Mean and Weak Convergence of Some Orthogonal Fourier Expansions by Using A p Theory

The study of the convergence of Snf in Lp(dμ) (p 6= 2) has been discussed for several classes of orthogonal polynomials (c.f. Askey-Wainger [1], Badkov [2–4], Muckenhoupt [9–11], Newman-Rudin [13], Pollard [14–16], Wing [19]). For instance, in the case of Jacobi polynomials {P (α,β) n (x)}n=0 which are orthogonal in [−1, 1] with respect to the weight w(x) = (1−x)α(1+x)β , α, β ≥ −1/2, Pollard proved that |1/p − 1/2| −1. On the other hand, Mate, Nevai and Totik [8] obtained, in a general way, necessary conditions for the mean convergence of Fourier expansions: