Undergrad Proving Fourier Method: Decompose Wave Forms with Sines & Cosines

[davidge]

Is it possible to show that every kind of possible wave form can be decomposed into a sum of sines and cosines? If so, how is it done?

 

[FactChecker]

You are asking to prove that every possible wave form can be represented as the limit of it's Fourier series. First you have to specify what type of limit convergence and what type of wave form you are talking about. The statement is not true for pointwise convergence and all continuous, periodic functions.

From section 6 of https://mat.iitm.ac.in/home/mtnair/public_html/FS-kesavan.pdf, we have
"A basic question that can be asked is the following: does the Fourier series of a continuous 2π-periodic function, f, converge to f(t) at every point t ∈ [−π, π]? Unfortunately, the answer is ‘No!’".

The statement is true if we specify "almost everywhere" and functions in L¹. See the section "Absolutely Convergent Fourier Series" in https://sites.math.washington.edu/~burke/crs/555/555_notes/fourier.pdf.
The definition of "almost everywhere" and L¹ are subjects in Real Analysis related to Lebesgue measure, and Lebesgue integration.