Special bases in function spaces (Q2760180)

The authors present in a unified and systematic fashion the most important systems of functions. Special properties established by various mathematicians are given. Concerning the unconditionality in \(L_p\) the authors ``present the sketch of a very general argument that many natural orthonormal systems with a dyadic structure are unconditional bases in \(L_p \) spaces''. A first important result is the following:NEWLINENEWLINELet \(\varphi\) be a bounded function on \(\mathbb{R}\), decreasing on \((0,\infty)\) such that \(\int^\infty_0 \varphi(x) \log(x+1) dx<\infty\) and let \(\psi\) and \(\psi^*\) be two functions on \(\mathbb{R}\) such that the system \((\psi_{jk})_{j,k \in\mathbb{Z}}\) is an unconditional basis in \(L_2(\mathbb{R})\) with biorthogonal functionals \(\psi^*_{jk}\). If \(|\psi(x) |\), \(|\psi^*(x) |\leq\varphi (x)\) for all \(x\in\mathbb{R}\), then \((\psi_{jk})_{j,k\in/\mathbb{Z}}\) is an unconditional basis in \(L_p(\mathbb{R})\) for \(1<p< \infty\). NEWLINENEWLINEParticular attention is paid to function spaces on compact smooth manifolds. Also, the case of bases with vector coefficients in spaces of vector-valued functions is considered.NEWLINENEWLINEFor the entire collection see [Zbl 0970.46001].