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![[parent]](http://images.planetmath.org:8080/images/uparrow.png) Viewing Correction to 'Fourier coefficients'
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Expressing the Fourier coefficients by jduchon
Correction id: 4506
Filed on: 2004-06-11 08:03:23
Status: Accepted on 2004-09-22 12:04:20
Type: Erratum
Correction text:
If you choose to define the Fourier coefficients as coordinates in a
*general* orthonormal basis, then it is definitely not correct to write
them as \hat{f}(k) where \hat{f} is the Fourier transform of f. The correct
formula in this general case is \int f(x) \bar{\phi_k (x)} dx. In the
particular case of \phi_k (x) = e^{ik.x} (normalized) this amounts to
the value at k (an integer) of the Fourier transform of the function
defined on R^N, equal to f on the box, and vanishing outside the box.
You certainly may call this \hat{f}(k) if you wish, but the Fourier
transform of the periodic (then nonintegrable if nonzero) function f
is defined in the theory of distributions: it is a sum of point measures
at the integers, whose masses are the Fourier coefficients.
But I insist that this formula f = \sum \hat{f}(k) \phi_k makes sense
only in a particular case, not for any orthonormal basis of L^2 (real
or complex, this too you should choose). |
No comment from object owner mathcam.
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