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Lower bounds for \(n\)-term approximations. (English. Russian original) Zbl 1043.41018

Math. Notes 70, No. 4, 574-576 (2001); translation from Mat. Zametki 70, No. 4, 636-638 (2001).
Let \(X\) be a real normed space, \(\Phi\subset X\) a subset in \(X\) (a dictionary), and \(f\in X\). The \(n\)-term approximation of an element \(f\) with respect to the dictionary \(\Phi\) is defined as \(e_n(f,\Phi,X)\equiv\inf_{P\in\Sigma_n}\| f-P\|_X\), \(\Sigma_n\equiv\left\{\sum_{j=1}^na_jx_j,a_j\in\mathbb R,x_j\in\Phi\right\}\), \(n\in\mathbb N\). Let \(\mathbb X\) be the ‘one-parametric family’ \[ \mathbb X\equiv\{\chi_t\}_{t\in[0,1]},\quad\chi_t(x)= \begin{cases} 0 &\text{if\quad \(0\leq x<t\),}\\ 1 &\text{if\quad \(t\leq x\leq1\).}\end{cases} \] The author proves that there exists an absolute positive constant \(C\) such that for a complete arbitrary orthonormal system (o.n.s.) \(\Phi\subset L^2(0,1)\) the inequality \(e_n(\mathbb X,\Phi, L^2(0,1))\geq C^{-n}\) holds, \(n\in\mathbb N\). The problem of finding the exact value of \(C\) in this theorem remains open. However, it follows from the proof that this constant is ‘not too large’. If \(\Phi\) is a uniformly bounded complete o.n.s: \(\Phi = \{\varphi_j\}_{j=1}^\infty\subset L^2(0,1)\), \(\|\varphi_j\|_{L^\infty}(0,1)\leq M\), \(j\in\mathbb N,\) then for \(n\in\mathbb N\) we have \(e_n(\mathbb X,\Phi, L^2(0,1))\geq C_M/\sqrt{n}>0. \) The accuracy of this estimate can be verified by the example of trigonometric systems, for which \(e_n(\mathbb X, T, L^2) \leq Cn^{-1/2}\).

MSC:

41A45 Approximation by arbitrary linear expressions
41A30 Approximation by other special function classes
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46B20 Geometry and structure of normed linear spaces
42C30 Completeness of sets of functions in nontrigonometric harmonic analysis
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