TITLE

# Simultaneous Approximation by Algebraic Polynomials

AUTHOR(S)
Kopotun, K.
PUB. DATE
January 1996
SOURCE
Constructive Approximation;1996, Vol. 12 Issue 1, p67
SOURCE TYPE
DOC. TYPE
Article
ABSTRACT
Some estimates for simultaneous polynomial approximation of a function and its derivatives are obtained. These estimates are exact in a certain sense. In particular, the following result is derived as a corollary: For f âˆˆ C[sup r][-1, 1], m âˆˆ N, and any n â‰¥ max{m + r - 1, 2r + 1}, an algebraic polynomial P[sub n] of degree â‰¤ n exists that satisfies f[sup (k)] (x) - P[sup (k), sub n](f,x) â‰¤ C(r, m) Î“[sub nrmk] (x)[sup r - k] Ï‰[sup m] (f[sup (r)], Î“[sub nrml] x)), for 0 â‰¤ k â‰¤ r and x âˆˆ [-1, 1], where Ï‰[sup v](f[sup (k)], Î´) denotes the usual vth modulus of smoothness of f[sup (k)], and Î“[sub nrmk] (x) := n[sup -1] âˆš 1 - xÂ², (1 - xÂ²)[sup (r - k + 1)/(r - k + m)] (1/nÂ²) [sup (m - 1) / (r - k + m), if x âˆˆ [-l + n[sup -2], 1 - n[sup -2]] if x âˆˆ [-1, -l + n[sup -2]] âˆª [1 - n[sup -2], 1]. Moreover, for no 0 â‰¤ k â‰¤ r can (1 - xÂ²)(r - k +1)/(r - k + m)(1/nÂ²)[sup (m - 1)/(r - k + m)] be replaced by (1 - xÂ²)[sup &&alpha[sub k]macr; n[sup 2Î±[sub k] - 2, with Î±[sub k] > (r - k + 1)/(r - k + m).
ACCESSION #
8858727

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