The discussion revolves around the relationship between the Fourier Transform and the Fourier Series, particularly focusing on the transition from the frequency domain to the time domain and the characteristics of each mathematical tool. The scope includes theoretical aspects and conceptual clarifications.
Participants generally agree on the basic definitions and relationships between Fourier Series and Fourier Transforms, but there is no explicit consensus on the extent of their applications or implications.
The discussion does not resolve potential ambiguities regarding the definitions of integrability or the specific conditions under which the Fourier Series and Transform are applicable.
That is one possible application, although there are many others. That is the general idea though.romsofia said:Is it going from the frequency domain to the time domain?
Yes. The Fourier series expresses a function as an infinite series of discrete terms. The Fourier transform uses the same idea, except it converts the function to a continuous spectrum of terms. That's why the equation switches from a Sum, to an Integral.romsofia said:Also, is there a relationship between the Fourier series and transform?
zhermes said:That is one possible application, although there are many others. That is the general idea though.
Yes. The Fourier series expresses a function as an infinite series of discrete terms. The Fourier transform uses the same idea, except it converts the function to a continuous spectrum of terms. That's why the equation switches from a Sum, to an Integral.
mathman said:Fourier series represent functions which are defined on a finite interval and periodic over the rest of the real line. Fourier integrals represent functions which are defined (and integrable in some sense, usually L1 or L2) over the entire real line.