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 ${\left(e^{-i{x}_j(\cdot )}\right)}_{j\in \mathbb{Z}}$ is a Riesz basis for $L_2[-\pi ,\pi ]$. Given such a sequence and a parameter $0<h\le 1$, we consider interpolation of functions $g\in W_2^k(\mathbb{R})$ at the set $(h{x}_j{)}_{j\in \mathbb{Z}}$ via translates of the Gaussian kernel. Existence is shown of an interpolant of the form $$I^{hX}(g)(x):=\sum _{j\in \mathbb{Z}}{a}_j{e}^{-(x-h{x}_j{)}^2},x\in \mathbb{R},$$ which is continuous and square-integrable on $\mathbb{R}$, and satisfies the interpolatory condition $I^{hX}(g)(h{x}_j)=g(h{x}_j),j\in \mathbb{Z}$. Moreover, use of the parameter $h$ gives approximation rates of order $h^k$. Namely, there is a constant independent of $g$ such that $\Vert I^{hX}(g)-g{\Vert }_{L_2(\mathbb{R})}\le C{h}^k|g{|}_{W_2^k(\mathbb{R})}$. 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.

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