Müntz Linear Transforms of Brownian Motion

We consider a class of Volterra linear transforms of Brownian motion associated to a sequence of Muntz Gaussian spaces and determine explicitly their kernels; the kernels take a simple form when expressed in terms of Muntz-Legendre polynomials. These are new explicit examples of progressive Gaussian enlargement of a Brownian filtration. We give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Muntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process. This completes some already obtained partial answers to the aforementioned problems in the infinite dimensional case.