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Journal Article

Bounds for Nearly Best Approximations

Rudolf Wegmann
Proceedings of the American Mathematical Society
Vol. 52, No. 1 (Oct., 1975), pp. 252-256
DOI: 10.2307/2040140
Stable URL: http://www.jstor.org/stable/2040140
Page Count: 5

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Topics: Approximation, Mathematical functions, Mathematical theorems, Convexity, Diameters, Polynomials
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Bounds for Nearly Best Approximations
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Abstract

Let X be a uniformly convex space and ψ be the inverse function of the modulus of convexity δ(·). Assume here that ψ is a concave function. Let V be a linear subspace of X and let f in X be such that $\|f\| = 1 = \min\{\|f - \nu\|: \nu \epsilon V\}$. Then for $0 < \delta \leq 1$ and for ν in V with |f - ν| ≤ 1 + δ, it follows that |ν| ≤ K · ψ(δ). Let T be a compact Hausdorff-space and V a finite-dimensional subspace of C(T, X). When V has the interpolation property (Pm) with dim V = m · dim X, then the same type of estimate as above holds.

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