Approximation Rates for Interpolation of Sobolev Functions via Gaussians and Allied Functions

Abstract

A \Riesz-basis sequence for $L_2[-\pi,\pi]$ is a strictly increasing sequence $X:=(x_j)_{j\in\mathbb{Z}}$ in $\mathbb{R}$ such that the set of functions $\left(e^{-ix_j(\cdot)}\right)_{j\in\mathbb{Z}}$ is a Riesz basis for $L_2[-\pi,\pi]$. Given such a sequence and a parameter $0<h\leq1$, we consider interpolation of functions $g\in W_2^k(\mathbb{R})$ at the set $(hx_j)_{j\in\mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\underset{j\in\mathbb{Z}}{\sum}a_je^{-(x-hx_j)^2},\quad x\in\mathbb{R},$$ which is continuous and square-integrable on $\mathbb{R}$, and satisfies the interpolatory condition $I^{hX}(g)(hx_j)=g(hx_j),j\in\mathbb{Z}$. Moreover, use of the parameter $h$ gives approximation rates of order $h^k$. Namely, there is a constant independent of $g$ such that $\|I^{hX}(g)-g\|_{L_2(\mathbb{R})}\leq Ch^k|g|_{W_2^k(\mathbb{R})}$. Interpolation using translates of certain functions other than the Gaussian, so-called regular interpolators, is also considered and shown to exhibit the same approximation rates.