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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
DOI: https://doi.org/10.1090/S0002-9939-1972-0316717-6
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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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0316717-6
Keywords: Best uniform approximation
Article copyright: © Copyright 1972 American Mathematical Society

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