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An elementary simultaneous approximation theorem


Author: Theodore Kilgore
Journal: Proc. Amer. Math. Soc. 118 (1993), 529-536
MSC: Primary 41A28; Secondary 41A65, 42A10
DOI: https://doi.org/10.1090/S0002-9939-1993-1129881-X
MathSciNet review: 1129881
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Abstract: We will give an elementary and direct proof that for $ f \in {C^q}[ - 1,1]$ there exists a sequence of polynomials $ {P_n}$ of degree at most $ n\;(n > 2q)$ such that for $ k = 0, \ldots ,q$

$\displaystyle \vert{f^{(k)}}(x) - P_n^{(k)}(x)\vert \leqslant {M_{q,k}}{\left( {\frac{{\sqrt {1 - {x^2}} }} {n}} \right)^{q - k}}{E_{n - q}}({f^{(q)}}),$

with $ {M_{q,k}}$ depending only upon $ q$ and $ k$. Moreover $ {f^{(q)}}( \pm 1) = P_n^{(q)}( \pm 1)$.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1993-1129881-X
Article copyright: © Copyright 1993 American Mathematical Society

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