We show that any -dimensional linear subspace of admits an orthonormal system such that the norm of the square variation operator is as small as possible. When applied to the span of the trigonometric system, we obtain an orthonormal system of trigonometric polynomials with a operator that is considerably smaller than the associated operator for the trigonometric system itself.
"Orthonormal systems in linear spans." Anal. PDE 7 (1) 97 - 115, 2014. https://doi.org/10.2140/apde.2014.7.97