Mean Convergence of Orthogonal Series

Abstract

j1 about the mean convergence of Fourier series in Lv spaces has been extended by Harry Pollard to series of classical orthogonal polynomials [2; 3; 4; S].In particular, it was shown that the Legendre expansion of a function f£Lv on ( -1, 1) converges in the mean of order ptof if 4/3<p<i, but that this is false for any p>4 (or l^p<4/3).The question of the status of the limiting values p=4 and p = 4/3 was raised but not settled by Pollard, and is repeated in the reviews