On Fourier Series on the Torus and Fourier Transforms

The question of the representability of a continuous function on R-d in the form of the Fourier integral of a finite Borel complex-valued measure on R-d is reduced in this article to the same question for a simple function. This simple function is determined by the values of the given function on the integer lattice R-d d. For d = 1, this result is already known: it is an inscribed polygonal line. The article also describes applications of the obtained theorems to multiple trigonometric Fourier series.