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Title: On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms (English)
Author: Hong, Nguyen Thanh
Author: Tuan, Trinh
Author: Thao, Nguyen Xuan
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940
Volume: 58
Issue: 4
Year: 2013
Pages: 473-486
Summary lang: English
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Category: math
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Summary: We deal with several classes of integral transformations of the form $$ \label {generalformula} f(x)\rightarrow D\int _{\mathbb R_+^2} \frac 1u ({\rm e}^{-u\cosh (x+v)}+{\rm e}^{-u\cosh (x-v)}) h(u)f(v) {\rm d}u {\rm d} v, $$ where $D$ is an operator. In case $D$ is the identity operator, we obtain several operator properties on $L_p(\mathbb R_+)$ with weights for a generalized operator related to the Fourier cosine and the Kontorovich-Lebedev integral transforms. For a class of differential operators of infinite order, we prove the unitary property of these transforms on $L_2(\mathbb R_+)$ and define the inversion formula. Further, for an other class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type. (English)
Keyword: convolution
Keyword: Hölder inequality
Keyword: Young's theorem
Keyword: Watson's theorem
Keyword: unitary
Keyword: Fourier cosine
Keyword: Kontorovich-Lebedev
Keyword: transform
Keyword: integro-differential equation
MSC: 33C10
MSC: 44A35
MSC: 45E10
MSC: 45J05
MSC: 47A30
MSC: 47B15
idZBL: Zbl 06221241
idMR: MR3083524
DOI: 10.1007/s10492-013-0023-5
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Date available: 2013-07-18T15:20:53Z
Last updated: 2015-09-07
Stable URL: http://hdl.handle.net/10338.dmlcz/143341
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