ON THE CONVERGENCE OF FOURIER–LAPLACE SERIES

Abstract

In the present paper we prove the following theorem. For any  $\epsilon > 0$ there exists a measurable set $G\subset S^3$ with measure mes $G>4\pi-\epsilon$ , such that for each $f(x)\in L^1 (S^3)$ there is a function $g(x)\in L^1 (S^3)$, coinciding with $f (x)$  on $G$ with the following properties. Its Fourier–Laplace series converges to $g(x)$ in metrics $L^1 (S^3)$ and the inequality holds $$\sup\limits_N \parallel\sum\limits^N_{n=1}Y_n[g,(0,\phi)]\parallel_{L^1 (S^3)}\ll 3\parallel g\parallel _{L^1 (S^3)}\leq 12\parallel f\parallel.$$