Pointwise Recovery Formula of a Function from its Generalized Fourier Transform

Abstract The article addresses the integrability of the Fourier transform defined on the space of bounded variation functions vanishing at infinity. Explicit formulas for some involved integrals are shown. In particular, we recover the function from its Generalized Fourier Transform, which offers advantages for numerical approximations of the Fourier transform function. Moreover, an optimal condition for integrability in the Henstock-Kurzweil sense of the classic Fourier transform defined on the Lebesgue spaces is given. The condition represents an extension of the Heisenberg’s Uncertainty Principle beyond the traditional mathematical formalism in Quantum Mechanics.