Definition of Fourier Transform on $\frac{2\pi}{L}\mathbb{Z}^{d}$
The usual inversion formula for the Fourier transform is: $$f(x) = \frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}e^{ip\cdot x}\hat{f}(p)dp$$ What is the analogous formula when $p$ ranges in a infinite lattice $\frac{2\pi}{L}\mathbb{Z}^{d}$, where $L>1$ is fixed? I know the integral becomes a sum, but what about the pre-factor $(2\pi)^{-d}$?
Solution 1:
The term $2 \pi$ will be included but it usually is inside the coefficients of the series. See this lecture.
http://www.math.toronto.edu/courses/apm346h1/20151/AJ.html