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Best approximation in a Banach space by a convergent sequence of subspaces of fixed dimension. (English. Russian original) Zbl 0649.41012
Sov. Math., Dokl. 35, No. 3, 483-486 (1987); translation from Dokl. Akad. Nauk SSSR 294, No. 1, 23-26 (1987).
Let X be a Banach space. Let \(S^ m_ n\) denote the space of \(2\pi\)- periodic splines of order m with n nodes, \(m<\infty\) and let \(S_ n^{\infty}\) be the n-dimensional space of trigonometric polynomials of minimal degree. For a fixed subspace \(M_ n\) of dimension n and an \(x\in X\) we consider the function \[ \delta_ x(M_ n,x;\epsilon)=\inf \{\| x-s\| -\| x-s^*(x)\|:s\in M_ n,\quad \| s- s^*(x)\| =\epsilon \}:[0,\infty)\to [0,\infty), \] where \(s^*(x)\) is an element of best approximation for \(x\in X\) in \(M_ n\). The author obtains estimates for \(\xi_ d(S^ m_ n,x;\epsilon)\), \(X=C\), \(L_ p\), where C denotes the space of continuous \(2\pi\)-periodic functions with the uniform norm. He also obtains relations of the strong uniqueness type for approximation by the subspaces \(S^ m_ n\), \(1\leq m\leq \infty\) with a constant which in a certain sense cannot be decreased for \(m=\infty\) and \(X=C\).
Reviewer: S.M.Mazhar

41A50 Best approximation, Chebyshev systems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A15 Spline approximation
42A10 Trigonometric approximation