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 $f_2$?

Added: More particularly, I am interested in maximizing $\int_{\mathbb{R}} f(t) \log{\frac{1}{|t|}} \, dt$ among all extremal ($f(0) = \widehat{f}(0) = 1$) functions meeting the stated Cohn-Elkies conditions.