(C, 1) means of orthonormal expansions for exponential weights

Let s(m)[f] denote the mth partial sum of the orthonormal expansion of f: R --> R with respect to the orthonormal polynomials for the weight W-2(x) = exp(-\x\(alpha)), alpha > 1. We show that for some C independent of f and n , [GRAPHICS] where phi(n)(x):= (\1-\x/a(n)parallel to + n(-2/3)) and a, denotes the nth Mhaskar-Rahmanov-Saff number for Q(x) = 1/2\x\(alpha ). The novelty is the presence of the factor phi(n)(-2/3), which is large c lose to +/- a(n): that factor was absent in the classic results of G. Freud . Related results are proved for more general exponential weights on (-1, 1 ) or R. (C) 2000 Academic Press.