On the Fourier cosine-Kontorovich-Lebedev generalized convolution transforms

Abstract

We deal with several classes of integral transformations of the form

$$f(x) \to D\int_{\mathbb{R}_ + ^2 } {\frac{1} {u}} \left( {e^{ - u\cosh (x + v)} + e^{ - u\cosh (x - v)} } \right)h(u)f(v)dudv,$$

, where D is an operator. In case D is the identity operator, we obtain several operator properties on L _p (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(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.