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 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.
Wavelets associated with Poisson probability distribution
Definition
For each positive integer n the Poisson wavelet ψ n ( t ) {\displaystyle \psi _{n}(t)} is defined by
ψ n ( t ) = { ( t − n n ! ) t n − 1 e − t for t ≥ 0 0 for t < 0. {\displaystyle \psi _{n}(t)={\begin{cases}\left({\frac {t-n}{n!}}\right)t^{n-1}e^{-t}&{\text{ for }}t\geq 0\\0&{\text{ for }}t<0.\end{cases}}}
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) = pn(t). Then we have
p n ( t ) = t n n ! e − t . {\displaystyle p_{n}(t)={\frac {t^{n}}{n!}}e^{-t}.}
The Poisson wavelet ψ n ( t ) {\displaystyle \psi _{n}(t)} is now given by
ψ n ( t ) = − d d t p n ( t ) . {\displaystyle \psi _{n}(t)=-{\frac {d}{dt}}p_{n}(t).}
Basic properties
ψ n ( t ) {\displaystyle \psi _{n}(t)} is the backward difference of the values of the Poisson distribution:
ψ n ( t ) = p n ( t ) − p n − 1 ( t ) . {\displaystyle \psi _{n}(t)=p_{n}(t)-p_{n-1}(t).}
The "waviness" of the members of this wavelet family follows from
∫ − ∞ ∞ ψ n ( t ) d t = 0. {\displaystyle \int _{-\infty }^{\infty }\psi _{n}(t)\,dt=0.}
The Fourier transform of ψ n ( t ) {\displaystyle \psi _{n}(t)} is given
Ψ ( ω ) = − i ω ( 1 + i ω ) n + 1 . {\displaystyle \Psi (\omega )={\frac {-i\omega }{(1+i\omega )^{n+1}}}.}
The admissibility constant associated with ψ n ( t ) {\displaystyle \psi _{n}(t)} is
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