ON THE MEAN CONVERGENCE OF FOURIER SERIES IN LEGENDRE POLYNOMIALS

Abstract

In this paper we study the convergence of Fourier series in Legendre polynomials in the space Lp, if 1 ≤ p ≤ 4/3 or 4 ≤ p < ∞ (i.e. in the case when the Lebesgue constants are unbounded). The fundamental result consists in the fact that with the improvement of the differential-difference properties of the function, the convergence is less affected by the growth of the Lebesgue constant (1 ≤ p ≤ 4/3). For functions with sufficiently good differential-difference properties the partial sums of the Fourier-Legendre series give an approximation in the Lp (1 < p ≤ 4/3) metric of an order as good as the best.