Universality Properties of a Double Series by the Generalized Walsh System

Abstract

We consider a question on existence of a double series by the generalized Walsh system, which is universal in weighted 𝐿¹_𝜇[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=1𝑐_𝑛,𝑘𝜓_𝑛(𝑥)𝜓_𝑘(𝑦) with the ∑^∞_𝑛,𝑘=1|𝑐_𝑛,𝑘|^𝑞<∞ for all 𝑞>2, which is universal in 𝐿¹_𝜇[0,1]² concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.

1. Introduction

Let 𝑋 be a Banach space.

Definition 1. A series ^∞_𝑘=1𝑓_𝑘,𝑓_𝑘∈𝑋 (1) 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=1𝑓_𝜎(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=1𝑓_𝑛𝑘 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]).

Let 𝑎≥2 be a fixed integer and 𝜔_𝑎=𝑒^{2𝜋𝑖/𝑎}. Recall the following definitions (see [11]).

The Rademacher system of order 𝑎 is defined inductively as follows. For 𝑛=0 let 𝜑₀(𝑥)=𝜔^𝑘_𝑎𝑘if𝑥∈𝑎,𝑘+1𝑎,𝑘=0,1,…,𝑎−1, (2) and for 𝑛≥1 let 𝜑_𝑛(𝑥+1)=𝜑_𝑛(𝑥)=𝜑₀𝑎^𝑛𝑥. (3)

The generalized Walsh system of order 𝑎 is defined by 𝜓₀(𝑥)=1, (4) and if 𝑛=𝛼₁𝑎^𝑛1+⋯+𝛼_𝑠𝑎^𝑛𝑠, where 𝑛₁>⋯>𝑛_𝑠,  0≤𝛼_𝑗<𝑎, 𝑗=1,2,…,𝑠, then 𝜓_𝑛(𝑥)=𝜑^𝛼1_𝑛1(𝑥)⋅…⋅𝜑^𝛼𝑠_𝑛𝑠(𝑥). (5)

We denote the generalized Walsh system of order 𝑎 by Ψ_𝑎. 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 𝐿¹_𝜇[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 of  Ψ_𝑎,  𝑎≥2, which will be useful later.(i)Each 𝑛th Rademacher function has period 𝑎^−𝑛 and 𝜑_𝑛(𝑥)=const∈Ω_𝑎=1,𝜔_𝑎,𝜔²_𝑎,…,𝜔_𝑎^𝑎−1, (6) if 𝑥∈Δ^(𝑘)_𝑛+1=[𝑘/𝑎^𝑛+1,(𝑘+1)/𝑎^𝑛+1),  𝑘=0,…,𝑎^𝑛+1−1, 𝑛=1,2,….(ii)  𝜑_𝑛(𝑥)^𝑘=𝜑_𝑛(𝑥)^𝑚,∀𝑛,𝑘∈𝒩,𝑚=𝑘(mod𝑎). (7)(iii)𝜓_𝑛(𝑥) is a finite product of the Rademacher functions with values in Ω_𝑎. (iv)  𝜓_{𝑎𝑘+𝑗}(𝑥)=𝜑_𝑘(𝑥)⋅𝜓_𝑗(𝑥),if0≤𝑗≤𝑎^𝑘−1. (8)(v)Ψ_𝑎, 𝑎≥2 is a complete orthonormal system in 𝐿²[0,1) and it is a basis in 𝐿^𝑝[0,1] for 𝑝>1.

The rectangular and spherical partial sums of the double series ^∞_𝑘,𝜈=1𝑐_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦,𝑥,𝑦∈𝑇=[0,1)² (9) will be denoted by 𝑆_𝑛,𝑚𝑥,𝑦=^𝑛𝑚_𝑘=1𝜈=1𝑐_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦,𝑆_𝑅𝑥,𝑦=_{𝜈2+𝑘2≤𝑅2}𝑐_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦. (10)

If 𝑔(𝑥,𝑦) is a continuous function on 𝑇=[0,1]², then we set 𝑔𝑥,𝑦_𝐶=max_{𝑥,𝑦∈𝑇}𝑔𝑥,𝑦. (11)

3. Main Results

Let us denote the generalized Walsh system of order 𝑎 by Ψ_𝑎, 𝑎≥2. These are the main results of the paper.

Theorem 3. There exists a double series of the form ^∞_𝑛,𝑘=1𝑐_𝑛,𝑘𝜓_𝑛(𝑥)𝜓_𝑘𝑦𝑤𝑖𝑡ℎ^∞_𝑛,𝑘=1𝑐_𝑛,𝑘^𝑞<∞∀𝑞>2 (12) with the following property: for any number 𝜀>0 a weighted function 𝜇(𝑥,𝑦) satisfying 0<𝜇𝑥,𝑦≤1,𝑥,𝑦∈𝑇∶𝜇𝑥,𝑦≠1<𝜀 (13) can be constructed so that the series (12) is universal in 𝐿¹_𝜇(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 𝐿¹_𝜇(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 numbers 0<𝜀<1, 𝑁₀>2  (𝑁₀∈𝒩) and a step function 𝑓(𝑥)=^𝑞_𝑠=1𝛾_𝑠⋅𝜒_Δ𝑠(𝑥), (14) where Δ_𝑠 is an interval of the form Δ_𝑚^(𝑖)=[(𝑖−1)/2^𝑚,𝑖/2^𝑚], 1≤𝑖≤2^𝑚, there exist a measurable set 𝐸⊂[0,1] and a polynomial 𝑃(𝑥) of the form 𝑃(𝑥)=^𝑁_𝑘=𝑁0𝑐_𝑘𝜓_𝑘(𝑥) (15) which satisfy the following conditions:(1)𝑃(𝑥)=𝑓(𝑥)𝑜𝑛𝐸, (16)(2)|𝐸|>(1−𝜀), (17)(3)^𝑁_𝑘=𝑁0𝑐_𝑘^2+𝜀<𝜀, (18)(4)max_{𝑁0≤𝑚<𝑁𝑒}^𝑚_𝑘=𝑁0𝑐_𝑘𝜓_𝑘(𝑥)𝑑𝑥<𝜀+_𝑒𝑓(𝑥)𝑑𝑥, (19)
for every measurable subset 𝑒 of 𝐸.

Then applying this Lemma we get the next one.

Lemma 6. For any numbers 𝛾≠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 𝑃𝑥,𝑦=^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦, (20)
with the following properties: (1)|𝐸|>1−𝛿, (21)(2)^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠^2+𝛿<𝛿, (22)(3)𝑃𝑥,𝑦=𝛾⋅𝜒_Δ𝑥,𝑦𝑓𝑜𝑟𝑥,𝑦∈𝐸, (23)(4)max_{𝑁≤𝑛,𝑚≤𝑀𝑒}^𝑛,𝑚_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦+max_{√√2𝑁≤𝑅≤2𝑀𝑒2𝑁2≤𝑘2+𝑠2≤𝑅2}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝛾⋅|Δ|,𝑑𝑥𝑑𝑦≤16⋅ (24)for every measurable subset 𝑒 of 𝐸.

Proof. We apply Lemma 5, setting 𝑓(𝑥)=𝛾⋅𝜒_Δ1(𝑥),𝑁₀𝛿=𝑁,𝜀=2. (25) Then we can define a measurable set 𝐸₁⊂[0,1] and a polynomial 𝑃₁(𝑥) of the form 𝑃₁(𝑥)=^𝑁1_𝑘=𝑁𝑎_𝑘𝜓_𝑘(𝑥) (26) which satisfy the following conditions:  (1⁰) 𝑃₁(𝑥)=𝛾⋅𝜒_Δ1(𝑥)for𝑥∈𝐸₁, (27) (2⁰) 𝐸₁𝛿>1−2, (28) (3⁰) ^𝑁1_𝑘=𝑁𝑎_𝑘^2+𝛿<𝛿, (29) (4⁰) max_{𝑁≤𝑛≤𝑁1𝑒1}^𝑛_𝑘=𝑁𝑎_𝑘𝜓_𝑘(𝑥)𝑑𝑥≤2⋅𝛾⋅Δ₁, (30)for every measurable subset 𝑒₁ of 𝐸₁.
Set 𝑀₀=2⋅𝑁²₁+1 (31) and apply Lemma 5 again, setting 𝑓𝑦=𝜒_Δ2𝑦,𝑁₀=𝑀₀𝛿,𝜀=2. (32) Then we can define a measurable set 𝐸₂⊂[0,1] and a polynomial 𝑃₂(𝑦) of the form 𝑃₂𝑦=^𝑀_𝑠=𝑀0𝑏_𝑠𝜓_𝑠𝑦, (33) which satisfy the following conditions: (1⁰⁰) 𝑃₂𝑦=𝜒_Δ2𝑦for𝑦∈𝐸₂, (34) (2⁰⁰) 𝐸₂𝛿>1−2, (35) (3⁰⁰) ^𝑀_𝑠=𝑀0𝑏_𝑠^2+𝛿<𝛿, (36) (4⁰⁰) max_{𝑀0≤𝑚≤𝑀𝑒2}^𝑚_𝑠=𝑀0𝑏_𝑠𝜓_𝑠𝑦𝑑𝑦≤2⋅Δ₂, (37) for every measurable subset 𝑒₂ of 𝐸₂.
Set 𝐸=𝐸₁×𝐸₂,𝑃𝑥,𝑦=𝑃₁(𝑥)⋅𝑃₂(𝑥)=^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦, (38) where 𝑐_𝑘,𝑠=𝑎_𝑘⋅𝑏_𝑠,if𝑁≤𝑘≤𝑁₁,𝑀₀𝑐≤𝑠≤𝑀,_𝑘,𝑠=0,forother𝑘,𝑠. (39) By (1⁰)–(3⁰),   (1⁰⁰)–(3⁰⁰), and (38), (39), we obtain |𝐸|>1−𝛿,_{𝑀𝑘,𝑠=𝑁}𝑐_𝑘,𝑠^2+𝛿=^𝑁1_𝑘=𝑁𝑎_𝑘^2+𝛿⋅^𝑀_𝑠=𝑀0𝑏_𝑠^2+𝛿𝑃<𝛿,𝑥,𝑦=𝛾⋅𝜒_Δ𝑥,𝑦for𝑥,𝑦∈𝐸. (40) Thus, the statements (1)–(3) of Lemma 6 are satisfied. Now we will check the fulfillment of statement (4).
Let 𝑁²+𝑀²₀<𝑅²<𝑁²₁+𝑀², then for some 𝑚₀>𝑀₀ we have 𝑚₀<𝑅<𝑚₀+1 and from (31) it follows that 𝑅²−𝑁²₁>(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 _{𝑒𝑁2+𝑀2≤𝑘2+𝑠2≤𝑅2}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦≤𝑑𝑥𝑑𝑦_{𝑒𝑁1𝑚𝑘=𝑁0−1𝑠=𝑀0}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦+max_{𝑁<𝑛≤𝑁1𝑒}^𝑛_𝑘=𝑁𝑐_𝑘,𝑚0𝜓_𝑘(𝑥)⋅𝜓_𝑚0𝑦≤𝑑𝑥𝑑𝑦_{𝑒1𝑁1𝑘=𝑁}𝑎_𝑘𝜓_𝑘(𝑥)𝑑𝑥⋅_𝑒2^𝑚0−1_𝑠=𝑀0𝑏_𝑠𝜓_𝑠𝑦+𝑏𝑑𝑦_𝑚0⋅_𝑒2𝜓_𝑚0𝑦𝑑𝑦⋅max_{𝑁<𝑛≤𝑁1𝑒1}^𝑛_𝑘=𝑁𝑎_𝑘𝜓_𝑘(𝑥𝛾⋅|Δ|.)𝑑𝑥≤12⋅ (41)
Similarly, for 𝑁≤𝑛≤𝑁₁,𝑀₀≤𝑚≤𝑀, we get _𝑒^𝑛,𝑚_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦≤4⋅𝛾⋅|Δ|. (42)
Lemma 6 is proved.

Lemma 7. For any numbers 𝜀>0, 𝑁>1 and a step function 𝑓𝑥,𝑦=^𝜈0_𝜈=1𝛾_𝜈⋅𝜒_Δ𝜈𝑥,𝑦, (43) there exists a measurable set 𝐸⊂𝑇 and a polynomial 𝑃(𝑥,𝑦) of the form 𝑃𝑥,𝑦=^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦, (44) which satisfy the following conditions: (1⁰)𝑃𝑥,𝑦=𝑓𝑥,𝑦𝑓𝑜𝑟𝑥,𝑦∈𝐸, (45) (2⁰)|𝐸|>1−𝜀, (46) (3⁰)^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠^2+𝜀<𝜀, (47)  (4⁰)max_{𝑁≤𝑛,𝑚<𝑀𝑒}^𝑛,𝑚_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦+max_{√√2𝑁≤𝑅≤2𝑀𝑒2𝑁2≤𝑘2+𝑠2≤𝑅2}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦≤2⋅_𝑒𝑓𝑥,𝑦𝑑𝑥𝑑𝑦+𝜀, (48) for every measurable subset 𝑒 of 𝐸.

Proof. Without any loss of generality, we assume that max_{1≤𝜈≤𝜈0}𝛾_𝜈⋅Δ_𝜈<𝜀,32 (49)Δ_𝜈,1≤𝜈≤𝜈₀ are the constancy rectangular domain of 𝑓(𝑥,𝑦), that is, where the function 𝑓(𝑥,𝑦) is constant.
Given an integer 1≤𝜈≤𝜈₀, by applying Lemma 6 with 𝛿=𝜀/16𝜈₀, we find that there exists a measurable set 𝐸_𝜈⊂𝑇 and a polynomial 𝑃_𝜈(𝑥,𝑦) of the form 𝑃_𝜈𝑥,𝑦=^𝑀𝜈_{𝑘,𝑠=𝑁𝜈}𝑐^(𝜈)_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦 (50) with the following properties: 𝐸_𝜈𝜀>1−2^𝜈_{𝑀𝜈𝑘,𝑠=𝑁𝜈},𝑐^(𝜈)_𝑘,𝑠^2+𝜀<𝜀𝜈₀,𝑃_𝜈𝑥,𝑦=𝛾_𝜈⋅𝜒_Δ𝜈𝑥,𝑦for𝑥,𝑦∈𝐸_𝜈,max_{𝑁𝜈≤𝑛,𝑚≤𝑀𝜈𝑒}^𝑛,𝑚_{𝑘,𝑠=𝑁𝜈}𝑐^(𝜈)_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦+max_{√2𝑁𝜈√≤𝑅≤2𝑀𝜈𝑒2𝑁2𝜈≤𝑘2+𝑠2≤𝑅2}𝑐^(𝜈)_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝛾𝑑𝑥𝑑𝑦≤16⋅_𝜈⋅Δ_𝜈<𝜀2, (51)(52)(53)(54) for every measurable subset 𝑒 of 𝐸_𝜈 (see (49)).
Then we can take 𝑁₁=𝑁,𝑛_𝜈=𝑀_𝜈−1+1,1≤𝜈≤𝜈₀. (55) Set 𝐸=^𝜈0_𝜈=1𝐸_𝜈,𝑃𝑥,𝑦=^𝜈0_𝜈=1𝑃_𝜈𝑥,𝑦=^𝑀_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦,𝑀=𝑀_𝜈0, (56)(57) where 𝑐_𝑘,𝑠=𝑐^(𝜈)_𝑘,𝑠,for𝑁_𝜈≤𝑘,𝑠≤𝑀_𝜈,1≤𝜈≤𝜈₀,𝑐_𝑘,𝑠=0,forother𝑘,𝑠. (58)(59) From (51)–(53) and (56)–(58) we obtain 𝑃|𝐸|𝑥,𝑦=𝑓𝑥,𝑦for𝑥,𝑦∈𝐸,>1−𝜀,_{𝑀𝑘,𝑠=𝑁}𝑐_𝑘,𝑠^2+𝜀<^𝜈0_𝜈=1^𝑀𝜈_{𝑘,𝑠=𝑁𝜈}𝑐^(𝜈)_𝑘,𝑠^2+𝜀<𝜀. (60)
Then,   let 𝑅∈[√2𝑁,√2𝑀], then for some 𝜈^′,1≤𝜈^′≤𝜈₀ we have √2𝑁_𝜈′≤𝑅≤√2𝑁_𝜈′+1, consequently from (57) and (58) we have _{2𝑁2≤𝑘2+𝑠2≤𝑅2}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦=^𝜈′−1_𝜈=1𝑃_𝜈𝑥,𝑦+_{2𝑁2𝜈′≤𝑘2+𝑠2≤𝑅2}𝑐^(𝜈′)_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦. (61)
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 _{𝑒2𝑁2≤𝑘2+𝑠2≤𝑅2}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦𝑑𝑥𝑑𝑦≤_𝑒^𝜈′−1_𝜈=1𝑃_𝜈+𝑥,𝑦𝑑𝑥𝑑𝑦_{𝑒2𝑁2𝜈′≤𝑘2+𝑠2≤𝑅2}𝑐^(𝜈′)_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦≤𝑑𝑥𝑑𝑦_𝑒𝑓𝜀𝑥,𝑦𝑑𝑥𝑑𝑦+2. (62) Similarly, for any 𝑒⊂𝐸 we have max_{𝑁≤𝑛,𝑚≤𝑀𝑒}^𝑛,𝑚_{𝑘,𝑠=𝑁}𝑐_𝑘,𝑠𝜓_𝑘(𝑥)⋅𝜓_𝑠𝑦≤𝑑𝑥𝑑𝑦_𝑒𝑓𝜀𝑥,𝑦𝑑𝑥𝑑𝑦+2. (63) Lemma 7 is proved.

4. Proofs of the Theorems

Theorem 3 is proved similarly [10, Theorem  3], but for maintenance of integrity of this paper, here we will give the proof.

Proof of Theorem 3. Let 𝑓_𝑠𝑥,𝑦^∞_𝑠=1,𝑥,𝑦∈𝑇 (64) 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 {𝐸_𝑠}^∞_𝑠=1 of sets and a sequence of polynomials 𝑃_𝑠𝑥,𝑦=∑^𝑁𝑠−1_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦,1=𝑁₀<𝑁₁<⋯<𝑁_𝑠<⋯,𝑠=1,2,…, (65) which satisfy the following conditions: 𝑃_𝑠𝑥,𝑦=𝑓_𝑠𝑥,𝑦,𝑥,𝑦∈𝐸_𝑠,𝐸_𝑠>1−2^−2(𝑠+1),𝐸_𝑠⊂𝑇,_{𝑁𝑠−1𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈^2+2−2𝑠<2^−2𝑠,max_{𝑁𝑠−1≤𝑛,𝑚<𝑁𝑠𝑒}^𝑛,𝑚_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝑑𝑥𝑑𝑦+max_{√2𝑁𝑠−1√≤𝑅≤2𝑁𝑠𝑒2𝑁2𝑠−1≤𝑘2+𝜈2≤𝑅2}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝑑𝑥𝑑𝑦≤2⋅_𝑒𝑓_𝑠𝑥,𝑦𝑑𝑥𝑑𝑦+2^−2(𝑠+1), (66)(67)(68)(69) for every measurable subset 𝑒 of 𝐸_𝑠.
Denote ^∞_𝑘,𝜈=1𝑐_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦=^∞_𝑠=1^𝑁𝑠−1_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦, (70) where 𝑐_𝑘,𝜈=𝑐^(𝑠)_𝑘,𝜈,for𝑁_𝑠−1≤𝑘,𝜈<𝑁_𝑠,𝑠=1,2,…. (71)
For an arbitrary number 𝜀>0 we set Ω_𝑛=^∞_𝑠=𝑛𝐸_𝑠,𝑛=1,2,…,𝐸=Ω_𝑛0=^∞_𝑠=𝑛0𝐸_𝑠,𝑛₀=log_1/1𝜀+1,𝐵=^∞_𝑛=𝑛0Ω_𝑛=Ω_𝑛0^∞_𝑛=𝑛0+1Ω_𝑛⧵Ω_𝑛−1. (72) It is obvious (see (67) and (72)) that |𝐵|=1 and |𝐸|>1−𝜀.
We define a function 𝜇(𝑥,𝑦) in the following way: 𝜇=();𝜇𝑥,𝑦1,for𝑥,𝑦∈𝐸∪𝑇⧵𝐵_𝑛,for𝑥,𝑦∈Ω_𝑛⧵Ω_𝑛−1,𝑛≥𝑛₀+1, (73) where 𝜇_𝑛=2^2𝑛⋅^𝑛_𝑠=1ℎ_𝑠⁻¹,ℎ_𝑠=𝑓_𝑠𝐶+max_{𝑁𝑠−1≤𝑛,𝑚<𝑁𝑠}^𝑛,𝑚_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦_𝐶+max_{√2𝑁𝑠−1√≤𝑅≤2𝑁𝑠2𝑁2𝑠−1≤𝑘2+𝜈2≤𝑅2}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦_𝐶+1. (74) From (68) and (70)–(74) we obtain the following: (A)  0<𝜇(𝑥,𝑦)≤1,𝜇(𝑥,𝑦) is a measurable function and 𝑥,𝑦∈𝑇∶𝜇𝑥,𝑦≠1<𝜀. (75) (B) Consider ∑^∞_𝑘,𝜈=1|𝑐_𝑘,𝜈|^𝑞<∞forall𝑞>2.
Hence, obviously we have (see (68) and (70)) lim_{{}min𝑘,𝜈→∞}𝑐_𝑘,𝜈=0. (76) It follows from (72)–(74) that for all 𝑠≥𝑛₀ and 𝑁_𝑠−1≤𝑛,𝑚<𝑁_{𝑠𝑇⧵Ω𝑠}^𝑛,𝑚_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇=𝑥,𝑦𝑑𝑥𝑑𝑦_{∞𝑛=𝑠+1Ω𝑛⧵Ω𝑛−1𝑛,𝑚𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇_𝑛≤𝑑𝑥𝑑𝑦_{∞𝑛=𝑠+1}2^−2𝑛_𝑇^𝑛,𝑚_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦ℎ_𝑠⁻¹<1𝑑𝑥𝑑𝑦32^−2𝑠. (77) Analogously for all 𝑠≥𝑛₀ and √2𝑁_𝑠−1≤𝑅≤√2𝑁_𝑠 we have _{𝑇⧵Ω𝑠2𝑁2𝑠−1≤𝑘2+𝜈2≤𝑅2}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦1×𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<32^−2𝑠. (78) By (65) and (72)–(74) for all 𝑠≥𝑛₀ we have _𝑇𝑃_𝑠𝑥,𝑦−𝑓_𝑠𝜇=𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦_Ω𝑠𝑃_𝑠𝑥,𝑦−𝑓_𝑠𝜇+𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦_{𝑇⧵Ω𝑠}𝑃_𝑠𝑥,𝑦−𝑓_𝑠𝜇=𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦_{∞𝑛=𝑠+1Ω𝑛⧵Ω𝑛−1}𝑃_𝑠𝑥,𝑦−𝑓_𝑠𝑥,𝑦𝜇_𝑛≤𝑑𝑥𝑑𝑦_{∞𝑛=𝑠+1}2^−2𝑛_𝑇𝑓_𝑠𝑥,𝑦+^𝑁𝑠−1_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦ℎ_𝑠⁻¹<1𝑑𝑥𝑑𝑦32^−2𝑠<2^−2𝑠. (79) By (69) and (72)–(77) for all 𝑁_𝑠−1≤𝑛,𝑚<𝑁_𝑠 and 𝑠≥𝑛₀+1 we obtain _𝑇^𝑛,𝑚_{𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦_{Ω𝑠𝑛,𝑚𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦_{𝑇⧵Ω𝑠𝑛,𝑚𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇<𝑥,𝑦𝑑𝑥𝑑𝑦_{𝑠𝑛=𝑛0+1Ω𝑛⧵Ω𝑛−1𝑛,𝑚𝑘,𝜈=𝑁𝑠−1}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦⋅𝜇_𝑛+1𝑑𝑥𝑑𝑦32^−2𝑠<^𝑠_𝑛=𝑛0+12^−2(𝑠+1)+2⋅_{Ω𝑛⧵Ω𝑛−1}𝑓_𝑠𝑥,𝑦𝑑𝑥𝑑𝑦⋅𝜇_𝑛+132^−2𝑠=2^−2(𝑠+1)⋅^𝑠_𝑛=𝑛0+1𝜇_𝑛+_Ω𝑠𝑓_𝑠𝜇+1𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦32^−2𝑠<2⋅_𝑇𝑓_𝑠𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦+2^−2𝑠. (80) Analogously for all 𝑠≥𝑛₀ and √2𝑁_𝑠−1≤𝑅≤√2𝑁_𝑠 we have (see (78)) _{𝑇2𝑁2𝑠−1≤𝑘2+𝜈2≤𝑅2}𝑐^(𝑠)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⋅_𝑇𝑓_𝑠𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦+2^−2𝑠. (81) Now we will show that the series (70) is universal in 𝐿¹_𝜇(𝑇) concerning subseries with respect to convergence by both spherical and rectangular partial sums.
Let 𝑓(𝑥,𝑦)∈𝐿¹_𝜇(𝑇), that is, _𝑇𝑓𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<∞. (82)
It is easy to see that we can choose a function 𝑓_𝑛1(𝑥,𝑦) from the sequence (64) such that _𝑇𝑓𝑥,𝑦−𝑓_𝑛1𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⁻²,𝑛₁>𝑛₀+1. (83) Hence, we have _𝑇𝑓_𝑛1𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⁻²+_𝑇𝑓𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦. (84) From (80) and (83) we get _𝑇𝑓𝑥,𝑦−𝑃_𝑛1𝜇≤𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦_𝑇𝑓𝑥,𝑦−𝑓_𝑛1𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦+_𝑇𝑓_𝑛1𝑥,𝑦−𝑃_𝑛1𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⋅2⁻². (85) Assume that numbers 𝑛₁<𝑛₂<⋯<𝑛_𝑞−1 are chosen in such a way that the following condition is satisfied: _𝑇𝑓𝑥,𝑦−^𝑗_𝑠=1𝑃_𝑛𝑠𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⋅2^−2𝑗,1≤𝑗≤𝑞−1. (86) Now we choose a function 𝑓_𝑛𝑞(𝑥,𝑦) from the sequence (64) such that _𝑇𝑓𝑥,𝑦−^𝑞−1_𝑠=1𝑃_𝑛𝑠𝑥,𝑦−𝑓_𝑛𝑞𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⋅2^−2𝑞,𝑛_𝑞>𝑛_𝑞−1. (87) This with (86) implies _𝑇𝑓_𝑛𝑞𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2^−2𝑞+2⋅2^{−2(𝑞−1)}=9⋅2^−2𝑞. (88) Hence and from (65) and (79)–(81) we obtain _𝑇𝑓_𝑛𝑞𝑥,𝑦−𝑃_𝑛𝑞𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2^−2𝑛𝑞, (89) where 𝑃_𝑛𝑞𝑥,𝑦=^{𝑁𝑛𝑞−1}_{𝑘,𝜈=𝑁𝑛𝑞−1}𝑐^(𝑛𝑞)_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦,max_{𝑁𝑛𝑞−1≤𝑛,𝑚<𝑁𝑛𝑞𝑇}^𝑛,𝑚_{𝑘,𝜈=𝑁𝑛𝑞−1}𝑐^(𝑛𝑞)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<19⋅2^−2𝑞. (90)(91) Analogously we have max_{√2𝑁𝑛𝑞−1√≤𝑅≤2𝑁𝑛𝑞𝑇2𝑁2𝑛𝑞−1≤𝑘2+𝜈2≤𝑅2}𝑐^(𝑛𝑞)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<19⋅2^−2𝑞. (92) In quality subseries of the theorem we will take ^∞_𝑞=1𝑃_𝑛𝑞𝑥,𝑦=^∞_𝑞=1^{𝑁𝑛𝑞−1}_{𝑘,𝜈=𝑁𝑛𝑞−1}𝑐^(𝑛𝑞)_𝑘,𝜈𝜓_𝑘(𝑥)𝜓_𝜈𝑦. (93) From (87) and (88) we have _𝑇𝑓𝑥,𝑦−^𝑞_𝑠=1𝑃_𝑛𝑠𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦≤_𝑇𝑓𝑥,𝑦−^𝑞−1_𝑠=1𝑃_𝑛𝑠𝑥,𝑦−𝑓_𝑛𝑞+𝑥,𝑦×𝜇𝑥,𝑦𝑑𝑥𝑑𝑦_𝑇𝑓_𝑛𝑞𝑥,𝑦−𝑃_𝑛𝑞𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<2⋅2^−2𝑞. (94)
Let 𝑛 and 𝑚 be arbitrary natural numbers. Then for some natural number 𝑞 we have 𝑁_𝑛𝑞−1≤min{𝑛,𝑚}<𝑁_𝑛𝑞. (95) Taking into account (89) and (93) for rectangular partial sums 𝑆_𝑛,𝑚(𝑥,𝑦) of (91) we get _𝑇𝑆_𝑛,𝑚𝜇≤𝑥,𝑦−𝑓𝑥,𝑦𝑥,𝑦𝑑𝑥𝑑𝑦_𝑇𝑓𝑥,𝑦−^𝑞_𝑠=1𝑃_𝑛𝑠𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦+max_{𝑁𝑛𝑞−1≤𝑛,𝑚<𝑁𝑛𝑞𝑇}^𝑛,𝑚_{𝑘,𝜈=𝑁𝑛𝑞−1}𝑐^(𝑛𝑞)_𝑘,𝜈𝜓_𝑘(𝑥)⋅𝜓_𝜈𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<21⋅2^−2𝑞. (96) Analogously for √2𝑁_𝑛𝑞−1≤𝑅≤√2𝑁_𝑛𝑞 we have _𝑇𝑆_𝑅𝑥,𝑦−𝑓𝑥,𝑦𝜇𝑥,𝑦𝑑𝑥𝑑𝑦<21⋅2^−2𝑞, (97) where 𝑆_𝑅(𝑥,𝑦) is the spherical partial sums of (91).
From (96) and (97) we conclude that the series (70) is universal in 𝐿¹_𝜇(𝑇) concerning subseries with respect to convergence by both spherical and rectangular partial sums (see Definition 2).
Theorem 3 is proved.

Remark 8. We can prove Theorem 4 by the same method used in the proof of Theorem 3.