2
votes
0answers
25 views

Poisson summation formula for the Casimir effect

I'm studying the Casimir Effect at finite temperature. To calculate the Helmoltz free energy in the canonical ensemble I need to sum a particular series. In some scientific papers it is suggested to ...
6
votes
1answer
80 views

How to make sense of Fourier series for a distribution?

In particular if I have an array of numbers say, $\{c_m\}_{m\in\mathbb{Z}^n}$. Under what conditions can we say that these are the Fourier coefficients of a distribution? [For examples Bessel's ...
1
vote
0answers
46 views

What is the difference between these two kernel definitions?

I am reading my graft and the document of David Haussler about Convolution Kernels on Discrete Structures, UCSC-CRL-99-10. My graft and the other document The terminology seems to differ. The ...
2
votes
0answers
199 views

Is there a particular meaning to the sum of Fourier coefficients $a_{n^2}$?

The formula $$\sum_{n=-\infty}^{+\infty} e^{-in^2x}$$ does not converge in any function space but it is perfectly valid in $\mathcal{D}'(\mathbb{R})$. When applied on a test function $\psi(x) = ...
2
votes
2answers
688 views

Sum over cosines = dirac delta - how to get the coefficients?

Given this formula: $$\sum\limits_{n=0}^\infty a_n \cos(n \pi x / d) = \delta(x-x_0)$$ Where $0 \leq x \leq d$. How can one calculate the coeffciients $a_n$? I googled and searched all kinds of ...
1
vote
1answer
447 views

What is the sum of only half the exponential terms that give the Dirac comb?

The following infinite sum of exponential terms gives a Dirac comb: $$ \sum_{n=-\infty}^\infty e^{i n x} = 2 \pi \sum_{n=-\infty}^\infty \delta(x - 2 \pi n) $$ Of course the sum doesn't strictly ...
4
votes
2answers
137 views

Function $f$ such that Fourier-series converges uniformly, but the series of the derivatives are divergent

I am studying Fourier-transformation right now, and I am asking if there exists a function $f$ such that is Fourier-series converges uniformly, the Fourier-series of $f'$ only in $L_2$ and that $f''$ ...