Greedy approximation in certain subsystems of the Schauder system. (English) Zbl 1177.42025

A Schauder basis for a Banach space \(X\) is a countable set \(\Psi:=\{\psi_n\mid n\in\mathbb{N}\}\subset X\) with respect to which each \(f\) in \(X\) can be represented by a unique series \(\sum_n \mathcal{C}_n(f)\psi_n\) that converges to \(f\) in the norm of \(X\).
Let \(\sigma:\mathbb{N}\rightarrow\mathbb{N}\) be a bijection for which \(|\mathcal{C}_{\sigma(n)}(f)|\geq|\mathcal{C}_{\sigma(n+1)}(f)|\), then \[ G_m(f)=\sum_{n=1}^m\mathcal{C}_{\sigma(n)}\psi_{\sigma(n)} \] is the \(m\)th greedy approximant of \(f\) with respect to the basis \(\Psi\) and the permutation \(\sigma\).
It is known, that there are functions in \(L^p[0,1]\) (\(1\leq p<2\)), for which a sequence of corresponding greedy approximants diverges in measure.
Although the greedy aproximants of \(f\) may diverge, the author proves that there always will be a continuous function \(g\), arbitrarily close to \(f\) in measure, such that the sequence of greedy approximants of \(g\) converges uniformly to \(g\).


42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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