Generalized convolution with two parameters for the Hankel transform and related integral operators

ABSTRACT

The present research is devoted to integral operators related to generalized convolutions (polyconvolutions) for the Hankel transform $\left(H_{\nu }f\right)\left(x\right)$.

The generalized convolution defined by the Parseval type equality $h\left(t\right)=H_{\nu }^{-1}\left[\right(H_{\mu }f\left)\right(x\left)\right(H_{\mu +\nu }g\left)\right(x\left)\right]\left(t\right)$ is considered in spaces $L_1({\mathbb{R}}_{+},\sqrt{t}\mathrm{d}t)$ and $L_2({\mathbb{R}}_{+},t\mathrm{d}t)$. Existence conditions and boundary properties are found.

Watson's-type theorems for convolution operators are proved. The integral operators with unsymmetrical kernels are studied. Some examples of solving integral equations are given.

KEYWORDS:

• Hankel transform
• convolution
• convolution transform
• Watson's theorem
• integral equations

AMS SUBJECT CLASSIFICATION:

• 44A05
• 44A35
• 45A05
• 45P05

Disclosure statement

No potential conflict of interest was reported by the author.