Universality properties of double series by generalized Walsh system

In this paper we consider a question on existence of double series by generalized Walsh system, which are universal in weighted $L_\mu^1[0,1]^2$ spaces. In particular, we construct a weighted function $\mu(x,y)$ and a double series by generalized Walsh system of the form $$\sum_{n,k=1}^\infty c_{n,k}\psi_n(x)\psi_k(y)\ \ \mbox{with} \ \ \sum_{n,k=1}^\infty \left | c_{n,k} \right|^q<\infty\ \mbox{for all}\ q>2,$$ which is universal in $L_\mu^1[0,1]^2$ concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.


Introduction
Let X be a Banach space.
is said to be universal in X with respect to rearrangements, if for any f ∈ X the members of (1) can be rearranged so that the obtained series Note, that for one-dimensional case there are many papers are devoted to the question on existence of various types of universal series in the sense of convergence almost everywhere and on a measure ( see [1] - [9], [11]).
The first usual universal in the sense of convergence almost everywhere trigonometric series were constructed by D.E.Menshov [1] and V.Ya.Kozlov [2]. The series of the form a k cos kx + b k sin kx was constructed just by them such that for any measurable on [0, 2π] function f (x) there exists the growing sequence of natural numbers n k such that the series having the sequence of partial sums with numbers n k converges to f (x) almost everywhere on [0, 2π]. Note that in this result, when f (x) ∈ L 1 [0,2π] , it is impossible to replace convergence almost everywhere by convergence in the metric L 1 [0,2π] . This result was distributed by A.A.Talalian on arbitrary orthonormal complete systems . He also established (see [3]), that if {φ n (x)} ∞ n=1 -the normalized basis of space L p [0,1] , p > 1, then there exists a series of the form which has property: for any measurable function f (x) the members of series (2) can be rearranged so that the again received series converge on a measure on [0,1] to f (x). In [4] O. P. Dzangadze these results are transferred to two-dimensional case.
W. Orlicz [5] observed the fact that there exist functional series that are universal with respect to rearrangements in the sense of a.e. convergence in the class of a.e. finite measurable functions. It is also useful to note that even Rieman proved that every convergent numerical series which is not absolutely convergent is universal with respect to rearrangements in the class of all real numbers.
Let µ(x), 0 < µ(x) ≤ 1, x ∈ [0, 1] be a measurable on [0, 1] function and let L 1 In [6] - [9] it is proved the existence of universal one-dimensional series by trigonometric and classical Walsh system with respect to rearrangements and subseries. Some results for two-dimensional case for classical Walsh system was obtained in [11]. In this paper we consider this problems for double series by generalized Walsh system

Preliminary Notes
Let a denote a fixed integer, a ≥ 2 and put ω a = e 2πi a . Now we will give the definitions of generalized Rademacher and Walsh systems (see [12] ).

Definition 2.1 The Rademacher system of order a is defined by
and for n ≥ 0 ϕ n (x + 1) = ϕ n (x) = ϕ 0 (a n x).

Definition 2.2 The generalized Walsh system of order a is defined by
and if n = α 1 a n 1 + ... + α s a ns where n 1 > ... > n s , then Let's denote the generalized Walsh system of order a by Ψ a , a ≥ 2. Note that Ψ 2 is the classical Walsh system. The basic properties of the generalized Walsh system of order a are obtained by H.E.Chrestenson, R. Pely, J. Fine, W. Young, C. Vatari, N. Vilenkin and others (see [12]- [17]).
First we present some properties of Ψ a system (see Definition 2.1). Property 1. Each nth Rademacher function has period 1 a n and Property 2. It is clear, that for any integer n the Walsh function ψ n (x) consists of a finite product of Rademacher functions and accepts values from Ω a .
The rectangular and spherical partial sums of the double series ∞ k,ν=1 c k,ν ψ k (x)ψ ν (y) will be denoted by If g(x, y) is a continuous function on T = [0, 1] 2 , then we set |g(x, y)|.

Main Results
Let's denote the generalized Walsh system of order a by Ψ a , a ≥ 2. These are the main results of the paper.
with the following property: for any number ε > 0 a weighted function µ(x, y) can be constructed so that the series (5) is universal in L 1 µ (T ) concerning subseries with respect to convergence in the sense of both spherical and rectangular partial sums. (5) with the following property: for any number ε > 0 a weighted function µ(x, y) with (6) can be constructed, so that the series (5) is universal in L 1 µ (T ) concerning rearrangements with respect to convergence in the sense of both spherical and rectangular partial sums.

Theorem 3.2 There exists a double series of the form
Repeating the reasoning of the proof of Lemma 2 in [10] we'll receive the following lemma: Lemma 3.3 For any given numbers 0 < ε < 1, N 0 > 2 (N 0 ∈ N ) and a step function which satisfy the conditions: for every measurable subset e of E.
Then applying this Lemma we get next one: For any numbers γ = 0, 0 < δ < 1, N > 1 and for any square ∆ = ∆ 1 × ∆ 2 ⊂ T there exists a measurable set E ⊂ T and a polynomial P (x, y) of the form with the following properties: for every measurable subset e of E.
Proof . We apply Lemma 3.3, setting Then we can define a measurable set E 1 ⊂ [0, 1] and a polynomial P 1 (x) of the form which satisfy the conditions: and apply Lemma 3.3 again, setting Then we can define a measurable set E 2 ⊂ [0, 1] and a polynomial P 2 (y) of the form which satisfy the conditions: where and c k,s = 0, f or other k and s.
Lemma 3.5 For any numbers ε > 0, N > 1 and a step function there exists a measurable set E ⊂ T and a polynomial P (x, y) of the form which satisfy the following conditions: |f (x, y)|dxdy + ε, for every measurable subset e of E.
Proof . Without any loss of generality, we assume that (∆ ν , 1 ≤ ν ≤ ν 0 are the constancy rectangular domian of f (x, y), i.e. where the function f (x, y) is constant). Given an integer 1 ≤ ν ≤ ν 0 , by applying Lemma 3.4 with δ = ε 16ν 0 , we find that there exists a measurable set E ν ⊂ T and a polynomial P ν (x, y) of the form with the following properties: for every measurable subset e of E ν (see (11)). Then we can take where and c k,s = 0, f or other k and s. (18) and (19) we have

Proofs of the theorems
The Theorem 3.1 is proved similarly Theorem 3 in [11], but for maintenance of integrity of this paper, here we will give the proof : Proof of Theorem 3.1. Let be a sequence of all step functions, values and constancy interval endpoints of which are rational numbers. Applying Lemma 3.5 consecutively, we can find a sequence {E s } ∞ s=1 of sets and a sequence of polynomials which satisfy the conditions: for every measurable subset e of E s . Denote ∞ k,ν=1 where For an arbitrary number ε > 0 we set It is obvious ( see (23), (27) ) that |B| = 1 and |E| > 1 − ε.
Hence, obviously we have (see (24) and (26) Analogously for all s ≥ n 0 and By (21), (27) -(29) for all s ≥ n 0 we have By (25) and (27)  µ n + + Ωs |f s (x, y)|µ(x, y)dxdy Analogously for all s ≥ n 0 and Now we'll show that the series (26) is universal in L 1 µ (T ) concerning subseries with respect to convergence by both spherical and rectangular partial sums.
It is easy to see that we can choose a function f n 1 (x, y) from the sequence (20) such that T |f (x, y) − f n 1 (x, y)| µ(x, y)dxdy < 2 −2 , n 1 > n 0 + 1.
Let n and m be arbitrary natural numbers. Then for some natural number q we have N nq−1 ≤ min{n, m} < N nq .
Taking into account (40) and (42) for rectangular partial sums S n,m (x, y) of (41) we get k,ν ψ k (x) · ψ ν (y) µ(x, y)dxdy Analogously for where S R (x, y) the spherical partial sums of (41). From (44) and (45) we conclude that the series (26) is universal in L 1 µ (T ) concerning subseries with respect to convergence by both spherical and rectangular partial sums (see Definition 1.2).