A Littlewood-Paley type theorem on orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions and its corollary

TL;DR

This paper establishes a Littlewood-Paley type theorem for orthogonal projections onto subspaces of piecewise polynomial functions, enabling norm estimates and deriving bounds for Kolmogorov widths of Besov classes with mixed Hölder conditions.

Contribution

It introduces a novel Littlewood-Paley type inequality for orthoprojectors on piecewise polynomial subspaces, linking function norms and widths in Besov spaces.

Findings

Provides upper bounds for norms of functions in $L_p$ via projections.

Derives estimates for Kolmogorov widths of Besov classes.

Establishes a Littlewood-Paley type theorem for polynomial subspace projections.

Abstract

The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $ I^d. $ This assertion provides an upper estimate for the norms of the functions in $ L_p(I^d) $ via corresponding norms of projections onto subspaces of piecewise polynomial multivariable functions. These relationships are used to obtain upper estimates of the Kolmogorov widths of Besov classes of non-periodic functions meeting the mixed Hoelder conditions.