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.
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