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Proceedings of the American Mathematical Society

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On the continuity of best polynomial approximations

Author: S. J. Poreda
Journal: Proc. Amer. Math. Soc. 36 (1972), 471-476
MSC: Primary 30A82; Secondary 41A10
MathSciNet review: 0316717
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Abstract: Suppose f is a continuous complex valued function defined on a compact set E in the plane and $ {p_n}(f,E)$ is the polynomial of degree n of best uniform approximation to f on E. If a polynomial $ {q_n}$ of degree n approximates f on E ``almost'' as well as $ {p_n}(f,E)$, then $ {q_n}$ is ``almost'' $ {p_n}(f,E)$. Sharp estimates, one for the real and one for the general case, are found for $ {\left\Vert {{q_n} - {p_n}(f,E)} \right\Vert _E}$ in terms of the quantity $ ({\left\Vert {f - {q_n}} \right\Vert _E} - {\left\Vert {f - {p_n}(f,E)} \right\Vert _E})$, where $ {\left\Vert \cdot \right\Vert _E}$ denotes the uniform norm on E.

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  • [1] G. Meinardus, Approximation of functions: Theory and numerical methods, Springer, Berlin, 1964; English transl., Springer Tracts in Natural Philosophy, vol. 13, Springer-Verlag, New York, 1967. MR 31 #547; MR 36 #571. MR 0217482 (36:571)
  • [2] T. S. Motzkin and J. L. Walsh, On the derivative of a polynomial and Chebyshev approximation, Proc. Amer. Math. Soc. 4 (1953), 76-87. MR 15, 701. MR 0060640 (15:701f)
  • [3] V. I. Smirnov and N. A. Lebedev, Functions of a complex variable, ``Nauka", Moscow, 1964; English transl., M.I.T. Press, Cambridge, Mass., 1968. MR 30 #2152; MR 37 #5369.
  • [4] G. Freud, Eine Ungleichung für Tschebyscheffsche Approximations-polynome, Acta Sci. Math. (Szeged) 19 (1958), 162-164. MR 21 #251. MR 0101440 (21:251)

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Keywords: Best uniform approximation
Article copyright: © Copyright 1972 American Mathematical Society

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