On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Hong, Nguyen Thanh, Tuan, Trinh, and Thao, Nguyen Xuan. "On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms." Applications of Mathematics 58.4 (2013): 473-486. <http://eudml.org/doc/260609>.

@article{Hong2013,
abstract = {We deal with several classes of integral transformations of the form \[ f(x)\rightarrow D\int \_\{\mathbb \{R\}\_+^2\} \frac\{1\}\{u\} (\{\rm e\}^\{-u\cosh (x+v)\}+\{\rm e\}^\{-u\cosh (x-v)\}) h(u)f(v) \{\rm d\}u \{\rm d\} v, \] 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.},
author = {Hong, Nguyen Thanh, Tuan, Trinh, Thao, Nguyen Xuan},
journal = {Applications of Mathematics},
keywords = {convolution; Hölder inequality; Young's theorem; Watson's theorem; unitary; Fourier cosine; Kontorovich-Lebedev; transform; integro-differential equation; convolution; Hölder inequality; Young's theorem; Watson's theorem; Fourier cosine; Kontorovich-Lebedev transform; integro-differential equation},
language = {eng},
number = {4},
pages = {473-486},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms},
url = {http://eudml.org/doc/260609},
volume = {58},
year = {2013},
}

TY - JOUR
AU - Hong, Nguyen Thanh
AU - Tuan, Trinh
AU - Thao, Nguyen Xuan
TI - On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms
JO - Applications of Mathematics
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 4
SP - 473
EP - 486
AB - We deal with several classes of integral transformations of the form \[ f(x)\rightarrow D\int _{\mathbb {R}_+^2} \frac{1}{u} ({\rm e}^{-u\cosh (x+v)}+{\rm e}^{-u\cosh (x-v)}) h(u)f(v) {\rm d}u {\rm d} v, \] 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.
LA - eng
KW - convolution; Hölder inequality; Young's theorem; Watson's theorem; unitary; Fourier cosine; Kontorovich-Lebedev; transform; integro-differential equation; convolution; Hölder inequality; Young's theorem; Watson's theorem; Fourier cosine; Kontorovich-Lebedev transform; integro-differential equation
UR - http://eudml.org/doc/260609
ER -