I had a talk this week about the construction of white noise test functions. It was nice actually. I was forced to articulate what I know and present it in a more coherent way. Concepts that were fuzzy before has become clearer now. I still have remaining objectives to answer but the process is the reward. Next week, I will be presenting my results in a colloquium. I will frame the results around a central question which is a variation of a certain Lie algebra of white noise operators. This is done by replacing the number operator by a generalized number operator, called conservation operator corresponding to an operator S. Doing this changes the dimension of the base Lie algebra depending on the properties of S. One result that I have is that the linear independence of the orbits of S characterizes the dimension of the Lie algebra. The question then becomes: which operators have a linearly independent orbit? One example is the the unilateral shift. It has a very simple structure...
Embracing Uncertainty