Universality systems in \(L^p\), \(1\leq p< 2\) (Q1603029)

Let \(\{\varphi_k(x): k= 1,2,\dots\}\), \(x\in [0,1]\), be an ONS and \(\{c_k\}\) a sequence of real numbers. The series \((*)\) \(\sum c_k\varphi_k(x)\) is said to be universal in \(L^p\) (respectively, in \(H= \bigcap_{1\leq p<2} L^p\)) if, for any function \(f(x)\in L^p\) (\(\in H\), respectively), there exists an increasing sequence \(\{n_k\}\) of natural numbers such that the subsequence \(S_{n_k}(x)= \sum^{n_k}_{j=1} c_j\varphi_j(x)\) of the partial sums of series \((*)\) converges to \(f(x)\) in the metric of \(L^p\) (in all metrics of \(L^p\), \(1\leq p< 2\), respectively). We also say that the ONS \(\{\varphi_k(x)\}\) is a universality system in \(L^p\) (\(H\), respectively) if there exists a series \((*)\) with the properties detailed above. It is known that the trigonometric system is not a universality system in any \(L^p\), \(p\geq 1\). On the other hand, it turns out that there exist ONS \(\{\varphi_k(x)\}\) which are universality systems in both \(H\) and any \(L^p\), where \(1\leq p< 2\). The author introduces the notions of property \(\Gamma_q\), \(q> 2\) (\(\Gamma\), respectively) and characterizes the universality property of an ONS \(\{\varphi_k(x)\}\) in terms of property \(\Gamma_q\) (\(\Gamma\), respectively).