Sampling and interpolation sequences in spaces of analytic functions are among subjects widely studied in modern analysis for the last few decades, after Carleson's interpolation theorem. Considerable efforts have been done to investigate these sequences, namely in Bergman and Bargmann-Fock space (also, weighted Bergman/Fock spaces) since 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. In fact, such problems were considered in the case of Bargmann-Fock space (resp. Hardy space -only interpolation-) by Brekk and Seip (resp. Nikolski, Vasyunin, and others around 1970's). Recently, with C. Cruz, A. Hartmann and K. Kellay we were investigating these questions in the situation of Bergman spaces  of unit disk for which the underlying geometry is more intricate (pseudo-hyperbolic geometry). For the results, a sufficient uniqueness condition (non zero sets) was obtained, as a first result, through understanding a key hyperbolic radii for distributing weighted Dirac point mass (on nodes). Moreover, a necessary/sufficient condition in both sampling and interpolation cases, respectively expressed by covering and separation of hyperbolic disks with critical radius (slightly bigger or smaller). Unexpectedly, our results might apply even for bounded multiplicities, in some sense. These results will be the subject of my talk, understanding first where the critical radius comes from and how to construct a (weighted) sub-harmonic function with some prescribed singularities for an L²-Hörmander ∂ type problem (needed for interpolation). Finally, if time permits, I will sketch the proof of the sampling theorem in the uniform case.

This is a joint work with C. Cruz (Barcelona), A. Hartmann, and K. Kellay (Bordeaux).

https://univienna.zoom.us/j/64895816787?pwd=L0tHVnBPUkJFQVVSR3Y2QnhVRXRGZz09