∩Hμ′, a necessary and sufficient condition for S1(ϕ) to be dense in Cμ,c(c∈I) is nonmembership in the class (30)πμ=spant2n+μ+1/2:n∈Z+ of Müntz polynomials generated by pμ,n(t)=t2n+μ+1/2(n∈Z+). This is the content of Theorem 9 in Section 3. In Section 2 we introduce the concept and give a characterization of zero-supported Bμ′-distributions (Theorem 5), which is used in the proof of Theorem 9 and might be interesting in its own right.


## 1.1. RBFNNs

The radial basis function (RBF) method is nowadays one of the primary tools for interpolating multidimensional scattered data. Its simple form and ability to accurately approximate an underlying function have made the method increasingly popular in several different types of applications, some of which include cartography, medical imaging, the numerical solution of partial differential equations, and neural networks (see, e.g., [1] and references therein).Radial basis function neural networks (RBFNNs) as such were introduced in the 1980s by Broomhead and Lowe [2] and soon applied to problems of supervised learning such as regression, classification, and time series prediction [3, 4]. This type of network falls within the general class of nonlinear, single hidden layer feedforward neural networks. Given d∈N, the family of RBFNNs consists of all those functions v:Rd→R of the form(1)vx=∑i=1mwiKx-ziσi=∑i=1mwigx-ziσi,where (i)
m∈N is the number of kernel nodes in the hidden layer(ii)
(w1,…,wm)∈Rm is the vector of weights from the ith kernel node to the output nodes(iii)
x∈Rd is an input vector(iv)
K is a radially symmetric kernel function of a unit in the hidden layer(v)
zi∈Rd and σi∈R are the centroid and smoothing factor (or width) of the ith kernel node (i∈N,1≤i≤m), respectively(vi)
g:[0,∞[→R is the so-called activation function, which characterizes the kernel shape, often a GaussianThe smoothing factors may be the same in all kernel nodes of a RBFNN or may vary across them. Park and Sandberg [5, 6] proved that under mild conditions on the kernel K (or the activation function g) both classes of RBFNNs (with either the same or varying smoothing factors across nodes) have the universal approximation property, meaning that they are dense in suitable spaces of continuous or integrable functions. Chen and Chen [7] considered RBFNNs with a continuous activation function ϕ in the hidden layer defining a tempered distribution in R and proved that the necessary and sufficient condition for such networks to uniformly approximate every continuous function on compacta is that ϕ is not an even polynomial. Nonpolynomiality is straightforwardly seen to be a necessary condition for these approximations and has been found necessary and sufficient for other types of networks to possess the universal approximation property as well, cf. [8–12]. In this paper we aim to extend the result in [7] to RBFNNs of Hankel translates. The precise meaning of this extension will be clarified in due course.


## 1.2. The Hankel Transformation and the Hankel Translation

LetI=]0,∞[. The Hankel integral transformation is usually defined by (2)hμφx=∫0∞φtJμxtdtx∈I, where Jμ(z)=z1/2Jμ(z)(z∈I) and Jμ denotes the Bessel function of the first kind and order μ∈R.Aiming to obtain a distributional extension ofhμ, Zemanian introduced new spaces of test and generalized functions. The space Hμ [13, 14] consists of all those smooth, complex-valued functions φ=φ(x)(x∈I) such that (3)νμ,pφ=max0≤k≤psupx∈I1+x2px-1Dkx-μ-1/2φx<∞p∈Z+. When topologized by the family of norms {νμ,p}p∈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. For μ∈R and a∈I, Zemanian [15] also introduced the space Bμ,a of all those smooth functions φ=φ(x)(x∈I) such that φ(x)=0(x>a) and(4)δμ,nφ=max0≤k≤nsupx∈Ix-1Dkx-μ-1/2φx<∞n∈Z+.Endowed with the topology generated by the family of seminorms {δμ,k}k∈Z+, Bμ,a becomes a Fréchet space. The strict inductive limit Bμ of the family {Bμ,a}a∈I satisfies Bμ⊂Hμ, with continuous embedding. Since Bμ is dense in Hμ, it turns out that Hμ′ can be regarded as a subspace of Bμ′, the dual of Bμ.The study of the Hankel #-convolution in spaces of generalized functions was initiated by Sousa Pinto [16], only on compactly-supported distributions and for μ=0. In a series of papers [17–19], Betancor and the author investigated systematically the generalized #-convolution in wider spaces of distributions, allowing μ>-1/2. In this context, the Hankel convolution φ#ϕ∈Hμ of φ,ϕ∈Hμ is defined as the function (5)φ#ϕx=∫0∞φyτxϕydyx∈I, where the Hankel translate τxϕ∈Hμ of ϕ∈Hμ is given by (6)τxϕy=∫0∞ϕzDμx,y,zdzx,y∈I. Here, for x,y,z∈I, (7)Dμx,y,z=∫0∞t-μ-1/2JμxtJμytJμztdt=z2-x-y2μ-1/2x+y2-z2μ-1/223μ-1π1/2Γμ+1/2xyzμ-1/2,x-y<z<x+y0,otherwise is the so-called Delsarte kernel. Note that Dμ(x,y,z)≥0, suppDμ(x,y,·)=x-y,x+y, Dμ(x,y,z) is symmetric in x,y,z, and(8)∫0∞Dμx,y,zzμ+1/2dz=cμ-1xyμ+1/2,where cμ=2μΓ(μ+1). Therefore, for any ϕ∈Hμ we have (9)τxϕy=τyϕxx,y∈I.The formula (10)hμτyϕx=x-μ-1/2Jμxyhμϕxx,y∈I and the exchange formula (11)hμφ#ϕx=x-μ-1/2hμφxhμϕxx∈I hold pointwise. The Hankel translation is defined on Hμ′ by transposition. The Hankel convolution f#φ∈Hμ′ of f∈Hμ′ and φ∈Hμ is defined [19, Definition 3.1] by (12)f#φx=f,τxφx∈I. The formulas(13)hμ′τyfx=x-μ-1/2Jμxyhμ′fxy∈Iand(14)hμ′f#φx=x-μ-1/2hμφxhμ′fxhold in the sense of equality in Hμ′ (cf. [19, Proposition 3.5]).The spaceO of all those smooth functions θ=θ(x)(x∈I) with the property that to every k∈Z+ there corresponds nk∈Z+ satisfying(15)supx∈I1+x2-nkx-1Dkθx<∞was characterized as the space of multipliers of Hμ and Hμ′ [20, Theorems 2.3 and 2.9].The generalized Hankel transformationhμ′ establishes an isomorphism between xμ+1/2O and the subspace Oμ,#′ of Hμ′ consisting of the Hankel convolution operators on Hμ and Hμ′, which is a homeomorphism under the natural topologies of xμ+1/2O and Oμ,#′ [19, Propositions 4.2 and 5.2]. The distribution given by(16)δμ,φ=cμlimx→0+x-μ-1/2φxφ∈Hμsatisfies δμ∈Oμ,#′ and f#δμ=δμ#f=f(f∈Hμ′), cf. [19, Proposition 4.7] and [21, Proposition 3].For the operational rules of the Hankel transformation and further properties of the Hankel translation and Hankel convolution that will be required, in particular those involving the Bessel differential operator(17)Sμ=Sμ,x=x-μ-1/2Dxx2μ+1Dxx-μ-1/2=Dx2-4μ2-14x2, the reader is mainly referred to [14, 17, 19]. Here we will highlight the following [14, Equation 5.5(8)]:(18)hμ′Sμfx=-x2hμ′fxf∈Hμ′.Ifd∈N and ϕ(x)=ϕ0(x) (a.e. x∈Rd) is an integrable radial function, then its d-dimensional Fourier transform (19)ϕ^ξ=2π-d/2∫Rdϕxe-ix·ξdxξ∈Rd is also radial and becomes a 1-dimensional Hankel transform of order d/2-1 [22, Theorem IV.3.3]: (20)ϕ^ξ=Φ0ξ=ξ-d-1/2∫0∞ϕ0ssd-1/2Jd/2-1ξsdsξ∈Rd. Actually, since (21)J-1/2z=2π1/2cos⁡zz∈I, it turns out that, on radial univariate—even—functions, the Fourier transformation, which reduces to a Fourier-cosine transformation, coincides with the Hankel transform of order μ=-1/2; similarly, the Hankel translation and Hankel convolution of order μ=-1/2 can be seen to coincide (modulo a multiplicative constant) with the usual translation and convolution on R (cf. [23, Example 3.2]). Thus for 2μ+2∉N the Hankel translation and the Hankel convolution provide strict generalizations of the usual translation and convolution operators, inasmuch as arbitrary orders μ≥-1/2 are allowed.


## 1.3. RBFNNs of Hankel Translates

Motivated by the fact that the Hankel transformation is best adapted to deal with radial functions, Arteaga and the author [24–27] have proved that the Hankel transformation and the Hankel convolution are suitable tools for the description and analysis of a RBF interpolation scheme by functions of the form (22)ux=∑i=1nαiτaiϕx+∑j=0m-1βjpμ,jxx∈I, where μ≥-1/2, ϕ is a complex function defined on I (the so-called basis function), pμ,j(x)=x2j+μ+1/2(j∈Z+,0≤j≤m-1) is a Müntz monomial, τz=τμ,z(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.In analogy to the standard case (1), we set the family S1(ϕ)=Sμ,1(ϕ) of RBFNNs of Hankel translates of order μ>-1/2 to consist of all those functions v:I→R which can be represented as(23)vx=∑i=1mwiτziλσiϕxx∈I,where m∈N is the number of kernel nodes in the hidden layer, for i∈N, 1≤i≤m, wi∈R is the weight from the ith kernel node to the output node, and zi,σi∈I are, respectively, the centroid and the smoothing factor of the ith kernel node. Further, ϕ is a kernel function of a unit in the hidden layer which, in this case, coincides with the activation function and, as above, τz(z∈I) denotes the Hankel translation operator, while (λrϕ)(t)=ϕ(rt)(r,t∈I) is a dilation operator. Note that, for d=1 and μ=-1/2, (23) becomes (1).An investigation on the universal approximation capabilities of a closely related class of RBFNNs defined on the nonnegative real axis has been carried out in several papers by Arteaga and the author [28–30]. It should be remarked that the results in the present paper can be derived neither from [24–27], where only the interpolation problem is addressed, nor from [28–30], where RBFNNs are constructed using the Bessel-Kingman hypergroup translation (or Delsarte translation) instead of the Hankel one, and where the universal approximation property, which is studied mainly in spaces of integrable functions, requires in turn integrability of the basis function.


## 1.4. Objectives

In the sequel we assumeμ>-1/2 and consider the following spaces:(i)
Givenc∈I, Cμ,c will denote the linear space of all those continuous functions u on [0,c] such that the limit(24)limz→0+z-μ-1/2uzexists and is finite. When endowed with the norm(25)uμ,∞,c=supz∈0,cz-μ-1/2uzu∈Cμ,c,Cμ,c becomes a Banach space. In fact, the map (26)u↦z-μ-1/2uzis an isometry fromCμ,c onto C[0,c], the space of all continuous functions on [0,c] with the uniform norm.(ii)
The linear spaceCμ consists of all those continuous functions u on I such that the limit (24) exists and is finite. Endowed with the topology generated by the family of seminorms {ρμ,n}n∈N, where (27)ρμ,nu=supz∈0,nz-μ-1/2uz<∞u∈Cμ,n∈N,Cμ becomes a Fréchet space. Note that sequential convergence in Cμ is equivalent to convergence in Cμ,c for all c∈I.(iii)
The spaceEμ consists of all those smooth functions u on I such that the limits (28)limz→0+z-1Dmz-μ-1/2uzm∈Z+exist and are finite. Endowed with the topology generated by the family of seminorms{κμ,n,m}(n,m)∈N×Z+, where (29)κμ,n,mu=supz∈0,nz-1Dmz-μ-1/2uz<∞u∈Eμ,n∈N,m∈Z+,Eμ becomes a Fréchet space.Our aim here is to find necessary and sufficient conditions on the basis functionϕ for the family of RBFNNs S1(ϕ) to have the universal approximation property. More precisely, the above mentioned result in [7] is extended to the Hankel setting in the following way. Given ϕ∈Cμ∩Hμ′, a necessary and sufficient condition for S1(ϕ) to be dense in Cμ,c(c∈I) is nonmembership in the class (30)πμ=spant2n+μ+1/2:n∈Z+ of Müntz polynomials generated by pμ,n(t)=t2n+μ+1/2(n∈Z+). This is the content of Theorem 9 in Section 3. In Section 2 we introduce the concept and give a characterization of zero-supported Bμ′-distributions (Theorem 5), which is used in the proof of Theorem 9 and might be interesting in its own right.


## 2. Zero-Supported Hankel Distributions

Definition 1.
SupposeΛ∈Bμ′. We say that the support of Λ is {0}, in symbols suppΛ={0}, if Λφ=0 for all φ∈Bμ with suppx-μ-1/2φ(x)⊂I, equivalently, if Λφ=0 for all φ∈Bμ such that φ(x)=0(x∈]0,a[), for some a∈I.Proposition 2.
AssumesuppΛ={0} and f∈Eμ satisfies x-μ-1/2f(x)=1(x∈]0,a[), for some a∈I. Then x-μ-1/2f(x)Λ=Λ.Proof.
Letφ∈Bμ. Since suppx-μ-1/2fφ-xμ+1/2φ(x)⊂I, necessarily (31)Λ,fφ-xμ+1/2φx=0. Therefore, (32)fΛ,φ=Λ,fφ=Λ,xμ+1/2φx=xμ+1/2Λ,φ. This yields the desired conclusion.Recall thatΛ∈Bμ′ if, and only if, the restrictions of Λ to every Bμ,a(a∈I) are continuous. By (4), this means that to each a∈I there corresponds C>0 and N∈Z+ such that(33)Λ,φ≤Cδμ,Nφφ∈Bμ,a.Definition 3.
If, in (33), one N will do for all a∈I (not necessarily with the same C), then the smallest such N is called the order of Λ. Otherwise, Λ is said to have infinite order.Remark 4.
Note that everyΛ∈Bμ′ with suppΛ={0} has finite order. Indeed, fix r,b∈I with r<b, and choose ψ∈Bμ,b such that x-μ-1/2ψ(x)=1(x∈]0,r[). By Proposition 2, x-μ-1/2ψ(x)Λ=Λ. Now (33) yields c1>0 and N∈Z+ satisfying (34)Λ,φ≤c1δμ,Nφφ∈Bμ,b. On the other hand, the Leibniz formula gives c2>0 such that (35)δμ,Nx-μ-1/2ψxφx≤c2δμ,Nφφ∈Bμ. Thus (36)Λ,φ=x-μ-1/2ψxΛ,φ=Λ,x-μ-1/2ψxφx≤c1δμ,Nx-μ-1/2ψxφx≤c1c2δμ,Nφφ∈Bμ, as asserted.Theorem 5.
AssumeΛ∈Bμ′, suppΛ={0}, and Λ has order N. Then there are constants αj(j∈Z+,0≤j≤N) such that (37)Λ=∑j=0NαjΛj. Here Λj is the functional defined by(38)Λj,φ=-1jcμ+jlimx→0+x-1Djx-μ-1/2φxφ∈Bμ,with cμ+j=2μ+jΓ(μ+j+1)(j∈Z+,0≤j≤N) (cf. (8)). Conversely, every distribution of this form has {0} for its support, unless αj=0(j∈Z+,0≤j≤N).Proof.
It is clear thatsuppΛj={0}(j∈Z+). This establishes the converse.
To prove the nontrivial half of the theorem, consider aφ∈Bμ that satisfies (39)Λj,φ=0j∈Z+,0≤j≤N. Our objective is to prove that Λφ=0. Since (40)ΛN,φ=-1Ncμ+Nlimx→0+x-1DNx-μ-1/2φx=0, given ε>0, there exists 0<δ<1 such that 0<x<δ implies x-1DNx-μ-1/2φx<ε. The mean value theorem yields ξ1∈]0,x[ such that (41)x-1DN-1x-μ-1/2φx=Dt-1DN-1t-μ-1/2φtt=ξ1x≤εx20<x<δ. If i∈Z+, 0≤i≤N, an induction process then shows that, for some ξi∈]0,x[, (42)x-1DN-ix-μ-1/2φx=Dt-1DN-it-μ-1/2φtt=ξix≤εx2i0<x<δ.
Chooseψ∈Bμ,1 such that x-μ-1/2ψ(x)=1(x∈]0,a[) for some 0<a<1, and define (43)ψrx=xr-μ-1/2ψxrx∈I,r>0. Fix 0<r<δ and n∈Z+. By the Leibniz formula, (44)x-1Dnx-μ-1/2ψrφx=∑k=0nnkx-1Dn-kψrxx-1Dkx-μ-1/2φxx∈I. On the other hand [14, Equation 5.2(6)], (45)x-1Dn-kψrx=rμ+1/2x-1Dn-kx-μ-1/2ψxr=rμ+1/2x-2n-k-μ-1/2∑j=0n-kbk,jxjDjψxr=r-2n-kxr-2n-k-μ-1/2∑j=0n-kbk,jxrjDjψxr=r-2n-kt-1Dn-kt-μ-1/2ψtt=x/rx∈I,0≤k≤n for suitable bk,j∈R(k,j∈Z+,0≤j≤n-k), with bk,n-k=1. Consequently, (46)max0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ψrφx≤max0≤n≤N∑k=0nnksupx∈0,rx-1Dn-kψrxsupx∈0,rx-1Dkx-μ-1/2φx≤ε2Nmax0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ψxmax0≤k≤Nsupx∈0,rxr2N-k≤ε2Nmax0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ψx.
SinceΛ has order N, there is a constant C>0 such that (47)Λ,ϕ≤Cmax0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ϕxϕ∈Bμ,1. And since ψr(x)=1(x∈]0,ar[), from Proposition 2 we infer (48)Λ,φ=ψrΛ,φ=Λ,ψrφ≤Cmax0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ψrφx≤εC2Nmax0≤n≤Nsupx∈Ix-1Dnx-μ-1/2ψx. The arbitrariness of ε>0 shows that 〈Λ,φ〉=0. Hence Λ vanishes on the intersection of the null spaces of the functionals Λj(j∈Z+,0≤j≤N), and the desired representation follows from [31, Lemma 3.9].Remark 6.
Note that the functionals (38) can be written (modulo constant factors) as derivatives of the identity for the Hankel convolution (16). In fact, we have (49)Λj,φ=-1jcμ+jlimx→0+x-1Djx-μ-1/2φx=-1jcμ-1cμ+jδμ,xμ+1/2x-1Djx-μ-1/2φx=cμ-1cμ+jx-μ-1/2Dx-1jxμ+1/2δμ,φφ∈Bμ,j∈Z+.


## 3. Nonpolynomiality of the Activation Function

We begin by establishing two auxiliary results.Lemma 7.
Letr,a∈I. (i)
The dilation operatorλr:Cμ→Cμ, defined by (λrϕ)(x)=ϕ(rx)(x∈I), is continuous.(ii)
The translation operatorτa:Cμ→Cμ, defined by (50)τaϕx=∫0∞Dμx,a,zϕzdzx∈I,is continuous.(iii)
Ifϕ∈Cμ, then S1(ϕ)⊂Cμ.Proof.
Givenϕ∈Cμ, it is apparent that λrϕ∈Cμ; indeed, the function (λrϕ)(x)=ϕ(rx)(x∈I) is clearly continuous, and the limit (51)limx→0+x-μ-1/2λrϕx=rμ+1/2limx→0+rx-μ-1/2ϕrx=rμ+1/2limz→0+z-μ-1/2ϕz exists and is finite. Similarly, (52)λrϕx=ϕrx=rxμ+1/2rx-μ-1/2ϕrxx∈I; therefore, for any c∈I,(53)supx∈0,cx-μ-1/2λrϕx=rμ+1/2supx∈0,crx-μ-1/2ϕrx=rμ+1/2supz∈0,rcz-μ-1/2ϕzϕ∈Cμ.Equation (53) proves continuity of λr and establishes (i).
To prove (ii), first of all we pickϕ∈Cμ and show that τaϕ is well defined. In fact, using (8) we may write(54)τaϕx≤∫0x+aDμx,a,zϕzdz≤supz∈0,x+az-μ-1/2ϕz∫0∞Dμx,a,zzμ+1/2dz=cμ-1xaμ+1/2supz∈0,x+az-μ-1/2ϕzx∈I.
Next, we want to see thatτaϕ is continuous on I. To this end, fix x0∈I. Since (55)limx→x0∫0∞Dμx,y,t-Dμx0,y,ttμ+1/2dt=0y∈I, given ε>0 there exists δ=δ(a,x0,ε)>0 such that x∈I and x-x0<δ imply (56)∫0∞Dμx,a,t-Dμx0,a,ttμ+1/2dt<ε. Therefore, for x-x0<δ we obtain (57)τaϕx-τaϕx0≤∫0∞Dμx,a,t-Dμx0,a,tϕtdt≤supt∈0,a+x0+δt-μ-1/2ϕt∫0∞Dμx,a,t-Dμx0,a,ttμ+1/2dt≤εsupt∈0,a+x0+δt-μ-1/2ϕt.
Since the functionf(x)=x-μ-1/2ϕ(x)(x∈I) is continuous at a, given ε>0 there exists δ=δ(a,ε)>0 such that z∈I and z-a<δ imply z-μ-1/2ϕz-a-μ-1/2ϕa<ε. Moreover, if 0<x<δ and z∈I with z-a≥δ>x, then Dμ(x,a,z)=0. Again by (8), for 0<x<δ we thus have (58)cμxa-μ-1/2τaϕx-a-μ-1/2ϕa=cμxa-μ-1/2∫0∞Dμx,a,zϕzdz-cμxa-μ-1/2a-μ-1/2ϕa∫0∞Dμx,a,zzμ+1/2dz≤cμxa-μ-1/2∫0∞Dμx,a,zz-μ-1/2ϕz-a-μ-1/2ϕazμ+1/2dz=cμxa-μ-1/2∫z-a<δDμx,a,zz-μ-1/2ϕz-a-μ-1/2ϕazμ+1/2dz≤ε. Consequently, limx→0+x-μ-1/2(τaϕ)(x)=cμ-1ϕ(a), thus proving that τaϕ∈Cμ. Now, the continuity of the translation operator can be deduced from (54): given c∈I, (59)supx∈0,cx-μ-1/2τaϕx≤cμ-1aμ+1/2supz∈0,c+az-μ-1/2ϕzϕ∈Cμ.
Finally, part (iii) derives immediately from (i) and (ii). The proof is complete.In what follows,ϕ∈Cμ∩Hμ′ will mean that the function ϕ defines an absolutely regular Hμ′-distribution (cf. [32, Theorem 3.7]). In other words, ϕφ is an integrable function on ]0,∞[ whenever φ∈Hμ, and (60)ϕ,φ=∫0∞ϕxφxdxφ∈Hμ.Lemma 8.
Assumeϕ∈Cμ∩Hμ′ and let r,a∈I. The following holds:(61)Sμλrϕ=r2λrSμϕ;(62)Sμτaϕ=τaSμϕ;(63)S1ϕ=spanτaλrϕ:a,r∈I=spanλrτaϕ:r,a∈I.Proof.
The identity in (61) may be easily verified by means of the operational rule (18). Equation (62) is a consequence of the fact that Sμ commutes with Hankel translations on Hμ′ (cf. [33]). Finally, (63) derives from (23) and the identity (64)τaλrϕ=r-μ-1/2λrτarϕ, which can be checked through (13).Theorem 9.
Letϕ∈Cμ∩Hμ′ and c>0. Then S1(ϕ)=span{τa(λrϕ):a,r∈I} is dense in Cμ,c if, and only if, ϕ∉πμ.Proof.
First supposeϕ∈πμ is such that deg[z-μ-1/2ϕ(z)]=m, so that Sμm+1ϕ=0. A combination of (61) and (62) gives (65)Sμm+1λrτaϕ=r2m+1λrτaSμm+1ϕ=0r,a∈I. Bearing in mind (63) and [24, Theorem 2.19], it turns out that, for every f∈S1(ϕ), one has f∈πμ and deg[z-μ-1/2f(z)]≤m. The subspace formed by such functions being finite-dimensional (hence closed), it cannot be dense in Cμ,c, which prevents S1(ϕ) from being dense in Cμ,c.
For the converse, suppose thatS1(ϕ) is not dense in Cμ,c. By the Hahn-Banach and Riesz representation theorems, there is a nonzero Radon measure γ, with suppγ=[0,c] and ∫0ctμ+1/2dγ(t)<∞, such that(66)frx=γ,τxλrϕ=∫0cτxλrϕtdγt=0x,r∈I.
The Hankel-Stieltjes transform(67)hμ′γt=∫0cJμxtdγxt∈I of the measure γ gives rise to a multiplier of Hμ. Indeed, (68)t-1Dtkt-μ-1/2hμ′γt=∫0ct-1Dtkxt-μJμxtxμ+1/2dγx=∫0c-x2kxt-μ-kJμ+kxtxμ+1/2dγxt∈I,k∈Z+. Hence (69)t-1Dtkt-μ-1/2hμ′γt≤c2kcμ+k-1∫0cxμ+1/2dγx<∞t∈I,k∈Z+. By (15), this proves that t-μ-1/2(hμ′γ)(t)∈O or, equivalently, γ∈Oμ,#′.
Givenr∈I, we want to show that fr∈Hμ′ and fr=(λrϕ)#γ. To this end, choose m∈N so that (70)∫0∞ϕx1+x2mxμ+1/2dx<∞ [32, Theorem 3.7], and pick an arbitrary φ∈Hμ. Then(71)∫0cdγt∫0∞λrϕxτtφxdx≤∫0cdγt∫0∞λrϕxdx∫x-t∞Dμx,t,z1+z2mz-μ-1/2φz1+z2mzμ+1/2dz≤cμ-1νμ,mφ∫0ctμ+1/2dγt∫0∞λrϕx1+x-t2mxμ+1/2dx.At this point we apply the following special case of Peetre’s inequality (cf. [34, Lemma 5.2]) (72)11+x-t2≤21+t21+x2 to infer(73)∫0ctμ+1/2dγt∫0∞λrϕx1+x-t2mxμ+1/2dx≤2m1+c2m∫0ctμ+1/2dγt∫0∞λrϕx1+x2mxμ+1/2dx.Now(74)∫0∞λrϕx1+x2mxμ+1/2dx=∫0∞ϕrx1+x2mxμ+1/2dx=∫0∞ϕu1+u/r2murμ+1/2dur=r-μ-3/2∫0∞ϕu1+u/r2muμ+1/2du.If 0<r≤1, then(75)11+u/r2m≤11+u2m.If r≥1, then(76)11+u/r2m=r2mr2+u2m≤r2m1+u2m.A combination of (71), (73), (74), (75), and (76) finally yields (77)∫0cdγt∫0∞λrϕxτtφxdx≤cμ-1νμ,mφ2m1+c2mCr,m∫0ctμ+1/2dγt∫0∞ϕu1+u2muμ+1/2du<∞, where C(r,m) denotes a suitable positive constant.
Consequently,fr∈Hμ′ and the Fubini theorem, along with [21, Definition 2 and Proposition 3], can be applied to write (78)fr,φ=∫0∞frxφxdx=∫0∞φxdx∫0cτxλrϕtdγt=∫0cdγt∫0∞λrϕxτtφxdx=γ,λrϕ#φ=γ#λrϕ,φφ∈Hμ, thus showing that fr=γ#(λrϕ)=(λrϕ)#γ.
Invoking [19, Definition 4.4] and taking into account (66) lead to (79)λrϕ,γ#λrφ=λrϕ#γ,λrφ=fr,λrφ=0φ∈Hμ,r∈I, or, equivalently from (14),(80)1r2hμ′ϕtr,t-μ-1/2hμφtrhμ′γt=0φ∈Hμ,r∈I.Since γ≠0, there exist t0∈I and 0<δ<t0 such that (81)t-μ-1/2hμ′γt≠0t-t0<δ. Given R>0, let t∗=t0/R. Then (82)Rt-μ-1/2hμ′γRt≠0t-t∗<δR. Choose any hμφ∈Bμ with (83)suppt-μ-1/2hμφt⊂t∗-δR,t∗+δR, and set (84)hμψt=RhμφtRt-μ-1/2hμ′γRtt∈I. Clearly, hμψ∈Bμ and (85)suppt-μ-1/2hμψt⊂t∗-δR,t∗+δR. Equation (80) allows us to conclude (86)hμ′ϕ,hμφ=∫0∞hμ′ϕtRt-μ-1/2hμψthμ′γRtdtR=∫0∞hμ′ϕtRt-μ-1/2hμψtRhμ′γtdtR2=1R2hμ′ϕtR,t-μ-1/2hμψtRhμ′γt=0. In other words, supphμ′ϕ={0}. From Remark 4 and Theorem 5 we get that hμ′ϕ∈spanΛj:j∈Z+, which means [24, Theorem 2.19] that ϕ∈πμ.
The proof is now complete.


## 4. Final Remarks

(i)
RBFNNs of Hankel translates, as defined in this paper, admit only one-dimensional inputs. In order to allow for multidimensional inputs one should consider the multidimensional Hankel translation, defined by iteration of the one-dimensional translation operator with respect to each of the variables while the others are kept fixed (see, e.g., [35] and references therein). The proof of the above results for the multidimensional case could well be the subject of a forthcoming paper. (ii)
According to Theorem9 and [32, Theorem 3.7 and Corollary 3.10], any continuous function σ∉πμ for which there exists k∈Z+ so that (1+x2)-kσ(x) is bounded on I, or (1+x2)-kxμ+1/2σ(x) is integrable on I, can be used as an activation function yielding universal approximation. A paradigmatic example is the Gaussian (87)σx=xμ+1/2exp⁡-x22x∈I. (iii)
By considering RBFNNs of Hankel translates, a new parameterμ is introduced which in practice leaves a greater variety of manageable kernels at our disposal. This could be useful in handling mathematical models built upon a class of radial basis functions depending on the order μ whose performance might be improved by finely tuning μ, without increasing the number of centroids [36, 37].



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