Continuous or analytic functions with this property of sinc function This question is motivated by my previous post in SE (math.stackexchange.com).
Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere), or continuous,
function, $f(x)$ on $\mathbb{R}$ such that $$\int_{-\infty}^\infty f(x) dx=\int_{-\infty}^\infty f(x)^2 dx=\sum_{-\infty}^\infty f(n) =\sum_{-\infty}^\infty f(n)^2 $$

Consider the function $f:\mathbb R\to\mathbb R$ which vanishes outside of $[-1,1]$ and such that for all $x \in [-1,1]$ has $$
f(x) = \frac{5 x^4}{4}+\frac{1}{2}
   \sqrt{\frac{37}{6}} x^3-\frac{9
   x^2}{4}-\frac{1}{2} \sqrt{\frac{37}{6}}
   x+1.
$$
Then $$\int_\mathbb Rf(x)\,\mathrm dx=\int_\mathbb Rf(x)^2\,\mathrm dx=\sum_{n\in\mathbb Z}f(n)=\sum_{n\in\mathbb Z}f(n)^2=1.$$
Moreover, the functions with the same support which on $[-1,1]$ coincide with the polynomials $$
\begin{array}{l}
 -\frac{1}{32} (x-1)^2 (x+1) \left(x
   \left(\left(11+\sqrt{3245}\right)
   x+\sqrt{3245}-29\right)-32\right); \\\\
 \frac{(x-1)^3 (x+1) \left(x \left(7
   \left(\sqrt{2657109}-923\right) x+5
   \sqrt{2657109}-8431\right)-3392\right)}
   {3392}; \\\\
 -\frac{1}{64} (x-1)^4 (x+1) \left(\left(3
   \sqrt{3855}-175\right) x^2+2
   \left(\sqrt{3855}-98\right) x-64\right);
   \\\\
 \frac{(x-1)^5 (x+1) \left(3
   \left(\sqrt{3040433}-2550\right) x^2+2
   \left(\sqrt{3040433}-4571\right)
   x-3008\right)}{3008}; \\\\
 -\frac{(x-1)^6 (x+1) \left(x \left(25
   \left(3 \sqrt{622687}-6365\right) x+51
   \sqrt{622687}-202885\right)-67328\right
   )}{67328};
\end{array}
$$ have the same properties and are $C^1$, $C^2$, $C^3$, $\dots$, respectively, and it seems one can go on indefinitely. There are no other polynomials satisfying the conditions and of degree less than or equal to theirs. The unique $C^k$ polynomial of degree $4+k$ satisfying the conditions is the product of $(x-1)^{k+1}(x+1)$ and a degree $2$ factor: one should be able to describe this last factor explicitly somehow...
