Orthonormal polynomial bases in function spaces (Q1196309)

Let \(\Phi=(\Phi_ k)_{k=0}^ \infty\) be an orthonormal system of trigonometric polynomials on the unit circle \(T\) and let \(v_ n=\max_{k\leq n}\deg\Phi_ k\). The main problem consists in estimating the sequence \((v_ n)\) from above. It is shown by construction that there exists a system \(\Phi\) such that \(v_ n\leq{4\over 3}n\), and \(\Phi\) is a Schauder basis in \(C(T)\) (also in \(L_ 1(T))\), and an unconditional basis in \(\text{Re} H_ 1\) (also in \(L_ p(T)\) for \(1<p<\infty\)). There is also given an estimation from below.