Müntz linear transforms of Brownian motion

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We consider a class of linear Volterra transforms of Brownian motion associated to a sequence of Müntz Gaussian spaces and determine explicitly their kernels; some interesting links with Müntz-Legendre polynomials are provided. This gives new explicit examples of progressive Gaussian enlargement of the Brownian filtration. By exploiting a link to stationarity, we give a necessary and sufficient condition for the existence of kernels of infinite order associated to an infinite dimensional Müntz Gaussian space; we also examine when the transformed Brownian motion remains a semimartingale in the filtration of the original process.