The Complete Asymptotic Evaluation for Mellin Convolution Operators

Abstract

For a set of kernels $K_{w}:\left( 0,\infty \right) \rightarrow \mathbb {R}$ ( $w>0$) which satisfies the condition $\int _{0}^{\infty }K_{w}\left( t\right) \frac{\mathrm{d}t}{t}=1$ the singular integral of Mellin convolution type is defined by

$$\begin{aligned} T_{w}\left( f\right) \left( s\right) =\int _{0}^{\infty }K_{w}\left( t\right) f\left( st\right) \frac{\mathrm{d}t}{t},s>0 \end{aligned}$$

where f belongs to the domain of $T_{w}$. In the paper we find the complete asymptotic evaluation for the above operator. As applications we find the complete asymptotic evaluation for the Mellin–Gauss–Weierstrass operator, for the Mellin–Abel operator, for the Hadamard type operator, for the moment operator and for the modified Mellin–Abel–Poisson operator and a new type of the Mellin convolution operator.