Abstract
For μ ≥ −1/2, the authors have developed elsewhere a scheme for interpolation by Hankel translates of a basis function Φ in certain spaces of continuous functions Y n (n ∈ ℕ) depending on a weight w. The functions Φ and w are connected through the distributional identity t 4n(h μ′Φ)(t) = 1/w(t), where h μ′ denotes the generalized Hankel transform of order μ. In this paper, we use the projection operators associated with an appropriate direct sum decomposition of the Zemanian space ℋ μ in order to derive explicit representations of the derivatives S μ mΦ and their Hankel transforms, the former ones being valid when m ∈ ℤ + is restricted to a suitable interval for which S μ mΦ is continuous. Here, S μ m denotes the mth iterate of the Bessel differential operator S μ if m ∈ ℕ, while S μ 0 is the identity operator. These formulas, which can be regarded as inverses of generalizations of the equation (h μ′Φ)(t) = 1/t 4n w(t), will allow us to get some polynomial bounds for such derivatives. Corresponding results are obtained for the members of the interpolation space Y n.
Highlights
The method of radial basis function interpolation has seen substantial developments, both theoretical and computational, and in applications; compare [1,2,3] and references therein
A radially symmetric function in Euclidean space Rd can be identified with a function on the positive real axis
[5, 6], the authors have benefited from the hypergroup structure in order to provide a new approach to the problem of radial basis function interpolation, which extends the usual scheme
Summary
The method of radial basis function interpolation has seen substantial developments, both theoretical and computational, and in applications; compare [1,2,3] and references therein. [5, 6], the authors have benefited from the hypergroup structure in order to provide a new approach to the problem of radial basis function interpolation, which extends the usual scheme. The space O consists of all those smooth, complex-valued functions θ on I such that θψ ∈ Hμ whenever ψ ∈ Hμ and the linear operator ψ → θψ is a continuous mapping of Hμ into itself. Our main results are, where a Hankel inversion formula is presented in a general setting and specialized to basis distributions and members of the interpolation space Yn. Section 4 contains some auxiliary results of a rather technical nature. For the operational rules of the Hankel transformation that eventually might be required, both in the classical and the generalized senses, the reader is mainly referred to [10]
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