On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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On Sidon-Telyakovskii-type conditions for the integrability of multiple trigonometric series

Authors:

P V Zaderei E N Pelagenko O V Ivashchuk

Institution:

(1) Kiev National University of Technology and Design, Kiev, Ukraine

Abstract:

For a trigonometric series

${\sum\limits_{k = 0}^\infty {a_{k} } }{\sum\limits_{l \in kV\backslash {\left( {k - 1} \right)}V} {e^{{i{\left( {l,x} \right)}}} } },\quad \quad a_{k} \to 0,\quad \quad k \to \infty ,$

defined on −π, π)^m, where V is a certain polyhedron in R ^m, we prove that

${\int\limits_{T^{m} } {{\left| {{\sum\limits_{k = 0}^\infty {a_{k} } }{\sum\limits_{l \in kV\backslash {\left( {k - 1} \right)}V} {e^{{i{\left( {l,x} \right)}}} } }} \right|}} }\,dx \leq C{\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}} }{\left| {\Delta A_{k} } \right|}$

if the coefficients a _k satisfy the following Sidon-Telyakovskii-type conditions:

$A_{k} \to 0,\quad {\left| {\Delta a_{k} } \right|} \leq A_{k} \quad \forall k \geq 0,\quad {\sum\limits_{k = 0}^\infty {{\left( {k + 1} \right)}{\left| {\Delta A_{k} } \right|}} } < \infty \,.$

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 5, pp. 579–585, May, 2008.

Keywords:

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