(C, 1) Means of Orthonormal Expansions for Exponential Weights Original Research Article

Let sm[f] denote the mth partial sum of the orthonormal expansion of f: R→R with respect to the orthonormal polynomials for the weight W2(x)=exp(−|x|α), α>1. We show that for some C independent of f and n,1n ∑m=1n sm[f] Wφ−2/3nL∞(R)⩽C ‖fW‖L∞(R) whereφn(x)≔1−xan+n−2/3 and an denotes the nth Mhaskar–Rahmanov–Saff number for Q(x)=12 |x|α. The novelty is the presence of the factor φ−2/3n, which is large close to ±an: 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.