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. 
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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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