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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

##### MSC:
 41A50 Best approximation, Chebyshev systems 41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) 41A15 Spline approximation 42A10 Trigonometric approximation
##### Keywords:
trigonometric polynomials of minimal degree; estimates