Boundedness in the mean of orthonormalized polynomials

Abstract

For the polynomials {p_n(t)} ^∞₀ , orthonormalized on [−1, 1] with weightp(t) = (1−t)^α (1+t)^β∏ ^m_v=1 , we obtain necessary and sufficient conditions for boundedness of the sequences of norms: 1)$\parallel (1 - t)^\mu p_n \parallel _{L^r (y_m ,1)}$ 2)$\parallel (1 + t)^\mu p_n \parallel _{L^r ( - 1,y_0 )}$ and 3)$\parallel (t - x_v )^\mu p_n \parallel _{L^r (y_{v - 1} ^{,y_v } )}$ with the conditions that$1 \leqslant r< \infty ,\alpha ,\beta , \delta _\nu > - 1(\nu = 1\overline {, m),} - 1< y_0< x_1< ...< y_m< x_m< 1,H(t) > 0$ on [−1, 1] and ω(H,δ)δ⁻¹ε L²(0, 2), whereω(H,δ) is the modulus of continuity in C(−1, 1) of function H.