Lebesgue constants on compact manifolds

TL;DR

This paper establishes sharp asymptotic estimates for Fourier projection norms on various compact manifolds, extending classical results and solving a longstanding problem on convergence rates of Fourier sums.

Contribution

It provides the first sharp asymptotic estimates for Lebesgue constants on several compact manifolds, generalizing classical results from the circle and sphere.

Findings

Sharp asymptotic estimates for Fourier projection norms on compact manifolds.

Extension of classical results from circle and sphere to broader manifolds.

Solution to Kolmogorov's problem on convergence rates of Fourier sums.

Abstract

Sharp asymptotic for norms of Fourier projections on two-point homogeneous manifolds (the real sphere, the real, complex and quaternionic projective spaces and the Cayley elliptic plain) are established. These results extend sharp asymptotic estimates found by Fejer on the circle (in 1910) and then by Gronwall on the two-dimensional sphere (in 1914). As an application of these results we give solution of the problem of Kolmogorov on sharp asymptotic for the rate of convergence of Fourier sums on a wide range of sets of multiplier operators.