Boundedness in the mean of orthonormalized polynomials

V. M. Badkov

Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 759-770

Résumé

For the polynomials $\{p_n(t)\}_0^\infty$, orthonormalized on $[-1,1]$ with weight $p(t)=(1-t)^\alpha(1+t)^\beta\Pi_{\nu=1}^m|t-x_\nu|^{\delta_\nu}H(t)$, we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1) $\|(1-t)^\mu p_n\|_{L^r(y_m,1)}$, 2) $\|(1+t)^\mu p_n\|_{L^r(-1,y_0)}$ and 3) $\||t-x_\nu|^\mu p_n\|_{L^r(y_{\nu-1},y_\nu}$ with the conditions that $1\le r<\infty$, $\alpha$, $\beta$, $\delta_\nu>-1$ ($\nu=\overline{1,m}$), $-1<y_0<x_1<\dots<y_m<x_m<1$, $H(t)>0$ on $[-1,1]$ and $\omega(H,\delta)\delta^{-1}\in L^2(0,2)$, where $\omega(H,\delta)$ is the modulus of continuity in $C(-1,1)$ of function $H$.