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. In another context, the term refers to a certain wavelet which involves a form of the Poisson integral kernel. 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.
For each positive integern the Poisson wavelet is defined by 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 t and, for each non-negative integern, let Prob = pn. Then we have The Poisson wavelet is now given by
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 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 in the time domain, the function f can be reconstructed as follows:
The Poisson wavelet is defined by the function This can be expressed in the form
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 in, 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
The "waviness" of the function follows from
The Fourier transform of is given by
The admissibility constant is
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
Relation with Poisson kernel
The function can be expressed as an n-th derivative as follows: Writing the function in terms of the Poisson integral kernel as we have Thus can be interpreted as a function proportional to the derivatives of the Poisson integral kernel.
