Abstract
In this paper, the Lp-convergence of Grünwald interpolation Gn(f,x) based on the zeros of Jacobi polynomials J (α,β)n (x)(−1<α,β<1) is considered. Lp-convergence (0<p<2) of Grünwald interpolation Gn(f,x) is proved for p·Max(α,β)<1. Moreover, Lp-convergence (p>0) of Gn(f,x) is obtained for −1<α,β≤0. Therefore, the results of [1] and [3–5] are improved.
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References
Grünwald, G., On the Theory of Interpolation, Acta Math., 75(1943), 219–245.
Wang Renhong, Approximation of Unbounded Function, Academic Press (Chinese), Beijing, 1983.
Min Guohua, On the Grünwald Interpolation and its Application, J. Math. Research & Exposition (Chinese), 9(1989), 442–447.
—, On the L1-convergence of Grünwald Interpolation Operator, Adv. Math., (Chinese) 18(1989), 498–503.
Min Guohua, Grünwald Interpolation Operator Based on the Zeros of Jacobi Polynomials, Appl. Math., (Chinese) (accepted).
Szegö, G., Orthogonal Polynomials, Amer. Math. Soc. Coll. Publ., New York, 1967.
Nevai, P., Orthogonal Polynomials, Memoirs Amer. Math. Soc., 213(1979).
Natanson, I.P., Constructive Function Theory, Vol. II, Ungar., New York, 1964.
Shen Xiechang, Polynomial Interpolation (−)—Lagrange Interpolation, Adv. Math., (Chinese) 12(1983), 193–214.
Натансон, Г. Н., Нзв Вшсших Учсб, Зав. Матем., 64(1967). 67–74.
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Guohua, M. On Lp-convergence of Grunwald interpolation. Approx. Theory & its Appl. 8, 28–37 (1992). https://doi.org/10.1007/BF02836336
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DOI: https://doi.org/10.1007/BF02836336