Cardinal interpolation with general multiquadrics

This paper studies the cardinal interpolation operators associated with the general multiquadrics, ? _{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 ? _{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 ? _p spaces as well as prove decay rates for L _{α, c} using a detailed analysis of the derivatives of its Fourier transform, \(\widehat {L_{\alpha ,c}}\).