An elementary simultaneous approximation theorem
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- by Theodore Kilgore PDF
- Proc. Amer. Math. Soc. 118 (1993), 529-536 Request permission
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 $k = 0, \ldots ,q$ \[ |{f^{(k)}}(x) - P_n^{(k)}(x)| \leqslant {M_{q,k}}{\left ( {\frac {{\sqrt {1 - {x^2}} }} {n}} \right )^{q - k}}{E_{n - q}}({f^{(q)}}),\] with ${M_{q,k}}$ depending only upon $q$ and $k$. Moreover ${f^{(q)}}( \pm 1) = P_n^{(q)}( \pm 1)$.References
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Additional Information
- © Copyright 1993 American Mathematical Society
- 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