Random zero sets of analytic functions and traces of functions in Fock spaces

• Autores: Jeremiah Buckley
• Directores de la Tesis: Francesc Xavier Massaneda Clarés (dir. tes.), Joaquim Ortega Cerdà (dir. tes.)
• Lectura: En la Universitat de Barcelona ( España ) en 2013
• Idioma: inglés
• Tribunal Calificador de la Tesis: Joaquim Bruna i Floris (presid.), Marta Sanz Solé (secret.), Mikhail Sodin (voc.)
• Materias:
  • Matemáticas
    • Análisis y análisis funcional
      • Funciones de una variable compleja
      • Espacios de Hilbert
    • Probabilidad
      • Procesos estocásticos
• Enlaces
  • Tesis en acceso abierto en:  TDX  Dipòsit Digital de la UB 
• Resumen

  • Interpolating and sampling sequences in spaces of functions are classical subjects in complex and harmonic analysis. A sequence of points is said to be interpolating if, given any collection of values, we can find a function from the space which takes these values on the points of the sequence, and a sequence of points is said to be sampling if it is possible to recover a function from the space knowing the values of the function on the sequence. In Fock spaces these sequences have been characterised in terms of a Beurling type density, that is, interpolating sequences are those sequences whose density is less than a certain critical value, and sampling sequences are those sequences whose density is greater than the same critical value. A critical sequence, that is a sequence whose density is exactly the critical value, is almost an interpolating sequence and almost a sampling sequence. In this thesis we have charaterised completely the trace of functions in these Fock spaces on critical sequences in terms of the discrete Beurling-Ahlfors transform. We also study random point processes in the complex plane and in the unit disc. These random point processes are the zero sets of analytic functions. These functions can be constructed through random linear combinations of elements of a basis for a space of functions. The distribution of the zero set of the function formed by taking a basis for the classical Bargmann-Fock space is well known, and depends on a translation-invariance inherent to the space. We have generalised these ideas to inhomogeneous Fock spaces, where no such invariance exists. In particular we see that the expected number of points is related to a certain measure associated to the space. We also study asymptotic normality and a ‘hole theorem’, that calculates the probability that there are no points in a disc of radius r. We study analogous processes in the unit disc, and on the real line. We calculate the variance of the process in the disc, and we prove a ‘hole theorem’ for large values of the ‘intensity’ of the process. In the real line we study the probability of a large gap for a process that is invariant under translations.