On the \(m\)-term approximation in \(L^1(0, 1)\) in the Haar system
Let \(X\) be a Banach space and \(\varphi=\{\varphi_{k}\}_{k=1}^{\infty}\) be a normalized basis for \(X.\) Suppose that for some natural number \(m\) and each \(x\in X\) there exist numbers \(\widehat{\alpha}_{1,}\widehat{\alpha}_{2} ,\dots,\widehat{\alpha}_{m}\) and \(\widehat{k}_{1},\dots, \widehat{k}_{2},\dots,\widehat{k}_{m}\) for which the following equality holds: \[ \inf_{{\alpha_{1},\alpha_{2},\dots,\alpha_{m}, k_{1} ,k_{2},\dots,k_{m}}}\left\| x-\sum_{i=1}^{m}\alpha_{i}\varphi_{k_{i} }\right\| _{X}=\left\| x-\sum_{i=1}^{m}\widehat{\alpha}_{i}\varphi_{\widehat {k}_{i}}\right\| . \] Denote \(G_{1}^{(m)}(x,\varphi,X)=\sum_{i=1}^{m}\widehat{\alpha}_{i} \varphi_{\widehat{k}_{i}}\) the best \(m\)-term approximant of \(x\) in the system \(\varphi,\) and let \(G_{k+1}^{(m)}(x,\varphi,X)= G_{k}^{(m)}(x,\varphi,X)+G_{k}^{(m)}(x-G_{k} ^{(m)}(x,\varphi,X),\varphi,X)\), \(k=1,2,3,\dots\) The method of approximation of the element \(x\) by the sequence \( \{ G_{k}^{(m)}(x,\varphi,X)\} _{k=1}^{\infty}\) is called the \(m\)-term \(X\)-greedy algorithm of \(x\) in the system \(\varphi.\) In this paper one considers the \(m\)-term \(L^{1}\)-greedy algorithm in the normalized Haar system in \(L^{1}(0,1).\) The following result is proved: for each positive integer \(m,\) there exists a function \(f_{m}\in L^{1}(0,1)\) for which there is no \(m\)-term \(L^{1}\)-greedy algorithm in the Haar system converging to \(f_{m}.\)
scientific article; zbMATH DE number 5155486