Algebraic Properties of Generalized Trigonometric Function Transforms

The generalized trigonometric functions called Ateb-functions are considered. On this basis, a generalization of the Fourier transform is constructed and called the Ateb-transform. From the operator theory point of view, the Ateb-transform is considered as a formalism of the convolution algebra in which multiplication is defined by means of hypergroups of the generalized shift operator. In this formal approach, the algebraic structure is presented, and its properties are developed. The eigenvalue problem for the differential equation of nonlinear oscillation is investigated. Some properties are illustrated numerically. The application of this approach for modeling vibration motion is considered.