We obtain new convolutions for quadratic-phase Fourier integral operators (which include, as subcases, e.g., the fractional Fourier transform and the linear canonical transform). The structure of these convolutions is based on properties of the mentioned integral operators and takes profit of weight-functions associated with some amplitude and Gaussian functions. Therefore, the fundamental properties of that quadraticphase Fourier integral operators are also studied (including a Riemann-Lebesgue type lemma, invertibility results, a Plancherel type theorem and a Parseval type identity).As applications, we obtain new Young type inequalities, the asymptotic behaviour of some oscillatory integrals, and the solvability of convolution integral equations.

No abstract

The main purpose of this paper is to present three new convolutions for the offset linear canonical transform, with the Hermite weights, and to illustrated their potential applications. In view of this, new factorization theorems are obtained and new Young's convolution inequalities will be introduced. Within the more applied side, the way to design filters (including multiplicative filters in the time domain) is also discussed in the last section.

In this paper, we consider a nonlinear differential system with initial and multi - point boundary conditions. The existence of solutions is proved by using the Banach contraction principle or the Krasnoselskii?s fixed point theorem. Furthermore, the existence of positive solutions is also obtained by applying the Guo-Krasnoselskii?s fixed point theorem in cones. As a consequence of the Guo-Krasnosellskii?s fixed point theorem, the multiplicity of positive solutions is established.