On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms. (English) Zbl 1289.44005

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

Appl. Math., Praha 58, No. 4, 473-486 (2013).

Summary: We deal with several classes of integral transformations of the form \[ f(x)\rightarrow D\int _{\mathbb {R}_{+}^2}\frac {1}{u}\left ( \mathrm{e}^{-u \cosh (x+v)}+\mathrm{e}^{-u \cosh (x-v)}\right )h(u)f(v) \mathrm{d}u \mathrm{d}v, \] where \(D\) is an operator. In the case that \(D\) 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 another class of differential operators of finite order, we apply these transformations to solve a class of integro-differential problems of generalized convolution type.

MSC:

44A35	Convolution as an integral transform
45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
33C10	Bessel and Airy functions, cylinder functions, \({}_0F_1\)
45J05	Integro-ordinary differential equations
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

convolution; Hölder inequality; Young’s theorem; Watson’s theorem; Fourier cosine; Kontorovich-Lebedev transform; integro-differential equation

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