On the mean convergence of Fourier series in Legendre polynomials

In this paper we study the convergence of Fourier series in Legendre polynomials in the space $L_p$, if $1\leqslant p\leqslant4/3$ or $4\leqslant p<\infty$ (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 ($1\leqslant p\leqslant4/3$). For functions with sufficiently good differential-difference properties the partial sums of the Fourier–Legendre series give an approximation in the $L_p$ ($1<p\leqslant4/3$) metric of an order as good as the best.