Fourier coefficient

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Fourier coefficient

For a function $f \in L^{2} ([- π, π])$ , the Fourier coefficient $\hat{f} (n)$ of $f$ is given by

\hat{f} (n) = \frac{1}{2 π} \int_{- π}^{π} f (t) e^{- i n t} d t

And the $N$ th partial Fourier sum is

S_{N} f (x) = \sum_{| n | \leq N} \hat{f} (n) e^{i n x} = \sum_{| n | \leq N} ⟨ f, \frac{e^{i n x}}{\sqrt{2 π}} ⟩ \frac{e^{i n x}}{2 π}

Note

this follows from all elements of hilbert spaces with orthonormal bases can be written as sums of the basis elements and Hilbert spaces are bijective with ell-2

File	Last Modified
Cesaro-Fourier mean	2025-07-15
fourier functions form an orthonormal set	2025-07-15
fourier partial sums are given by the Dirichlet kernel	2025-07-15
Fourier series	2025-07-15
Functional Analysis Lecture 15	2025-07-15

Fourier coefficient

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