Petter Brändén: Strip preservers and Fourier transforms with only real zeros.

Abstract: The topic of this talk is polynomials and transcendental entire functions with all zeros in a strip. Two questions are addressed: 1). Which operators preserve the property of having zeros in a prescribed strip (or even better shrink the strip width)? 2). Which Fourier transforms have only real zeros? The two questions are intertwined as revealed by work by Pólya and de Bruijn. We give a complete answer to the first question for complex polynomials and entire functions. The question for real functions is considerably harder, but we are able to show that a large class of operators shrink the strip width, thus extending results of Pólya and de Bruijn. The progress on the first question leads to progress on the second. We extend a theorem of de Bruijn and Ilieff and produce a large class of Fourier transforms with only real zeros. This is joint work with Matthew Chasse (KTH).