Some properties of finite Fourier transformations and their application to the summation of a class of trigonometric series

Fourier transforms on finite intervals (finite Fourier transforms) are considered here, not as Fourier coefficients, but as functions of a continuous variable. The tables of properties of finite Fourier exponential, Fourier sine, and Fourier cosine transformations are composed. The use of these properties is illustrated by means of an extension of Graf's theorem for Bessel functions, a series of the parabolic cylinder functions, and spectral relationships for Chebyshev and Legendre polynomials.