This comes from a scattering problem. Consider the usual non singular Bessel functions of the first kind, $J_n(x)$. It is known that their zeros are countable, and all zeros are distinct. My question is the following: is there a lower bound to $$ f(N)=\sup_{x>0}\inf_{n\in\mathbb{N},n\leq N}|J_n(x)| $$ It is clear $f(N)\to0$ as $N\to\infty$, so it must depend on $N$.
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$\begingroup$ Is it a correct expression? It means we first take any x, then for this x find inf in n as function of x, and then take sup over x? Or it is better to change sup and inf? $\endgroup$– SergeiCommented Oct 30, 2014 at 12:59
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$\begingroup$ @Sergei It is the correct expression. If you invert the sup and inf, the answer is known: $n\to n^{1/3}\sup_{x>0} |J_n(x)|$ is a monotone function, growing from $0$ to $0.675..$ as $n\to \infty$ from a paper of Landau, J. London Math. Soc. (2) 61 (2000) 197-215. $\endgroup$– usernameCommented Oct 30, 2014 at 15:07
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$\begingroup$ Thank you. What I missed. Take such $x=a$ that $J_1(a)=0$. Then inf mod over all n will be zero, not so? $\endgroup$– SergeiCommented Oct 30, 2014 at 15:52
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The results of the mathematical experiment done with Maple (The DirectSearch package should be downloaded and installed in your Maple.)
f := proc (N) options operator, arrow; DirectSearch:-Search(proc (x) options operator, arrow; min(seq(BesselJ(n, x), n = 1 .. N)) end proc, {0 < x}, maximize) end proc:
seq(evalf(-ln(f(N)[1])/(ln(N)*N)), N = [2, 10, 20, 40, 100, 1000, 10^4, 5*10^4]);
$$.5564145611, .3883486929, .4924980215, .5720147732, .6487478567, .7617480189, .8208958156, .8474981352 $$ suggest the dependence $$\log(f(N))\asymp -N\log(N),\,N\to\infty. $$
