Fourier coefficient

[[concept]]

Topics

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Fourier coefficient

For a function $f\in L^2([-\pi ,\pi ])$, the Fourier coefficient $\hat{f}(n)$ of $f$ is given by $\hat{f}(n)=\frac{1}{2\pi }{\int }_{-\pi }^{\pi }f(t)e^{-int}dt$ And the $N$th partial Fourier sum is $S_{N}f(x)={\sum }_{|n|\le N}\hat{f}(n)e^{inx}={\sum }_{|n|\le N}⟨f,\frac{e^{inx}}{\sqrt{2\pi }}⟩\frac{e^{inx}}{2\pi }$

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

References

References

See Also

Mentions

Mentions

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