Convergence of Biorthogonal Series in the System of Contractions and Translations of Functions in the Spaces

P. A. Terekhin

Matematičeskie zametki, Tome 83 (2008) no. 5, pp. 722-740

Résumé

We obtain conditions for the convergence in the spaces $L^p[0,1]$, $1\le p\infty$, of biorthogonal series of the form $$ f=\sum_{n=0}^\infty(f,\psi_n)\varphi_n $$ in the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$. The proposed conditions are stated with regard to the fact that the functions belong to the space $\mathfrak L^p$ of absolutely bundle-convergent Fourier–Haar series with norm $$ \|f\|_p^\ast=|(f,\chi_0)| +\sum_{k=0}^\infty 2^{k(1/2-1/p)} \biggl(\mspace{2mu}\sum_{n=2^k}^{2^{k+1}-1} |(f,\chi_n)|^p\biggr)^{1/p}, $$ where $(f,\chi_n)$, $n=0,1,\dots$, are the Fourier coefficients of a function $f\in L^p[0,1]$ in the Haar system $\{\chi_n\}_{n\ge 0}$. In particular, we present conditions for the system $\{\varphi_n\}_{n\ge 0}$ of contractions and translations of a function $\varphi$ to be a basis for the spaces $L^p[0,1]$ and $\mathfrak L^p$.

Détail

Mots-clés : biorthogonal series
Keywords: system of contractions and translations of a function, the space $L^p[0,1]$, bundle convergence of Fourier–Haar series, Haar function, wavelet theory.