On the completeness and other properties of some function systems in \(L_p\), \(0<p<\infty\)
Let \(\{\varphi_n\}^\infty_{n= 1}\) be a system of functions in \(L_1(a,b)\) (resp. in \(L_2(a,b)\cap L_p(a,b)\), \(0< p<1\)), where \(-\infty\leq a< b\leq+\infty\). Let \(\{\varphi_{n_k}\}\) be a subsystem of \(\{\varphi_n\}^\infty_{n= 1}\) satisfying suitable conditions. It is proved that for any positive integer \(N\), \(\{\varphi_{n_k}\}^\infty_{k=N}\) is a representation system in \(L_1(a,b)\) (resp. in \(L_p(a,b)\)), i.e. for every \(f\in L_1(a,b)\) (resp. for every \(f\in L_p(a,b)\)) there exists a series \(\sum^\infty_{k=N} c_k\varphi_{n_k}\) such that \(\lim_{m\to\infty} \left\| f-\sum^m_{k=N} c_k\varphi_{n_k}\right\|= 0\) (the norm in \(L_1(a,b)\) resp. in \(L_p(a,b)\)).
scientific article; zbMATH DE number 1214031
scientific article