Parametric Discrete Fourier Transforms and Their Applications

The paper discusses direct and inverse parametric discrete Fourier transforms of two types, which have a "control" parameter over a variable responsible for frequency or time. The basic properties of parametric discrete exponential functions are presented. The concept of periodicity has been expanded, from which classical periodicity follows as a special case. The main theorems of parametric discrete Fourier transform are proven. The operations of increasing the interval for determining a discrete finite signal are considered. The representation of a system of discrete exponential Fourier functions through systems of parametric exponential functions has been studied. The change in the Fourier spectrum with increasing interval by the operations of stretching and lengthening a discrete finite signal is considered.