Fourier Transform of a Fourier Series (www.johndcook.com)

A recent post explored the Fourier transform of periodic functions like sine and cosine, demonstrating how distribution theory allows for a broader understanding of Fourier transforms—even for functions without classical transforms. Traditionally, the Fourier transform requires functions to diminish to zero at infinity, which excludes constant functions. However, distribution theory broadens this capability, accommodating functions that grow, provided their growth is polynomial rather than exponential. The significance of this development lies in unifying the concepts of Fourier series and Fourier transforms. By applying new conventions, the Fourier transform of constants can yield results such as the Dirac delta function, while sine and cosine functions transform into sums and differences of delta functions, respectively. This approach leads to the conclusion that the Fourier transform of a Fourier series is essentially a series of delta functions, aligned with integer shifts, with the coefficients reflecting those of the original Fourier series. This advancement presents a compelling intersection of classical analysis and modern mathematical frameworks, offering new tools for researchers in the AI and machine learning community who utilize Fourier analysis in signal processing and data representation.