Power series, uncertainty inequalities and real zeros of Fourier transforms (Q1917644)

Let \(f(x)\) be a summable function and \(\widehat f(u)\) its Fourier transform. In 1940, A. Beurling claimed that \(f\) and \(\widehat f\) cannot both decay at infinity ``so quickly'' that the following double integral \[ \int^\infty_{-\infty} \int^\infty_{-\infty} |f(x)\widehat f(u)|e^{|xu|} dx du \] converges. A statement of this kind is an instance of an uncertainty inequality. In the paper, the author explores another kind of uncertainty inequality which is related, but not equivalent, to Beurling's result. Let us consider the even function \(f(x)\) that decays rapidly on the real line together with \(\widehat f(u)\), \[ |f(x)|\leq e^{-\alpha|x|^p};\;p>1,\;\alpha>0,\quad |\widehat f(u)|\leq e^{-\beta|u|^q};\;q> 1,\;\beta>0.\tag{1} \] Theorem. If a function \(f\) satisfies (1) with \(1/p+1/q<1\) then it is identically zero. Conjecture. Let \(f(z)\) be an entire function, strongly critical with \(p>2\), monotone decreasing on the positive real axis, even and positive on the whole real axis. Then \(\widehat f\) has infinitely many real zeros.