A generalization of a standard inequality for Hardy space \(H_1\) (Q5950260)

Let \(H_{p}\) \((p>0)\) denote the Hardy spaces of functions \(G=G(z)\) analytic on the unit disk with the norm NEWLINE\[NEWLINE\|G\|_{p}=\begin{cases} \left(\frac{1}{2\pi}\sup_{r<1}{\int_{0}^{2\pi}}|G(re^{i\theta})|^{p} d\theta\right)^{\frac{1}{p}},&p<\infty,\\ \sup_{|z|<1}|G(z)|, & p=\infty.\end{cases}NEWLINE\]NEWLINE The well-known Hardy's theorem says that \({\sum_{k=0}^{+\infty}}\frac{|a_{k}|}{k+1}\leq\pi\|f\|_{1}\) where \(a_{k}\), \(k=0,1,2,\ldots\), are the coefficients of the function \(f\in H_{1}\) with respect to the usual basis, i.e., \(f(z)=\sum_{k=0}^{+\infty} a_{k}z^{k}\). In this paper the author constructs a new orthonormal basis for the spaces \(H_{p}\); the basis is given by the formulae NEWLINE\[NEWLINEB_{k}(z)=\frac{(1-|z_{k}|^{2})^{\frac{1}{2}}} {1-\overline{z_{k}}z} \phi_{k}(z),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\phi_{k}(z)=\prod_{j=0}^{k-1} \frac{z-z_{j}}{1-\overline{z_{k}}z},\quad \phi_{0}(z)=1NEWLINE\]NEWLINE where \(z_{0}=0\), \(|z_{k}|<1\) and \(z_{k}=z_{k+nm}\) for \(k=1,2,\ldots\) and \(m=0,1,2,\ldots\); \(n\) is a fixed positive integer. The main result of the paper is that the following Hardy-type inequality holds: if \(f(z)= \sum_{k=0}^{+\infty}a_{k}B_{k}(z)\in H_{1}\) then \(\sum_{k=0}^{+\infty} \frac{|a_{k}|}{k+1}\leq C\|f\|_{1}\) where \(C\) is a constant which depends on \(z_{k}\) and \(n\). A similar inequality holds for the functions \(f\in H_{p}\) with \(p\in(1,2]\). This inequality is used to estimate the worst-case error of a least-squares algorithm for an impulse input signal.