On the Fourier cosine—Kontorovich-Lebedev generalized convolution transforms

Hong, Nguyen Thanh ; Tuan, Trinh ; Thao, Nguyen Xuan

Applications of Mathematics, Tome 58 (2013) no. 4, pp. 473-486.

Résumé

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.

Détail

DOI : 10.1007/s10492-013-0023-5

Classification : 33C10, 44A35, 45E10, 45J05, 47A30, 47B15
Keywords: convolution; Hölder inequality; Young's theorem; Watson's theorem; unitary; Fourier cosine; Kontorovich-Lebedev; transform; integro-differential equation