Cardinal Interpolation With General Multiquadrics
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 $$\mathscr{I}_{α,c}\mathbf{y}(x) = \sum_{j\in\mathbb{Z}^d}y_jL_{α,c}(x-j),\quad\mathbf{y}=(y_j)_{j\in\mathbb{Z}^d},\quad x\in\mathbb{R}^d,$$ where $L_{α,c}$ is a fundamental function formed by integer translates of $ϕ_{α,c}$ which satisfies the interpolatory condition $L_{α,c}(k) = δ_{0,k},\; k\in\mathbb{Z}^d$. 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 $\mathscr{I}_{α,c}$ acting on $\ell_p$ spaces as well as prove decay rates for $L_{α,c}$ using a detailed analysis of the derivatives of its Fourier transform, $\widehat{L_{α,c}}$.