Fourier coefficient

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Fourier coefficient

For a function f∈L2([−π,π]), the Fourier coefficient f^(n) of f is given by

f^(n)=12π∫−ππf(t)e−intdt

And the Nth partial Fourier sum is

SNf(x)=∑|n|≤Nf^(n)einx=∑|n|≤N⟨f,einx2π⟩einx2π
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

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