On 26 Feb, 23:38, Rune Allnor <all...@tele.ntnu.no> wrote:
> On 26 Feb, 20:54, "tkremund98" <tkremun...@hotmail.com> wrote:
>
> > This is just a question about Gibb's effect. Is Gibb's Effect is present in
> > stationary data?
>
> Gibbs' effect has nothing to do with whether data are
> stationary, but with properties of the Fourier transform
> around discontinuities.
>
> The problem is that the Fourier transform converges
> in the mean square sense, i.e. that the L2 reconstruction
> error (view with fixed-width font)
>
> * * * * * * * * * * * * *n-1
> e_n^2 = integral |f(t) - sum A_n exp(j w_n t)|^2 dt
> * * * * * * * * * * * * *k=0
>
> vanishes as n -> infinity. However, this convergence
> in the mean does not ensure *pointwise* convergence,
> since pointwise convergence is expressed in terms of
> the L1 norm:

This should be the L_inf norm:

n-1
E_n = max { |f(t) - sum A_n exp(j w_n t)| }
k=0

> This discrepancy between pointwise and mean convergence
> becomes particularly important at dicontinuities in f(t),
> which is what causes Gibbs' phenomenon.
>
> Rune