Geodesic
Representation of measurable functions by series with respect to Walsh subsystems
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2009), pp. 51-62

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For every sequence $\{\omega(n)\}_{n\in\mathbb N}$ that tends to infinity we construct a “quasiquadratic” representation spectrum $\Lambda=\{n^2+o(\omega(n))\}_{n\in\mathbb N}$: for each almost everywhere finite measurable function $f(x)$ there exists a series in the form $\sum_{k\in\Lambda}a_kw_k(x)$ that converges almost everywhere to this function, where $\{w_k(x)\}_{k\in\mathbb N}$ is the Walsh system. We also find representation spectra in the form $\{n^l+o(n^l)\}_{n\in\mathbb N}$, where $l\in\{2^k\}_{k\in\mathbb N}$.
Keywords: Walsh system, orthogonal series, representation theorems, expansion spectrum. 
M. A. Nalbandyan. Representation of measurable functions by series with respect to Walsh subsystems. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2009), pp. 51-62. http://geodesic.mathdoc.fr/item/IVM_2009_10_a5/
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