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Boundedness in the mean of orthonormalized polynomials

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Abstract

For the polynomials {pn(t)} 0 , orthonormalized on [−1, 1] with weightp(t) = (1−t)α (1+t)β v=1 m , 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.

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Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 759–770, May, 1973.

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Badkov, V.M. Boundedness in the mean of orthonormalized polynomials. Mathematical Notes of the Academy of Sciences of the USSR 13, 453–459 (1973). https://doi.org/10.1007/BF01147477

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Keywords

  • Orthonormalized Polynomial