On singularities of summable functions

A. B. Gulisashvili

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XI, Tome 113 (1981), pp. 76-96

Résumé

Let $\Phi=\{\varphi_n\}$ be a family of $L$-functions on $[0,1]$ endowed with the Lebesgue measure, satisfying the Bessel inequality. Given any sequence $\{\nu_n\}$ of non-negative real numbers we denote $\|f\|_{S(2,\nu)}=\{\sum c^*_n(f;\Phi)^2\nu_n\}^{1/2}$, $f\in L^1$, where $c^*_n$ is a non-increasing rearrangement of the sequence $\{|c_n(f;\Phi)|\}$ and $c_n(f;\Phi)$ are the Fourier coefficients of $f$ with respect to the family $\Phi$. We prove that if $\nu_n\to0$ then $$ \inf_{T_\omega\in G_1}\|T_\omega f-P_\Delta f\|_{S(2,\nu)}=\inf_{T_r\in G_2}\|T_rf\|_{S(2,\nu)}=0. $$ Here $G_1$ denotes the group of all Lebesque measure preserving automorphisms $\omega$ of $[0,1]$, $T_\omega f=f\circ\omega$, $f\in L^1$, and $G_2$ denotes the group of all real measurable unimodular functions $r$ on $[0,1]$, $T_rf=r\cdot f$, $f\in L^1$. Moreover, $P_\Delta f=\int_0^1f\,dt$, $f\in L^1$. In the case when $\nu_n\to\infty$ and $\Phi$ is a complete orthogonal family in $L^2$, we prove several theorems on the localization of the property $\|f\|_{S(2,\nu)}=\infty$ on measurable subsets of $[0,1]$.