Druhý seminář z matematické analýzy: Multiple sampling and Interpolation in standard weighted Bergman spaces of unit disk (Driss Aadi)

• ustav
• 10.10.2023

Na semináři z matematické analýzy vedeném prof. Englišem bude přednášet Driss Aadi na téma Multiple sampling and Interpolation in standard weighted Bergman spaces of unit disk.

Seminář se koná v úterý 10. 10. 2023, od 9:30 do 10:30 v Knihovně prof. Smítala (dveře č. 5)

At the seminar in mathematical analysis led by Prof. Engliš, Driss Aadi will give a lecture on Multiple sampling and Interpolation in standard weighted Bergman spaces of unit disk.

The seminar will seminar will take place on Tuesday October 10 on 9:30 - 10:30 in Professor Smital's Library (room no.5).

Abstract:
Sampling and interpolation sequences in space of analytic functions are among subjects widely studied in modern analysis for the last four decades, after Carleson's interpolation theorem. Besides, their theoretical considerations they find many applications concrete applications as in Gabor analysis signal processing, telecommunications, and etc. One of the most considerable work in space of analytic functions (after L. Carleson) goes back to K. Seip who completely characterized sampling and interpolation for both Bergman and Bargman-Fock space by mean of Landau type density tools around 1990's. Similarly to Hermite interpolation, a natural idea, is instead of looking at the values of a function not only at some samples (nodes) but also at its derivatives up to a certain order in the interpolation/sampling nodes. This leads to multiple interpolations/sampling. Actually, the same problem was considered in the case of Bargman-Fock space (resp. Hardy space only for interpolation) by Brekk and Seip (resp. Nikolski, Vasyunin, and Volberg others around 1970's). Recently, with Cruz, Hartman and Kellay, we were considering the situation of Bergman spaces of unit disk for which the underlying geometry is more intricate (pseudo-hyperbolic geometry). The main results obtained were a uniqueness condition through understanding a key hyperbolic radii for covering and separation, Also a necessary/sufficient condition in both the sampling and interpolation cases, with a small gap. We mention that our results may applied for bounded multiplicities, as a weak alternative for Seip's characterization of sampling/interpolation sequences (presence of a Beurling-Landau type densities, which are difficult to check in general, these results will be the subject of my talk, stemming from the paper \cite{ACHK} in https://doi.org/10.1016/j.jfa.2023.109865.