Tao Qian, University of Macau, Macau

Boundary phase behavior of Blaschke products

 

Abstract

It is well known that the phase derivative of a Möbius transform on the boundary is, by module a multiple constant, the Poisson kernel. Therefore, it, in particular, has the positivity property. While finite Blaschke products are trivial, the talk will show that this positive phase derivative property can be extended to infinite Blaschke products through the Wolff-Julia-Caratheodory theorem in 1930's with the formulation of non tangential boundary limit. The talk then includes a quick survey on impacts of this result to signal analysis, especially to DSP (digital signal processing). The talk finishes with providing information of latest studies on finding analytic functions with positive boundary phase derivatives.