Cardinal interpolation with general multiquadrics

Abstract

Abstract. This paper studies the cardinal interpolation operators associated with the general multiquadrics, φα,c(x) = ( x 2 + c 2 ) α , x ∈ R d . These operators take the formwhere Lα,c is a fundamental function formed by integer translates of φα,c which satisfies the interpolatory condition Lα,c(kWe consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter c → ∞. In the univariate case, we consider the norm of the operator Iα,c acting on p spaces as well as prove decay rates for Lα,c using a detailed analysis of the derivatives of its Fourier transform, Lα,c.

Cardinal interpolation with general multiquadrics: convergence rates

“…Cardinal interpolation finds its origins in the work of Schoenberg ([41] and references therein), who studied interpolation at the integer lattice using splines. Subsequent investigations ensued involving cardinal functions associated with radial basis functions (RBFs), including much work by Buhmann [6][7][8], Baxter, Riemenschneider, and Sivakumar [3,[35][36][37][38]42], and the authors [18]. Some of the RBFs considered in those works are the thin plate spline, the Gaussian kernel, and the Hardy multiquadric.…”

Regular families of kernels for nonlinear approximation

“…Proof. The estimate is well-known for the Gaussian, and may be found in [37,Theorem 5.2], while for the multiquadrics, the estimate in dimension 1 for general α is given by [22,Proposition 7] (though for α = 1/2, the estimate is in [36]), and the proof of the bound in higher dimensions follows the same line of reasoning upon using the bounds provided in [23,Corollary 4.7].…”

“…the general multiquadrics may be found in [22,23] for a broad range of exponents α, whereas the particular cases of α = ±1/2, 1, −k + 1/2, k ∈ N were considered previously [4,10,12].…”

“…we also note that the estimate may be obtained in a straightforward manner from Section 4 of [22]). Therefore, the p-th power of the L p norm in question is bounded by…”

Convergence Properties of Spline-Like Cardinal Interpolation Operators Acting on $$\ell ^p$$ ℓ p Data

“…The case that a = 1/2 is the subject of [4], while the case a = −1 is the Poisson kernel above. More recently, these interpolators were studied in [6]. This time the Fourier transforms are given by constant multiples ofφ c (ξ ) = |ξ | −(a+1/2) K a+1/2 (c|ξ |) where a / ∈Ñ = N ∪ {0} ∪ {−k − 1/2 : k ∈ N}.…”