Georgia Institute of Technology College of Sciences

In this talk we discuss some nonlinear transformations between moment sequences. One of these transformations is the following: if (a_n)_n is a non-vanishing Hausdorff moment sequence then the sequence defined by 1/(a_0 ... a_n) is a Stieltjes moment sequence. Our approach is constructive and use Euler's idea of developing q-infinite products in power series. Some others transformations will be considered as well as some relevant moment sequences and analytic functions related to them. We will also propose some conjectures about moment transformations defined by means of continuous fractions.

Using integral formulas based on Green's theorem and in particular a lemma of Uchiyama, we give simple proofs of comparisons of different BMO norms without using the John-Nirenberg inequality while we also give a simple proof of the strong John-Nirenberg inequality. Along the way we prove the inclusions of BMOA in the dual of H^1 and BMO in the dual of real H^1. Some difficulties of the method and possible future directions to take it will be suggested at the end.

It is well-known that every Schur function on the bidisk can be written as a sum involving two positive semidefinite kernels. Such decompositions, called Agler decompositions, have been used to answer interpolation questions on the bidisk as well as to derive the transfer function realization of Schur functions used in systems theory. The original arguments for the existence of such Agler decompositions were nonconstructive and the structure of these decompositions has remained quite mysterious. In this talk, we will discuss an elementary proof of the existence of Agler decompositions on the bidisk, which is constructive for inner functions. We will use this proof as a springboard to examine the structure of such decompositions and properties of their associated reproducing kernel Hilbert spaces.