Two Extensions of the Sard–Schoenberg Theory of Best Approximation
Christian H. Reinsch · SIAM Journal on Numerical Analysis · 1974
A linear functional $J(f)$ defined on $C^{m - 1} [a,b]$ can be approximated by appropriate linear combinations of function values $f(x_i )$ at discrete points $x_1 , \cdots ,x_n \in [a,b]$. 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.