Poisson summation and Parsevals identity

[broegger]

I've heard something about Poisson summation in relation to Fourier analysis, but I can't seem to find any good info on the subject... Can anyone explain what "Poisson summation" is?

Furthermore, I would like to know exactly what "Parsevals identity" states and how it is applied.

Thanks.

 

[matt grime]

These kinds of indentities relate the coefficients of the series to the function, usually in terms of integrals.

I can't recall the exact wording of hte relevant theorems, but if you go to

www.dpmms.cam.ac.uk

go to people, find tom koerner's link and onhis personal web page shld be some very good notes on Fourier analysis

 

[M98Ranger]

Poisson summation is a mathematical formula that relates the Fourier transform of a periodic function to the sum of the function's values at integer multiples of the period. In other words, it allows us to express a function in terms of its Fourier series, which is a sum of sines and cosines of different frequencies. This is useful in Fourier analysis because it allows us to simplify the calculation of Fourier transforms and understand the behavior of a function in the frequency domain.

Parseval's identity, on the other hand, is a theorem that states the total energy of a function in the time domain is equal to the total energy of its Fourier transform in the frequency domain. In other words, it relates the integral of a function squared to the integral of its Fourier transform squared. This is useful in many fields, including signal processing and physics, where it allows us to analyze the energy distribution of a signal or system.

Both Poisson summation and Parseval's identity are important tools in Fourier analysis and are used to simplify calculations and gain insight into the properties of functions and signals. I hope this helps clarify these concepts for you.