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Journal of Function Spaces and Applications
VolumeΒ 2012, Article IDΒ 930967, 16 pages
http://dx.doi.org/10.1155/2012/930967
Research Article

Pointwise Approximation of Functions from 𝐿𝑝(𝑀)𝛽 by Linear Operators of Their Fourier Series

Faculty of Mathematics, Computer Science and Econometrics, University of Zielona GΓ³ra, ul. Szafrana 4a, 65-516 Zielona GΓ³ra, Poland

Received 1 September 2010; Accepted 24 November 2010

Academic Editor: HenrykΒ Hudzik

Copyright Β© 2012 WΕ‚odzimierz Łenski and Bogdan Szal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We show the results, corresponding to theorem of Lal (2009), on the rate of pointwise approximation of functions from the pointwise integral Lipschitz classes by matrix summability means of their Fourier series as well as the theorems on norm approximations.

1. Introduction

Let 𝐿𝑝(1≀𝑝<∞)be the class of all 2πœ‹-periodic real-valued functions integrable in the Lebesgue sense with 𝑝th power over 𝑄=[βˆ’πœ‹,πœ‹] with the normβ€–π‘“β€–βˆΆ=‖𝑓(β‹…)‖𝐿𝑝=ξ‚΅ξ€œπ‘„||||𝑓(𝑑)𝑝𝑑𝑑1/𝑝,(1.1) and consider the trigonometric Fourier seriesπ‘Žπ‘†π‘“(π‘₯)∢=π‘œ(𝑓)2+βˆžξ“πœˆ=1ξ€·π‘Žπœˆ(𝑓)cos𝜈π‘₯+π‘πœˆ(ξ€Έ,𝑓)sin𝜈π‘₯(1.2) and conjugate one𝑆𝑓(π‘₯)∢=βˆžξ“πœˆ=1ξ€·π‘πœˆ(𝑓)cos𝜈π‘₯βˆ’π‘Žπœˆ(𝑓)sin𝜈π‘₯(1.3) with the partial sums π‘†π‘˜π‘“ and ξ‚π‘†π‘˜π‘“, respectively. We know that if π‘“βˆˆπΏ, then1𝑓(π‘₯)∢=βˆ’πœ‹ξ€œπœ‹0πœ“π‘₯1(𝑑)2cot𝑑2𝑑𝑑=limπœ–β†’0𝑓(π‘₯,πœ–),(1.4) where1𝑓(π‘₯,πœ–)∢=βˆ’πœ‹ξ€œπœ‹πœ–πœ“π‘₯1(𝑑)2cot𝑑2𝑑𝑑(1.5) withπœ“π‘₯(𝑑)∢=𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯βˆ’π‘‘)(1.6) exists for almost all π‘₯ [1, Th. (3.1)IV].

Let 𝐴∢=(π‘Žπ‘›,π‘˜) be an infinite lower triangular matrix of real numbers such thatπ‘Žπ‘›,π‘˜β‰₯0whenπ‘˜=0,1,2,…,𝑛,π‘Žπ‘›,π‘˜=0whenπ‘˜>𝑛,π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜=1,where𝑛=0,1,2,…,(1.7) and let the 𝐴-transformationsof (π‘†π‘˜π‘“) and (ξ‚π‘†π‘˜π‘“) be given by𝑇𝑛,𝐴𝑓(π‘₯)∢=π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜π‘†π‘˜ξ‚π‘‡π‘“(π‘₯)(𝑛=0,1,2,…),𝑛,𝐴𝑓(π‘₯)∢=π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚π‘†π‘˜π‘“(π‘₯)(𝑛=0,1,2,…),(1.8) respectively. Denote, for π‘š=0,1,2,…,𝑛,𝐴𝑛,π‘š=π‘šξ“π‘˜=0π‘Žπ‘›,π‘˜,𝐴𝑛,π‘š=π‘›ξ“π‘˜=π‘šπ‘Žπ‘›,π‘˜.(1.9)

We define two classes of sequences (see [2]).

A sequence π‘βˆΆ=(𝑐𝑛) of nonnegative numbers tending to zero is called the Rest Bounded Variation Sequence, or briefly π‘βˆˆRBVS, if it has the propertyβˆžξ“π‘˜=π‘š||π‘π‘˜βˆ’π‘π‘˜+1||≀𝐾(𝑐)π‘π‘š(1.10) for all natural numbers π‘š, where 𝐾(𝑐) is a constant depending only on 𝑐.

A sequence π‘βˆΆ=(𝑐𝑛) of nonnegative numbers will be called the Head Bounded Variation Sequence, or briefly π‘βˆˆHBVS, if it has the propertyπ‘šβˆ’1ξ“π‘˜=0||π‘π‘˜βˆ’π‘π‘˜+1||≀𝐾(𝑐)π‘π‘š(1.11) for all natural numbers π‘š, or only for all π‘šβ‰€π‘ if the sequence 𝑐 has only finite nonzero terms and the last nonzero terms is 𝑐𝑁.

Now, we define another class of sequences.

Followed by Leindler [3], a sequence π‘βˆΆ=(𝑐𝑛) of nonnegative numbers tending to zero is called the Mean Rest Bounded Variation Sequence, or briefly π‘βˆˆMRBVS, if it has the propertyβˆžξ“π‘˜=π‘š||π‘π‘˜βˆ’π‘π‘˜+1||1≀𝐾(𝑐)π‘š+1π‘šξ“π‘˜β‰₯π‘š/2π‘π‘˜(1.12) for all natural numbers π‘š, where 𝐾(𝑐) is a constant depending only on 𝑐.

Analogously, a sequence π‘βˆΆ=(π‘π‘˜) of nonnegative numbers will be called the Mean Head Bounded Variation Sequence, or briefly π‘βˆˆMHBVS, if it has the propertyπ‘›βˆ’π‘šβˆ’1ξ“π‘˜=0||π‘π‘˜βˆ’π‘π‘˜+1||1≀𝐾(𝑐)π‘š+1π‘›ξ“π‘˜=π‘›βˆ’π‘šπ‘π‘˜,(1.13) for all positive integer π‘š<𝑛, where the sequence 𝑐 has only finite nonzero terms and the last nonzero term is 𝑐𝑛 and where 𝐾(𝑐) is a constant depending only on 𝑐.

It is clear that (see [4])RBVS⊊MRBVS,HBVS⊊MHBVS.(1.14)

Consequently, we assume that the sequence (𝐾(𝛼𝑛))βˆžπ‘›=0 is bounded, that is, that there exists a constant 𝐾 such that𝛼0≀𝐾𝑛≀𝐾(1.15) holds for all 𝑛, where 𝐾(𝛼𝑛) denotes the sequence of constants appearing in the inequalities (1.12) or (1.13) for the sequences 𝛼𝑛=(π‘Žπ‘›,π‘˜)π‘›π‘˜=0.

Now, we can give the conditions to be used later on. We assume that, for all 𝑛 and 0β‰€π‘š<𝑛,π‘›βˆ’1ξ“π‘˜=π‘š||π‘Žπ‘›,π‘˜βˆ’π‘Žπ‘›,π‘˜+1||1β‰€πΎπ‘š+1π‘šξ“π‘˜β‰₯π‘š/2π‘Žπ‘›,π‘˜,π‘›βˆ’π‘šβˆ’1ξ“π‘˜=0||π‘Žπ‘›,π‘˜βˆ’π‘Žπ‘›,π‘˜+1||1β‰€πΎπ‘š+1π‘›ξ“π‘˜=π‘›βˆ’π‘šπ‘Žπ‘›,π‘˜,(1.16) where 𝐾 is the same as above, hold if 𝛼𝑛=(π‘Žπ‘›,π‘˜)π‘›π‘˜=0 belong to MRBVS or MHBVS, for 𝑛=0,1,2,…, respectively.

As a measure of approximation of functions by the above means, we use the generalized pointwise moduli of continuity of 𝑓 in the space 𝐿𝑝 defined for 𝛽β‰₯0 by the formulas𝑀π‘₯,𝛽𝑓(𝛿)𝐿𝑝1∢=𝛿1+π›½π‘ξ€œπ›Ώ0ξ‚΅||πœ“π‘₯|||||𝑑(𝑑)sin2|||𝛽𝑝𝑑𝑑1/𝑝,𝑀π‘₯,𝛽𝑓(𝛿)𝐿𝑝1∢=𝛿1+π›½π‘ξ€œπ›Ώ0ξ‚΅||πœ‘π‘₯|||||𝑑(𝑑)sin2|||𝛽𝑝𝑑𝑑1/𝑝,(1.17) whereπœ‘π‘₯(𝑑)∢=𝑓(π‘₯+𝑑)+𝑓(π‘₯βˆ’π‘‘)βˆ’2𝑓(π‘₯).(1.18) It is clear that, for 𝛽>𝛼β‰₯0,𝑀π‘₯,𝛽𝑓(𝛿)𝐿𝑝≀𝑀π‘₯,𝛼𝑓(𝛿)𝐿𝑝,𝑀π‘₯,𝛽𝑓(𝛿)𝐿𝑝≀𝑀π‘₯,𝛼𝑓(𝛿)𝐿𝑝.(1.19) It is easily seen that 𝑀π‘₯,0𝑓(β‹…)𝐿𝑝=𝑀π‘₯𝑓(β‹…)𝐿𝑝 and 𝑀π‘₯,0𝑓(β‹…)𝐿𝑝=𝑀π‘₯𝑓(β‹…)𝐿𝑝 are the classical pointwise moduli of continuity.

The deviation 𝑇𝑛,π΄π‘“βˆ’π‘“ with special form of matrix 𝐴 was estimated in the norm of 𝐿𝑝 by Lal [5, Theorem  2, page 347] as follows.

Theorem A. If π‘“βˆˆπΏπ‘π›½ξƒ―(πœ”)=π‘“βˆˆπΏπ‘βˆΆπœ”π‘“(𝛿)πΏπ‘π›½βˆΆ=sup0≀|𝑑|β‰€π›Ώξ‚»ξ€œπœ‹0||πœ‘π‘₯||(𝑑)𝑝|||π‘₯sin2|||𝛽𝑝𝑑π‘₯1/𝑝,β‰€πœ”(𝛿)(1.20)πœ”(𝑑)𝑑isadecreasingfunctionof𝑑,(1.21)ξƒ―βˆ«0πœ‹/(𝑛+1)𝑑||πœ‘π‘₯||(𝑑)ξ‚Άπœ”(𝑑)𝑝sin𝛽𝑝𝑑𝑑𝑑1/𝑝=𝑂(𝑛+1)βˆ’1ξ€Έ,(1.22)ξƒ―ξ€œπœ‹πœ‹/(𝑛+1)ξ‚΅π‘‘βˆ’π›Ύ||πœ‘π‘₯||(𝑑)ξ‚Άπœ”(𝑑)𝑝sin𝛽𝑝𝑑𝑑𝑑1/𝑝=𝑂((𝑛+1)𝛾)ξ‚΅10<𝛾<𝑝,(1.23) then β€–β€–β€–β€–1𝑛+1π‘›ξ“πœˆ=01π‘ƒπœˆπœˆξ“π‘˜=0π‘πœˆβˆ’π‘˜π‘†π‘˜β€–β€–β€–β€–π‘“βˆ’π‘“πΏπ‘ξ‚€(=𝑂𝑛+1)𝛽+(1/𝑝)πœ”ξ‚€1,𝑛+1(1.24) where 𝑃𝑛=βˆ‘π‘›πœˆ=0π‘πœˆ,(π‘πœˆ) is a nonnegative and nonincreasing sequence, and the function πœ” of modulus of continuity type will be defined in the next section.

In this note we show that the conditions (1.21), (1.22), and (1.23) are superfluous when we use the pointwise modulus of continuity.

In our theorems, we will consider the pointwise deviations 𝑇𝑛,π΄ξ‚π‘‡π‘“βˆ’π‘“,𝑛,π΄ξ‚π‘“π‘“βˆ’, 𝑇𝑛,π΄ξ‚π‘“βˆ’π‘“(β‹…,2πœ‹/(𝑛+2)) with the matrix whose rows belong to the classes of sequences MRBVS and MHBVS. Consequently, we also give some results on norm approximation.

We will write 𝐼1β‰ͺ𝐼2 if there exists a positive constant 𝐾, sometimes depending on some parameters, such that 𝐼1≀𝐾𝐼2.

2. Statement of the Results

Let us define, for a fixed π‘₯, a function 𝑀π‘₯ (or 𝑀π‘₯) of modulus of continuity type on the interval [0,2πœ‹], that is, a nondecreasing continuous function having the following properties:𝑀π‘₯(0)=0,𝑀π‘₯𝛿1+𝛿2≀𝑀π‘₯𝛿1ξ€Έ+𝑀π‘₯𝛿2ξ€Έ,ξ€·or𝑀π‘₯(𝑀0)=0,π‘₯𝛿1+𝛿2≀𝑀π‘₯𝛿1ξ€Έ+𝑀π‘₯𝛿2,ξ€Έξ€Έ(2.1) for any 0≀𝛿1≀𝛿2≀𝛿1+𝛿2≀2πœ‹. It is easy to conclude that the function π›Ώβˆ’1𝑀π‘₯(𝛿) nondecreases in 𝛿. Let𝐿𝑝𝑀π‘₯𝛽=ξ€½π‘“βˆˆπΏπ‘βˆΆπ‘€π‘₯,𝛽𝑓(𝛿)𝐿𝑝≀𝑀π‘₯ξ€Ύ,𝐿(𝛿)𝑝𝑀π‘₯𝛽=ξ€½π‘“βˆˆπΏπ‘βˆΆξ‚π‘€π‘₯,𝛽𝑓(𝛿)𝐿𝑝≀𝑀π‘₯ξ€Ύ,𝐿(𝛿)𝑝(𝑀)𝛽=ξ€½π‘“βˆˆπΏπ‘βˆΆβ€–β€–π‘€β‹…,𝛽𝑓(𝛿)𝐿𝑝‖‖𝐿𝑝,𝐿≀𝑀(𝛿)𝑝𝑀𝛽=ξ€½π‘“βˆˆπΏπ‘βˆΆβ€–β€–ξ‚π‘€β‹…,𝛽𝑓(𝛿)𝐿𝑝‖‖𝐿𝑝≀,𝑀(𝛿)(2.2) where 𝑀π‘₯,𝑀π‘₯𝑀, and 𝑀 are also the functions of modulus of continuity type. It is clear that, for 𝛽>𝛼β‰₯0,𝐿𝑝𝑀π‘₯ξ€Έπ›ΌβŠ‚πΏπ‘ξ€·π‘€π‘₯𝛽,𝐿𝑝(𝑀)π›ΌβŠ‚πΏπ‘(𝑀)𝛽,𝐿𝑝𝑀π‘₯ξ€Έπ›ΌβŠ‚πΏπ‘ξ€·ξ‚π‘€π‘₯𝛽,πΏπ‘ξ€·ξ‚π‘€ξ€Έπ›ΌβŠ‚πΏπ‘ξ€·ξ‚π‘€ξ€Έπ›½.(2.3)

Now, we can formulate our main results on the degrees of pointwise summability.

Theorem 2.1. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽 with 𝛽<1βˆ’(1/𝑝). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS is such that 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ||𝑇𝑛,𝐴||𝑓(π‘₯)βˆ’π‘“(π‘₯)=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.4) for considered π‘₯.

Theorem 2.2. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽 with 𝛽<1βˆ’(1/𝑝). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS is such that 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ||𝑇𝑛,𝐴||𝑓(π‘₯)βˆ’π‘“(π‘₯)=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.5) for considered π‘₯.

Theorem 2.3. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽 with 𝛽<2βˆ’(1/𝑝). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS is such that 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then |||𝑇𝑛,𝐴𝑓𝑓(π‘₯)βˆ’π‘₯,2πœ‹ξ‚|||𝑛+2=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.6) for considered π‘₯.

Theorem 2.4. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽 with 𝛽<2βˆ’(1/𝑝). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS is such that 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then |||𝑇𝑛,𝐴𝑓𝑓(π‘₯)βˆ’π‘₯,2πœ‹ξ‚|||𝑛+2=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.7) for considered π‘₯.

Theorem 2.5. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽, and ξ‚»ξ€œ02πœ‹/(𝑛+2)1𝑑𝑀π‘₯(𝑑)sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘žξ‚€=𝑂(𝑛+1)𝛽𝑀π‘₯ξ‚€πœ‹π‘›+1(2.8) holds with π‘ž=𝑝(π‘βˆ’1)βˆ’1 when 𝛽>0 or with π‘ž=1 when 𝛽=0. If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS is such that 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ||𝑇𝑛,𝐴||𝑓(π‘₯)βˆ’π‘“(π‘₯)=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.9) for considered π‘₯ such that 𝑓(π‘₯) exists.

Theorem 2.6. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽, and (2.8) holds with π‘ž=𝑝(π‘βˆ’1)βˆ’1 when 𝛽>0 or with π‘ž=1 when 𝛽=0. If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS and 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ||𝑇𝑛,𝐴||𝑓(π‘₯)βˆ’π‘“(π‘₯)=𝑂(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹,𝑛+1(2.10) for considered π‘₯ such that 𝑓(π‘₯) exists.

Consequently, we formulate the results on norm approximation.

Theorem 2.7. Let π‘“βˆˆπΏπ‘(𝑀)𝛽 with 𝛽<1βˆ’(1/𝑝),(π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS is such that 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖𝑇𝑛,𝐴‖‖𝑓(β‹…)βˆ’π‘“(β‹…)𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)π‘€ξ‚€πœ‹.𝑛+1(2.11)

Theorem 2.8. Let π‘“βˆˆπΏπ‘(𝑀)𝛽 with 𝛽<1βˆ’(1/𝑝),(π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS is such that 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖𝑇𝑛,𝐴‖‖𝑓(β‹…)βˆ’π‘“(β‹…)𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)π‘€ξ‚€πœ‹.𝑛+1(2.12)

Theorem 2.9. Let π‘“βˆˆπΏπ‘(𝑀)𝛽 with 𝛽<1βˆ’(1/𝑝),(π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS is such that 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖‖𝑇𝑛,𝐴𝑓𝑓(β‹…)βˆ’β‹…,2πœ‹ξ‚β€–β€–β€–π‘›+2𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)ξ‚π‘€ξ‚€πœ‹.𝑛+1(2.13)

Theorem 2.10. Let π‘“βˆˆπΏπ‘(𝑀)𝛽 with 𝛽<1βˆ’(1/𝑝),(π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS is such that 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖‖𝑇𝑛,𝐴𝑓𝑓(β‹…)βˆ’β‹…,2πœ‹ξ‚β€–β€–β€–π‘›+2𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)ξ‚π‘€ξ‚€πœ‹.𝑛+1(2.14)

Theorem 2.11. Let π‘“βˆˆπΏπ‘(𝑀)𝛽, and ξ‚»ξ€œ02πœ‹/(𝑛+2)1𝑑𝑀(𝑑)sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘žξ‚€=𝑂(𝑛+1)π›½ξ‚π‘€ξ‚€πœ‹π‘›+1(2.15) holds with π‘ž=𝑝(π‘βˆ’1)βˆ’1. If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS and 𝐴𝑛,𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖𝑇𝑛,𝐴‖‖𝑓(β‹…)βˆ’π‘“(β‹…)𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)ξ‚π‘€ξ‚€πœ‹.𝑛+1(2.16)

Theorem 2.12. Let π‘“βˆˆπΏπ‘(𝑀π‘₯)𝛽, and (2.15) holds with π‘ž=𝑝(π‘βˆ’1)βˆ’1. If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS is such that 𝐴𝑛,π‘›βˆ’2𝜏=𝑂(𝜏/(𝑛+1)), where 𝜏=[πœ‹/𝑑](2πœ‹/(𝑛+2)β‰€π‘‘β‰€πœ‹), then ‖‖𝑇𝑛,𝐴‖‖𝑓(β‹…)βˆ’π‘“(β‹…)𝐿𝑝=𝑂(𝑛+1)𝛽+(1/𝑝)ξ‚π‘€ξ‚€πœ‹.𝑛+1(2.17)

Remark 2.13. In the case 𝑝β‰₯1(speciallyif𝑝=1), we can suppose that the expression π‘‘βˆ’π›½π‘€π‘₯(𝑑) nondecreases in 𝑑 instead of the assumption 𝛽<1βˆ’(1/𝑝).

Remark 2.14. Under additional assumptions π‘Žπ‘›π‘›=𝑂(1/𝑛),𝛽=0, and 𝑀π‘₯(𝑑)=𝑂(𝑑𝛼)(0<𝛼≀1), the degree of approximation in Theorem 2.1 is following 𝑂(𝑛(1/𝑝)βˆ’π›Ό). The same degree of approximation we obtain in Theorem 2.2 under the assumption π‘Žπ‘›0=𝑂(1/𝑛). In case of the remaining theorems, we can use the same remarks.

Remark 2.15. If we consider the classical modulus of continuity πœ”0𝑓(𝛿)𝐿𝑝 of the function 𝑓 instead of the modulus 𝑀, then the condition ‖𝑀⋅,𝛽𝑓(𝛿)πΏπ‘β€–πΏπ‘β‰€πœ”0𝑓(𝛿)𝐿𝑝 holds for every function π‘“βˆˆπΏπ‘ and thus 𝐿𝑝(𝑀)𝛽=𝐿𝑝. The same remark for conjugate functions holds too.

Remark 2.16. Our theorems will be also true if we consider function 𝑓 from 𝐿𝑝𝛽(πœ”) with the following norm: β€–π‘“β€–πΏπ‘π›½βˆΆ=‖𝑓(β‹…)‖𝐿𝑝𝛽=ξ‚΅ξ€œπ‘„||||𝑓(𝑑)𝑝|||𝑑sin2|||𝛽𝑝𝑑𝑑1/𝑝.(2.18)

Remark 2.17. We can observe that, taking π‘Žπ‘›π‘˜βˆ‘=1/(𝑛+1)π‘›πœˆ=π‘˜π‘πœˆβˆ’π‘˜/π‘ƒπœˆ, we obtain the mean considered by Lal [5], and if (π‘πœˆ) is monotonic with respect to 𝜈, then (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS. Therefore, if (π‘πœˆ) is a nonincreasing sequence such that π‘ƒπœβˆ‘π‘›πœˆ=πœπ‘ƒπœˆβˆ’1=𝑂(𝜏), then, from Theorem 2.1, we obtain the corrected form of the result of Lal [5] (i.e., with condition {∫0πœ‹/(𝑛+1)(|πœ‘π‘₯(𝑑)|/πœ”(𝑑))𝑝sin𝛽𝑝𝑑𝑑𝑑}1/𝑝=𝑂π‘₯((𝑛+1)βˆ’1/𝑝) instead of (1.22)).

3. Auxiliary Results

We begin this section by some notations following Zygmund [1].

It is clear thatξ‚π‘†π‘˜1𝑓(π‘₯)=βˆ’πœ‹ξ€œπœ‹βˆ’πœ‹ξ‚π·π‘“(π‘₯+𝑑)π‘˜π‘†(𝑑)𝑑𝑑,π‘˜π‘“1(π‘₯)=πœ‹ξ€œπœ‹βˆ’πœ‹π‘“(π‘₯+𝑑)π·π‘˜ξ‚π‘‡(𝑑)𝑑𝑑,𝑛,𝐴1𝑓(π‘₯)=βˆ’πœ‹ξ€œπœ‹βˆ’πœ‹π‘“(π‘₯+𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·π‘˜π‘‡(𝑑)𝑑𝑑,𝑛,𝐴1𝑓(π‘₯)=πœ‹ξ€œπœ‹βˆ’πœ‹π‘“(π‘₯+𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜(𝑑)𝑑𝑑,(3.1) whereξ‚‹π·π‘˜(𝑑)=π‘˜ξ“πœˆ=0sinπœˆπ‘‘=cos(𝑑/2)βˆ’cos((2π‘˜+1)𝑑/2),𝐷2sin(𝑑/2)π‘˜1(𝑑)=2+π‘˜ξ“πœˆ=1cosπœˆπ‘‘=sin((2π‘˜+1)𝑑/2).2sin(𝑑/2)(3.2) Hence,𝑇𝑛,𝐴𝑓𝑓(π‘₯)βˆ’π‘₯,2πœ‹ξ‚1𝑛+2=βˆ’πœ‹ξ€œ02πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·π‘˜+1(𝑑)π‘‘π‘‘πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜ξ‚π‘‡(𝑑)𝑑𝑑,𝑛,𝐴1𝑓(π‘₯)βˆ’π‘“(π‘₯)=πœ‹ξ€œπœ‹0πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜(𝑑)𝑑𝑑,(3.3) whereξ‚‹π·βˆ˜π‘˜(𝑑)=cos((2π‘˜+1)𝑑/2),𝑇2sin(𝑑/2)𝑛,𝐴1𝑓(π‘₯)βˆ’π‘“(π‘₯)=πœ‹ξ€œπœ‹0πœ‘π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜(𝑑)𝑑𝑑.(3.4) Now, we formulate some estimates for the conjugate Dirichlet kernel.

Lemma 3.1 (see [1]). If 0<|𝑑|β‰€πœ‹/2, then ||ξ‚‹π·βˆ˜π‘˜||β‰€πœ‹(𝑑)2,||𝐷|𝑑|π‘˜||β‰€πœ‹(𝑑),|𝑑|(3.5) and, for any real 𝑑, we have ||ξ‚‹π·π‘˜||≀1(𝑑)2||ξ‚‹π·π‘˜(π‘˜+1)|𝑑|,π‘˜||(𝑑)β‰€π‘˜+1.(3.6)

Lemma 3.2 (cf. [2, 6]). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS, then |||||π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜|||||𝑑(𝑑)=π‘‚βˆ’1𝐴𝑛,π‘›βˆ’2πœξ‚,(3.7) and if (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS, then |||||π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜|||||𝑑(𝑑)=π‘‚βˆ’1𝐴𝑛,πœξ€Έ,(3.8) for 2πœ‹/π‘›β‰€π‘‘β‰€πœ‹(𝑛=2,3,…), where 𝜏=[πœ‹/𝑑].

Lemma 3.3 (see [7]). If (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MHBVS, then |||||π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜|||||𝑑(𝑑)=π‘‚βˆ’1𝐴𝑛,π‘›βˆ’2πœξ‚,(3.9) and if (π‘Žπ‘›,π‘˜)π‘›π‘˜=0∈MRBVS, then |||||βˆžξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜|||||𝑑(𝑑)=π‘‚βˆ’1𝐴𝑛,πœξ€Έ,(3.10) for 2πœ‹/π‘›β‰€π‘‘β‰€πœ‹(𝑛=2,3,…), where 𝜏=[πœ‹/𝑑].

4. Proofs of the Results

4.1. Proof of Theorem 2.1

As usual,𝑇𝑛,𝐴1𝑓(π‘₯)βˆ’π‘“(π‘₯)=πœ‹ξ€œ02πœ‹/(𝑛+2)πœ‘π‘₯(𝑑)βˆžξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜(+1𝑑)π‘‘π‘‘πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)πœ‘π‘₯(𝑑)βˆžξ“π‘˜=0π‘Žπ‘›,π‘˜π·π‘˜(𝑑)𝑑𝑑=𝐼1+𝐼2,||𝑇𝑛,𝐴||≀||𝐼𝑓(π‘₯)βˆ’π‘“(π‘₯)1||+||𝐼2||.(4.1) By the HΓΆlder inequality ((1/𝑝)+(1/π‘ž)=1) and Lemma 3.1, for 𝛽<1βˆ’(1/𝑝),||𝐼1||≀(𝑛+1)πœ‹ξ€œ02πœ‹/(𝑛+2)||πœ‘π‘₯||≀(𝑑)𝑑𝑑(𝑛+1)πœ‹ξ‚»ξ€œ02πœ‹/(𝑛+2)||πœ‘π‘₯||(𝑑)sin𝛽𝑑2𝑝𝑑𝑑1/π‘ξ‚»ξ€œ02πœ‹/(𝑛+2)ξ‚Έ1sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ(𝑛+1)1βˆ’π›½βˆ’(1/𝑝)𝑀π‘₯ξ‚€2πœ‹ξ‚ξ‚»ξ€œπ‘›+202πœ‹/(𝑛+2)1π‘‘π›½ξ‚„π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ𝑀π‘₯ξ‚€2πœ‹ξ‚π‘›+2≀2𝑀π‘₯ξ‚€πœ‹ξ‚.𝑛+1(4.2) Using Lemma 3.2 and the HΓΆlder inequality ((1/𝑝)+(1/π‘ž)=1),||𝐼2||β‰ͺ1πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)||πœ‘π‘₯||(𝑑)𝑑𝐴𝑛,𝜏1𝑑𝑑β‰ͺξ€œπœ‹(𝑛+1)πœ‹2πœ‹/(𝑛+2)||πœ‘π‘₯||(𝑑)𝑑2≀1π‘‘π‘‘ξ€œπœ‹(𝑛+1)πœ‹πœ‹/(𝑛+2)||πœ‘π‘₯||(𝑑)sin𝛽(𝑑/2)𝑑2sinπ›½πœ‹(𝑑/2)π‘‘π‘‘β‰€π›½βˆ’1ξ€œ(𝑛+1)πœ‹πœ‹/(𝑛+1)||πœ‘π‘₯||(𝑑)sin𝛽(𝑑/2)𝑑2+𝛽=πœ‹π‘‘π‘‘π›½βˆ’1ξƒ―ξ‚Έ1𝑛+1𝑑2+π›½ξ€œπ‘‘0||πœ‘π‘₯||(𝑒)sin𝛽𝑒2ξ‚Ήπ‘‘π‘’πœ‹π‘‘=πœ‹/(𝑛+1)ξ€œ+(2+𝛽)πœ‹πœ‹/(𝑛+1)βˆ«π‘‘0||πœ‘π‘₯||(𝑒)sin𝛽(𝑒/2)𝑑𝑒𝑑3+𝛽≀1π‘‘π‘‘πœ‹3ξ€œ(𝑛+1)πœ‹0||πœ‘π‘₯||(𝑒)sin𝛽𝑒2+πœ‹π‘‘π‘’π›½βˆ’1(2+𝛽)ξƒ―ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)ξƒ¬π‘‘βˆ’1βˆ«π‘‘0||πœ‘π‘₯||(𝑒)sin𝛽(𝑒/2)𝑑𝑒𝑀π‘₯(𝑑)𝑝𝑑𝑑1/π‘β‹…ξ‚»ξ€œπœ‹πœ‹/(𝑛+1)𝑀π‘₯(𝑑)𝑑2+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰€1πœ‹2ξ‚»1(𝑛+1)πœ‹ξ€œπœ‹0ξ‚€||πœ‘π‘₯||(𝑒)sin𝛽𝑒2𝑝𝑑𝑒1/𝑝+πœ‹π›½βˆ’1(2+𝛽)⎧βŽͺ⎨βŽͺβŽ©ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)βŽ‘βŽ’βŽ’βŽ’βŽ£ξ‚†π‘‘βˆ’1βˆ«π‘‘0ξ€·||πœ‘π‘₯(||𝑒)sin𝛽(𝑒/2)𝑝𝑑𝑒1/𝑝𝑀π‘₯⎀βŽ₯βŽ₯βŽ₯⎦(𝑑)π‘βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/π‘β‹…ξ‚»ξ€œπœ‹πœ‹/(𝑛+1)𝑀π‘₯(𝑑)𝑑2+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ1𝑀𝑛+1π‘₯1(πœ‹)+ξ‚»ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)𝑑𝛽𝑝𝑑𝑑1/𝑝𝑀π‘₯(πœ‹/(𝑛+1))ξ‚»ξ€œπœ‹/(𝑛+1)πœ‹πœ‹/(𝑛+1)1𝑑1+π›½ξ‚„π‘žξ‚Όπ‘‘π‘‘1/π‘ž.(4.3) Since 𝛽+(1/𝑝)>0, we have||𝐼2||β‰ͺ1𝑀𝑛+1π‘₯𝑑(πœ‹)+(βˆ’1βˆ’π›½)π‘ž+1ξ‚Ή(βˆ’1βˆ’π›½)π‘ž+1πœ‹π‘‘=πœ‹/(𝑛+1)ξƒ°1/π‘žπ‘€π‘₯ξ‚€πœ‹ξ‚β‰ͺ1𝑛+1𝑀𝑛+1π‘₯(πœ‹)+(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹ξ‚π‘›+1β‰ͺ(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹ξ‚.𝑛+1(4.4)

Collecting these estimates, we obtain the desired result.

4.2. Proof of Theorem 2.2

The proof is the same as the proof of Theorem 2.1, the only difference is that, in place of 𝐴𝑛,𝜏, we have to write the quantity 𝐴𝑛,π‘›βˆ’2𝜏 for which we suppose the same order.

4.3. Proof of Theorem 2.3

We start with the obvious relations𝑇𝑛,𝐴𝑓𝑓(π‘₯)βˆ’π‘₯,2πœ‹ξ‚1𝑛+1=βˆ’πœ‹ξ€œ02πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›π‘˜ξ‚‹π·π‘˜+1(𝑑)π‘‘π‘‘πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›π‘˜ξ‚‹π·βˆ˜π‘˜ξ‚πΌ(𝑑)=1+ξ‚πΌβˆ˜2,|||𝑇𝑛,𝐴𝑓𝑓(π‘₯)βˆ’π‘₯,2πœ‹ξ‚|||≀||𝐼𝑛+21||+||ξ‚πΌβˆ˜2||.(4.5) By the HΓΆlder inequality ((1/𝑝)+(1/π‘ž)=1) and Lemma 3.1, we have||𝐼1||≀(𝑛+1)2ξ€œ02πœ‹/(𝑛+2)𝑑||πœ“π‘₯||(𝑑)𝑑𝑑≀(𝑛+1)2ξ‚»ξ€œ02πœ‹/(𝑛+2)||πœ“π‘₯||(𝑑)sin𝛽𝑑2𝑝𝑑𝑑1/π‘ξ‚»ξ€œ02πœ‹/(𝑛+2)𝑑sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ(𝑛+1)2βˆ’π›½βˆ’(1/𝑝)𝑀π‘₯ξ‚€2πœ‹ξ‚ξ‚»ξ€œπ‘›+202πœ‹/(𝑛+2)𝑑1βˆ’π›½ξ€»π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ𝑀π‘₯ξ‚€2πœ‹ξ‚β‰ͺ𝑀𝑛+2π‘₯ξ‚€πœ‹ξ‚π‘›+1(4.6) for 𝛽<2βˆ’(1/𝑝).

Using Lemma 3.3 and the HΓΆlder inequality ((1/𝑝)+(1/π‘ž)=1), we get||ξ‚πΌβˆ˜2||β‰ͺ1πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)||πœ“π‘₯||(𝑑)𝑑𝐴𝑛,𝜏1𝑑𝑑β‰ͺξ€œπœ‹(𝑛+1)πœ‹2πœ‹/(𝑛+2)||πœ“π‘₯||(𝑑)𝑑2≀1π‘‘π‘‘ξ€œπœ‹(𝑛+1)πœ‹πœ‹/(𝑛+1)||πœ“π‘₯||(𝑑)𝑑2β‰€πœ‹π‘‘π‘‘π›½βˆ’1ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)||πœ“π‘₯||(𝑑)sin𝛽(𝑑/2)𝑑2+𝛽1π‘‘π‘‘β‰€πœ‹3ξ€œ(𝑛+1)πœ‹0||πœ“π‘₯||(𝑒)sin𝛽𝑑2+πœ‹π‘‘π‘’π›½βˆ’1(2+𝛽)ξƒ―ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)ξƒ¬π‘‘βˆ’1βˆ«π‘‘0||πœ“π‘₯||(𝑒)sin𝛽(𝑒/2)𝑑𝑒𝑀π‘₯ξƒ­(𝑑)𝑝𝑑𝑑1/π‘ξ‚»ξ€œπœ‹πœ‹/(𝑛+1)𝑀π‘₯(𝑑)𝑑2+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ1πœ‹2ξ‚»1(𝑛+1)πœ‹ξ€œπœ‹0ξ‚€||πœ“π‘₯||(𝑒)sin𝛽𝑒2𝑝𝑑𝑒1/𝑝+πœ‹π›½βˆ’1(2+𝛽)⎧βŽͺ⎨βŽͺβŽ©ξ€œπ‘›+1πœ‹πœ‹/(𝑛+1)βŽ‘βŽ’βŽ’βŽ’βŽ£ξ‚†π‘‘βˆ’1βˆ«π‘‘0ξ€·||πœ“π‘₯||(𝑒)sin𝛽(𝑒/2)𝑝𝑑𝑒1/𝑝𝑀π‘₯⎀βŽ₯βŽ₯βŽ₯⎦(𝑑)π‘βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/π‘β‹…ξ‚»ξ€œπœ‹πœ‹/(𝑛+1)𝑀π‘₯(𝑑)𝑑2+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘ž.(4.7) Since 𝛽+(1/𝑝)>0, we have||ξ‚πΌβˆ˜2||β‰ͺ1𝑀𝑛+1π‘₯𝑀(πœ‹)+π‘₯ξ‚€πœ‹ξ‚ξƒ―ξ‚Έπ‘‘π‘›+1(βˆ’1βˆ’π›½)π‘ž+1ξ‚Ή(βˆ’1βˆ’π›½)π‘ž+1πœ‹π‘‘=πœ‹/(𝑛+1)ξƒ°1/π‘žβ‰ͺ(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹ξ‚,𝑛+1(4.8) and our proof is complete.

4.4. Proof of Theorem 2.4

The proof is the same as the proof of Theorem 2.3, the only difference is that, in place of 𝐴𝑛,𝜏, we have to write the quantity 𝐴𝑛,π‘›βˆ’2𝜏 for which we suppose the same order.

4.5. Proof of Theorem 2.5

We start with the obvious relations𝑇𝑛,𝐴1𝑓(π‘₯)βˆ’π‘“(π‘₯)=πœ‹ξ€œ02πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜+1(𝑑)π‘‘π‘‘πœ‹ξ€œπœ‹2πœ‹/(𝑛+2)πœ“π‘₯(𝑑)π‘›ξ“π‘˜=0π‘Žπ‘›,π‘˜ξ‚‹π·βˆ˜π‘˜ξ‚πΌ(𝑑)𝑑𝑑=∘1+ξ‚πΌβˆ˜2,||𝑇𝑛,𝐴||≀||𝐼𝑓(π‘₯)βˆ’π‘“(π‘₯)∘1||+||ξ‚πΌβˆ˜2||.(4.9) Lemma 3.1 gives||ξ‚πΌβˆ˜1||β‰ͺ1πœ‹ξ€œ02πœ‹/(𝑛+2)||πœ“π‘₯||(𝑑)π‘‘π‘‘π‘‘β‰€πœ‹π›½βˆ’1ξ€œ02πœ‹/(𝑛+2)||πœ“π‘₯||(𝑑)sin𝛽(𝑑/2)𝑑1+𝛽𝑑𝑑=πœ‹π›½βˆ’1ξ‚Έ1𝑑1+π›½ξ€œπ‘‘0||πœ“π‘₯||(𝑒)sin𝛽𝑒2𝑑𝑒2πœ‹/(𝑛+2)𝑑=0+πœ‹π›½βˆ’1ξ€œ(1+𝛽)02πœ‹/(𝑛+2)π‘‘βˆ’1βˆ«π‘‘0||πœ“π‘₯||(𝑒)sin𝛽(𝑒/2)𝑑𝑒𝑑1+𝛽𝑑𝑑.(4.10) If 𝛽>0 and π‘ž>1, then, by the HΓΆlder inequality ((1/𝑝)+(1/π‘ž)=1),||ξ‚πΌβˆ˜1||β‰€πœ‹π›½βˆ’1𝑛+22πœ‹1+π›½ξ€œ02πœ‹/(𝑛+2)||πœ“π‘₯||(𝑒)sin𝛽𝑒2𝑑𝑒+πœ‹π›½βˆ’1(ξƒ―ξ€œ1+𝛽)02πœ‹/(𝑛+2)ξƒ¬π‘‘βˆ’1βˆ«π‘‘0||πœ“π‘₯(||𝑒)sin𝛽(𝑒/2)𝑑𝑒𝑑1/𝑝𝑀π‘₯ξƒ­(𝑑)𝑝𝑑𝑑1/π‘β‹…ξ‚»ξ€œ02πœ‹/(𝑛+2)𝑀π‘₯(𝑑)𝑑(1/π‘ž)+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰€πœ‹π›½βˆ’1𝑛+22πœ‹π›½ξ‚»π‘›+2ξ€œ2πœ‹02πœ‹/(𝑛+2)ξ‚€||πœ“π‘₯||(𝑒)sin𝛽𝑒2𝑝𝑑𝑒1/𝑝+πœ‹π›½βˆ’1(⎧βŽͺ⎨βŽͺβŽ©ξ€œ1+𝛽)02πœ‹/(𝑛+2)βŽ‘βŽ’βŽ’βŽ’βŽ£ξ‚†π‘‘βˆ’1βˆ«π‘‘0ξ€·||πœ“π‘₯||(𝑒)sin𝛽(𝑒/2)𝑝𝑑𝑒1/𝑝𝑑1/𝑝𝑀π‘₯⎀βŽ₯βŽ₯βŽ₯⎦(𝑑)π‘βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/π‘β‹…ξ‚»ξ€œ02πœ‹/(𝑛+2)𝑀π‘₯(𝑑)𝑑(1/π‘ž)+π›½ξ‚Ήπ‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ𝑀π‘₯ξ‚€2πœ‹ξ‚+ξ‚»ξ€œπ‘›+202πœ‹/(𝑛+2)π‘‘π›½π‘βˆ’1𝑑𝑑1/π‘ξ‚»ξ€œ02πœ‹/(𝑛+2)1𝑑𝑀π‘₯(𝑑)sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘žβ‰ͺ𝑀π‘₯ξ‚€2πœ‹ξ‚+𝑛+2(𝑛+1)βˆ’π›½ξ‚»ξ€œ02πœ‹/(𝑛+2)1𝑑𝑀π‘₯(𝑑)sin𝛽(𝑑/2)π‘žξ‚Όπ‘‘π‘‘1/π‘ž,(4.11) or if 𝛽=0, then||ξ‚πΌβˆ˜1||β‰ͺ𝑀π‘₯ξ‚€2πœ‹ξ‚+ξ€œπ‘›+202πœ‹/(𝑛+2)𝑀π‘₯(𝑑)𝑑𝑑𝑑.(4.12) Therefore, using (2.8), we have||ξ‚πΌβˆ˜1||β‰ͺ𝑀π‘₯ξ‚€πœ‹ξ‚.𝑛+1(4.13) The same estimation as in the proof of Theorem 2.3 gives||ξ‚πΌβˆ˜2||β‰ͺ(𝑛+1)𝛽+(1/𝑝)𝑀π‘₯ξ‚€πœ‹ξ‚.𝑛+1(4.14) Collecting these estimates, we obtain the desired result.

4.6. Proof of Theorem 2.6

For the proof, we use the analogical remark as these in the proofs of Theorems 2.2 and 2.4.

4.7. Proofs of Theorems from 2.7 to 2.12

The proofs are similar to these above. We have only to use the generalized Minkowski inequality.

For example, in case of Theorem 2.7, we get‖‖𝑇𝑛,𝐴‖‖𝑓(β‹…)βˆ’π‘“(β‹…)𝐿𝑝≀‖‖𝐼1‖‖𝐿𝑝+‖‖𝐼2‖‖𝐿𝑝≀‖‖‖𝑀⋅,𝛽𝑓2πœ‹ξ‚π‘›+2𝐿𝑝‖‖‖𝐿𝑝+‖‖𝐼2‖‖𝐿𝑝β‰ͺ‖‖‖𝑀⋅,π›½π‘“ξ‚€πœ‹ξ‚π‘›+1𝐿𝑝‖‖‖𝐿𝑝+1‖‖𝑀𝑛+1β‹…,𝛽𝑓(πœ‹)𝐿𝑝‖‖𝐿𝑝+(𝑛+1)𝛽+(1/𝑝)‖‖‖𝑀⋅,π›½π‘“ξ‚€πœ‹ξ‚π‘›+1𝐿𝑝‖‖‖𝐿𝑝β‰ͺ1𝑛+1𝑀(πœ‹)+(𝑛+1)𝛽+(1/𝑝)π‘€ξ‚€πœ‹ξ‚ξ‚€π‘›+1=𝑂(𝑛+1)𝛽+(1/𝑝)π‘€ξ‚€πœ‹.𝑛+1(4.15)

This completes the proof of Theorem 2.7.

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