On the continuity of best polynomial approximations
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- by S. J. Poreda
- Proc. Amer. Math. Soc. 36 (1972), 471-476
- DOI: https://doi.org/10.1090/S0002-9939-1972-0316717-6
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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 ${\left \| {{q_n} - {p_n}(f,E)} \right \|_E}$ in terms of the quantity $({\left \| {f - {q_n}} \right \|_E} - {\left \| {f - {p_n}(f,E)} \right \|_E})$, where ${\left \| \cdot \right \|_E}$ denotes the uniform norm on E.References
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Bibliographic Information
- © Copyright 1972 American Mathematical Society
- 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