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5 papers
Convergence of Biorthogonal Series in the System of Contractions and Translations of Functions in the Spaces
$L^p[0,1]$
P. A. Terekhin Saratov State University named after N. G. Chernyshevsky
Abstract:
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
$\mathfrak L^p$.
Keywords:
biorthogonal series, system of contractions and translations of a function, the space $L^p[0,1]$, bundle convergence of Fourier–Haar series, Haar function, wavelet theory.
UDC:
517.51 Received: 19.04.2007
Revised: 11.11.2007
DOI:
10.4213/mzm4046
Fulltext:
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