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Cardinal interpolation with general multiquadrics

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Abstract

This paper studies the cardinal interpolation operators associated with the general multiquadrics, ϕ α, c (x)=(∥x∥2 + c 2)α, \(x\in \mathbb {R}^{d}\). These operators take the form

$$\mathcal{I}_{\alpha,c}\mathbf{y}(x) = \sum\limits_{j\in\mathbb{Z}^{d}}y_{j}L_{\alpha,c}(x-j),\quad\mathbf{y}=(y_{j})_{j\in\mathbb{Z}^{d}},\quad x\in\mathbb{R}^{d}, $$

where L α, c is a fundamental function formed by integer translates of ϕ α, c which satisfies the interpolatory condition \(L_{\alpha ,c}(k) = \delta _{0,k},\; k\in \mathbb {Z}^{d}\). We consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter \(c\to \infty \). In the univariate case, we consider the norm of the operator \(\mathcal {I}_{\alpha ,c}\) acting on ℓ p spaces as well as prove decay rates for L α, c using a detailed analysis of the derivatives of its Fourier transform, \(\widehat {L_{\alpha ,c}}\).

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA

    Keaton Hamm

  2. Department of Mathematics, Virginia Commonwealth University, Richmond, VA, 23284, USA

    Jeff Ledford

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  1. Keaton Hamm
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  2. Jeff Ledford
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Correspondence to Keaton Hamm.

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Communicated by: T. Lyche

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Hamm, K., Ledford, J. Cardinal interpolation with general multiquadrics. Adv Comput Math 42, 1149–1186 (2016). https://doi.org/10.1007/s10444-016-9457-0

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