# Heuristic for large $x$ behavior from small $q$ behavior of Fourier Transform

If I have a function $$h(\mathbf x)$$ which may be written

$$h(\mathbf x)= \int \frac{\text{d}^d\mathbf q}{(2\pi)^d} \, h(\mathbf q) e^{-i \mathbf q \cdot \mathbf x}$$

and assume spherical symmetry, is there way to determine the large $$x$$ behavior of $$h(x)$$ from the small $$q$$ behavior of $$h(q)$$ without computing the integral? Assume nice "physics" functions-- and further assume, if useful, that $$h(x)$$ has power-law or exponential asymptotics.

Motivation:

The motivation for this question is understanding the real-space asymptotics of various propagators/ correlators in QFT and statistical field theory.

Guesses:

Suppose for instance $$h(q) \sim q^{-k}$$ as $$q\to 0$$. Then the RHS scales as $$q^{d-k}$$ for small $$q$$ and I may expect that $$h(x) \sim x^{k-d}$$ for large $$x$$. I expect this doesn't work in general, since $$\frac{1}{q^2+m^2}$$ Fourier transforms to an exponential. For a better answer, perhaps there is a way to expand $$h(q)$$ in Laurent series near $$q=0$$ and Fourier transform term-by-term... and then attempt to sum the series to see exponential behavior?

If you know the derivatives of $$h(q)$$, you can exponentiate it to $$S_{d-2}\int_0^\pi d\theta\,\sin^{d-2}\theta\int \frac{d q}{(2\pi)^d}\, q^{d-1}\exp\left(\log h(q) - i qx\cos\theta\right)\,,$$ where $$S_{d-2}$$ is the volume of the $$d-2$$-dimensional sphere. After performing the integral in $$d\theta$$ (which doesn't require the knowledge of $$h(q)$$), you'll obtain an expression of the form $$I(x)=\int dq\,F(q) \,\exp\left(G(q) - i q x\right)\,.$$ At this point there is a systematic procedure for approximating this integral, which is called either Stationary phase approximation or Method of steepest descent.
At the lowest order you just have to expand the exponential to quadratic order around the point that has zero first derivative. Then you evaluate $$F(q)$$ at this point and evaluate the Gaussian integral.
\begin{aligned} I(x) &\sim \int dq F(q_0) \exp\left(G(q_0) - i q_0 x + \tfrac12 (q-q_0)^2 G''(q_0) \right) = \\&= F(q_0) \sqrt{\frac{2\pi}{x\, G''(q_0)}}\exp\left(G(q_0) - i q_0 x\right)\,. \end{aligned} This may not be exactly what you are looking for, but it's an idea.