Poisson summation formula

In summary: Using the properties of exponential functions, we can simplify this as:##f(x) = \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} \gamma_m \delta_{mn}\right) e^{inx} = \sum_{n=-\infty}^{\infty} \gamma_n e^{inx}##where ##\delta_{mn}## is the Kronecker delta function. Therefore, we have shown that ##f(x) =
  • #1
mr.tea
102
12

Homework Statement


let ##g## be a ##C^1## function such that the two series ##\sum_{-\infty}^{\infty} g(x+2n\pi)## and ##\sum_{n=-\infty}^{\infty} g'(x+2n\pi)## are uniformly convergent in the interval ##0\leq x \leq 2\pi ##. Show the Poisson summation formula:

##\sum_{n=-\infty}^{\infty} g(2n\pi) = \sum_{-\infty}^{\infty} \gamma _m##

where ##\gamma _m= \frac{1}{2\pi} \int_{-\infty}^{\infty} g(x)e^{-imx} dx ## is assumed to be convergent.
Hint: The numbers ##\gamma _m## are the Fourier coefficients of the ##2\pi##-periodic function ## u(x)= \sum_{-\infty}^{\infty} g(x+2n\pi)##

Homework Equations

The Attempt at a Solution


I have tried to use the hint, but arrived nowhere. Also, I am not sure why do I need the differentiated series ##\sum_{-\infty}^{\infty} g'(x+2n\pi)## ...

Thank you.
 
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  • #2


Dear student,

Thank you for your question. The Poisson summation formula is a powerful tool in mathematical analysis and has wide applications in various fields such as physics, engineering, and statistics. It relates the values of a function at integer points to the values of its Fourier coefficients. In this case, we are given that the two series ##\sum_{-\infty}^{\infty} g(x+2n\pi)## and ##\sum_{n=-\infty}^{\infty} g'(x+2n\pi)## are uniformly convergent in the interval ##0\leq x \leq 2\pi##. This means that the functions ##g(x+2n\pi)## and ##g'(x+2n\pi)## are well-behaved and do not have any large fluctuations or oscillations in this interval.

Now, the hint given in the problem suggests that we consider the function ##u(x) = \sum_{-\infty}^{\infty} g(x+2n\pi)##, which is a ##2\pi##-periodic function. This means that ##u(x+2\pi) = u(x)## for all values of ##x##. We can also express this function in terms of its Fourier series as:

##u(x) = \sum_{m=-\infty}^{\infty} \gamma_m e^{imx}##

where the coefficients ##\gamma_m## are given by ##\gamma_m = \frac{1}{2\pi} \int_{-\pi}^{\pi} u(x) e^{-imx} dx##.

Now, let us consider the function ##f(x) = \sum_{-\infty}^{\infty} g(2n\pi) e^{inx}##, which is also a ##2\pi##-periodic function. We can write this as:

##f(x) = \sum_{n=-\infty}^{\infty} g(2n\pi) e^{inx} = \sum_{n=-\infty}^{\infty} \left(\sum_{m=-\infty}^{\infty} \gamma_m e^{im(2n\pi)}\right) e^{inx} = \sum_{n=-\infty}^
 

What is the Poisson summation formula?

The Poisson summation formula is a mathematical formula that relates the Fourier transform of a function to its Fourier series. It allows for the conversion of a continuous function into a discrete series and vice versa.

Why is the Poisson summation formula important?

The Poisson summation formula has many practical applications in various fields such as physics, engineering, and signal processing. It allows for the simplification and analysis of complex functions and makes it easier to solve certain problems in these fields.

How is the Poisson summation formula derived?

The Poisson summation formula can be derived from the properties of Fourier transforms and series. It involves using the Dirac delta function and the periodicity of the Fourier transform to convert between continuous and discrete representations of a function.

What are some key properties of the Poisson summation formula?

The Poisson summation formula is linear, meaning that it can be applied to combinations of functions. It also has an inverse formula, allowing for the conversion between discrete and continuous representations of a function. Additionally, it has a convolution property, making it useful for solving convolution problems.

Are there any limitations to the Poisson summation formula?

The Poisson summation formula is limited to functions that are periodic or can be extended to be periodic. It also assumes that the function and its Fourier transform are well-behaved and have certain properties, such as being continuous and having a finite number of discontinuities.

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