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The solvability and explicit solutions of two integral equations via generalized convolutions. (English) Zbl 1197.45004
Authors’ abstract: This paper presents the necessary and sufficient conditions for the solvability of two integral equations of convolution type; the first equation generalizes from integral equations with the Gaussian kernel, and the second one contains the Toeplitz plus Hankel kernels. Furthermore, the paper shows that the normed rings on \(L^1(\mathbb R^d)\) are constructed by using the obtained convolutions, and an arbitrary Hermite function and appropriate linear combination of those functions are the weight-function of four generalized convolutions associating \(F\) and \(\check F\). The open question about Hermitian weight-function of generalized convolution is posed at the end of the paper.

MSC:
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
44A35 Convolution as an integral transform
46H05 General theory of topological algebras
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