# Approximating a Fourier transform

Suppose the Fourier transform $$\hat{f}(k)$$ (with $$k \in \mathbb{R}^d$$) is given, and one intends to get some information about its position-space counterpart $$f(x)$$. When the analytical calculation of the inverse Fourier transform of $$\hat{f}(k)$$ is not possible, one may still be able to extract useful information by specializing to specific regions of $$k$$ space; for instance, in statistical physics, it is often customary to study the "macroscopic" properties of, e.g., correlation functions, by examining the $$k\to 0$$ limit of their Fourier transforms. It appears to me that such a process is somewhat analogous to looking at the Taylor series of a Fourier transform, i.e., $$$$\hat{f}(k) = \hat{f}\big\rvert_{k=0} + k \partial_k\hat{f}\big\rvert_{k=0} + \ldots$$$$ If one truncates this series and then tries to perform on it the inverse Fourier transformation, $$\int \frac{dk}{2\pi} e^{ikx} \hat{f}_{\rm trunc}(k),$$ in some cases one might find that the result diverges as $$k\to\infty$$. However, in many theories, and especially in field theories, there is an upper cutoff for $$k$$ which determines the range of validity of that theory; such a cutoff often resolves the possible divergence of the inverse Fourier transform.

Question Does the position-space function that is obtained from the inverse transformation of the truncated Taylor series $$\hat{f}_{\rm trunc}$$, with some cutoff $$\Lambda$$, approximate the original function $$f(x)$$ in any sense? otherwise, is there a systematic way of obtaining such an approximate form from its Fourier transform $$\hat{f}(k)$$?

When you truncate the Taylor expansion around $$0$$, you are saying that you are interested in modes with long wavelength. These are often the modes that are long-lived, so that for long times they will approximately describe your system. In spirit, it is like doing a coarse graining: you forget about the fast microscopic dynamics and retain only macroscopoic information. In a more rigorous sense, one has $$|| \mathcal{F}^{-1} [\hat f_{trunc}](x) - f(x) ||_2 = || \hat f_{trunc}(k) - \hat f (k) ||_2$$, so if the approximation of your fourier transform is good in the $$L^2$$ sense so it will be the approximation of the position space $$f(x)$$.
• Thank you for your answer. I understand your point about the long-lived modes (having a background in stat phys). In fact, I think my question can be cast as when does the approximation in the Fourier space gets better by including more terms in the Taylor expansion. I also don't have an understanding of 'a better approximation in $L^2$ sense'. – SaMaSo Jan 15 at 18:47