On the Convergence of Absolute Summability for Functions of Bounded Variation in Two Variables

Abstract

By adopting some new ideas, we obtain the estimates of an absolute convergence for the functions of the bounded variation in two variables. Our results generalize the related results of Humphreys and Bojanic (1999) and Wang and Yu (2003) from one dimension to two dimensions and can be applied to several summability methods.

1. Introduction

Let 𝑓 be 2𝜋-periodic functions integrable on [−𝜋,𝜋]. Denote its Fourier series by 𝑆∼𝑎𝑓,𝑥⁰2+^∞_𝑘=1𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥, (1.1) where the Fourier coefficients 𝑎_𝑘 and 𝑏_𝑘 are defined as follows: 𝑎_𝑘1∶=𝜋^𝜋_−𝜋𝑓(𝑡)cos𝑘𝑡𝑑𝑡,𝑏_𝑘1∶=𝜋^𝜋_−𝜋𝑓(𝑡)sin𝑘𝑡𝑑𝑡. (1.2)

Denoted by 𝑆_𝑛(𝑓,𝑥) the 𝑛th partial sums of the Fourier series (1.1), that is, 𝑆_𝑛𝑎𝑓,𝑥∶=⁰2+^𝑛_𝑘=1𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥. (1.3)

When 𝑓 is of bounded variation, Bojanić [1] obtained the following result on the rate of the convergence of approximation by 𝑆_𝑛(𝑓,𝑥).

Theorem A. If a periodic function 𝑓 is of bounded variation on the interval [−𝜋,𝜋], then the following estimate holds for every 𝑥 and 𝑛=1,2,…: 𝑆_𝑛(𝑥≤3𝑓,𝑥−𝑓)𝑛^𝑛_𝑘=1𝑉₀^𝜋/𝑘𝜑_𝑥, (1.4) where 𝜑_𝑥(𝑡)∶=𝑓(𝑥+𝑡)+𝑓(𝑥−𝑡)−𝑓(𝑥+0)−𝑓(𝑥−0) and 𝑉(𝜑_𝑥,[0,𝑡]) is the total variation of 𝜑_𝑥 on [0,𝑡].

There are a lot of interesting generalizations that have been achieved by many authors (see [1–9]). Among them, Jenei [4] and Móricz [5] generalized Theorem A to the double Fourier series of functions of bounded variation; Humphreys and Bojanic [3], Wang and Yu [7] investigated the absolute convergence of Cesàro means of Fourier series.

Let 𝐶_𝛼∶=(𝑐^𝛼_𝑛𝑘) be the Cesàro matrix of order 𝛼, that is, 𝑐^𝛼_𝑛𝑘𝐴∶=^{𝛼−1𝑛−𝑘}𝐴^𝛼_𝑛,𝑘=0,1,…,𝑛, (1.5) where 𝐴^𝛼_𝑛Γ()∶=𝑛+𝛼+1Γ(𝛼+1)Γ(𝑛+1),𝑛=0,1,…. (1.6)

Then, the so-called Cesàro means of order 𝛼 of (1.1) is defined by (denote by 𝜎^𝛼_𝑛(𝑓,𝑥)): 𝜎^𝛼_𝑛=1𝑓,𝑥𝐴^𝛼_𝑛^𝑛_𝑘=0𝐴^𝛼−1_𝑛−𝑘𝑆_𝑘𝑓,𝑥. (1.7)

Humphreys and Bojanic [3] investigated the rate of the absolute convergence of Cesàro means of the series (1.1). Their result can be read as follows.

Theorem B. Let 𝑥∈[0,𝜋] and 𝑓 be 2𝜋-periodic functions of bounded variation on [−𝜋,𝜋]. Then for 𝛼>0 and 𝑛≥2, one has 𝑅^𝛼_𝑛𝑓,𝑥≤4𝛼𝑛𝜋^𝑛_𝑘=1𝑉₀^𝜋/𝑘𝜑_𝑥, (1.8) where 𝑅^𝛼_𝑛𝑓,𝑥∶=^∞_𝑘=𝑛+1𝜎^𝛼_𝑘𝑓,𝑥−𝜎^𝛼_𝑘−1,𝜑𝑓,𝑥_𝑥(𝑡)∶=𝑓(𝑥+𝑡)+𝑓(𝑥−𝑡)−𝑓(𝑥+0)−𝑓(𝑥−0) (1.9) and 𝑣𝑎𝑟^𝑏_𝑎(𝑓) be the total variation of 𝑓 on [𝑎,𝑏].

However, Wang and Yu [7] showed that Theorem A is not correct when 0<𝛼<1. In fact, they proved the following.

Theorem C. Suppose 0<𝛼<1, 𝑥∈[0,𝜋] and 𝑓 are 2𝜋-periodic functions of bounded variation on [−𝜋,𝜋]. Then for 𝑛≥2, one has 𝑅^𝛼_𝑛𝑓,𝑥≤100𝛼²𝑛^𝛼𝑛_𝑘=1𝑘^𝛼−1𝑉₀^𝜋/𝑘𝜑_𝑥, (1.10) and there exists a 2𝜋-periodic function 𝑓^∗ of bounded variation on [−𝜋,𝜋] and a point 𝑥∈[0,𝜋] such that 𝑅^𝛼_𝑛𝑓^∗>1,𝑥20000𝛼𝑛^𝛼𝑛_𝑘=1𝑘^𝛼−1𝑉₀^𝜋/𝑘𝜑_𝑥(𝑛≥8). (1.11)

Motivated by Theorems B and C, and the results of Jenei [4] and Móricz [5] on the double Fourier series, we will investigate the absolute convergence of a kind of very general summability of the double Fourier series. Our new results not only generalize Theorems B and C to the double case, but also can be applied to many other classical summability methods. We will present our main results on Section 2. Proofs will be given in Section 3. In Section 4, we will apply our results to some classical summability methods.

2. The Main Results

Let 𝑓(𝑥,𝑦) be a function periodic in each variable with period 2𝜋 and integrable on the two-dimensional torus 𝕋²=𝕋×𝕋 in Lebesgue’s sense, in symbol, 𝑓∈𝐿(𝕋²). The double Fourier series of a complex-valued function 𝑓∈𝐿(𝕋²) is defined by 𝑓𝑥,𝑦∼_{𝑘∈ℤ𝑙∈ℤ}𝑓(𝑘,𝑙)𝑒^{𝑖(𝑘𝑥+𝑙𝑦)}, (2.1) where the 𝑓(𝑘,𝑙) are the Fourier coefficients of 𝑓: 𝑓()1𝑘,𝑙∶=4𝜋²_𝕋²𝑓(𝑢,𝑣)𝑒^{−𝑖(𝑘𝑢+𝑙𝑣)}𝑑𝑢𝑑𝑣,(𝑘,𝑙)∈ℤ². (2.2)

Let 𝑅∶=[𝑎₁,𝑏₁]×[𝑎₂,𝑏₂] be a bounded and closed rectangle on the plane. A function 𝑓(𝑥,𝑦) defined on 𝑅 is said to be of bounded variation over 𝑅 in the sense of Hardy-Krause, in symbol, 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝑅), if(i) the total variation 𝑉(𝑓,𝑅) of 𝑓(𝑥,𝑦) over 𝑅 is finite, that is, 𝑉𝑓,𝑅∶=sup^𝑚𝑛_𝑗=1𝑘=1𝑓𝑥_𝑗,𝑦_𝑘−𝑓𝑥_𝑗−1,𝑦_𝑘−𝑓𝑥_𝑗,𝑦_𝑘−1+𝑓𝑥_𝑗,𝑦_𝑘<∞, (2.3) where the supremum is extended for all finite partitions 𝑎₁=𝑥₀<𝑥₁<⋯<𝑥_𝑚=𝑏₁,𝑎₂=𝑦₀<𝑦₁<⋯<𝑦_𝑛=𝑏₂ (2.4) of the intervals [𝑎₁,𝑏₁] and [𝑎₂,𝑏₂];(ii) the marginal functions 𝑓(⋅,𝑎₂) and 𝑓(𝑎₁,⋅) are of bounded variation over the intervals [𝑎₁,𝑏₁] and [𝑎₂,𝑏₂], respectively.

For any 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝕋), define 𝑓1𝑥,𝑦∶=4𝑓,𝜑𝑥+0,𝑦+0+𝑓𝑥−0,𝑦+0+𝑓𝑥+0,𝑦−0+𝑓𝑥−0,𝑦−0_𝑥𝑦(𝑢,𝑣)∶=𝑓𝑥+𝑢,𝑦+𝑣+𝑓𝑥−𝑢,𝑦+𝑣+𝑓𝑥+𝑢,𝑦−𝑣+𝑓𝑥−𝑢,𝑦−𝑣−4𝑓𝑥,𝑦. (2.5)

For convenience, write 𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦∶=𝑉𝜑_𝑥𝑦,[0,𝑠]×[0,𝑡]. (2.6)

For any double sequence {𝑎_𝑚𝑛}, define Δ₁₁𝑎_𝑚𝑛∶=𝑎_𝑚𝑛−𝑎_{𝑚−1,𝑛}−𝑎_{𝑚,𝑛−1}+𝑎_{𝑚−1,𝑛−1},Δ₁₀𝑎_𝑚𝑛∶=𝑎_𝑚𝑛−𝑎_{𝑚−1,𝑛},Δ₀₁𝑎_𝑚𝑛∶=𝑎_𝑚𝑛−𝑎_{𝑚,𝑛−1}. (2.7)

For any fourfold sequence {𝑎_{𝑚𝑛𝑗𝑘}}, write Δ₁₁𝑎_{𝑚𝑛𝑗𝑘}∶=𝑎_{𝑚𝑛𝑗𝑘}−𝑎_{𝑚,𝑛,𝑗+1,𝑘}−𝑎_{𝑚,𝑛,𝑗,𝑘+1}+𝑎_{𝑚,𝑛,𝑗+1,𝑘+1},Δ₀₁𝑎_{𝑚𝑛𝑗𝑘}∶=𝑎_{𝑚𝑛𝑗𝑘}−𝑎_{𝑚,𝑛,𝑗,𝑘+1},Δ₁₀𝑎_{𝑚𝑛𝑗𝑘}∶=𝑎_{𝑚,𝑛,𝑗,𝑘}−𝑎_{𝑚,𝑛,𝑗+1,𝑘}. (2.8)

A doubly infinite matrix 𝑇∶=(𝑡_{𝑚𝑛𝑗𝑘}) is said to be doubly triangular if 𝑡_{𝑚𝑛𝑗𝑘}=0 for 𝑗>𝑚 or 𝑘>𝑛. The 𝑚𝑛th term of the 𝑇-transform of the double Fourier series (2.1) is defined by 𝑇_𝑚𝑛𝑥,𝑦∶=^𝑚𝑛_𝜇=0𝜈=0𝑡_{𝑚𝑛𝜇𝜈}𝑆_𝜇𝜈𝑥,𝑦, (2.9) where 𝑆_𝜇𝜈(𝑥,𝑦) is the 𝜇𝜈th partial sum of (2.1), that is, 𝑆_𝜇𝜈𝑥,𝑦∶=^𝜇_|𝑘|^𝜈_=0|𝑙|=0𝑓(𝑘,𝑙)𝑒^{𝑖(𝑘𝑥+𝑙𝑦)}∶=^𝜇_|𝑘|^𝜈_=0|𝑙|=0𝐴_𝑘𝑙𝑥,𝑦. (2.10)

Write 𝑡_{𝑚𝑛𝑗𝑘}∶=^𝑚𝑛_{𝜇=𝑗𝜈=𝑘}𝑡_{𝑚𝑛𝑗𝑘},0≤𝑗≤𝑚,0≤𝑘≤𝑛,𝑡_{𝑚𝑛𝑗𝑘}𝑡̂=0,if𝑗>𝑚or𝑘>𝑛,_{(1,1)𝑚𝑛𝑗𝑘}∶=𝑡_{𝑚𝑛𝑖𝑗}−𝑡_{𝑚−1,𝑛,𝑖,𝑗}−𝑡_{𝑚,𝑛−1,𝑖,𝑗}+𝑡_{𝑚−1,𝑛−1,𝑖,𝑗}. (2.11)

Set 𝑅^(1,1)_𝑚𝑛𝑓;𝑥,𝑦∶=^∞∞_{𝑗=𝑚+1𝑘=𝑛+1}Δ₁₁𝑇_𝑗𝑘𝑥,𝑦. (2.12)

Our main results are the following Theorems 2.1 and 2.2.

Theorem 2.1. Let 𝑓(𝑥,𝑦) be a periodic function and 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝕋²). Assume that 𝑇 is a lower doubly triangular matrix satisfying 𝑡̂^(1,1)_{𝑚𝑛𝑘0}=𝑡̂^(1,1)_{𝑚𝑛0𝑙}=0,𝑘,𝑙=0,1,…,_{𝑚𝑛𝑘=1𝑙=1}𝑡̂^(1,1)_{𝑚𝑛𝑘𝑙}=𝑂(1), (2.13)(2.14) and there exist constants 𝛼,𝛽(0<𝛼,𝛽≤1) such that 𝑡̂^𝑚𝑛_𝑘=1𝑙=1^(1,1)_{𝑚𝑛𝑘𝑙}=𝑂𝑙𝑘𝑙sin𝑘𝑥sin𝑙𝑦^−1−𝛽𝑥𝑦^−𝛽,∈𝜋𝑥,𝑦0,𝑚×𝜋𝑛,𝑂𝑘,𝜋_−1−𝛼𝑥^−𝛼𝑦,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋0,𝑛,𝑂𝑘^−1−𝛼𝑙^−1−𝛽𝑥^−𝛼𝑦^−𝛽,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋. (2.15)
Then for 𝑛≥2, (𝑥,𝑦)∈𝕋², one has 𝑅^(1,1)_𝑚𝑛1𝑓;𝑥,𝑦≤𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑘=1𝑙=1𝑘^𝛼−1𝑙^𝛽−1𝑉^{(𝜋/𝑘)(𝜋/𝑙)}₀₀𝜑_𝑥𝑦. (2.16)

Theorem 2.2. Let 𝑓(𝑥,𝑦) and 𝑇 satisfy all the conditions of Theorem 2.1, except (2.15) is replaced by ^𝑚𝑛_𝑘=1𝑙=1Δ_𝑘^∗_𝑙^∗𝑡̂^(1,1)_{𝑚𝑛𝑘𝑙}𝑘^{𝜎(𝑘∗)}𝑙^{𝜎(𝑙∗)}=𝑂𝑚^{𝜏(𝑘∗)}𝑛^{𝜏(𝑙∗)}, (2.17) where 𝑘^∗,𝑙^∗=0 or 1, 𝑘^∗+𝑙^∗≥1, 𝜏(0)=𝜎(0)=0, 𝜏(1)=𝜎(1)+1=2. Then for 𝑛≥2, (𝑥,𝑦)∈𝕋², one has 𝑅^(1,1)_𝑚𝑛1𝑓;𝑥,𝑦≤𝐶𝑚𝑛^𝑚𝑛_𝑘=1𝑙=1𝑉^{(𝜋/𝑘)(𝜋/𝑙)}₀₀𝜑_𝑥𝑦. (2.18)

3. Proofs of Results

Lemma 3.1 (see [10]). If 𝑔(𝑥,𝑦) is continuous on a rectangle 𝑅∶=[𝑎₁,𝑏₁]×[𝑎₂,𝑏₂] and 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝑅), then(i)𝑔(𝑥,𝑦) is integrable with respect to 𝑓(𝑥,𝑦) over 𝑅 in the sense of Riemann-Stieltjes integral;(ii)𝑓(𝑥,𝑦) is integrable with respect to 𝑔(𝑥,𝑦) over 𝑅 in the sense of Riemann-Stieltjes integral and _𝑅𝑓𝑥,𝑦𝑑_𝑥𝑑_𝑦𝑔𝑥,𝑦=_𝑅𝑔𝑥,𝑦𝑑_𝑥𝑑_𝑦𝑓+𝑥,𝑦_𝑏1𝑎1𝑓𝑥,𝑏₂𝑑_𝑥𝑔𝑥,𝑏₂−^𝑏1_𝑎1𝑓𝑥,𝑎₂𝑑_𝑥𝑔𝑥,𝑎₂+^𝑏2_𝑎2𝑓𝑏₁,𝑦𝑑_𝑦𝑔𝑏₁,𝑦−^𝑏2_𝑎2𝑓𝑎₁,𝑦𝑑_𝑦𝑔𝑎₁𝑏,𝑦−𝑓₁,𝑏₂𝑔𝑏₁,𝑏₂+𝑓𝑎₁,𝑏₂𝑔𝑎₁,𝑏₂+𝑓𝑎₂,𝑏₁𝑔𝑎₂,𝑏₁−𝑓𝑎₁,𝑎₂𝑔𝑎₁,𝑎₂. (3.1)

Lemma 3.2 (see [10]). If 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝑅) and 𝑔(𝑥,𝑦) is absolutely continuous on 𝑅, that is, 𝑔𝑥,𝑦=^𝑥_𝑎1^𝑦_𝑎2ℎ(𝑢,𝑣)𝑑𝑢𝑑𝑣+𝐶,𝑥,𝑦∈𝑅, (3.2) for some function ℎ(𝑥,𝑦)∈𝐿(𝑅) and constant 𝐶, then _𝑅𝑓𝑥,𝑦𝑑_𝑥𝑑_𝑦𝑔𝑥,𝑦=_𝑅𝑓𝑥,𝑦ℎ𝑥,𝑦𝑑𝑥𝑑𝑦. (3.3)

Proof of Theorem 2.1. By the definition of 𝑇_𝑚𝑛(𝑥,𝑦) (see (2.9)), we have 𝑇_𝑗𝑘𝑥,𝑦=^𝑗_𝑖1^𝑘_=0𝑖2=0𝑡_{𝑗𝑘𝑖1𝑖2}^𝑖1_𝜇^𝑖₌₀²_|𝜈|=0𝐴_𝜇𝜈=𝑥,𝑦_{𝑗𝜇𝑘=0|𝜈|=0}𝐴_𝜇𝜈𝑥,𝑦𝑡_{𝑗𝑘|𝜇||𝜈|}. (3.4)
Therefore, Δ₁₁𝑇_𝑗𝑘𝑥,𝑦=^𝑗_𝜇^𝑘_=0|𝜈|=0𝐴_𝜇𝜈𝑥,𝑦𝑡_{𝑗𝑘|𝜇||𝜈|}−^𝑗−1_𝜇^𝑘_=0|𝜈|=0𝐴_𝜇𝜈𝑥,𝑦𝑡_{𝑗−1,𝑘,|𝜇|,|𝜈|}−^𝑗_𝜇=0^𝑘−1_|𝜈|=0𝐴_𝜇𝜈𝑥,𝑦𝑡_{𝑗,𝑘−1,|𝜇|,|𝜈|}+^𝑗−1_𝜇=0^𝑘−1_|𝜈|=0𝐴_𝜇𝜈𝑥,𝑦𝑡_{𝑗−1,𝑘−1,|𝜇|,|𝜈|}=^𝑗_𝜇^𝑘_=0|𝜈|=0𝐴_𝜇𝜈𝑥,𝑦𝑡̂_𝜇𝜈|^(1,1)_𝑗𝑘|(=by2.11)_{𝑗𝑘𝜇=1𝜈=1}𝐴_𝜇𝜈𝑥,𝑦+𝐴_−𝜇,𝜈𝑥,𝑦+𝐴_𝜇,−𝜈𝑥,𝑦+𝐴_{−𝜇,−𝜈}𝑥,𝑦𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}by(2.13). (3.5)
Noting that 𝐴_𝜇𝜈𝑥,𝑦+𝐴_−𝜇,𝜈𝑥,𝑦+𝐴_𝜇,−𝜈𝑥,𝑦+𝐴_{−𝜇,−𝜈}=𝑓𝑒𝑥,𝑦𝜇,𝜈^{𝑖(𝜇𝑥+𝜈𝑦)}+𝑓−𝜇,𝜈𝑒^{𝑖(−𝜇𝑥+𝜈𝑦)}+𝑓𝜇,−𝜈𝑒^{𝑖(𝜇𝑥−𝜈𝑦)}+𝑓−𝜇,−𝜈𝑒^{𝑖(−𝜇𝑥−𝜈𝑦)}=14𝜋²_𝕋²𝑓(𝑠,𝑡)cos𝜇(𝑠−𝑥)cos𝜈𝑡−𝑦𝑑𝑠𝑑𝑡, (3.6) we have Δ₁₁𝑇_𝑗𝑘=1𝑥,𝑦4𝜋^2𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈𝕋2}𝑓()()=1𝑠,𝑡cos𝜇𝑠−𝑥cos𝜈𝑡−𝑦𝑑𝑠𝑑𝑡4𝜋^2𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}^𝜋₀𝜑_𝑥𝑦()=1𝑠,𝑡cos𝜇𝑠cos𝜈𝑡𝑑𝑠𝑑𝑡4𝜋^2𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}𝜇𝜈^𝜋₀𝜑_𝑥𝑦(𝑠,𝑡)𝑑_𝑠𝑑_𝑡sin𝜇𝑠sin𝜈𝑡, (3.7) where in the last equality, Lemma 3.2 is applied.
By (3.7) and Lemma 3.1, we have 𝑅^(1,1)_𝑚𝑛=1𝑓;𝑥,𝑦4𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+1}^𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}𝜇𝜈^𝜋₀𝜑_𝑥𝑦(𝑠,𝑡)𝑑_𝑠𝑑_𝑡=1sin𝜇𝑠sin𝜈𝑡4𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+1}^𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}𝜇𝜈^𝜋₀sin𝜇𝑠sin𝜈𝑡𝑑_𝑠𝑑_𝑡𝜑_𝑥𝑦()≤1𝑠,𝑡4𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+1}^𝜋₀^𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}𝑑𝜇𝜈sin𝜇𝑠sin𝜈𝑡_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦=14𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+10}^𝜋/𝑗₀^𝜋/𝑘+₀^{𝜋/𝑗𝜋}_𝜋/𝑘+^𝜋_𝜋/𝑗0^𝜋/𝑘+^𝜋_𝜋/𝑗^𝜋_𝜋/𝑘×^𝑗𝑘_𝜇=1𝜈=1𝑡̂^(1,1)_{𝑗𝑘𝜇𝜈}𝑑𝜇𝜈sin𝜇𝑠sin𝜈𝑡_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦=∶𝐼₁+𝐼₂+𝐼₃+𝐼₄. (3.8)
By (2.14), we have (Denote by 𝜒_{(𝑎1,𝑏1)×(𝑎2,𝑏2)}(𝑥,𝑦) and 𝜒_{(𝑎1,𝑏1)}(𝑥) the characteristic functions of (𝑎₁,𝑏₁)×(𝑎₂,𝑏₂) and (𝑎₁,𝑏₁), respectively.)𝐼₁=14𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+10}^𝜋/𝑗₀^{𝜋/𝑘𝑗𝑘}_𝜇=1𝜈=1𝑡̂_{𝑗𝑘𝜇𝜈}𝑑𝜇𝜈sin𝜇𝑠sin𝜈𝑡_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤14𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+10}^𝜋/𝑗₀^{𝑗𝜋/𝑘𝑘}_𝜇=1𝜈=1𝑡̂_{𝑗𝑘𝜇𝜈}𝑠𝑡𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶^∞∞_{𝑗=𝑚+1𝑘=𝑛+10}^𝜋/𝑗₀^𝜋/𝑘𝑠𝑡𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶₀^{𝜋/(𝑚+1)}₀^{∞𝜋/(𝑛+1)∞}_{𝑗=𝑚+1𝑘=𝑛+1}𝑠𝑡𝜒_{(0,𝜋/𝑗)×(0,𝜋/𝑘)}(𝑠,𝑡)𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶₀^{𝜋/(𝑚+1)}₀^{𝜋/(𝑛+1)}𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶𝑉^{(𝜋/(𝑚+1))(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦≤𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑗=1𝑘=1𝑗^𝛼−1𝑘^𝛽−1𝑉^{(𝜋/𝑗)(𝜋/𝑘)}₀₀𝜑_𝑥𝑦. (3.9)
For 𝐼₂, by the first inequality of (2.15), 𝐼₂=14𝜋^2∞∞_{𝑗=𝑚+1𝑘=𝑛+10}^{𝜋/𝑗𝜋}_𝜋/𝑘^𝑗𝑘_𝜇=1𝜈=1𝑡̂_{𝑗𝑘𝜇𝜈}𝑑𝜇𝜈sin𝜇𝑠sin𝜈𝑡_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶^∞∞_{𝑗=𝑚+1𝑘=𝑛+1}𝑘^−1−𝛽₀^{𝜋/𝑗𝜋}_𝜋/𝑘𝑠𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶^∞∞_{𝑗=𝑚+1𝑘=𝑛+1}𝑘^−1−𝛽₀^{𝜋/𝑗𝜋/(𝑛+1)}_𝜋/𝑘+₀^{𝜋/𝑗𝜋}_{()𝜋/𝑛+1}𝑠𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶₀^{𝜋/(𝑚+1)}₀^{∞𝜋/(𝑛+1)}_𝑗=𝑚+1𝑠𝜒_{(0,𝜋/𝑗)}(𝑠)^∞_{()𝑘>𝜋/𝑡}𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦+𝐶₀^{𝜋/(𝑚+1)𝜋}₍₎^∞_{𝜋/𝑛+1𝑗=𝑚+1}𝑠𝜒_{(0,𝜋/𝑗)}(𝑠)^∞_𝑘=𝑛+1𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶₀^{𝜋/(𝑚+1)}₀^{𝜋/(𝑛+1)}𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦+𝐶𝑛^−𝛽₀^{𝜋/(𝑚+1)𝜋}_{()𝜋/𝑛+1}𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶𝑉^{(𝜋/(𝑚+1))(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦+𝐶𝑛^𝑛−𝛽_𝑘=10^{𝜋/(𝑚+1)}₍₎^𝜋/𝑘_𝜋/𝑘+1𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶𝑉^{(𝜋/(𝑚+1))(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦+𝐶𝑛^𝑛−𝛽_𝑘=1𝑘^𝛽−1𝑉^{(𝜋/(𝑚+1))(𝜋/𝑘)}₀₀𝜑_𝑥𝑦≤𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑗=1𝑘=1𝑗^𝛼−1𝑘^𝛽−1𝑉^{(𝜋/𝑗)(𝜋/𝑘)}₀₀𝜑_𝑥𝑦. (3.10)
Similarly, we also have 𝐼₃≤𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑗=1𝑘=1𝑗^𝛼−1𝑘^𝛽−1𝑉^{(𝜋/𝑗)(𝜋/𝑘)}₀₀𝜑_𝑥𝑦. (3.11)
By (2.15), we deduce that 𝐼₄≤𝐶^∞∞_{𝑗=𝑚+1𝑘=𝑛+1}^𝜋_𝜋/𝑗^𝜋_𝜋/𝑘𝑗^−1−𝛼𝑘^−1−𝛽𝑠^−𝛼𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶^∞∞_{𝑗=𝑚+1𝑘=𝑛+1}^{𝜋/(𝑚+1)}_𝜋/𝑗^{𝜋/(𝑛+1)}_𝜋/𝑘+^𝜋_{()𝜋/𝑚+1}^{𝜋/(𝑛+1)}_𝜋/𝑘+^{𝜋/(𝑚+1)}_𝜋/𝑗^𝜋_{()𝜋/𝑛+1}+^𝜋_{()𝜋/𝑚+1}^𝜋_{()𝜋/𝑛+1}×𝑗^−1−𝛼𝑘^−1−𝛽𝑠^−𝛼𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶₀^{𝜋/(𝑚+1)}₀^{𝜋/(𝑛+1)∞}_{()𝑗>𝜋/𝑠}𝑗^−1−𝛼𝑠^−𝛼∞_{()𝑘>𝜋/𝑡}𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦+𝐶₀^{𝜋/(𝑚+1)𝜋}_{()𝜋/𝑛+1}^∞_{()𝑗>𝜋/𝑠}𝑗^−1−𝛼𝑠^−𝛼∞_𝑘=𝑛+1𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦+𝐶^𝜋_{()𝜋/𝑚+10}^{𝜋/(𝑛+1)∞}_𝑗=𝑚+1𝑗^−1−𝛼𝑠^−𝛼∞_{()𝑘>𝜋/𝑡}𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦+𝐶^𝜋_{()𝜋/𝑚+1}^𝜋_{()𝜋/𝑛+1}^∞_𝑗=𝑚+1𝑗^−1−𝛼𝑠^−𝛼∞_𝑘=𝑛+1𝑘^−1−𝛽𝑡^−𝛽𝑑_𝑠𝑑_𝑡𝑉^𝑠𝑡₀₀𝜑_𝑥𝑦≤𝐶𝑉^{(𝜋/(𝑚+1))(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦+𝐶𝑛^𝑛−𝛽_𝑘=1𝑘^𝛽−1𝑉^{(𝜋/(𝑚+1))(𝜋/𝑘)}₀₀𝜑_𝑥𝑦+𝐶𝑚^𝑚−𝛼_𝑗=1𝑗^𝛼−1𝑉^{(𝜋/𝑗)(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦+𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑗=1𝑘=1𝑗^𝛼−1𝑘^𝛽−1𝑉^{(𝜋/𝑗)(𝜋/𝑘)}₀₀𝜑_𝑥𝑦𝐶≤𝐶𝑚^𝛼𝑛^𝛽𝑚𝑛_𝑗=1𝑘=1𝑗^𝛼−1𝑘^𝛽−1𝑉^{(𝜋/(𝑚+1))(𝜋/(𝑛+1))}₀₀𝜑_𝑥𝑦. (3.12)
We get (2.16) by combining (3.8)–(3.12).

Proof of Theorem 2.2. Set 𝐷_𝑖(𝑡)=^𝑖_𝑗=1()((𝑡sin𝑖𝑡=cos1/2𝑡−cos𝑖+1/2))2sin(1/2)𝑡,𝑖=1,2,…. (3.13)
By Abel’s transformation, we have 𝑡̂^𝑚𝑛_𝑘=1𝑙=1^(1,1)_{𝑚𝑛𝑘𝑙}𝑘𝑙sin𝑘𝑠sin𝑙𝑡=^𝑚_𝑘=1sin𝑘𝑠𝑘^𝑛_𝑙=1Δ₀₁𝑡̂^(1,1)_{𝑚𝑛𝑘𝑙}𝑙𝐷_𝑙(𝑡)=^𝑛_𝑙=1sin𝑙𝑡𝑙^𝑚_𝑘=1Δ₁₀𝑡̂^(1,1)_{𝑚𝑛𝑘𝑙}𝑘𝐷_𝑘(𝑠)=^𝑚𝑛_𝑘=1𝑙=1Δ₁₁𝑡̂^(1,1)_{𝑚𝑛𝑘𝑙}𝐷𝑘𝑙^𝑘(𝑠)𝐷_𝑙(𝑡). (3.14)
Therefore, by the following well-known inequalities: |𝐷_𝑖(𝑡)|=𝑂(𝑡⁻¹), |sin𝑡|≤|𝑡|, and condition (2.17), we see that (2.15) holds with 𝛼=𝛽=1, and thus we get Theorem 2.2 from Theorem 2.1.

4. Applications

4.1. Cesàro’s Means

Let 𝑇=(𝐶,𝛾,𝛿)=𝐶_𝛾×𝐶_𝛿 be the double Cesàro matrix of order (𝛾,𝛿) with 𝛾,𝛿>−1, that is, 𝑇 has entries 𝑡_{𝑚𝑛𝑗𝑘}=𝐶^𝛾_𝑚𝑗×𝐶^𝛿_𝑛𝑘=𝐴^𝛾−1_𝑚−𝑗𝐴^𝛿−1_𝑛−𝑘𝐴^𝛾_𝑚𝐴^𝛿_𝑛. (4.1)

Then for 0≤𝑗≤𝑚−1, 0≤𝑘≤𝑛−1, 𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}=Δ₁₁𝑡_{𝑚−1,𝑛−1,𝑗,𝑘}=^𝑚−1_𝑢=𝑗𝐴^𝛾−1_{𝑚−1−𝑢}𝐴^𝛾_𝑚−1−^𝑚_𝑢=𝑗𝐴^𝛾−1_𝑚−𝑢𝐴^𝛾_𝑚^𝑛−1_𝑣=𝑘𝐴^𝛿−1_{𝑛−1−𝑣}𝐴^𝛿_𝑛−1−^𝑛_𝑣=𝑘𝐴^𝛿−1_𝑛−𝑣𝐴^𝛿_𝑛=𝐴^𝛾_𝑚−𝑗𝐴^𝛾_𝑚−𝐴^𝛾_{𝑚−1−𝑗}𝐴^𝛾_𝑚−1𝐴^𝛿_𝑛−𝑘𝐴^𝛿_𝑛−𝐴^𝛿_{𝑛−1−𝑘}𝐴^𝛿_𝑛−1=𝑗𝑘𝐴^𝛾−1_𝑚−𝑗𝐴^𝛿−1_𝑛−𝑘𝑚𝑛𝐴^𝛾_𝑚𝐴^𝛿_𝑛,𝑡̂^(1,1)_{𝑚𝑛𝑚𝑛}=𝑡_{𝑚𝑛𝑚𝑛}=1𝐴^𝛼_𝑚𝐴^𝛽_𝑛. (4.2)

In particular, 𝑡̂^(1,1)_{𝑚𝑛0𝑘}=𝑡̂^(1,1)_{𝑚𝑛𝑗0}=0,𝑗,𝑘=0,1,…. (4.3)

In other words, 𝑇 satisfies condition (2.13) of Theorem 2.1.

By the well-known inequality (e.g., see [11]) 𝐴^𝛾_𝑛=𝑛^𝛾Γ1𝛾+11+𝑂𝑛, (4.4) we deduce that 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}=^𝑚_𝑗=1𝑗𝐴_𝑚^𝛾−1𝑚𝐴^𝛾_𝑚+1𝐴^𝛾_𝑚^𝑛_𝑘=1𝑘𝐴^𝛿−1_𝑛−𝑘𝑛𝐴^𝛿_𝑛+1𝐴^𝛿_𝑛=𝑂𝑚^−2𝑚/2_𝑗=1𝑗+𝑂𝑚^−𝛾𝑚_𝑗=𝑚/2𝐴^𝛾−1_𝑚−𝑗+1𝐴^𝛾_𝑚×𝑂𝑛^−2𝑛/2_𝑘=1𝑘+𝑂𝑛^−𝛿𝑛_𝑘=𝑛/2𝐴^𝛿−1_𝑛−𝑘+1𝐴^𝛿_𝑛=𝑂(1),𝛾,𝛿>0, (4.5) which means condition (2.14) of Theorem 2.1.

For 0<𝛾<1, we have ^𝑛_𝑘=1𝐴^𝛾−1_𝑛−𝑘𝑛𝐴^𝛾_𝑛1sin𝑘𝑥=Im𝑛𝐴^𝛾_𝑛^𝑛_𝑘=1𝐴^𝛾−1_𝑛−𝑘𝑒^𝑖𝑘𝑥𝑒=Im^𝑖𝑛𝑥𝑛𝐴^𝛾_𝑛^𝑛−1_𝑘=0𝐴_𝑘^𝛾−1𝑒^{−𝑖𝑘𝑥}=∶Im𝐽₁+𝐽₂, (4.6) where 𝐽₁𝑒∶=^𝑖𝑛𝑥𝑛𝐴^𝛾_𝑛1−𝑒^{−𝑖𝑥−𝛾},𝐽₂𝑒∶=^𝑖𝑛𝑥𝑛𝐴^𝛾_𝑛^∞_𝑘=𝑛𝐴_𝑘^𝛾−1𝑒^{−𝑖𝑘𝑥}. (4.7)

By (4.4), we observe that 𝐽₁=𝑂𝑛^−1−𝛾𝑥^−𝛾. (4.8)

Since 𝐴_𝑘^𝛾−1 decreases monotonically to 0, 𝐽₂ converges for 0<𝑥≤𝜋, and 𝐽₂𝐴=𝑂^𝑛𝛾−1𝑛𝐴^𝛾_𝑛1−𝑒^{−𝑖𝑥−1}=𝑂𝑛⁻²𝑥⁻¹=𝑂𝑛^−1−𝛾𝑥^−𝛾 (4.9) for 𝑡∈[(𝜋/𝑛),𝜋].

By (4.4)–(4.9), we obtain that ^𝑛_𝑘=1𝐴^𝛾−1_𝑛−𝑘𝑛𝐴^𝛾_𝑛sin𝑘𝑥=𝑂𝑛^−1−𝛾𝑥^−𝛾𝜋,𝑥∈𝑛,𝜋,0<𝛾≤1. (4.10)

On the other hand, we have ^𝑛_𝑘=1𝐴^𝛾−1_𝑛−𝑘𝑛𝐴^𝛾_𝑛sin𝑘𝑥=𝑂(𝑥)^𝑛_𝑘=1𝑘𝐴^𝛾−1_𝑛−𝑘𝑛𝐴^𝛾_𝑛(𝑥)𝜋=𝑂,𝑥∈0,𝑛,𝛾>0. (4.11)

For 𝛾≥1, we have that ^𝑛−1_𝑘=1Δ𝐴^𝛾−1_𝑛−𝑘𝑛𝐴^𝛾_𝑛=1𝑛𝐴^𝛾_𝑛^𝑛−1_𝑘=1𝐴^𝛾−1_𝑛−𝑘−𝐴^𝛾−1_{𝑛−𝑘−1}+1𝑛𝐴^𝛾_𝑛=𝑂𝑛⁻². (4.12)

Now, by (4.10) and (4.11), we deduce that 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}≤𝑗𝑘sin𝑗𝑥sin𝑘𝑦^𝑚_𝑗=1𝐴^𝛾−1_𝑚−𝑗𝑚𝐴^𝛾_𝑚sin𝑗𝑥^𝑛_𝑘=1𝐴^𝛿−1_𝑛−𝑘𝑛𝐴^𝛿_𝑛=𝑂𝑘sin𝑘𝑦_−1−𝛿𝑥𝑦^−𝛿,∈𝜋𝑥,𝑦0,𝑚×𝜋𝑛,𝑂𝑗,𝜋_−1−𝛾𝑥^−𝛾𝑦,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋0,𝑛,𝑂𝑗^−1−𝛾𝑘^−1−𝛿𝑥^−𝛾𝑦^−𝛿,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋 (4.13) for 0<𝛾, 𝛿≤1.

When 𝛾>1, 0<𝛿≤1, by (4.10) and (4.11) again, 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}≤𝑗𝑘sin𝑗𝑥sin𝑘𝑦^𝑚_𝑗=1𝐴^𝛾−1_𝑚−𝑗𝑚𝐴^𝛾_𝑚sin𝑗𝑥^𝑛_𝑘=1𝐴^𝛿−1_𝑛−𝑘𝑛𝐴^𝛿_𝑛sin𝑘𝑦=𝑂𝑘^−1−𝛿𝑥𝑦^−𝛿, (4.14) for (𝑥,𝑦)∈[0,(𝜋/𝑚)]×[(𝜋/𝑛),𝜋]. By (4.10)–(4.12), we have 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}≤𝑗𝑘sin𝑗𝑥sin𝑘𝑦^𝑚_𝑗=1Δ𝐴^𝛾−1_𝑚−𝑗𝑚𝐴^𝛾_𝑚𝐷_𝑗(𝑥)^𝑛_𝑘=1𝐴^𝛿−1_𝑛−𝑘𝑛𝐴^𝛿_𝑛=𝑂𝑗sin𝑘𝑦₋₂𝑥⁻¹𝑦,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋0,𝑛,𝑂𝑗⁻²𝑘^−1−𝛿𝑥⁻¹𝑦^−𝛿,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋. (4.15)

Similarly, we have 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}=𝑂𝑘𝑗𝑘sin𝑗𝑥sin𝑘𝑦⁻²𝑥𝑦⁻¹,∈𝜋𝑥,𝑦0,𝑚×𝜋𝑛,𝑂𝑗,𝜋_−1−𝛾𝑥^−𝛾𝑦,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋0,𝑛,𝑂𝑗^−1−𝛾𝑘⁻²𝑥^−𝛾𝑦⁻¹,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋, (4.16) for 𝛿>1,0<𝛾≤1,and 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}=𝑂𝑘𝑗𝑘sin𝑗𝑥sin𝑘𝑦⁻²𝑥𝑦⁻¹,∈𝜋𝑥,𝑦0,𝑚×𝜋𝑛,𝑂𝑗,𝜋₋₂𝑥⁻¹𝑦,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋0,𝑛,𝑂𝑗⁻²𝑘⁻²𝑥⁻¹𝑦⁻¹,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋, (4.17) for 𝛾,𝛿>1.

Setting 𝛾∶=𝛾,0<𝛾≤1,1,𝛾>1,𝛿∶=𝛿,0<𝛿≤1,1,𝛿>1, (4.18) then we have 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}=𝑂𝑘𝑗𝑘sin𝑗𝑥sin𝑘𝑦⁻¹⁻𝛿𝑥𝑦⁻¹,𝑥,𝑦∈0,𝜋𝑚×𝜋𝑛,𝑂𝑗,𝜋₋₁₋𝛾𝑥⁻¹𝑦,𝑥,𝑦∈𝜋𝑚×𝜋,𝜋0,𝑛,𝑂𝑗⁻¹⁻𝛾𝑘⁻¹⁻𝛿𝑥⁻¹𝑦⁻¹,∈𝜋𝑥,𝑦𝑚×𝜋,𝜋𝑛,𝜋, (4.19) by (4.13)–(4.17).

From (4.19), we have the following corollary of Theorem 2.1, which generalizes Theorem B from one dimension to two dimensions.

Theorem 4.1. Let 𝑓(𝑥,𝑦) be a periodic function and 𝑓(𝑥,𝑦)∈𝐵𝑉_𝐻(𝕋²) and 𝑇=(𝐶,𝛾,𝛿) be the double Cesàro matrix of order (𝛾,𝛿) with 𝛾,𝛿>0. Then for 𝑛≥2, (𝑥,𝑦)∈𝕋², one has 𝑅^(1,1)_𝑚𝑛1𝑓;𝑥,𝑦≤𝐶𝑚𝛾𝑛𝛿^𝑚𝑛_𝑘=1𝑙=1𝑘𝛾₋₁𝑙𝛿₋₁𝑉^{(𝜋/𝑘)(𝜋/𝑙)}₀₀𝜑_𝑥𝑦. (4.20)

4.2. Bernstein-Rogosinski’s Means

The so-called Bernstein-Rogosinski means of Fourier series (1.1) are defined by 𝐵_𝑛1𝑓,𝑥∶=2𝑆_𝑛1𝑓,𝑥+𝜋2𝑛+1+𝑆_𝑛1𝑓,𝑥−𝜋.2𝑛+1 (4.21)

Now, we introduce the following Bernstein-Rogosinski means of double Fourier series (2.1): 𝐵_𝑚𝑛1𝑓,𝑥∶=4𝑆_𝑚𝑛𝜋𝑓;𝑥+𝜋2𝑚+1,𝑦+2𝑛+1+𝑆_𝑚𝑛𝜋𝑓;𝑥+𝜋2𝑚+1,𝑦−2𝑛+1+𝑆_𝑚𝑛𝜋𝑓;𝑥−𝜋2𝑚+1,𝑦+2𝑛+1+𝑆_𝑚𝑛𝜋𝑓;𝑥−𝜋2𝑚+1,𝑦−.2𝑛+1 (4.22)

Direct calculations yield that 𝐵_𝑚𝑛𝑓,𝑥=^𝑚𝑛_𝑗=0𝑘=0cos𝑗𝜋2𝑚+1cos𝑘𝜋Δ2𝑛+1¹¹𝑆_𝑗𝑘=𝑥,𝑦_{𝑚−1𝑗=0𝑛−1𝑘=0}𝑗cos2𝑚+1𝜋−cos𝑗+1𝜋𝑘2𝑚+1cos2𝑛+1𝜋−cos𝑘+1𝜋𝑆2𝑛+1^𝑗𝑘+𝑥,𝑦_{𝑚−1𝑗=0}𝑗cos2𝑚+1𝜋−cos𝑗+1𝜋2𝑚+1cos𝑛𝜋𝑆2𝑛+1^𝑗𝑛+𝑥,𝑦_{𝑛−1𝑘=0}𝑘cos2𝑛+1𝜋−cos𝑘+1𝜋2𝑛+1cos𝑚𝜋𝑆2𝑚+1^𝑚𝑘𝑥,𝑦+cos𝑚𝜋2𝑚+1cos𝑛𝜋𝑆2𝑛+1^𝑚𝑛𝑥,𝑦, (4.23)

Thus, 𝐵_𝑚𝑛(𝑓,𝑥) can be regarded as a 𝑇-transformation of Fourier series (1.1) with 𝑇 of entries defined as 𝑡_{𝑚𝑛𝑗𝑘}=𝑗cos2𝑚+1𝜋−cos𝑗+1𝜋𝑘2𝑚+1cos2𝑛+1𝜋−cos𝑘+1𝜋𝑡2𝑛+1,0≤𝑗<𝑚,0≤𝑘<𝑛,_{𝑚𝑛𝑗𝑛}=𝑗cos2𝑚+1𝜋−cos𝑗+1𝜋2𝑛+1cos𝑛𝜋𝑡2𝑛+1,0≤𝑗<𝑚,_{𝑚𝑛𝑚𝑘}=𝑘cos2𝑛+1𝜋−cos𝑘+1𝜋2𝑛+1cos𝑚𝜋𝑡2𝑚+1,0≤𝑘<𝑛,_{𝑚𝑛𝑚𝑛}=cos𝑚𝜋2𝑚+1cos𝑛𝜋.2𝑛+1 (4.24)

By direct calculations, we have 𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}=𝑗cos𝑗2𝑚−1𝜋−cos𝜋𝑘2𝑚+1cos𝑘2𝑛−1𝜋−cos𝜋𝑡̂2𝑛+1,0≤𝑗<𝑚,0≤𝑘<𝑛,_{(1,1)𝑚𝑛𝑗𝑛}=𝑗cos𝑗2𝑚−1𝜋−cos𝜋2𝑚+1cos𝑛𝜋𝑡̂2𝑛+1,0≤𝑗<𝑚,_{(1,1)𝑚𝑛𝑚𝑘}=𝑘cos𝑘2𝑛−1𝜋−cos𝜋2𝑛+1cos𝑚𝜋𝑡̂2𝑚+1,0≤𝑘<𝑛,_{(1,1)𝑚𝑛𝑚𝑛}=cos𝑚𝜋2𝑚+1cos𝑛𝜋.2𝑛+1 (4.25)(4.26)(4.27)(4.28)

Therefore, (4.26) and (4.27) imply (2.13), while from (4.25)–(4.28), we get 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}≤^𝑚−1_𝑗=1^𝑛−1_𝑘=1𝑗cos𝑗2𝑚−1𝜋−cos𝜋𝑘2𝑚+1cos𝑘2𝑛−1𝜋−cos𝜋+2𝑛+1_{𝑚−1𝑗=1}𝑗cos𝑗2𝑚−1𝜋−cos𝜋2𝑚+1cos𝑛𝜋+2𝑛+1_{𝑛−1𝑘=1}𝑘cos𝑘2𝑛−1𝜋−cos𝜋2𝑛+1cos𝑚𝜋2𝑚+1+cos𝑚𝜋2𝑚+1cos𝑛𝜋2𝑛+1≤4^𝑚−1_𝑗=1^𝑛−1_𝑘=1sin2𝑚𝑗4𝑚²𝑗−1𝜋sin4𝑚²−1𝜋sin2𝑛𝑘4𝑛²𝑘−1𝜋sin4𝑛²𝜋−1+2^𝑚−1_𝑗=1sin2𝑚𝑗4𝑚²𝑗−1𝜋sin4𝑚²−1𝜋+2^𝑛−1_𝑘=1sin2𝑛𝑘4𝑛²𝑘−1𝜋sin4𝑛²−1𝜋+1≤4𝜋^4𝑚−1_𝑗=1^𝑛−1_𝑘=12𝑚𝑗²4𝑚²−1²2𝑛𝑘²4𝑛²−1²+2𝜋^2𝑚−1_𝑗=12𝑚𝑗²4𝑚²−1²+2𝜋^2𝑛−1_𝑘=12𝑛𝑘²4𝑛²−1²(1),+1=𝑂 (4.29) which implies (2.14).

Finally, we verify that 𝑇 satisfies condition (2.17); thus 𝑇 satisfies all the conditions of Theorem 2.2. By direct calculations, we deduce that ^𝑚𝑛_𝑗=1𝑘=1Δ₀₁𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑘=^𝑚_𝑗=1𝑗cos𝑗2𝑚−1𝜋−cos𝜋2𝑚+1+cos𝑚𝜋×2𝑚+1_{𝑛−1𝑘=1}cos(𝑘/(2𝑛−1))𝜋−cos(𝑘/(2𝑛+1))𝜋𝑘−cos((𝑘+1)/(2𝑛−1))𝜋−cos((𝑘+1)/(2𝑛+1))𝜋+1𝑘+1𝑛cos𝑛𝜋2𝑛+1=𝑂^𝑛−1_𝑘=1cos(𝑘/(2𝑛−1))𝜋−cos(𝑘/(2𝑛+1))𝜋𝑘−cos((𝑘+1)/(2𝑛−1))𝜋−cos((𝑘+1)/(2𝑛+1))𝜋𝑘+1+𝑛⁻²=𝑂(1)^𝑛−1_𝑘=11𝑘𝑘cos𝑘2𝑛−1𝜋−cos𝜋−2𝑛+1cos𝑘+12𝑛−1𝜋−cos𝑘+1𝜋(1)2𝑛+1+𝑂^𝑛−1_𝑘=11𝑘−1𝑘+1cos𝑘+12𝑛−1𝜋−cos𝑘+1𝜋=2𝑛+1_{𝑛−1𝑘=1}1𝑘sin2𝑛𝑘𝜋4𝑛²()𝜋−1−sin2𝑛𝑘+14𝑛²−1sin𝑘𝜋4𝑛²+−1sin𝑘𝜋4𝑛²()𝜋−1−sin𝑘+14𝑛²()𝜋−1sin2𝑛𝑘+14𝑛²+−1_{𝑛−1𝑘=1}1𝑘()()𝜋𝑘+1sin2𝑛𝑘+14𝑛²()𝜋−1sin𝑘+14𝑛²(1)−1=𝑂^𝑛−1_𝑘=11𝑘sin𝑛𝜋4𝑛²−1sin𝑘𝜋4𝑛²(1)−1+𝑂^𝑛−1_𝑘=11𝑘²()𝜋sin2𝑛𝑘+14𝑛²()𝜋−1sin𝑘+14𝑛²(1)−1=𝑂^𝑛−1_𝑘=1𝑛⁻³=𝑂𝑛⁻². (4.30)

Analogously, ^𝑚𝑛_𝑗=1𝑘=1Δ₁₀𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑗=𝑂𝑚⁻²,_{𝑚𝑛𝑗=1𝑘=1}Δ₁₁𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑗=𝑂𝑚⁻²𝑛⁻². (4.31)

Now, (4.30) and (4.31) imply (2.17).

4.3. Riesz’s Means

Let {𝑝_𝑛} and {𝑞_𝑛} be positive sequences such that 𝑃_𝑛=𝑝₀+𝑝₁+⋯+𝑝_𝑛→∞, 𝑄_𝑛 = 𝑞₀+𝑞₁+⋯+𝑞_𝑛 → ∞, and let 𝑇∶=(𝑡_{𝑚𝑛𝑗𝑘}) be a lower triangular matrix with entries 𝑡_{𝑚𝑛𝑗𝑘} = 𝑝_𝑗𝑞_𝑘/𝑃_𝑚𝑄𝑛, 𝑗=0,1,…𝑚, 𝑘=0,1,…,𝑛; 𝑚,𝑛=1,2,…. Then the 𝑇-transformation of Fourier series (2.1) is known as Riesz’s mean.

Proposition 4.2. Let {𝑝_𝑛} and {𝑞_𝑛} be positive nondecreasing sequences such that (one says 𝐴_𝑛∼𝐵_𝑛, if there are two positive constants 𝐶₁ and 𝐶₂ such that 𝐶₁≤(𝐴_𝑛/𝐵_𝑛)≤𝐶₂.)𝑛𝑝_𝑛∼𝑃_𝑛,𝑛𝑞_𝑛∼𝑄_𝑛,_𝑛𝑘=1𝑝_𝑘𝑘=𝑂𝑝_𝑛,^𝑛_𝑘=1𝑞_𝑘𝑘=𝑂𝑞_𝑛. (4.32)(4.33)
Then 𝑇 satisfies all the conditions of Theorem 2.2.

Proof. Direct calculations yield that (set 𝑃₋₁=𝑄₋₁=0) 𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}=^𝑚_𝑖=𝑗𝑝_𝑖𝑃_𝑚−^𝑛−1_𝑖=𝑗𝑝_𝑖𝑃_𝑚−1^𝑛_𝑖=𝑘𝑞_𝑖𝑄_𝑛−^𝑛−1_𝑖=𝑘𝑞_𝑖𝑄_𝑛−1=𝑝_𝑚𝑞_𝑛𝑃_𝑗−1𝑄_𝑘−1𝑃_𝑚𝑃_𝑚−1𝑄_𝑛𝑄_𝑛−1𝑡̂,0≤𝑗<𝑚,0≤𝑘<𝑛,_{(1,1)𝑚𝑛𝑚𝑘}=𝑝_𝑚𝑃_𝑚^𝑛_𝑖=𝑘𝑞_𝑖𝑄_𝑛−^𝑛−1_𝑖=𝑘𝑞_𝑖𝑄_𝑛−1=𝑝_𝑚𝑞_𝑛𝑄_𝑘−1𝑃_𝑚𝑄_𝑛𝑄_𝑛−1𝑡̂,0≤𝑘<𝑛,_{(1,1)𝑚𝑛𝑗𝑛}=^𝑚_𝑖=𝑗𝑝_𝑖𝑃_𝑚−^𝑛−1_𝑖=𝑗𝑝_𝑖𝑃_𝑚−1𝑞_𝑛𝑄_𝑛=𝑝_𝑚𝑞_𝑛𝑃_𝑗−1𝑃_𝑚𝑃_𝑚−1𝑄_𝑛𝑡̂,0≤𝑗<𝑚,_{(1,1)𝑚𝑛𝑚𝑛}=𝑝_𝑚𝑞_𝑛𝑃_𝑚𝑄_𝑛. (4.34)(4.35)(4.36)
It follows from (4.35) that 𝑇 satisfies condition (2.13).
Since {𝑝_𝑛} and {𝑞_𝑛} are nondecreasing, by (4.32), (4.34)–(4.36), we have 𝑡̂^𝑚𝑛_𝑗=1𝑘=1^(1,1)_{𝑚𝑛𝑗𝑘}=𝑝_𝑚𝑞_𝑛𝑃_𝑚𝑃_𝑚−1𝑄_𝑛𝑄^𝑚_𝑛−1^𝑛_𝑗=1𝑘=1𝑃_𝑗−1𝑄_𝑘−1=𝑂𝑚𝑛𝑝_𝑚𝑞_𝑛𝑃_𝑚𝑄_𝑛=𝑂(1). (4.37)
Hence, 𝑇 satisfies condition (2.14).
Since {𝑝_𝑛} and {𝑞_𝑛} are nondecreasing, by (4.32)–(4.36), we have ^𝑚𝑛_𝑗=1𝑘=1Δ₀₁𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑘=𝑝_𝑚𝑃_𝑚𝑃_𝑚−1^𝑚−1_𝑗=1𝑃_𝑗−1+𝑝_𝑚𝑃_𝑚×𝑞_𝑛𝑄_𝑛𝑄_𝑛−1^𝑛−1_𝑘=1𝑄_𝑘−1𝑘−𝑄_𝑘+𝑞𝑘+1𝑛𝑃^𝑛_𝑛(1)𝑞=𝑂^𝑛𝑄_𝑛𝑄_𝑛−1^𝑛−1_𝑘=1𝑄_𝑘−1𝑘()+𝑘+1^𝑛−1_𝑘=2𝑞_𝑘𝑛𝑘+1+𝑂⁻²(1)1=𝑂𝑛𝑄_𝑛−1^𝑛−1_𝑘=1𝑞_𝑘𝑘+𝑛⁻²𝑞=𝑂^𝑛−1𝑛𝑄_𝑛−1=𝑂𝑛⁻². (4.38)
Similarly, ^𝑚𝑛_𝑗=1𝑘=1Δ₁₀𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑗=𝑂𝑚⁻²,_{𝑚𝑛𝑗=1𝑘=1}Δ₁₁𝑡̂^(1,1)_{𝑚𝑛𝑗𝑘}𝑗=𝑂𝑚⁻²𝑛⁻². (4.39)
Now, by (4.38) and (4.39), we show that 𝑇 also satisfies condition (2.17).

Acknowledgments

A Project Supported by Scientific Research Fund of Zhejiang Provincial Education Department (Y201223607). Research of the second author is supported by NSF of China (10901044), Qianjiang Rencai Program of Zhejiang Province (2010R10101), SRF for ROCS, SEM, and Program for excellent Young Teachers in HZNU.