Question

Explain what truncating a Fourier series expansion and Fourier Integral does

Answer 1

Fourier transform and Fourier series are two manifestations of a similar idea, namely, to write general functions as "super positions" (whether integrals or sums) of some special class of functions.Fourier transform is the limit of the Fourier series of a function with the period approaches to infinity, so the limits of integration change from one period to (−∞,∞)(−∞,∞).It is then used to represent a general, non-periodic function by a continuous superposition or integral of complex exponentials and The Fourier series is used to represent a periodic function by a discrete sum of complex exponentials.

A periodic function provably can be expressed as a "discrete" superposition of exponentials,where it would not be possible to use the Fourier transform..A non-periodic is a function does not admit an expression as discrete superposition of exponentials, but only a continuous superposition, namely, the integral that shows up in Fourier inversion for Fourier transforms for the limit in L1(−∞,∞)L1(−∞,∞).It shown that the Fourier series coefficients of a periodic function are sampled values of the Fourier transform of one period of the function.

let G,μ are the variables.We can define the Fourier transform of a function f∈L1(G):

f(χ)ˆ=∫ Gf(x)χ(x)...d μ(x)

f(χ)ˆ is a bounded continuous function that vanishes at infinity on Gˆ.

If G=R then Gˆ=R and we have the regular Fourier transform.

If G=S1 then Gˆ=Z and we have the Fourier series