Fourier transform of a rational function with spike at origin

Question

Consider a rational function $f(x) = \frac{p(x)}{q(x)}$, where both $p(x)$ and $q(x)$ are polynomial functions of the multivariate $x = (x_1, x_2,..., x_n) \in \mathbb{R}^n$. Also, let us say that the degree of $p(x)$ is very small compared to the degree of $q(x)$, and all the variables make an appearance at least once in $q(x)$, so that $f(x)$ has sufficiently fast decay outside an open set containing the origin. My question is, does it make sense to talk about the Fourier transform of $f(x)$?

Answer 1

The answer to your question is yes, but it involves deep work of two distinguished mathematicans. As indicated in a comment above, the key notion is that of a tempered distribution due to L. Schwartz. All polynomials are tempered distributions and the Fourier transform can be applied to such distributions. For details see his monograph "Théorie des Distributions". The second ingredient is the deep result of Lojasiewicz which states that you can divide tempered distributions by polynomial. It follows that every rational function is a tempered distribtion. You can find some examples of explicit computations in the literature, e.g., in the multi-volume series by Gelfand and Silov.

Let me add that I am surprised that this has been migrated from MO. Although it is not research level in the sense that it is an old result, it is at a much higher level than many queries on MO which also involve known results.