# Universality Properties of a Double Series by the Generalized Walsh System

Authors: Sergo A. Episkoposian
Journal: International Journal of Analysis
 (2013)

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## Abstract

We consider a question on existence of a double series by the generalized Walsh system, which is universal in weightedLμ1[0,1]2 spaces. In particular, we construct a weighted function μ(x,y) and a double series by generalized Walsh system of the form ∑n,k=1∞‍cn,kψn(x)ψk(y) with the ∑n,k=1∞‍|cn,k|q<∞ for  all q>2, which is universal in Lμ1[0,1]2 concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.


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## Body

## 1. Introduction

LetX be a Banach space.Definition 1.
A series(1)∑k=1∞‍fk,fk∈X
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 ∑k=1∞fσ(k) converges to f by norm of X.Definition 2.
The series (1) is said to be universal (in X) concerning subseries, if for any f∈X, it is possible to choose a subseries ∑k=1∞fnk from (1), which converges to the f by norm of X.Note that for one-dimensional case there are many papers that 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 [1–10]).Leta≥2 be a fixed integer and ωa=e2πi/a. Recall the following definitions (see [11]).The Rademacher system of ordera is defined inductively as follows. For n=0 let
(2)φ0(x)=ωakifx∈[ka,k+1a),k=0,1,…,a-1,
and for n≥1 let
(3)φn(x+1)=φn(x)=φ0(anx).The generalized Walsh system of ordera is defined by
(4)ψ0(x)=1,
and if n=α1an1+⋯+αsans, where n1>⋯>ns,  0≤αj<a, j=1,2,…,s, then
(5)ψn(x)=φn1α1(x)·…·φnsαs(x).We denote the generalized Walsh system of ordera by Ψa. Not that Ψ2 is the classical Walsh system. The basic properties of the generalized Walsh system of order a have been obtained by Chrestenson, Fine, Watari, Young, Vilenkin, and others (see [11–16]).In [6–9], the existence of universal one-dimensional series by trigonometric and the classical Walsh system with respect to rearrangements and subseries in some weighted space Lμ1[0,1]. Some results for two-dimensional case for the classical Walsh system were obtained in [10] is proved. In this paper we consider the universality properties of a double series by the generalized Walsh system.


## 2. Preliminary Notes

Now we list some properties ofΨa,  a≥2, which will be useful later.(i)
Eachnth Rademacher function has period a-n and
(6)φn(x)=const∈Ωa={1,ωa,ωa2,…,ωaa-1},
if x∈Δn+1(k)=[k/an+1,(k+1)/an+1),  k=0,…,an+1-1, n=1,2,….(ii)
(7)(φn(x))k=(φn(x))m,∀n,k∈𝒩,m=k(moda).(iii)
ψn(x) is a finite product of the Rademacher functions with values in Ωa.(iv)
(8)ψak+j(x)=φk(x)·ψj(x),if0≤j≤ak-1.(v)
Ψa, a≥2 is a complete orthonormal system in L2[0,1) and it is a basis in Lp[0,1] for p>1.The rectangular and spherical partial sums of the double series(9)∑k,ν=1∞‍ck,νψk(x)ψν(y),(x,y)∈T=[0,1)2
will be denoted by
(10)Sn,m(x,y)=∑k=1n‍∑ν=1m‍ck,νψk(x)ψν(y),SR(x,y)=∑ν2+k2≤R2‍ck,νψk(x)ψν(y).Ifg(x,y) is a continuous function on T=[0,1]2, then we set
(11)∥g(x,y)∥C=max(x,y)∈T|g(x,y)|.


## 3. Main Results

Let us denote the generalized Walsh system of ordera by Ψa, a≥2. These are the main results of the paper.Theorem 3.
There exists a double series of the form(12)∑n,k=1∞‍cn,kψn(x)ψk(y)with∑n,k=1∞‍|cn,k|q<∞∀q>2
with the following property: for any number ε>0 a weighted function μ(x,y) satisfying
(13)0<μ(x,y)≤1,|{(x,y)∈T:μ(x,y)≠1}|<ε
can be constructed so that the series (12) is universal in Lμ1(T) concerning subseries with respect to convergence in the sense of both spherical and rectangular partial sums.Theorem 4.
There exists a double series of the form (12) with the following property: for any number ε>0 a weighted function μ(x,y) with (13) can be constructed, so that the series (12) is universal in Lμ1(T) concerning rearrangements with respect to convergence in the sense of both spherical and rectangular partial sums.Repeating the reasoning of the proof of [17, Lemma  2] we will receive the following lemma.Lemma 5.
For any given numbers0<ε<1, N0>2  (N0∈𝒩) and a step function
(14)f(x)=∑s=1q‍γs·χΔs(x),
where Δs is an interval of the form Δm(i)=[(i-1)/2m,i/2m], 1≤i≤2m, there exist a measurable set E⊂[0,1] and a polynomial P(x) of the form
(15)P(x)=∑k=N0N‍ckψk(x)
which satisfy the following conditions:(1)
(16)P(x)=f(x)onE,(2)
(17)|E|>(1-ε),(3)
(18)∑k=N0N‍|ck|2+ε<ε,(4)
(19)maxN0≤m<N[∫e‍|∑k=N0m‍ckψk(x)|dx]<ε+∫e‍|f(x)|dx,
for every measurable subsete of E.Then applying this Lemma we get the next one.Lemma 6.
For any numbersγ≠0, 0<δ<1, N>1 and for any square Δ=Δ1×Δ2⊂T, there exists a measurable set E⊂T and a polynomial P(x,y) of the form
(20)P(x,y)=∑k,s=NM‍ck,sψk(x)·ψs(y),
with the following properties:(1)
(21)|E|>1-δ,(2)
(22)∑k,s=NM‍|ck,s|2+δ<δ,(3)
(23)P(x,y)=γ·χΔ(x,y)for(x,y)∈E,(4)
(24)maxN≤n¯,m¯≤M[∫‍∫e‍|∑k,s=Nn¯,m¯‍ck,sψk(x)·ψs(y)|dxdy]+max2N≤R≤2M[∫∫e|∑2N2≤k2+s2≤R2‍ck,sψk(x)·ψs(y)|dxdy]≤16·|γ|·|Δ|,for every measurable subset e of E.Proof.
We apply Lemma5, setting
(25)f(x)=γ·χΔ1(x),N0=N,ε=δ2.
Then we can define a measurable set E1⊂[0,1] and a polynomial P1(x) of the form
(26)P1(x)=∑k=NN1‍akψk(x)
which satisfy the following conditions: (10) (27)P1(x)=γ·χΔ1(x)forx∈E1,(20) (28)|E1|>1-δ2,(30) (29)∑k=NN1‍|ak|2+δ<δ,(40) (30)maxN≤n¯≤N1[∫e1‍|∑k=Nn¯‍akψk(x)|dx]≤2·|γ|·|Δ1|,for every measurable subset e1 of E1.
Set(31)M0=2·(N12+1)
and apply Lemma 5 again, setting
(32)f(y)=χΔ2(y),N0=M0,ε=δ2.
Then we can define a measurable set E2⊂[0,1] and a polynomial P2(y) of the form
(33)P2(y)=∑s=M0M‍bsψs(y),
which satisfy the following conditions:(100) (34)P2(y)=χΔ2(y)fory∈E2,(200) (35)|E2|>1-δ2,(300) (36)∑s=M0M‍|bs|2+δ<δ,(400) (37)maxM0≤m¯≤M[∫e2‍|∑s=M0m¯‍bsψs(y)|dy]≤2·|Δ2|, for every measurable subset e2 of E2.
Set(38)E=E1×E2,P(x,y)=P1(x)·P2(x)=∑k,s=NM‍ck,sψk(x)·ψs(y),
where
(39)ck,s=ak·bs,ifN≤k≤N1,M0≤s≤M,ck,s=0,forotherk,s.
By (10)–(30),   (100)–(300), and (38), (39), we obtain
(40)|E|>1-δ,∑k,s=NM‍|ck,s|2+δ=∑k=NN1‍|ak|2+δ·∑s=M0M‍|bs|2+δ<δ,P(x,y)=γ·χΔ(x,y)for(x,y)∈E.
Thus, the statements (1)–(3) of Lemma 6 are satisfied. Now we will check the fulfillment of statement (4).
LetN2+M02<R2<N12+M2, then for some m0>M0 we have m0<R<m0+1 and from (31) it follows that R2-N12>(m0-1)2.
Consequently taking relations(40),(400), and (38), (39) for any measurable set e⊂E  (e=e1×e2,e1⊂E1,e2⊂E2) we obtain
(41)∫‍∫e‍|∑N2+M2≤k2+s2≤R2‍ck,sψk(x)·ψs(y)|dxdy≤∫‍∫e‍|∑k=NN1‍∑s=M0m0-1‍ck,sψk(x)·ψs(y)|dxdy+maxN<n≤N1[∫‍∫e‍|∑k=Nn‍ck,m0ψk(x)·ψm0(y)|dxdy]≤[∫e1‍|∑k=NN1‍akψk(x)|dx]·[∫e2‍|∑s=M0m0-1‍bsψs(y)|dy]+|bm0|·[∫e2‍|ψm0(y)|dy]·maxN<n≤N1[∫e1‍|∑k=Nn‍akψk(x)|dx]≤12·|γ|·|Δ|.
Similarly, forN≤n¯≤N1,M0≤m¯≤M, we get
(42)∫‍∫e‍|∑k,s=Nn¯,m¯‍ck,sψk(x)·ψs(y)|dxdy≤4·|γ|·|Δ|.
Lemma6 is proved.Lemma 7.
For any numbersε>0, N>1 and a step function
(43)f(x,y)=∑ν=1ν0‍γν·χΔν(x,y),
there exists a measurable set E⊂T and a polynomial P(x,y) of the form
(44)P(x,y)=∑k,s=NM‍ck,sψk(x)·ψs(y),
which satisfy the following conditions:(10)(45)P(x,y)=f(x,y)for(x,y)∈E,(20)(46)|E|>1-ε,(30)(47)∑k,s=NM‍|ck,s|2+ε<ε,(40)(48)maxN≤n¯,m¯<M[∫‍∫e‍|∑k,s=Nn¯,m¯‍ck,sψk(x)·ψs(y)|dxdy]+max2N≤R≤2M[∑2N2≤k2+s2≤R2‍∫‍∫e‍|∑2N2≤k2+s2≤R2‍ck,sψk(x)·ψs(y)∑2N2≤k2+s2≤R2‍|dxdy∑2N2≤k2+s2≤R2‍]≤2·∫‍∫e‍|f(x,y)|dxdy+ε, for every measurable subset e of E.Proof.
Without any loss of generality, we assume that(49)max1≤ν≤ν0(|γν|·|Δν|)<ε32,Δν,1≤ν≤ν0 are the constancy rectangular domain of f(x,y), that is, where the function f(x,y) is constant.
Given an integer1≤ν≤ν0, by applying Lemma 6 with δ=ε/16ν0, we find that there exists a measurable set Eν⊂T and a polynomial Pν(x,y) of the form
(50)Pν(x,y)=∑k,s=NνMν‍ck,s(ν)ψk(x)·ψs(y)
with the following properties:
(51)|Eν|>1-ε2ν,(52)∑k,s=NνMν‍|ck,s(ν)|2+ε<εν0,(53)Pν(x,y)=γν·χΔν(x,y)for(x,y)∈Eν,(54)maxNν≤n¯,m¯≤Mν[∫‍∫e‍|∑k,s=Nνn¯,m¯‍ck,s(ν)ψk(x)·ψs(y)|dxdy]+max2Nν≤R≤2Mν[∫‍∫e‍|∑2Nν2≤k2+s2≤R2‍ck,s(ν)ψk(x)·ψs(y)∑2Nν2≤k2+s2≤R2‍|dxdy∑2Nν2≤k2+s2≤R2‍]≤16·|γν|·|Δν|<ε2,
for every measurable subset e of Eν (see (49)).
Then we can take(55)N1=N,nν=Mν-1+1,1≤ν≤ν0.
Set
(56)E=⋂ν=1ν0‍Eν,(57)P(x,y)=∑ν=1ν0‍Pν(x,y)=∑k,s=NM‍ck,sψk(x)·ψs(y),M=Mν0,
where
(58)ck,s=ck,s(ν),forNν≤k,s≤Mν,1≤ν≤ν0,(59)ck,s=0,forotherk,s.
From (51)–(53) and (56)–(58) we obtain
(60)P(x,y)=f(x,y)for(x,y)∈E,|E|>1-ε,∑k,s=NM‍|ck,s|2+ε<∑ν=1ν0‍[∑k,s=NνMν‍|ck,s(ν)|2+ε]<ε.
Then,   letR∈[2N,2M], then for some ν′,1≤ν′≤ν0 we have 2Nν′≤R≤2Nν′+1, consequently from (57) and (58) we have
(61)∑2N2≤k2+s2≤R2‍ck,sψk(x)·ψs(y)=∑ν=1ν′-1‍Pν(x,y)+∑2Nν′2≤k2+s2≤R2‍ck,s(ν′)ψk(x)·ψs(y).
In view of the conditions (51)–(54) and the equality P(x,y)=f(x,y) on E, for any measurable set e⊂E we obtain
(62)∫‍∫e‍|∑2N2≤k2+s2≤R2‍ck,sψk(x)·ψs(y)|dxdy≤∫‍∫e‍|∑ν=1ν′-1‍Pν(x,y)|dxdy+∫‍∫e‍|∑2Nν′2≤k2+s2≤R2‍ck,s(ν′)ψk(x)·ψs(y)|dxdy≤∫‍∫e‍|f(x,y)|dxdy+ε2.
Similarly, for any e⊂E we have
(63)maxN≤n¯,m¯≤M[∫‍∫e‍|∑k,s=Nn¯,m¯‍ck,sψk(x)·ψs(y)|dxdy]≤∫‍∫e‍|f(x,y)|dxdy+ε2.
Lemma 7 is proved.


## 4. Proofs of the Theorems

Theorem3 is proved similarly [10, Theorem  3], but for maintenance of integrity of this paper, here we will give the proof.Proof of Theorem3.
Let(64){fs(x,y)}s=1∞,(x,y)∈T
be a sequence of all step functions, values, and constancy interval endpoints which are rational numbers. Applying Lemma 7 consecutively, we can find a sequence {Es}s=1∞ of sets and a sequence of polynomials
(65)Ps(x,y)=∑k,ν=Ns-1Ns-1‍ck,ν(s)ψk(x)ψν(y),1=N0<N1<⋯<Ns<⋯,s=1,2,…,
which satisfy the following conditions:
(66)Ps(x,y)=fs(x,y),(x,y)∈Es,(67)|Es|>1-2-2(s+1),Es⊂T,(68)∑k,ν=Ns-1Ns-1‍|ck,ν(s)|2+2-2s<2-2s,(69)maxNs-1≤n¯,m¯<Ns[∫‍∫e‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|dxdy]+max2Ns-1≤R≤2Ns[∫‍∫e‍|∑2Ns-12≤k2+ν2≤R2‍ck,ν(s)ψk(x)·ψν(y)∑2Ns-12≤k2+ν2≤R2‍|dxdy∑2Ns-12≤k2+ν2≤R2‍]≤2·∫‍∫e‍|fs(x,y)|dxdy+2-2(s+1),
for every measurable subset e of Es.
Denote(70)∑k,ν=1∞‍ck,νψk(x)ψν(y)=∑s=1∞‍[∑k,ν=Ns-1Ns-1‍ck,ν(s)ψk(x)ψν(y)],
where
(71)ck,ν=ck,ν(s),forNs-1≤k,ν<Ns,s=1,2,….
For an arbitrary numberε>0 we set
(72)Ωn=⋂s=n∞‍Es,n=1,2,…,E=Ωn0=⋂s=n0∞‍Es,n0=[log1/1ε]+1,B=⋃n=n0∞‍Ωn=Ωn0⋃‍(⋃n=n0+1∞‍Ωn∖Ωn-1).
It is obvious (see (67) and (72)) that |B|=1 and |E|>1-ε.
We define a functionμ(x,y) in the following way:
(73)μ(x,y)={1,for(x,y)∈E∪(T∖B);μn,for(x,y)∈Ωn∖Ωn-1,n≥n0+1,
where
(74)μn=[22n·∏s=1n‍hs]-1,hs=∥fs∥C+maxNs-1≤n¯,m¯<Ns∥∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)∥C+max2Ns-1≤R≤2Ns∥∑2Ns-12≤k2+ν2≤R2‍ck,ν(s)ψk(x)·ψν(y)∥C+1.
From (68) and (70)–(74) we obtain the following:(A)  0<μ(x,y)≤1,μ(x,y) is a measurable function and
(75)|{(x,y)∈T:μ(x,y)≠1}|<ε.(B) Consider ∑k,ν=1∞‍|ck,ν|q<∞forallq>2.
Hence, obviously we have (see (68) and (70))
(76)limmin{k,ν}→∞ck,ν=0.
It follows from (72)–(74) that for all s≥n0 and Ns-1≤n¯,m¯<Ns(77)∫‍∫T∖Ωs‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|μ(x,y)dxdy=∑n=s+1∞‍[∫‍∫Ωn∖Ωn-1‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|μndxdy]≤∑n=s+1∞‍2-2n[∫‍∫T‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|hs-1dxdy]<132-2s.
Analogously for all s≥n0 and 2Ns-1≤R≤2Ns we have
(78)∫‍∫T∖Ωs‍|∑2Ns-12≤k2+ν2≤R2‍ck,ν(s)ψk(x)·ψν(y)|×μ(x,y)dxdy<132-2s.
By (65) and  (72)–(74) for all s≥n0 we have
(79)∫∫T|Ps(x,y)-fs(x,y)|μ(x,y)dxdy=∫‍∫Ωs‍|Ps(x,y)-fs(x,y)|μ(x,y)dxdy+∫‍∫T∖Ωs‍|Ps(x,y)-fs(x,y)|μ(x,y)dxdy=∑n=s+1∞‍[∫‍∫Ωn∖Ωn-1‍|Ps(x,y)-fs(x,y)|μndxdy]≤∑n=s+1∞‍2-2n[∑k,ν=Ns-1Ns-1‍∫‍∫T‍(∑k,ν=Ns-1Ns-1‍|fs(x,y)|+∑k,ν=Ns-1Ns-1‍ck,ν(s)ψk(x)·ψν(y)∑k,ν=Ns-1Ns-1‍)hs-1dxdy]<132-2s<2-2s.
By (69) and (72)–(77) for all Ns-1≤n¯,m¯<Ns and s≥n0+1 we obtain
(80)∫‍∫T‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|μ(x,y)dxdy∫‍∫Ωs‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|μ(x,y)dxdy∫‍∫T∖Ωs‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|μ(x,y)dxdy<∑n=n0+1s‍[∫‍∫Ωn∖Ωn-1‍|∑k,ν=Ns-1n¯,m¯‍ck,ν(s)ψk(x)·ψν(y)|·μndxdy]+132-2s<∑n=n0+1s‍(2-2(s+1)+2·∫‍∫Ωn∖Ωn-1‍|fs(x,y)|dxdy)·μn+132-2s=2-2(s+1)·∑n=n0+1s‍μn+∫‍∫Ωs‍|fs(x,y)|μ(x,y)dxdy+132-2s<2·∫‍∫T‍|fs(x,y)|μ(x,y)dxdy+2-2s.
Analogously for all s≥n0 and 2Ns-1≤R≤2Ns we have (see (78))
(81)∫‍∫T‍|∑2Ns-12≤k2+ν2≤R2‍ck,ν(s)ψk(x)·ψν(y)|μ(x,y)dxdy<2·∫‍∫T‍|fs(x,y)|μ(x,y)dxdy+2-2s.
Now we will show that the series (70) is universal in Lμ1(T) concerning subseries with respect to convergence by both spherical and rectangular partial sums.
Letf(x,y)∈Lμ1(T), that is,
(82)∫‍∫T‍|f(x,y)|μ(x,y)dxdy<∞.
It is easy to see that we can choose a functionfn1(x,y) from the sequence (64) such that
(83)∫‍∫T‍|f(x,y)-fn1(x,y)|μ(x,y)dxdy<2-2,n1>n0+1.
Hence, we have
(84)∫‍∫T‍|fn1(x,y)|μ(x,y)dxdy<2-2+∫‍∫T‍|f(x,y)|μ(x,y)dxdy.
From (80) and (83) we get
(85)∫‍∫T‍|f(x,y)-Pn1(x,y)|μ(x,y)dxdy≤∫‍∫T‍|f(x,y)-fn1(x,y)|μ(x,y)dxdy+∫‍∫T‍|fn1(x,y)-Pn1(x,y)|μ(x,y)dxdy<2·2-2.
Assume that numbers n1<n2<⋯<nq-1 are chosen in such a way that the following condition is satisfied:
(86)∫‍∫T‍|f(x,y)-∑s=1j‍Pns(x,y)|μ(x,y)dxdy<2·2-2j,1≤j≤q-1.
Now we choose a function fnq(x,y) from the sequence (64) such that
(87)∫‍∫T‍|(f(x,y)-∑s=1q-1‍Pns(x,y))-fnq(x,y)|μ(x,y)dxdy<2·2-2q,nq>nq-1.
This with (86) implies
(88)∫‍∫T‍|fnq(x,y)|μ(x,y)dxdy<2-2q+2·2-2(q-1)=9·2-2q.
Hence and from (65) and (79)–(81) we obtain
(89)∫‍∫T‍|fnq(x,y)-Pnq(x,y)|μ(x,y)dxdy<2-2nq,
where
(90)Pnq(x,y)=∑k,ν=Nnq-1Nnq-1‍ck,ν(nq)ψk(x)ψν(y),(91)maxNnq-1≤n¯,m¯<Nnq[∑k,ν=Nnq-1n¯,m¯‍∫‍∫T‍|∑k,ν=Nnq-1n¯,m¯‍ck,ν(nq)ψk(x)·ψν(y)∑k,ν=Nnq-1n¯,m¯‍|μ(x,y)dxdy∑k,ν=Nnq-1n¯,m¯‍]<19·2-2q.
Analogously we have
(92)max2Nnq-1≤R≤2Nnq[∫‍∫T‍|∑2Nnq-12≤k2+ν2≤R2‍ck,ν(nq)ψk(x)·ψν(y)∑2Nnq-12≤k2+ν2≤R2‍|μ(x,y)dxdy∑2Nnq-12≤k2+ν2≤R2‍]<19·2-2q.
In quality subseries of the theorem we will take
(93)∑q=1∞‍Pnq(x,y)=∑q=1∞‍[∑k,ν=Nnq-1Nnq-1‍ck,ν(nq)ψk(x)ψν(y)].
From (87) and (88) we have
(94)∫‍∫T‍|f(x,y)-∑s=1q‍Pns(x,y)|μ(x,y)dxdy≤∫‍∫T‍|(f(x,y)-∑s=1q-1‍Pns(x,y))-fnq(x,y)|×μ(x,y)dxdy+∫‍∫T‍|fnq(x,y)-Pnq(x,y)|μ(x,y)dxdy<2·2-2q.
Letn¯ and m¯ be arbitrary natural numbers. Then for some natural number q we have
(95)Nnq-1≤min{n¯,m¯}<Nnq.
Taking into account (89) and (93) for rectangular partial sums Sn¯,m¯(x,y) of (91) we get
(96)∫‍∫T‍|Sn¯,m¯(x,y)-f(x,y)|μ(x,y)dxdy≤∫‍∫T‍|f(x,y)-∑s=1q‍Pns(x,y)|μ(x,y)dxdy+maxNnq-1≤n¯,m¯<Nnq[∑k,ν=Nnq-1n¯,m¯‍∫‍∫T‍|∑k,ν=Nnq-1n¯,m¯‍ck,ν(nq)ψk(x)·ψν(y)∑k,ν=Nnq-1n¯,m¯‍|μ(x,y)dxdy∑k,ν=Nnq-1n¯,m¯‍]<21·2-2q.
Analogously for 2Nnq-1≤R≤2Nnq we have
(97)∫‍∫T‍|SR(x,y)-f(x,y)|μ(x,y)dxdy<21·2-2q,
where SR(x,y) is the spherical partial sums of (91).
From (96) and (97) we conclude that the series (70) is universal in Lμ1(T) concerning subseries with respect to convergence by both spherical and rectangular partial sums (see Definition 2).
Theorem3 is proved.Remark 8.
We can prove Theorem4 by the same method used in the proof of Theorem 3.



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*Source: 424686-2013-04-22.xml*