Convergence in the mean and almost everywhere of Fourier series in polynomials orthogonal on an interval

V. M. Badkov

Sbornik. Mathematics, Tome 24 (1974) no. 2, pp. 223-256

Résumé

Let $\sigma_p=\{p_n(t)\}_{n=0}^\infty$ be the system of polynomials orthonormal on $[-1,1]$ with weight $$ p(t)=H(t)(1-t)^\alpha(1+t)^\beta\prod_{\nu=1}^m|t-x_\nu|^{\gamma_\nu}, $$ where $-1$, $\alpha,\beta,\gamma_\nu>-1$ ($\nu=1,\dots,m$), $H(t)>0$ on $[-1,1]$ and $\omega(H,\delta)\delta^{-1}\in L(0,2)$ ($\omega(H,\delta)$ is the modulus of continuity in $C(-1,\,1)$). Consider the class of functions $(qL)^r=\{f(t):q(t)f(t)\in L^r(-1,1)\}$, where $q(t)=(1-t)^A(1+t)^B\times\prod_{\nu=1}^m|t-x_\nu|^{\Gamma_\nu}.$ Let $S_n^{(p)}(f)=S_n^{(p)}(f,x)$ ($n=0,1,\dots$) denote the partial sums of the Fourier series of a function $f$ with repect to the system $\sigma_p$. In the paper, conditions are obtained on the exponents of the functions $p(t)$ and $q(t)$ and the exponent $r\in(1,\infty)$ that are necessary and sufficient for the boundedness in $(qL)^r$ of each of the operators $S_n^{(p)}(f,x)$ and $\sup_{n\geqslant0}\{|S_n^{(p)}(f,x)|\}$. Sufficient conditions for the convergence of the partial sums $S_n^{(p)}(f)$ to $f\in(qL)^r$ in the mean and almost everywhere in $(-1,\,1)$ are revealed as a consequence. It is proved that these conditions are best possible on the class $(qL)^r$ (for $\omega(H,\delta)\delta^{-1}\in L^2(0,2)$ in the case of convergence almost everywhere). Estimates of the polynomials $p_n(t)$ and necessary and sufficient conditions for their boundedness in the mean are also obtained. Bibliography: 26 titles.