Cardinal Interpolation With General Multiquadrics

Abstract

This paper studies the cardinal interpolation operators associated with the general multiquadrics, ${\varphi }_{\alpha ,c}(x)=(\Vert x{\Vert }^2+c^2{)}^{\alpha }$, $x\in {\mathbb{R}}^d$. These operators take the form $${\mathcal{I}}_{\alpha ,c}\mathbf{y}(x)=\sum _{j\in {\mathbb{Z}}^d}{y}_j{L}_{\alpha ,c}(x-j),\mathbf{y}=(y_j{)}_{j\in {\mathbb{Z}}^d},x\in {\mathbb{R}}^d,$$ where $L_{\alpha ,c}$ is a fundamental function formed by integer translates of ${\varphi }_{\alpha ,c}$ 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 ${\ell }_p$ spaces as well as prove decay rates for $L_{\alpha ,c}$ using a detailed analysis of the derivatives of its Fourier transform, $\widehat{L_{\alpha ,c}}$.

Current version contains corrected definition of modified Bessel function and corresponding proof of Lemma 1 from the published version

References in corpus (3)

• On the Convergence of Regular Families of Cardinal Interpolators
• A Novel Galerkin Method for Solving PDEs on the Sphere Using Highly Localized Kernel Bases
• Convergence properties of spline-like cardinal interpolation operators acting on $l^p$ data

Cited by in corpus (3)

• Regular Families of Kernels for Nonlinear Approximation
• Cardinal Interpolation With General Multiquadrics: Convergence Rates
• Recovering functions from the modulation spaces $\mathcal{F}W$