Georgia Institute of Technology College of Sciences

This talk discusses exponential frames and Riesz sequences in L^2 over a set of finite measure. (Roughly speaking, Frames and Riesz sequences are over complete bases and under complete bases, respectively). Intuitively, one would assume that the frequencies of an exponential frame can not be too sparse, while those of an exponential Riesz sequence can not be too dense. This intuition was confirmed in a very general theorem of Landau, which holds for all bounded sets of positive measure. Landau's proof involved a deep study of the eigenvalues of compositions of certain projection operators. Over the years Landaus technique, as well as some relaxed version of it, were used in many different setting to obtain results of a similar nature. Recently , joint with A. Olevskii, we found a surprisingly simple approach to Landau's density theorems, which provides stronger versions of these results. In particular, the theorem for Riesz sequences was extended to unbounded sets (for frames, such an extension is trivial). In this talk we will discuss Landau's results and our approach for studying questions of this type.

In this talk I will describe an Lp theory for outer measures, which could be used to connect two themes of Lennart Carleson's work: Carleson measures and time frequency analysis. This is joint work with Christoph Thiele.

In 1980, T. M. Wolff has given the following version of the ideal membership for finitely generated ideals in $H^{\infty}(\mathbb{D})$: \[\ensuremath{\mbox{If \,\,}\left\{ f_{j}\right\} _{j=1}^{n}}\subset H^{\infty}(\mathbb{D}),\, h\in H^{\infty}(\mathbb{D})\,\,\mbox{and }\]\[\vert h(z)\vert\leq\left(\underset{j=1}{\overset{n}{\sum}}\vert f_{j}(z)\vert^{2}\right)^{\frac{1}{2}}\,\mbox{for all \ensuremath{z\in\mathbb{D},}}\]then \[h^{3}\in\mathcal{I}\left(\left\{ f_{j}\right\} _{j=1}^{n}\right),\,\,\mbox{the ideal generated by \ensuremath{\left\{ f_{j}\right\} _{j=1}^{n}}in \ensuremath{H^{\infty}}\ensuremath{(\mathbb{D})}. }\]In this talk, we will give an analogue of the Wolff's ideal problem in the multiplier algebra on weighted Dirichlet space. Also, we will give a characterization for radical ideal membership.