Integral transforms of Hartley–Fourier cosine polyconvolution type

Abstract

In this paper, after introducing a new polyconvolution for the Hartley–Fourier cosine integral transforms, we consider an integral transformation of this polyconvolution type, namely , where are given functions and is some differential operator. We obtain the necessary and sufficient conditions for the unitary property and the inverse formula of in . A sequence of functions that converges to the original function in norm is defined. We further show that the operator is a bounded operator from to , here and is the conjugate exponent of . Besides showing some nice properties of the Watson and the Plancherel types of the operator , we demonstrate how to use it to solve a class of integro-differential equations and systems of two integro-differential equations in which some other convolutions on are also involved.

Keywords:

• convolutions
• polyconvolutions
• Fourier cosine transform
• Hartley transform
• integral equations
• Watson theorem
• Plancherel theorem

AMS Subject Classifications:

• 44A05; 44A15; 44A20; 44A35; 45E10

Notes

This research is funded by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED) under [grant number 101.02-2014.08].