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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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