The Riemann Ξ-function under repeated differentiation

Differentiation causes the small gaps between zeros of a given real entire function with order 1 to become larger and the larger gaps to become smaller. In this article, we show that for the Riemann Ξ-function, there exist 〈An〉 and 〈Cn〉 with Cn→0 such thatlimn→∞AnΞ(2n)(Cnz)=cosz uniformly on compact subsets of C. With our method, one can prove the same result for the analogues of the Riemann Ξ-function from automorphic L-functions. For some other Fourier transforms, we have the similar results as the Riemann Ξ-function.