On the weighted Lebesgue function of
Fourier–Jacobi series

Abstract. S.A. Agahanov and G.I. Natanson [1] established lower and upper bounds for the Lebesgue functions \(L_n^{(\alpha,\beta)}(x)\) of Fourier–Jacobi series on the interval \([-1,1]\). The bounds differ from each other only in a constant factor depending on Jacobi parameters \(\alpha\) and \(\beta\), so their result is of final character. The aim of this paper is to extend their estimation for the weighted Lebesgue functions \(L^{(\alpha,\beta),(\gamma,\delta)}_n(x)\) using Jacobi weights with parameters \(\gamma\) and \(\delta\). We shall also give sufficient conditions with respect to \(\alpha,\beta,\gamma\) and \(\delta\) for which the order of the weighted Lebesgue functions is \(\log{(n+1)}\) on the whole interval \([-1,1]\).

Key words and phrases. Lebesgue function, Fourier–Jacobi series.