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Generalized Convolution for the Kontorovich-Lebedev, Fourier Transforms and Applications to Acoustic Fields
Trinh Tuan
,
Nguyen Thanh Hong
,
Pham Van Hoang
Abstract
To study the boundedness of the acoustic field in $L_{p}(\mathbb {R}_{+} ; dx)$ and its asymptotic behavior, we introduce a new generalized convolution for the Kontorovich-Lebedev ($\mathcal {K}\mathcal {L}$) transform and the Fourier transforms, and a new representation of the acoustic field via this generalized convolution. Some properties of the new generalized convolution are obtained. Moreover, an analog of Watson’s theorem is established, in which we obtain the necessary and sufficient conditions for a class of generalized convolution transforms to be isomorphism-isometric.