“Poisson wavelet”的意思、由来-开放百科全书

In mathematics, in functional analysis, several different wavelets are known by the name Poisson wavelet. In one context, the term "Poisson wavelet" is used to denote a family of wavelets labeled by the set of positive integers, the members of which are associated with the Poisson probability distribution. These wavelets were first defined and studied by Karlene A. Kosanovich, Allan R. Moser and Michael J. Piovoso in 1995–96.^[1]^[2] In another context, the term refers to a certain wavelet which involves a form of the Poisson integral kernel.^[3] In a still another context, the terminology is used to describe a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel.^[4]

Wavelets associated with Poisson probability distribution

Definition

To see the relation between the Poisson wavelet and the Poisson distribution let X be a discrete random variable having the Poisson distribution with parameter (mean) t and, for each non-negative integer n, let Prob(X = n) = p_n(t). Then we have

Basic properties

Poisson wavelet transform

The Poisson wavelet family can be used to construct the family of Poisson wavelet transforms of functions defined the time domain. Since the Poisson wavelets satisfy the admissibility condition also, functions in the time domain can be reconstructed from their Poisson wavelet transforms using the formula for inverse continuous-time wavelet transforms.

If f(t) is a function in the time domain its n-th Poisson wavelet transform is given by

In the reverse direction, given the n-th Poisson wavelet transform

of a function f(t) in the time domain, the function f(t) can be reconstructed as follows:

Applications

Poisson wavelet transforms have been applied in multi-resolution analysis, system identification, and parameter estimation. They are particularly useful in studying problems in which the functions in the time domain consist of linear combinations of decaying exponentials with time delay.

Wavelet associated with Poisson kernel

Definition

Relation with Poisson kernel

The function

appears as an integral kernel in the solution of a certain initial value problem of the Laplace operator.

This is the initial value problem: Given any

, find a harmonic function

defined in the upper half-plane satisfying the following conditions:

The problem has the following solution: There is exactly one function

satisfying the two conditions and it is given by

where

and where "

" denotes the convolution operation. The function

is the integral kernel for the function

. The function

is the harmonic continuation of

into the upper half plane.

Properties

A class of complex wavelets associated with the Poisson kernel

Definition

The Poisson wavelet is a family of complex valued functions indexed by the set of positive integers and defined by^[4]^[5]

Relation with Poisson kernel

Thus

can be interpreted as a function proportional to the derivatives of the Poisson integral kernel.

Properties

References

1. ^{{cite journal|last1=Karlene A. Kosanovich, Allan R. Moser and Michael J. Piovoso|title=The Poisson wavelet transform|journal=Chemical Engineering Communications| date=1996| volume=146|issue=1|pages= 131–138}}
2. ^{{cite journal|last1=Karlene A. Kosanovich, Allan R. Moser and Michael J. Piovoso|title=A new family of wavelets: the Poisson wavelet transform|journal=Computers in Chemical Engineering|date=1997|volume=21|issue=6|pages=601–620}}
3. ^¹{{cite book|last1=Roland Klees, Roger Haagmans (editors)|title=Wavelets in the Geosciences|date=2000|publisher=Springer|location=Berlin|pages=18–20}}
4. ^¹{{cite book|last1=Abdul J. Jerri|title=The Gibbs Phenomenon in Fourier Analysis, Splines and Wavelet Approximations|date=1998|publisher=Springer Science+Business Media|location=Dordrech|isbn=978-1-4419-4800-7|pages=222–224}}
5. ^{{cite book|last1=Wojbor A. Woyczynski|title=Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets, Volume 1|date=1997|publisher=Springer Science & Business Media|isbn=9780817639242|page=223}}