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., \begin{equation} \hat{f}(k) = \hat{f}\big\rvert_{k=0} + k \partial_k\hat{f}\big\rvert_{k=0} + \ldots \end{equation} 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 $f(x)$.

  • $\begingroup$ 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'. $\endgroup$ – SaMaSo Jan 15 at 18:47

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