On tensor product approximation of analytic functions

Abstract

We prove sharp, two-sided bounds on sums of the form ∑<em>_{k∈&naturals;^d₀</em>_{\<em>&#119967;_a</em>(<em>T</em>)}} exp(− ∑<em>^d_j</em>₌₁ <em>a_jk_j</em>), where <em>&#119967;_a</em>(<em>T</em>) := {<em><b>k</b></em> ∈ <em>&naturals;^d₀</em> : ∑<em>^d_j</em>₌₁ <em>a_jk_j</em> ≤ <em>T</em>} and <em>a</em> ∈ <em>&Ropf;^d</em>₊. These sums appear in the error analysis of tensor product approximation, interpolation and integration of <em>d</em>-variate analytic functions. Examples are tensor products of univariate Fourier-Legendre expansions [6] or interpolation and integration rules at Leja points [13, 40, 41]. Moreover, we discuss the limit <em>d</em> &rarr; ∞, where we prove both, algebraic and sub-exponential upper bounds. As an application we consider tensor products of Hardy spaces, where we study convergence rates of a certain truncated Taylor series, as well as of interpolation and integration using Leja points.