Are there other analytic functions with this property of sinc function?

Question

This question is motivated by my previous post about sinc function.

Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire (i.e. analytic everywhere) function $f(x)$ on $\mathbb{R}$ such that $$\int_0^\infty f(x) dx=\int_0^\infty f(x)^2 dx$$ or $$\int_{-\infty}^\infty f(x) dx=\int_{-\infty}^\infty f(x)^2 dx.$$

If $f$ is required only to be continuous, then other examples are possible, e.g. the even extension of the following function: $$ f(x)=\left\{\begin{array}{ll} -2(5+\sqrt{65})x^2+(7+\sqrt{65})x-1 & 0\le x\le \frac{1}{2}\\ 2(5+\sqrt{65})x^2-(13+3\sqrt{65})x+4+\sqrt{65} & \frac{1}{2}\le x\le 1\\ \frac{1}{x^2} & x\ge 1 \end{array}\right. $$

As commented below, it turns out that there are easy answers to the above question. AD also showed a function below that also satisfies $$ \sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty f(n)^2=0. $$ In view of these answers, my question is now revised to:

Prove or disprove that $\frac{\sin x}{x}$ is the only nonzero entire function, $f(x)$ on $\mathbb{R}$ such that $$\int_{-\infty}^\infty f(x) dx=\int_{-\infty}^\infty f(x)^2 dx=\sum_{-\infty}^\infty f(n) =\sum_{-\infty}^\infty f(n)^2 $$

Answer 1

Similar to the suggestion of Zaricuse, that is take an entire function $f$ such that $\int_{-\infty}^\infty f(x) dx \ne0$ then solve for $$\int_{-\infty}^\infty af(x)dx = \int_{-\infty}^\infty (af(x))^2dx$$ Then $g(z)=af(z)$ solves half of the problem. To reach $$\sum g(n)=\sum g(n)^2$$ we may for example start with $f(z)=\sin (\pi z) \cdot h(z)$ where $h$ is an other integrable entire function.

Answer 2

In the spirit of Zaricuse's solution without the sum requirement, take any sufficiently well behaved functions f and g. Then you should be able to find a linear combination af+bg that satisfies both equations. If you let

$$if=\int_{-\infty}^\infty f(x) dx$$ $$if2=\int_{-\infty}^\infty f(x)^2 dx$$ $$sf=\sum_{n=1}^\infty f(n)$$ $$sf2=\sum_{n=1}^\infty f(n)^2$$

and similarly for fg and g, we have

$$a*if + b*ig=a^2*if2+2ab*ifg+b^2ig2$$ and
$$a*sf + b*sg=a^2*sf2+2ab*sfg+b^2sg2$$

which can be solved for a and b in most cases.

Added In response to the new request that the values of the two integrals and two sums all match, I just need enough knobs to turn. Define $g(k,x)=\exp(-kx^2)$ and take $f(x)=g(1,x)+ag(2,x)+bg(3,x)+cg(4,x)$ The nice thing about this $f$ is that $f^2$ is written in terms of $g(k,x)$, though k goes up to 8. The integral of $g(k,x)$ is just $\sqrt{\frac{\pi}{k}}$ and the sum is calculated by Wolfram Alpha as $\vartheta_3(0,\exp(-k))$. We can make a table:

$$\begin{array}{ccc}k&\int g(k,x)&\sum g(k,x)\\1&1.772453851&1.77264\\2&1.253314137&1.27134\\3&1.023326708&1.09959\\4&0.886226925&1.03663\\5&0.79266546&1.01348\\6&0.723601255&1.00496\\7&0.669924586&1.00182\\8&0.626657069&1.00067\end{array}$$

So the integral of f is $\sqrt{\pi}(1+a/\sqrt{2}+b/\sqrt{3}+c/\sqrt{4})$ The integral of f^2 is $\sqrt{\pi}(1/\sqrt{2}+2a/\sqrt{3}+(a^2+2b)/\sqrt{4}+(2c+2ab)/\sqrt{5}+(b^2+2ac)/\sqrt{6}+2bc/\sqrt{7}+c^2/\sqrt{8})$ with similar expressions for the sum in terms of theta3. We want to find a,b,c so that the integrals and sums all match. Unless my matrix of coefficients has a very unlikely dependence it will be available. Excel claims $a=-3.782590725, b=4.503400057, c=-1.83137936$ is very close to a solution, and there should be more.

Answer 3

The revised question has been answered in this post at MO. In particular, it was shown that $\frac{\sin ax}{ax}$ satisfies this equality for each $0<a\le \pi$.