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Integral transforms related to a generalized convolution. (English) Zbl 0970.44004

A general convolution transform of the Fourier cosine-sine type is investigated. The authors find necessary and sufficient conditions on the kernel function, which makes the mentioned transform a unitary transform on \(L_2(\mathbb{R})\). A special class of the Fourier sine kernels is defined. Watson and Plancherel type theorems are proved. Interesting examples of convolutions, which are associated with the Airy, Anger-Weber and modified Bessel special functions as kernels are demonstrated.

MSC:

44A35 Convolution as an integral transform
42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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