The domain of the Fourier integral

TL;DR

This paper investigates the mathematical properties of Fourier integrals within Hilbert spaces, highlighting the limitations of defining Fourier integrals in the space of square integrable functions due to the intersection with almost periodic functions.

Contribution

It clarifies the conditions under which Fourier integrals can be defined in Hilbert spaces, emphasizing the intersection issues between different function spaces.

Findings

Fourier integral is the scalar product of functions in different Hilbert spaces.

The intersection of these spaces contains only the zero element.

Fourier integral is not defined in the space of square integrable functions with nonzero norm.

Abstract

We consider the problem of determining the Fourier integral in the Hilbert space of square integrable functions. Fourier integral is the scalar product of two functions belonging to the Hilbert space of square integrable functions and the Hilbert space of almost periodic functions. Scalar product for different Hilbert spaces defined at the intersection of these spaces, which contains only one zero element. Therefore, the Fourier integral is not defined in the Hilbert space of square integrable functions with nonzero norm.