#### Vol. 7, No. 1, 2014

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Orthonormal systems in linear spans

### Allison Lewko and Mark Lewko

Vol. 7 (2014), No. 1, 97–115
##### Abstract

We show that any $N$-dimensional linear subspace of ${L}^{2}\left(\mathbb{T}\right)$ admits an orthonormal system such that the ${L}^{2}$ norm of the square variation operator ${V}^{2}$ is as small as possible. When applied to the span of the trigonometric system, we obtain an orthonormal system of trigonometric polynomials with a ${V}^{2}$ operator that is considerably smaller than the associated operator for the trigonometric system itself.

##### Keywords
orthogonal systems, square variation, maximal operator, Fourier analysis
##### Mathematical Subject Classification 2010
Primary: 42A61, 42B05, 42C05