  In this article, we study predictable projections of stochastic integrals
with respect to the conformal Brownian motion, extending the connection between
powers of the conformal Brownian motion and the corresponding Hermite
polynomials. As a consequence of this result, we then investigate the relation
between analytic functions and $L^p$-convergent series of Hermite polynomials.
Finally, our results are applied to Widder's representation for a class of
Brownian martingales, retrieving a characterization for the moments of Widder's
measure.
