Generalized Convolutions for the Fourier Integral Transforms and Applications

This paper provides some generalized convolutions for the Fourier integral transforms and treats the applications. Namely, there are six generalized convolutions with weight-function for the Fourier integral transforms. As for applications, the normed ring structures on $L^1(\mathbb R^d)$ are constructed, and the explicit solution in $L^1(\mathbb R^d)$ of the integral equations with the mixed Toeplitz–Hankel kernel are obtained.