Extremal functions for the 'packing density in dimension one'

Question

The $n = 1$ case of Theorem 3.1 of Cohn and Elkies's paper New upper bounds on sphere packings I amounts to the inequality $f(0) \geq 1$ for all ('admissible') functions $f$ on $\mathbb{R}$ satisfying

• $\widehat{f}(0) = 1$;
• $f \leq 0$ outside of $[-1,1]$; and
• $\widehat{f} \geq 0$ everywhere.

I am interested in the functions which attain the equality: $f(0) = \widehat{f}(0) = 1$. Can anything (everything?) be said about those functions, or even more particularly about the ones among them having the second condition strengthened to $\mathrm{supp}(f) \subset [-1,1]$?

Cohn and Elkies give $f_1(t) = \Big( \frac{\sin{\pi t}}{\pi t} \Big)^2$ and $f_2(t) = \frac{1}{1-t^2}\Big( \frac{\sin{\pi t}}{\pi t} \Big)^2$ as examples of extremal functions. Are there additional examples besides the convex hull of $f_1, \widehat{f_1}$ and $f_2$?

Answer 1

You didn't make the precise assumptions on $f$ explicit, but even in a fairly general situation, your $\widehat{f}_1$ is the only example that is supported by $[-1,1]$.

Given an $f$ with the stated properties, we can let $\widehat{g}=\widehat{f}^{1/2}$, so $f=g*g$, and if $\textrm{supp}\, f\subseteq [-1,1]$, then Titchmarsh's convolution theorem shows that $g$ is supported by $[-1/2,1/2]$. Since we're dealing with real valued functions $f,\widehat{f}$ here, they are even, and thus $$ 1=f(0) = \int_{-1/2}^{1/2} g(x)g(-x)\, dx = \int_{-1/2}^{1/2} g(x)^2\, dx \ge \left(\int_{-1/2}^{1/2} g(x)\, dx\right)^2 = \widehat{g}(0)^2=1 . $$ Equality in the Cauchy-Schwarz inequality means that the functions $1,g$ are linearly dependent, so $g=1$ (or $=-1$), and this gives us the triangular function $f(x)=1-|x|$ as the only possible example supported by $[-1,1]$.