Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces L
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Authors: | P A Terekhin |
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Institution: | 1. Saratov State University, Saratov, Russia
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Abstract: | We obtain conditions for the convergence in the spaces L p 0, 1], 1 ≤ p < ∞, of biorthogonal series of the form $$ f = \sum\limits_{n = 0}^\infty {(f,\psi _n )\phi _n } $$ in the system {? n } n≥0 of contractions and translations of a function ?. The proposed conditions are stated with regard to the fact that the functions belong to the space $ \mathfrak{L}^p $ of absolutely bundleconvergent Fourier-Haar series with norm $$ \left\| f \right\|_p^ * = \left| {f,\chi _0 } \right| + \sum\limits_{k = 0}^\infty {2^{k({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})} } \left( {\sum\limits_{n = 2^k }^{2^{k + 1} - 1} {\left| {f,\chi _n } \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where (f,χ n ), n = 0, 1, ..., are the Fourier coefficients of a function f ? L p 0, 1] in the Haar system {χ n } n≥0. In particular, we present conditions for the system {? n } n≥0 of contractions and translations of a function ? to be a basis for the spaces L p 0, 1] and $ \mathfrak{L}^p $ . |
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