A Fourier series of a function, $f(x)$ is the decomposition of that fuction into a sum of [sinusoids](Fourier%20analysis%20(Index).md#Sinusoids) that may be defined as $f(x)=\frac{1}{L}\sum_k e^{ikx}\widetilde{f}_k(x)$ where the exponentials terms, $e^{ikx}$ are referred to as [[Fourier mode]]s. These modes are weighted by complex [Fourier coefficients](Fourier%20series.md#Fourier%20Coefficients), $\widetilde{f}_k(x).$^7fe728 # Which functions can be written as a Fourier series     # Constructing a Fourier series with sums of sines and cosines One way of constructing a [Fourier series](Fourier%20series.md) of a [function](Fourier%20series.md#Which%20functions%20can%20be%20written%20as%20a%20Fourier%20series) is to start with the notion that a [Fourier series is built from a sum of sines and cosines](Fourier%20series#^7fe728) where we use the fact that both $\sin{x}$ and $\cos{x}$ have a period of $2\pi.$ Thus we need to construct a series that contains all the possible sines and cosines in a sum. %%This is from page 350 of Boas. But can we find a more rigorously written source that does a similar derivation of this ansatz and somehow proves the lemmas involved? E.g. should we show that all sines and cosines can be written as we do here? For a rigorous discussion of Fourier series consider using first volume of the analysis series by Elias Stein%% # Derivation of Fourier series from Fourier transforms The [Fourier series](Fourier%20series.md) is related to the [Fourier transform](Fourier%20transform.md) in that it may be viewed as discritized Fourier transform where as the Fourier transform is derived from [turning the sum into an integral.](Fourier%20transform.md#Derivation%20of%20the%20Fourier%20transform%20from%20the%20Fourier%20series) %%page 77 of the fft book illustrates this with an example but can you find a more rigorous intro to this?%% # Fourier coefficients # Completeness relation The sum of [[Fourier mode]]s in a given fourier series form a [completeness relation](Fourier%20mode.md#Completeness%20Relation). #MathematicalFoundations/Analysis/FourierAnalysis