Simultaneous Approximation by Algebraic Polynomials

Kopotun, K.
January 1996
Constructive Approximation;1996, Vol. 12 Issue 1, p67
Academic Journal
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 ∈ C[sup r][-1, 1], m ∈ N, and any n ≥ max{m + r - 1, 2r + 1}, an algebraic polynomial P[sub n] of degree ≤ n exists that satisfies f[sup (k)] (x) - P[sup (k), sub n](f,x) ≤ C(r, m) Γ[sub nrmk] (x)[sup r - k] ω[sup m] (f[sup (r)], Γ[sub nrml] x)), for 0 ≤ k ≤ r and x ∈ [-1, 1], where ω[sup v](f[sup (k)], δ) denotes the usual vth modulus of smoothness of f[sup (k)], and Γ[sub nrmk] (x) := n[sup -1] √ 1 - x², (1 - x²)[sup (r - k + 1)/(r - k + m)] (1/n²) [sup (m - 1) / (r - k + m), if x ∈ [-l + n[sup -2], 1 - n[sup -2]] if x ∈ [-1, -l + n[sup -2]] ∪ [1 - n[sup -2], 1]. Moreover, for no 0 ≤ k ≤ r can (1 - x²)(r - k +1)/(r - k + m)(1/n²)[sup (m - 1)/(r - k + m)] be replaced by (1 - x²)[sup &&alpha[sub k]macr; n[sup 2α[sub k] - 2, with α[sub k] > (r - k + 1)/(r - k + m).


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