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 $f_2$?

  • $\begingroup$ Is that $\widehat{f}_j$ in the examples? $\endgroup$ – Christian Remling Oct 8 '16 at 2:13
  • $\begingroup$ @ChristianRemling: It is the Fourier transform of the first of these functions, $f_1(t)$. This is the 'triangle function' with the vertices $(\pm 1, 0)$ and $(0,1)$, so it meets the three conditions (and is extremal). As for $\widehat{f_2}$, it of course doesn't count as its Fourier transform isn't non-negative. $\endgroup$ – Vesselin Dimitrov Oct 8 '16 at 2:16

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 $[-1,1]$.

  • 1
    $\begingroup$ Christian, this trick seems to work in high dimensions as well. Does it give you interesting bounds? $\endgroup$ – Paata Ivanishvili Oct 8 '16 at 13:40
  • 2
    $\begingroup$ @PaataIvanisvili: That's right. I think the general version is that if (in a similar situation) $f(0)\le c_d 2^{-d}$, with $c_d$ being the volume of the unit ball, then again $f=g*g$, $g=\chi_{B_{1/2}}$. I don't know if that's interesting. $\endgroup$ – Christian Remling Oct 9 '16 at 21:17

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