Representation of Functions by Walsh ’ s Series with Monotone Coe ffi cients

There exists a series in the Walsh system {φn} of the form ∞ ∑ i=1 aiφi, with |ai| ↘ 0, that possess the following properties: For any > 0 and any function f ∈ L1(0, 1) there exists set E ⊂ [0, 1] (|E| > 1 − ) and a sequence {δi}i=0, δi = 0 or 1, such that the series ∞ ∑ i=0 δiaiφi converges to f on E in the L1(0, 1)-metric and on [0, 1] E in the Lr([0, 1] E) metric for all r ∈ (0, 1).


Introduction
The problem of representing a function f by a series in classical and general orthonormal systems has a long history.
A question posed by Lusin in 1915 asks whether it be possible to find for every measurable function [0, 2π] a trigonometric series, with coefficient sequence converging to function.For real-valued functions, this question was given an affirmative answer by Men'shov in 1941.There are many other works (see Talalian, 1960;Men'shov, 1947Men'shov, , 1941;;Grigorian, 1999Grigorian, , 2003Grigorian, , 2000;;Ul'janov, 1972;Ivanov, 1989;Krotov, 1977;Kozlov, 1950) devoted to representations of functions by series in classical and general orthonormal systems and the existence of different types of universal series in the sense of convergence almost everywhere and by measure.
The papers by Men'shov (1947) and Kozlov (1950) were the first to construct some ordinary universal trigonometric series in the class of all measurable functions in the sense of a.e.convergence.Grigoryan (2009) proved the following important result: For any > 0 and any function f ∈ L[0, 1] there exists a sequence {δ i } ∞ i=0 , δ i = 0 or 1, such that the series with monotone coefficients ∞ i=0 δ i a i ϕ i converges to f in the L 1 (E)-metric.In Grigoryan's paper properties of series outside the E remained open.In this paper we succeed to ensure the convergence of the series to f outside E in weaker metric.Note that convergence is impossible in the same metric.
Let r be the periodic function, of least period 1, defined on [0, 1) by The Rademacher system, R = r n : n = 0, 1, . . ., is defined by the conditions and, in the ordering employed by Payley (see Golubov, Efimov, & Skvartsov, 1987;Paley, 1932), the n-th element of the Walsh system {ϕ n } is given by where ∞ k=0 n k 2 k is the unique binary expansion of n, with each n k either 0 or 1.In the present work we prove the following theorem: Theorem There exists a series in the Walsh system of the form that possess the following properties: For any > 0 and any function f ∈ L 1 (0, 1) there exists set E ⊂ [0, 1] (|E| > 1 − ) and a sequence {δ i } ∞ i=0 , δ i = 0 or 1, such that the series ∞ i=0 δ i a i ϕ i converges to f on E in the L 1 (0, 1)-metric and on [0, 1] E in the L r ([0, 1] E) metric for all r ∈ (0, 1).
The following problem remains open: is the Theorem true for the trigonometric system?

Basic Concepts and Terminology
We put and periodically extend these functions on R 1 with period 1.
By χ E (x) we denote the characteristic function of the set E, i.e.
Then, clearly and let for the natural numbers k ≥ 1, and j ] and numbers N 0 ∈ N, γ 0, ∈ (0, 1), r 2 ∈ (0, 1) be given.Then there exists a measurable set E ⊂ [0, 1] and a polynomial Q in the Walsh system {ϕ k } of the following form We define the polynomial Q(x) and the numbers c n , a i and b j in the following form: Taking into consideration the following equation and having the following relations ( 5)-( 8) and ( 10)-( 12), we obtain that the polynomial Q(x) has the following form: where Then let Clearly that (see ( 2) and ( 10)), Hence and from ( 9) for all r ∈ (0, r 2 ) we obtain By Bessel's inequality and by ( 13)-( 16) we have max Lemma 1 is proved.