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Two Extensions of the Sard-Schoenberg Theory of Best Approximation

Christian Reinsch
SIAM Journal on Numerical Analysis
Vol. 11, No. 1 (Mar., 1974), pp. 45-51
Stable URL: http://www.jstor.org/stable/2156429
Page Count: 7
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Two Extensions of the Sard-Schoenberg Theory of Best Approximation
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Abstract

A linear functional J(f) defined on Cm - 1 [ a, b ] can be approximated by appropriate linear combinations of function values f(xi) at discrete points x1, ⋯, xn ∈ [ a, b ]. The problem of best approximation with respect to a given class of functions was posed by Sard [11] and solved for special classes by Schoenberg [12]. We give a simplified proof of Schoenberg's result which immediately carries over to the periodic case. An example for its application is the attenuation factors in practical Fourier analysis. Another extension is possible if interpolation is replaced by smoothing.

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