# Lower Bound for Fourier Transform Expression

Let $$f,g \in L_2(\mathbb{R})$$, and let $$\hat{f}$$ and $$\hat{g}$$ denote their Fourier Transforms. I'm trying to find a lower bound for this sum:

$$\left(\int_{-\infty}^{\infty} x^2 \, f^2(x) \, dx \int_{-\infty}^{\infty} k^2 \, \widehat{g}^2(k) \, dk \right)+ \left(\int_{-\infty}^{\infty} k^2 \, \hat{f}^2(k) \, dk \int_{-\infty}^{\infty} x^2 \, g^2(x) \, dx \right).$$

(It comes up in a problem on PDE's I'm working on). I've written the same thing below using the $$L_2$$ norm:

$$\left(|| xf || \cdot ||k \hat{g}||\right) + \left(||k\hat{f}|| \cdot || xg|| \right)$$

My initial guess was to use the Uncertainty Principle, but that only provides a lower bound for the product of $$|| xf ||$$ and $$|| k \hat{f} ||$$, which doesn't seem to help here. Are there any identities about the Fourier Transform that might help here? Perhaps, identities about $$\int x^2 f^2 + \int k^2 \hat{f}^2$$? Any help/suggestions would be greatly appreciated. Thanks!

For any $$h\in L^2(\mathbb R)$$, let $$D(h)=\int_{\mathbb R} u^2|h(u)|^2du$$.

You're looking for a lower bound for $$D(f)D(\hat g)+D(\hat f )D(g)$$. It's worth noting that, without any constraint on $$f$$ and $$g$$, the lower bound is $$0$$ since you can plug in $$f=0$$ or $$g=0$$. In fact the whole expression scales linearly with $$\|f\|$$ or with $$\|g\|$$. So let's assume that we've normalized $$f$$ and $$g$$ so that $$\|f\|=\|g\|=1$$.

By the Uncertainty Principle, there is a universal constant $$C>0$$ such that for all $$h\in L^2(\mathbb R)$$ with $$\|h\|=1$$, $$D(h)D(\hat h)\geq C$$

The exact value of $$C$$ depends on the normalization convention taken for the Fourier transform (it can be $$\frac 1 {16\pi^2}$$ or $$\frac 1 {4\pi}$$). The equality is attained for Gaussian functions.

With this, $$D(f)D(\hat g)+D(\hat f )D(g)\geq C\left(\frac{D(f)}{D(g)}+\frac{D(g)}{D(f)}\right)$$

The right-hand side is minimal iff $$D(f)=D(g)$$. In that case, the expression to minimize is equal to $$2D(f)D(\hat f)$$, which, by the Uncertainty Principle, is lower-bounded by $$2C$$, with equality attained by Gaussian functions.

We conclude that the lower bound is 2C, with equality attained when $$f$$ and $$g$$ are both Gaussian functions.

• Wow! That's brilliant - thanks so much. Appreciate the help :) – chaad Sep 14 '20 at 2:16
• You're too kind! – Stefan Lafon Sep 14 '20 at 2:17