By way of concrete presentations, we construct two infinite-dimensional transforms at the crossroads Gaussian fields and reproducing kernel Hilbert spaces (RKHS), thus leading to a new Fourier transform in general setting processes. Our results serve unify existing tools from analysis.

The Hilbert and Fourier transforms are tools used for signal analysis in the time/frequency domains. The Hilbert transform is applied to casual continuous signals. The majority of the practical signals are discrete signals and they are limited in time. It appeared therefore the need to create numeric algorithms for the Hilbert transform. Such an algorithm is a numeric operator, named the Discre...

Let W be an operator weight taking values almost everywhere in the bounded positive invertible linear operators on a separable Hilbert space H. We show that if W and its inverse W−1 both satisfy a matrix reverse Hölder property introduced in [2], then the weighted Hilbert transform H : LW (R,H) → L 2 W (R,H) and also all weighted dyadic martingale transforms Tσ : LW (R,H)→ L 2 W (R,H) are bound...