Representation in \(L_ p\) by series of translates and dilates of one function
A system of functions \(\varphi_k\) is called an \(L^p\) representation system if given \(f\in L^p\) there are coefficients \(c_k\in R\) such that \(\sum^\infty_{k=1} c_k\varphi_k\) converges to \(f\) in \(L^p\) norm. It is called an unconditional \(L^p\) representation system if such a series can be found whose rearrangements still converge to \(f\) in \(L^p\) norm. The authors investigate the problem of finding when the system of translates of dilations of a given \(\varphi\), namely the system \(\varphi_{k,i} (t)\equiv \varphi(2^kt-i)\), \(0\leq i<2^k\), \(k=0,1,\dots\), is a representation system or an unconditional representation system. For \(\varphi\in L^q[0,1]\), \(1\leq q<\infty\), they find that if the Fourier transform \(\widehat{\varphi} (0)\neq 0\), then this system is an \(L^p\) representation system for all \(0<p\leq q\); for \(\varphi\in L^2[0,1]\), they find that if \(\varphi\neq 0\), then this system is an \(L^p\) representation system for all \(0<p<1\). The unconditional case is a bit more complicated. For \(\varphi\in L^1(R)\) if \(\widehat{\varphi} (0)\neq 0\), then this system is an unconditional \(L^1\) representation system. For \(\varphi\in L^p(R)\), \(1<p< \infty\), if \(\varphi\) decays rapidly enough, namely, if there is a \(\delta>0\) such that \(\varphi(t)=O (|t|^{-1-\delta})\) as \(|t|\to \infty\), then this system is an unconditional \(l^p\)-representation system. Multidimensional versions of these results are also discussed.
unconditional \(L^ p\) representation system
scientific article; zbMATH DE number 782141