Boundedness in the mean of orthonormalized polynomials

Abstract

For the polynomials {pn(t)} 0 ∞ , orthonormalized on [−1, 1] with weightp(t) = (1−t)α (1+t)β∏ 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 $$1 \leqslant r - 1(\nu = 1\overline {, m),} - 1 0$$ on [−1, 1] and ω(H,δ)δ−1e L2(0, 2), whereω(H,δ) is the modulus of continuity in C(−1, 1) of function H.