A scheme for interpolation by Hankel translates of a basis function
type: Item
instance of: scholarly article
title: A scheme for interpolation by Hankel translates of a basis function
zbMATH Open document ID: 1262.41001
author: Cristian Arteaga
author: Isabel Marrero
published in: Journal of Approximation Theory
publication date: 2012-12-06T00:00:00Z
review text: The authors develop a variational approach following \textit{W. Light} and \textit{H. Wayne} [Numer. Math. 81, No. 3, 415--450 (1999; Zbl 0918.65007)] for interpolation by Hankel tranforms of a basis function. Let \(I=(0,\infty)\), \(w\in C(I)\), \(w(x)>0\) be a weight; let \(\mu\geq -1/2\), \(m\in\mathbb{N}_0\), \(\pi_{\mu,m-1}\) the linear space of Müntz polynomials spanned by the monomials \(p_{\mu,n}(t)=t^{2n+\mu+1/2}\) \((t\in I,n\in\mathbb{Z}_+,0\leq n\leq m-1)\). Let \(S_{\mu}=x^{-\mu-1/2}Dx^{2\mu+1}Dx^{-\mu-1/2}\) denote the Bessel differential operator, \(J_{\mu}\) the Bessel function of the first kind and order \(\mu\). \(\mathcal{H}_{\mu}'\) is the dual to the Fréchet space \(\mathcal{H}_{\mu}\) of all smooth complex valued functions \(\varphi\) such that \[ \nu_{\mu,p}(\varphi)=\max_{0\leq k\leq p}\sup_{x\in I}\left|(1+x^2)^p(x^{-1}D)^kx^{-\mu-1/2}\varphi(x)\right|<\infty,\quad p\in\mathbb{Z}_+, \] with the Hankel integral transform \[ (h_{\mu}\varphi)(x)=\int_0^{\infty}\varphi(t)\mathcal{J}_{\mu}(xt)\, dt,\quad \mathcal{T}_{\mu}(z)=z^{1/2}J_{\mu}(z), \] as an automorphism on \(\mathcal{H}_{\mu}\). \(h_{\mu}'\) is defined by transposition. Let further \[ L_{\mu,l}^1=\left\{v:\int_0^{a}|v(t)|t^{\mu+1/2}\,dt<\infty\, (a>0)\right\}; \] \[ L_{\mu,w}^2=\left\{u:\left(\int_0^{\infty}|u(t)|^2w(t)t^{\mu+1/2}\,dt\right)^{1/2}=\| u\|_{\mu,w}<\infty\right\}; \] \[ Y=Y_0=\{f\in\mathcal{H}'_{\mu}:\, h_{\mu}'f\in L_{\mu,l}^1\cap L_{\mu,w}^2\}\text{ with the norm } \| f\|_Y=\| h_{\mu}'f\|_{\mu,w}; \] \[ Y_m=\{f\in\mathcal{H}'_{\mu}: S_{\mu}^mf\in Y\} \text{ with the seminorm } |f|_m=\| S_{\mu}^mf\|_Y. \] If \(1/w\in L_{\mu,l}^1\) and there exists \(\gamma\in\mathbb{R}\) for which \(1/w(t)=\mathcal{O}(t^{-\gamma})\) as \(t\to\infty,\) and \(n\in \mathbb{Z}_+\) is such that \(4n<4m+\gamma-\mu -3/2\), then it is possible to define a basis distribution \(\phi\in\mathcal{H}_{\mu}'\) from the identity \(t^{4m}(h_{\mu}'\phi)(t)=1/w(t)\). The authors prove also that \(Y_m\) is…
Mathematics Subject Classification ID: 41A05
Mathematics Subject Classification ID: 44A15
Mathematics Subject Classification ID: 46F12
Mathematics Subject Classification ID: 41A30
zbMATH DE Number: 6112920
zbMATH Keywords: basis function
zbMATH Keywords: Bessel-Kingman hypergroup
zbMATH Keywords: Bessel operator
zbMATH Keywords: Hankel convolution
zbMATH Keywords: Hankel translation
zbMATH Keywords: minimal norm interpolant
zbMATH Keywords: Sobolev embedding theorem
reviewed by: Alexei Lukashov
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: j.jat.2012.08.005
OpenAlex ID: W1990580282
cites work: Decomposition of conditionally positive definite functions on commutative hypergroups
cites work: Q4493275
cites work: Q4695653
cites work: The Hankel Convolution and the Zemanian Spaces βμ and β'μ
cites work: Q4285303
cites work: McKennon-type spaces for the Hankel transformation
cites work: Radial Basis Functions
cites work: A Hankel convolution complex inversion theory
cites work: A necessary and sufficient condition for the representation of a function as a Hankel-Stieltjes transform
cites work: Les espaces du type de Beppo Levi
cites work: A Generalised Hankel Convolution
cites work: Q4110059
cites work: Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines
cites work: Q3764614
cites work: Q3485377
cites work: Q5827353
cites work: Q3446085
cites work: Q5509277
cites work: Transference for hypergroups
cites work: Q3267544
cites work: Integral Equations Associated with Hankel Convolutions
cites work: Variation diminishing Hankel transforms
cites work: A Duchon framework for the sphere
cites work: \(L_{p}\)-error estimates for radial basis function interpolation on the sphere
cites work: Random walks with spherical symmetry
cites work: Q4379949
cites work: Spaces of distributions, interpolation by translates of a basis function and error estimates
cites work: Q3482292
cites work: Multivariate Interpolation and Conditionally Positive Definite Functions. II
cites work: Regular and absolutely regular Hankel-transformable distributions
cites work: Q4856847
cites work: Interpolation of scattered data: distance matrices and conditionally positive definite functions
cites work: On Functions Positive Definite Relative to the Orthogonal Group and the Representation of Functions as Hankel-Stieltjes Transforms
cites work: Q3160255
cites work: Q4340161
cites work: Q4486593
cites work: Characterization and construction of radial basis functions.
cites work: Operators on radial functions
cites work: Metric spaces and completely monontone functions
cites work: Metric Spaces and Positive Definite Functions
cites work: The Structure of the Algebra of Hankel Transforms and the Algebra of Hankel-Stieltjes Transforms
cites work: Q5640160
cites work: Q4365828
cites work: Q4348657
cites work: Generalized harmonic analysis and wavelet packets
cites work: Q4938102
cites work: Scattered Data Approximation
cites work: Local error estimates for radial basis function interpolation of scattered data
cites work: A Distributional Hankel Transformation
cites work: The Hankel Transformation of Certain Distributions of Rapid Growth
cites work: Q5571178
DOI: 10.1016/J.JAT.2012.08.005
description: scientific article; zbMATH DE number 6112920