On a problem of H. Shapiro (Q596821)

Let \(\mu\) be a real measure on the real line. Assume that its Poisson integral \(M(z)\), \(z\in \mathbb C\setminus \mathbb R\) satisfies \[ | M(x+iy)| \leq Ae^{-cy^\alpha},\;\;y\to \infty, \] for some constants \(A,c>0\) and \(0<\alpha\leq 1\). It is known that this estimate is equivalent (when \(\mu\) is absolutely continuous) with the condition that the support of the Fourier transform of \(\mu\) is disjoint from the interval \((-c,c)\). The authors show that for \(1/2<\alpha\leq 1\), the measure \(\mu\) must have many sign changes on both positive and negative rays. For \(0<\alpha\leq 1/2\), this is true for at least one of the rays, and not always true for both rays. The authors provide examples that show that their asymptotic bounds for the number of sign changes are sharp.