Lacunary subsets of orthonormal sets
The author studies the density of certain subsets in sets of functions \(\Phi =\{\phi_ k(x)\}^{\infty}_{k=1}\), which are orthonormal on [0,1]. One such type of subset is that of a lacunary set. These are defined as follows. An orthonormal set (ONS) of functions \(\Phi =\{\phi_ k(x)\}^{\infty}_{k=1}\), \(x\in [0,1]\), is said to be lacunary of order p \((p>2)\), or an \(S_ p\)-set, if for each polynomial P(x) with respect to this set the following estimate holds: (1) \(\| P(x)\|_ p\equiv P(x)\|_{L_ p}\leq C_{\Phi}\| P(x)\|_ 2.\) Using earlier work by Banach and Kashin, the author establishes several results about the density of lacunary subsets (and similar subsets) in orthonormal sets. One such result is the following theorem: Let \(p>2\) be an even integer, \(0<\delta \leq p-2\), \(\Phi =\{\phi_ n(x)\}^ N_{n=1}\) an ONS, \(x\in [0,1]\), with \(\| \phi_ n\|_{\tilde p}\leq M\), \(n=1,2,...,N\), where \(\tilde p=p+\delta\), \(0<M<\infty\). Then a subset \(\Phi '=\{\phi_{nk}\}^{N'}_{k=1}\) can be extracted from the set \(\Phi\), with \(N'\geq N^{\alpha (\delta)}\), \(\alpha (\delta)=2\delta /(p(p-2+\delta))\), such that for each polynomial \(P(x)=\sum^{N'}_{k- 1}a_ k\phi_{n_ k}(x)\) the following estimate holds: \[ (\sum^{N'}_{k=1}a^ 2_ k)^{1/2}\leq \| P\|_ p\leq c_{p,M,\delta}(\sum^{N'}_{k=1}a^ 2_ k)^{1/2}. \] The author also shows (based on work by Gaposhkin) that the estimates in this theorem are sharp.
scientific article; zbMATH DE number 4107391
scientific article