In this paper we introduce two generalized convolutions for the Fourier cosine, Fourier sine and Laplace integral transforms. Convolution properties and their applications to solving integral equations and systems of integral equations are considered. Keywords Fourier sine transform • Fourier cosine transform • Laplace transform Mathematics Subject Classification (2000) 33C10 • 44A35 • 45E10 • 45J05 • 47A30 • 47B15 1 Introduction Convolutions for integral transforms are studied in the early years of the 20th century, such as convolutions for the Fourier transform (see [2, 9, 13]), the Laplace transform (see [1, 2, 8, 13, 16-19]), the Mellin transform (see [8, 13]), the Hilbert transform (see [2, 3]), the Fourier cosine and sine transforms (see [5, 7, 13, 14]), and so on. These convolutions have many important applications in image processing, partial differential equations, integral equations, inverse heat problems (see [2-4, 8, 11-13, 15-18]).

We introduce several weighted L p (R +)-norm inequalities and integral transform related to the generalized convolution with a weight function for the Fourier cosine and Laplace transforms. Some applications of these inequalities to estimate the solutions of some partial differential equations are considered. We also obtained solutions of a class of the Toeplitz plus Hankel integro-differential equations in closed form.

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