Theorem about convergence of Fourier series
Wiener–Lévy theorem is a theorem in
Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after
Norbert Wiener and
Paul Lévy.
Norbert Wiener first proved Wiener's 1/f theorem,
[1] see
Wiener's theorem. It states that if f has absolutely convergent Fourier series and is never zero, then its inverse 1/f also has an absolutely convergent Fourier series.
Wiener–Levy theorem
Paul Levy generalized Wiener's result,
[2] showing that
Let
be an absolutely convergent Fourier series with
![{\displaystyle \|F\|=\sum \limits _{k=-\infty }^{\infty }|c_{k}|<\infty .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6b4990e247b84abc83e5ed00e9e51f77afeaf4d)
The values of
lie on a curve
, and
is an analytic (not necessarily single-valued) function of a complex variable which is regular at every point of
. Then
has an absolutely convergent Fourier series.
The proof can be found in the Zygmund's classic book
Trigonometric Series.
[3]
Example
Let
and
) is
characteristic function of discrete probability distribution. So
is an absolutely convergent Fourier series. If
has no zeros, then we have
![{\displaystyle H[F(\theta )]=\ln \left(\sum \limits _{k=0}^{\infty }p_{k}e^{ik\theta }\right)=\sum _{k=0}^{\infty }c_{k}e^{ik\theta },}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a610e07b7b1c013a221a8abe334feeb6cc2a63c5)
where
The statistical application of this example can be found in discrete pseudo
compound Poisson distribution
[4] and
zero-inflated model.
If a discrete r.v.
with
,
, has the probability generating function of the form
![{\displaystyle P(z)=\sum \limits _{i=0}^{\infty }P_{i}z^{i}=\exp \left\{\sum \limits _{i=1}^{\infty }\alpha _{i}\lambda (z^{i}-1)\right\},z=e^{ik\theta }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32d4a874175df3795d2b1073e5a6d235a627ee08)
where
,
,
, and
. Then
is said to have the discrete pseudo compound Poisson distribution, abbreviated DPCP.
We denote it as
.
See also
References