Some theorems for Preiss systems
Let \(X(\varphi)\) be a symmetric space of Lebesgue measurable functions on \(I=[0,1]\) with fundamental function \(\varphi(t)= \|\chi_{[0,t]} \|_X\), and let \(\{\psi_n\}^\infty_{n=0}\) be the Preiss orthonormal system on \(I^*=[0,1]^*\), defined by \(\psi_0(x)=1\), \(\psi_{n_k}(x)= \exp{2\pi ix_{k+1} \over p_{k+1}}\), \(\psi_n(x)= \prod^r_{k=0} (\psi_{m_k} (x))^{\alpha_k}\), where \(n=\sum^r_{k=0} \alpha_km_k\), \(\alpha_i=0,1, \dots,p_{k-1}\), \(m_0=1\), \(m_k=p_1, \dots,p_k\). If \(\{p_n\}\) is bounded then under specific assumptions a norm in \(X(\varphi)\) is built by means of the system \(\{\psi_n \}^\infty_{n=0}\), equivalent to the norm \(\|\cdot\|_X\). Also, a counterexample is constructed in the case of unbounded \(\{p_n\}\).
scientific article; zbMATH DE number 1960456