B-spline quasi-interpolation sampling representation and sampling recovery in Sobolev spaces of mixed smoothness

TL;DR

This paper develops B-spline quasi-interpolation sampling methods for Sobolev spaces with mixed smoothness, providing theoretical guarantees and optimality results for function recovery from samples.

Contribution

It introduces new sampling representations and establishes their optimality for recovering functions in Sobolev spaces of mixed smoothness.

Findings

Proved direct and inverse theorems for B-spline quasi-interpolation in Sobolev spaces.

Established estimates for approximation error in $L_q$-norm.

Demonstrated asymptotic optimality of sampling algorithms.

Abstract

We proved direct and inverse theorems on B-spline quasi-interpolation sampling representation with a Littlewood-Paley-type norm equivalence in Sobolev spaces $W^r_p$ of mixed smoothness $r$, established estimates of the approximation error of recovery in $L_q$-norm of functions from the unit ball $U^r_p$ in the spaces $W^r_p$ by linear sampling algorithms based on this representation, the asymptotic optimality of these sampling algorithms in terms of Smolyak sampling width $r^s_n(U^r_p, L_q)$ and sampling width $r_n(U^r_p, L_q)$.