CONVERGENCE IN THE MEAN AND ALMOST EVERYWHERE OF FOURIER SERIES IN POLYNOMIALS ORTHOGONAL ON AN INTERVAL

Abstract

Let be the system of polynomials orthonormal on with weight where , , on and ( is the modulus of continuity in ). Consider the class of functions , where Let denote the partial sums of the Fourier series of a function with respect to the system .In the paper, conditions are obtained on the exponents of the functions and and the exponent that are necessary and sufficient for the boundedness in of each of the operators and . Sufficient conditions for the convergence of the partial sums to in the mean and almost everywhere in are revealed as a consequence. It is proved that these conditions are best possible on the class (for in the case of convergence almost everywhere). Estimates of the polynomials and necessary and sufficient conditions for their boundedness in the mean are also obtained.Bibliography: 26 items.