This note is about Doob decomposition and the basics of Square integrable martingales Contents 1 Doob-Meyer Decomposition 1 2 Square Integrable Martingales 4 Brownian Motion and Stochastic Calculus Continuout Time Submartingales Usually it’s su¢ ce to only discuss submartingales by symmetry in de…nition and techniques are the same. 1 Doob-Meyer Decomposition Doob-meyer decomposition clears the ...

We prove that the HRT (Heil, Ramanathan, and Topiwala) conjecture holds for finite Gabor systems generated by square-integrable functions with certain behavior at infinity. These functions include functions ultimately decaying faster than any exponential function, as well as squareintegrable functions ultimately analytic and whose germs are in a Hardy field that is closed under translations. Tw...

We introduce an exponentially confining potential well that can be used as a model to describe the structure of strongly localized system. obtain approximate partial solution Schrödinger equation with this where we find lowest energy spectrum and corresponding wavefunctions. use tridiagonal representation approach method for obtaining finite series square-integrable functions written in terms B...