Direct form seminorms arising in the theory of interpolation by Hankel translates of a basis function
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Direct form seminorms arising in the theory of interpolation by Hankel translates of a basis function Cristian Arteaga · Isabel Marrero
Received: 11 January 2012 / Accepted: 10 April 2013 / Published online: 10 May 2013 © Springer Science+Business Media New York 2013
Abstract Certain spaces of functions which arise in the process of interpolation by Hankel translates of a basis function, as developed by the authors elsewhere, are defined with respect to a seminorm which is given in terms of the Hankel transform of each function. This kind of seminorm is called an indirect one. Here we discuss essentially two cases in which the seminorm can be rewritten in direct form, that is, in terms of the function itself rather than its Hankel transform. This is expected to lead to better estimates of the interpolation error. Keywords Basis function · Bessel operator · Direct form seminorm · Generalized surface spline · Generalized thin-plate spline · Hankel convolution · Indirect form seminorm · Minimal norm interpolant Mathematics Subject Classification (2010) Primary 46E35 · Secondary 41A05 · 44A35 · 46F12
1 Introduction The Hankel integral transformation is usually defined by ∞ ϕ(t)Jμ (xt)dt (x ∈ I ), (hμ ϕ)(x) =
(1.1)
0
Communicated by: Leslie Greengard Dedicated to Professor Fernando P´erez-Gonz´alez on the occasion of his sixtieth birthday. C. Arteaga · I. Marrero () Departamento de An´alisis Matem´atico, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain e-mail: [email protected] C. Arteaga e-mail: [email protected]
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C. Arteaga, I. Marrero
where I =]0, ∞[, Jμ (z) = z1/2 Jμ (z) (z ∈ I ), and Jμ denotes the Bessel function of the first kind and order μ ∈ R. Aiming to obtain a distributional extension of the Hankel transformation, Zemanian [16, 17] introduced new spaces of test and generalized functions. The space Hμ [16] consists of all those smooth, complex-valued functions ϕ = ϕ(x) (x ∈ I ) such that k p 2 −1 −μ−1/2 ϕ(x) < ∞ (p ∈ Z+ ). x D x νμ,p (ϕ) = max sup 1 + x 0≤k≤p x∈I
Here, and in what follows, D = Dx = d/dx. When topologized by the family of norms {νμ,p }p∈Z+ , Hμ becomes a Fr´echet space where hμ is an automorphism provided that μ ≥ − 1/2. Then the generalized Hankel transformation hμ , defined of H , is an automorphism of H when this latter by transposition on the dual Hμ μ μ space is endowed with either its weak∗ or its strong topology. Hirschman [11], Cholewinski [6] and Haimo [10] developed a convolution theory on Lebesgue spaces for a variant of the Hankel transformation closely connected to (1.1). For μ > −1/2, straightforward manipulations of the results in [6, 10, 11] 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 ∞ ϕ(y)(τx φ)(y)dy (x ∈ I ), (1.2) (ϕ#φ)(x) = 0
where the Hankel translate τx φ of φ is given by ∞ φ(z)Dμ (x, y, z)dz (τx φ)(y) =
(x, y ∈ I ).
(1.3)
0
Here
∞
Dμ (x, y, z) =
t −μ−1/2 Jμ (xt)Jμ (yt)J
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