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In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform.
In probability theory and statistics, the Poisson distribution (/ ˈ p w ɑː s ɒ n /) is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event. [1]
These samples can be deduced from one cycle of a periodic function which has Fourier series coefficients proportional to those samples by the Poisson summation formula: = (+) = = ^ (), The integrability of f {\displaystyle f} ensures the periodic summation converges.
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
This list of mathematical series contains formulae for finite and infinite sums. It can be used in conjunction with other tools for evaluating sums. It can be used in conjunction with other tools for evaluating sums.
In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of discrete values. The DTFT is often used to analyze samples of a continuous function. The term discrete-time refers to the fact that the transform operates on discrete data, often samples whose interval has units of time.
In mathematics, a Voronoi formula is an equality involving Fourier coefficients of automorphic forms, with the coefficients twisted by additive characters on either side. It can be regarded as a Poisson summation formula for non-abelian groups .
[1] [2] That identity is a form of the Poisson summation formula. Similarly, a Fourier series whose coefficients are samples of s ( t ) {\displaystyle s(t)} at constant intervals ( T ) is equivalent to a periodic summation of S ( f ) , {\displaystyle S(f),} which is known as a discrete-time Fourier transform .