Let $f(z)$ be an entire function defined by $f(z)=\prod_{n=1}^{\infty}\bigg(1-\frac{z^{2}}{a_{n}^{2}}\bigg),\qquad z\in \mathbb C$ where $\{a_{n}\}_{n=1}^{\infty}$ is a sequence of positive real numbers, determined so that the infinite product above defines an entire function. How can we compute the integral $\int_{-\infty}^{\infty}|f(x)|^{2}dx$ where $x$ is real. Or at least finding an upper bound for it (if it is finite)?
$L^{2}(\mathbb R)$- norm of entire function
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real-analysis
complex-analysis
convergence-divergence
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3At least, nobody here can think of a method. Which doesn't mean there isn't one. A couple of observations: As noted by Robert Israel, the convergence of the product has to do with the convergence of a sum, i.e., it has to do with the asymptotic behaviour of the sequence $(a_n)$. But by adding just one more factor to timur's example, I can destroy the finiteness of the integral, which shows that this does not relate well to asymptotic behaviours of $(a_n)$. I think this is likely a quite hard problem. – 2012-08-16
1 Answers
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I searched infinite products, http://mathworld.wolfram.com/InfiniteProduct.html. Consider $cos(x)=\prod_{n=1}^\infty \left(1-\frac{4x^2}{\pi^2(2n-1)^2}\right).$ Then $\int_{-\infty}^\infty |cos(x)|^2 dx=\infty.$ So I don't think an upper bound can be found.