Best approximation of sums of elements and a theorem of Newman and Shapiro
Let F be a normed space with norm \(\| \cdot \|_ F\), and let B be a subspace of F. For \(f\in F\) we set \((1)\quad E(f,B,F)=\inf_{g\in B} \| f-g\|_ F.\) We consider the following problem: find the condition on the elements \(f_ k\in F\), \(1\leq k\leq N\), \(N\geq 2\), under which we have the equality \((2)\quad E(\sum^{N}_{k=1}f_ k,B,F)=\sum^{N}_{k=1}E(f_ k,B,F).\) We obtain criteria for the equality (2) or for its integral analog to hold. As a consequence we present the known result of \textit{D. J. Newman} and \textit{H. S. Shapiro} [Duke Math. J. 30, No.4, 673-681 (1963; Zbl 0116.045)] on the validity of (2) in the case of the approximation of functions of m variables of the form \(\sum^{m}_{k=1}\psi_ k(x_ k)\) by generalized polynomials in the uniform metric. It is shown that the variant of the Newman-Shapiro theorem, in the case of the approximation in the integral metric, does not hold. We also show an analog of the Newman-Shapiro theorem for the approximation by entire functions of exponential type.
approximation by entire functions of exponential type
scientific article; zbMATH DE number 4177805
scientific article