Approximation on the complex sphere

TL;DR

This paper develops harmonic analysis tools on the complex sphere to establish approximation inequalities and sharp estimates for m-term approximations, integrating classical orthogonal polynomials and geometric analysis.

Contribution

It introduces new harmonic analysis elements on the complex sphere and derives sharp approximation estimates, combining polynomial, manifold, and geometric analysis techniques.

Findings

Established Bernstein's, Jackson's, and Kolmogorov's inequalities on the complex sphere.

Derived order sharp estimates for m-term approximations.

Synthesized results from orthogonal polynomials, harmonic analysis, and geometry.

Abstract

We develop new elements of harmonic analysis on the complex sphere on the basis of which Bernstein's, Jackson's and Kolmogorov's inequalities are established. We apply these results to get order sharp estimates of $m$-term approximations. The results obtained is a synthesis of new results on classical orthogonal polynomials, harmonic analysis on manifolds and geometric properties of Euclidean spaces.