Generalized Fourier Integrals

Classical Fourier analysis is based on the decomposition of time functions into pure sine waves or exponentials. In this paper, a more general method of decomposition involving periodic elementary time functions is developed. Explicit inversion formulas are obtained and a general expression for the inverse kernel is derived. In the special case of elementary signals having the form of square waves, it is found that the coefficients entering into the expression for the inverse kernel are given by a function which is closely related to the Mobius function. As an application, the impulsive response of an ideal filter which passes all square waves whose fundamental frequency is less than and stops all those whose fundamental frequency is greater than , is determined.