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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 964101, 11 pages
http://dx.doi.org/10.1155/2012/964101
Research Article

Trigonometric Approximation of Signals (Functions) Belonging to π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) Class by Matrix (𝐢1⋅𝑁𝑝) Operator

Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee 247667, India

Received 22 March 2012; Revised 24 April 2012; Accepted 3 May 2012

Academic Editor: JewgeniΒ Dshalalow

Copyright Β© 2012 Uaday Singh et al. 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

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2Ο€-periodic signals (functions) belonging to Lip(𝛼,π‘Ÿ) class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip(𝛼,π‘Ÿ)- class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip α and π‘Š(πΏπ‘Ÿ, πœ‰(𝑑)) classes by using CesΓ‘ro-NΓΆrlund (𝐢1⋅𝑁𝑝) summability without monotonicity condition on {𝑝𝑛}, which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).

1. Introduction

For a given signal (function) π‘“βˆˆπΏπ‘ŸβˆΆ=πΏπ‘Ÿ[0,2πœ‹], π‘Ÿβ‰₯1, let 𝑠𝑛(𝑓)=π‘ π‘›ξ‚€π‘Ž(𝑓;π‘₯)=02+π‘›ξ“π‘˜=1ξ€·π‘Žπ‘˜cosπ‘˜π‘₯+π‘π‘˜ξ€Έ=sinπ‘˜π‘₯π‘›ξ“π‘˜=0π‘’π‘˜(𝑓;π‘₯)(1.1) denote the partial sum, called trigonometric polynomial of degree (or order) 𝑛, of the first (𝑛+1) terms of the Fourier series of 𝑓. Let {𝑝𝑛} be a nonnegative sequence of real numbers such that π‘ƒπ‘›βˆ‘(=π‘›π‘˜=0π‘π‘˜β‰ 0)β†’βˆž as π‘›β†’βˆž and π‘ƒβˆ’1=0=π‘βˆ’1.

Define 𝑁𝑛(𝑓)=𝑁𝑛(𝑓;π‘₯)=π‘ƒπ‘›π‘›βˆ’1ξ“π‘˜=0π‘π‘›βˆ’π‘˜π‘ π‘˜(𝑓;π‘₯),βˆ€π‘›β‰₯0,(1.2) the NΓΆrlund (𝑁𝑝) means of the sequence 𝑠𝑛(𝑓) or Fourier series of 𝑓. The Fourier series of 𝑓 is said to be NΓΆrlund (𝑁𝑝) summable to 𝑠(π‘₯) if 𝑁𝑛(𝑓;π‘₯)→𝑠(π‘₯)asπ‘›β†’βˆž. The Fourier series of 𝑓 is called CesΓ‘ro-NΓΆrlund (𝐢1⋅𝑁𝑝) summable to 𝑆(π‘₯) if 𝑑𝑛𝐢𝑁(𝑓)=(𝑛+1)π‘›βˆ’1ξ“π‘˜=0π‘ƒπ‘˜π‘˜βˆ’1𝑖=0π‘π‘˜βˆ’π‘–π‘ π‘–(𝑓;π‘₯)βŸΆπ‘†(π‘₯)asπ‘›βŸΆβˆž.(1.3) We note that 𝑁𝑛(𝑓) and 𝑑𝑛𝐢𝑁(𝑓) are also trigonometric polynomials of degree (or order) 𝑛.

Some interesting applications of the CesΓ‘ro summability can be seen in [1, 2].

The πΏπ‘Ÿ-norm of signal 𝑓 is defined by β€–π‘“β€–π‘Ÿ=ξ‚΅1ξ€œ2πœ‹02πœ‹||||𝑓(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿβ€–(1β‰€π‘Ÿ<∞),π‘“β€–βˆž=sup[]π‘₯∈0,2πœ‹||||.𝑓(π‘₯)(1.4) A signal (function) 𝑓 is approximated by trigonometric polynomials 𝑇𝑛(𝑓) of degree 𝑛, and the degree of approximation 𝐸𝑛(𝑓) is given by 𝐸𝑛(𝑓)=Min𝑛‖‖𝑓(π‘₯)βˆ’π‘‡π‘›β€–β€–(𝑓)π‘Ÿ.(1.5) This method of approximation is called trigonometric Fourier approximation.

A signal (function) 𝑓 is said to belong to the class Lip𝛼 if |𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)|=𝑂(|𝑑|𝛼),0<𝛼≀1, and π‘“βˆˆLip(𝛼,π‘Ÿ) if ‖𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)β€–π‘Ÿ=𝑂(|𝑑|𝛼),0<𝛼≀1,π‘Ÿβ‰₯1.

For a positive increasing function πœ‰(𝑑) and π‘Ÿβ‰₯1, π‘“βˆˆLip(πœ‰(𝑑),π‘Ÿ) if ‖𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)β€–π‘Ÿ=𝑂(πœ‰(𝑑)), and π‘“βˆˆπ‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) if β€–[𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)]sin𝛽(π‘₯/2)β€–π‘Ÿ=𝑂(πœ‰(𝑑)),𝛽β‰₯0.

If 𝛽=0, then π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) reduces to Lip(πœ‰(𝑑),π‘Ÿ), and if πœ‰(𝑑)=𝑑𝛼(0<𝛼≀1), then Lip(πœ‰(𝑑),π‘Ÿ) class coincides with the class Lip(𝛼,π‘Ÿ). Lip(𝛼,π‘Ÿ)β†’Lip𝛼 for π‘Ÿβ†’βˆž.

We also write 1πœ™(π‘₯,𝑑)=πœ™(𝑑)=𝑓(π‘₯+𝑑)+𝑓(π‘₯βˆ’π‘‘)βˆ’2𝑓(π‘₯),𝐾(𝑛,𝑑)=2πœ‹(𝑛+1)π‘›ξ“π‘˜=0π‘ƒπ‘˜π‘˜βˆ’1𝑖=0𝑝𝑖sin(π‘˜βˆ’π‘–+1/2)𝑑,sin(𝑑/2)(1.6)𝜏=[1/𝑑], the greatest integer contained in 1/𝑑, π‘ƒπœ=𝑃[1/𝑑], Ξ”π‘π‘˜β‰‘π‘π‘˜βˆ’π‘π‘˜+1.

2. Known Results

Chandra [3] and Khan [4] have obtained the error estimates ‖𝑁𝑛(𝑓;π‘₯)βˆ’π‘“(π‘₯)β€–π‘Ÿ=𝑂(π‘›βˆ’π›Ό) in Lip(𝛼,π‘Ÿ) class using monotonicity conditions on the means generating sequence {𝑝𝑛}, which was generalized by Leindler [5] to almost monotone weights {𝑝𝑛} and by Mittal et al. [6] to general summability matrix. Further, Mittal et al. [7] have extended the results of Leindler [5] to general summability matrix, which in turn generalizes some results of Chandra [3] and Mittal et al. [6]. Recently, Lal [8] has determined the degree of approximation of the functions belonging to Lip𝛼 and π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) classes using CesΓ‘ro-NΓΆrlund (𝐢1⋅𝑁𝑝) summability with nonincreasing weights {𝑝𝑛}. He proved the following theorem.

Theorem 2.1. Let 𝑁𝑝 be a regular NΓΆrlund method defined by a sequence {𝑝𝑛} such that π‘ƒπœπ‘›ξ“π‘˜=πœπ‘ƒπ‘˜βˆ’1=𝑂(𝑛+1).(2.1)

Let π‘“βˆˆπΏ1[0,2πœ‹] be a 2Ο€-periodic function belonging to Lip𝛼(0<𝛼≀1), then the degree of approximation of 𝑓 by 𝐢1⋅𝑁𝑝 means of its Fourier series is given by sup[]π‘₯∈0,2πœ‹||𝑑𝑛𝐢𝑁||=‖‖𝑑(π‘₯)βˆ’π‘“(π‘₯)π‘›πΆπ‘β€–β€–βˆ’π‘“βˆž=⎧βŽͺ⎨βŽͺβŽ©π‘‚((𝑛+1)βˆ’π›Όπ‘‚ξ‚΅),0<𝛼<1,log(𝑛+1)πœ‹π‘’(𝑛+1),𝛼=1.(2.2)

Theorem 2.2. If 𝑓 is a 2Ο€-periodic function and Lebesgue integrable on [0, 2Ο€] and is belonging to π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) class, then its degree of approximation by 𝐢1⋅𝑁𝑝 means of its Fourier series is given by β€–β€–π‘‘π‘›πΆπ‘β€–β€–βˆ’π‘“π‘Ÿξ€·=𝑂(𝑛+1)𝛽+1/π‘Ÿπœ‰ξ€·(𝑛+1)βˆ’1,ξ€Έξ€Έ(2.3) provided πœ‰(𝑑) satisfies the following conditions: ξ‚»πœ‰(𝑑)π‘‘ξ‚Όξƒ―ξ€œbeadecreasingsequence,(2.4)01/(𝑛+1)𝑑||||πœ™(𝑑)sin𝛽𝑑ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘Ÿξ€·=𝑂(𝑛+1)βˆ’1ξ€Έ,ξƒ―ξ€œ(2.5)πœ‹1/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώ||||πœ™(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘Ÿξ€·(=𝑂𝑛+1)𝛿,(2.6) where 𝛿 is an arbitrary number such that 𝑠(1βˆ’π›Ώ)βˆ’1>0, π‘Ÿβˆ’1+π‘ βˆ’1=1, 1β‰€π‘Ÿβ‰€βˆž, conditions (2.5) and (2.6) hold uniformly in π‘₯.

Remark 2.3. In the proof of Theorem 2.1 of Lal [8, page 349], the estimate for 𝛼=1 is obtained as𝑂1𝑛+1+𝑂log(𝑛+1)πœ‹ξ‚Άξ‚΅π‘›+1=𝑂log𝑒𝑛+1+𝑂log(𝑛+1)πœ‹ξ‚Άξ‚΅π‘›+1=𝑂log(𝑛+1)πœ‹π‘’ξ‚Άπ‘›+1.(2.7) Since 1/(𝑛+1)≀log(𝑛+1)πœ‹/(𝑛+1), the 𝑒 is not needed in (2.2) for the case 𝛼=1 (cf. [9, page 6870]).

Remark 2.4. (i) The author has used monotonicity condition on sequence {𝑝𝑛} in the proof of Theorem 2.1 and Theorem 2.2, but not mentioned it in the statements. Further in condition (2.4), {πœ‰(𝑑)/𝑑} is a function of 𝑑 not a sequence.
(ii) The condition (2.5) of Theorem 2.2 leads to the divergent integral βˆ«πœ€1/(𝑛+1)π‘‘βˆ’(𝛽+1)𝑠𝑑𝑑 as πœ€β†’0 and 𝛽β‰₯0 [8, page 349]. Also in [8, pages 349-350], the author while writing the proof of Theorem 2.2 has used sin𝑑β‰₯2𝑑/πœ‹ in the interval [1/(𝑛+1),πœ‹], which is not valid for 𝑑=πœ‹.

3. Main Results

The observations of Remarks 2.3 and 2.4 motivated us to determine a proper set of conditions to prove Theorems 2.1 and 2.2 without monotonocity on {𝑝𝑛}. More precisely, we prove the following theorem.

Theorem 3.1. Let 𝑁𝑝 be the NΓΆrlund summability matrix generated by the nonnegative sequence {𝑝𝑛}, which satisfies (𝑛+1)𝑝𝑛𝑃=𝑂𝑛,βˆ€π‘›β‰₯0.(3.1) Then the degree of approximation of a 2Ο€-periodic signal (function) π‘“βˆˆLip𝛼 by 𝐢1⋅𝑁𝑝 means of its Fourier series is given by ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)∞=⎧βŽͺ⎨βŽͺβŽ©π‘‚(π‘›βˆ’π›Όπ‘‚ξ‚΅),0<𝛼<1,log𝑛𝑛,𝛼=1.(3.2)

Theorem 3.2. Let the condition (3.1) be satisfied. Then the degree of approximation of a 2Ο€-periodic signal (function) π‘“βˆˆπ‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) with 0≀𝛽≀1βˆ’1/π‘Ÿ by 𝐢1⋅𝑁𝑝 means of its Fourier series is given by ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)π‘Ÿξ‚€π‘›=𝑂𝛽+1/π‘Ÿπœ‰ξ‚€1𝑛,(3.3) provided positive increasing function πœ‰(𝑑) satisfies the condition (2.4) and ξƒ―ξ€œ0πœ‹/𝑛||||πœ™(𝑑)sin𝛽(𝑑/2)ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘Ÿξƒ―ξ€œ=𝑂(1),(3.4)πœ‹πœ‹/π‘›ξƒ©π‘‘βˆ’π›Ώ||||πœ™(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘Ÿξ€·π‘›=𝑂𝛿,(3.5) where 𝛿 is an arbitrary number such that 𝑠(π›½βˆ’π›Ώ)βˆ’1>0, π‘Ÿβˆ’1+π‘ βˆ’1=1, π‘Ÿβ‰₯1, conditions (3.4) and (3.5) hold uniformly in π‘₯.

Remark 3.3. For nonincreasing sequence {𝑝𝑛}, we have 𝑃𝑛=π‘›ξ“π‘˜=0π‘π‘˜β‰₯π‘π‘›π‘›ξ“π‘˜=01=(𝑛+1)𝑝𝑛,thatis,(𝑛+1)𝑝𝑛𝑃=𝑂𝑛.(3.6) Thus condition (3.1) holds for nonincreasing sequence {𝑝𝑛}; hence our Theorems 3.1 and 3.2 generalize Theorems 2.1 and 2.2, respectively.

Note 1. Using condition (2.4), we get(𝑛/πœ‹)πœ‰(πœ‹/𝑛)β‰€π‘›πœ‰(1/𝑛).

4. Lemmas

For the proof of our Theorems, we need the following lemmas.

Lemma 4.1 (see [10, 5.11]). If {𝑝𝑛} is nonnegative and nonincreasing sequence, then for 0β‰€π‘Ž<π‘β‰€βˆž,0β‰€π‘‘β‰€πœ‹ and for any 𝑛|||||π‘ξ“π‘˜=π‘Žπ‘π‘˜π‘’π‘–(π‘›βˆ’π‘˜)𝑑|||||=ξ‚»π‘‚ξ€·π‘ƒξ€·π‘‘βˆ’1𝑂𝑑,foranyπ‘Ž,βˆ’1π‘π‘Žξ€Έ,forπ‘Žβ‰₯π‘‘βˆ’1.(4.1)

Lemma 4.2 (see  [8, page 348]). For 0<π‘‘β‰€πœ‹/𝑛, 𝐾(𝑛,𝑑)=𝑂(𝑛).

Lemma 4.3. If {𝑝𝑛} is nonnegative sequence satisfying (3.1), then for πœ‹π‘›βˆ’1<π‘‘β‰€πœ‹, 𝑑𝐾(𝑛,𝑑)=π‘‚βˆ’2𝑑(𝑛+1)+π‘‚βˆ’1ξ€Έ.(4.2)

Proof. We have 1𝐾(𝑛,𝑑)=2πœ‹(𝑛+1)sin(𝑑/2)π‘›ξ“π‘˜=0π‘ƒπ‘˜π‘˜βˆ’1ξ“π‘Ÿ=0π‘π‘Ÿξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑=12πœ‹(𝑛+1)sin(𝑑/2)πœξ“π‘˜=0+π‘›ξ“π‘˜=𝜏+1𝑃ξƒͺξƒ©π‘˜π‘˜βˆ’1ξ“π‘Ÿ=0π‘π‘Ÿξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑ξƒͺ=𝐾1(𝑛,𝑑)+𝐾2(𝑛,𝑑),say.(4.3) Now, using (sin𝑑/2)βˆ’1β‰€πœ‹/𝑑, for 0<π‘‘β‰€πœ‹, we get ||𝐾1||ξ€·(𝑛,𝑑)=𝑂(𝑛+1)βˆ’1π‘‘βˆ’1ξ€Έπœξ“π‘˜=0ξƒ©π‘ƒπ‘˜π‘˜βˆ’1ξ“π‘Ÿ=0π‘π‘Ÿξƒͺξ‚΅=π‘‚πœπ‘‘βˆ’1𝑑(𝑛+1)=π‘‚βˆ’2ξ‚Ά.(𝑛+1)(4.4) Using (sin𝑑/2)βˆ’1β‰€πœ‹/𝑑, for 0<π‘‘β‰€πœ‹ and changing the order of summation, we have ||𝐾2||𝑑(𝑛,𝑑)=π‘‚βˆ’1(𝑛+1)βˆ’1ξ€Έ|||||π‘›ξ“π‘˜=𝜏+1π‘ƒπ‘˜π‘˜βˆ’1ξ“π‘Ÿ=0π‘π‘Ÿξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑|||||𝑑=π‘‚βˆ’1(𝑛+1)βˆ’1ξ€Έ|||||𝜏+1ξ“π‘Ÿ=0π‘π‘Ÿπ‘›ξ“π‘˜=𝜏+1π‘ƒπ‘˜βˆ’1ξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑+π‘›ξ“π‘Ÿ=𝜏+1π‘π‘Ÿπ‘›ξ“π‘˜=π‘Ÿπ‘ƒπ‘˜βˆ’1ξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑|||||.(4.5) Again using (sin𝑑/2)βˆ’1β‰€πœ‹/𝑑, for 0<π‘‘β‰€πœ‹, Lemma 4.1, (in view of 𝑃𝑛 being positive and 𝑃𝑛+1βˆ’1β‰€π‘ƒπ‘›βˆ’1 for all 𝑛β‰₯0) and π‘‘βˆ’1<𝜏+1, we get |||||𝜏+1ξ“π‘Ÿ=0π‘π‘Ÿπ‘›ξ“π‘˜=𝜏+1π‘ƒπ‘˜βˆ’1ξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑|||||β‰€βŽ›βŽœβŽœβŽπœ+1ξ“π‘Ÿ=0π‘π‘Ÿ|||||π‘›ξ“π‘˜=𝜏+1π‘ƒπ‘˜βˆ’1𝑒𝑖(π‘˜βˆ’π‘Ÿ)𝑑|||||βŽžβŽŸβŽŸβŽ ξ€·π‘‘=π‘‚βˆ’1π‘ƒπœ+1βˆ’1ξ€Έπœ+1ξ“π‘Ÿ=0π‘π‘Ÿξ‚€1=𝑂𝑑.(4.6) Using Abel’s transformation, we get π‘›ξ“π‘˜=π‘Ÿπ‘ƒπ‘˜βˆ’1ξ‚€1sinπ‘˜βˆ’π‘Ÿ+2𝑑=π‘›βˆ’1ξ“π‘˜=π‘Ÿξ€·Ξ”π‘ƒπ‘˜βˆ’1ξ€Έπ‘˜ξ“π‘—=0ξ‚€1sinπ‘˜βˆ’π‘—+2𝑑+π‘ƒπ‘›π‘›βˆ’1𝑗=0ξ‚€1sinπ‘˜βˆ’π‘—+2ξ‚π‘‘βˆ’π‘ƒπ‘Ÿβˆ’1π‘Ÿβˆ’1𝑗=0ξ‚€1sinπ‘˜βˆ’π‘—+2𝑑1=π‘‚π‘‘ξ‚ξƒ©π‘›βˆ’1ξ“π‘˜=π‘Ÿ||Ξ”π‘ƒπ‘˜βˆ’1||+π‘ƒπ‘›βˆ’1+π‘ƒπ‘Ÿβˆ’1ξƒͺξ‚€1=π‘‚π‘‘ξ‚ξ€·π‘ƒπ‘›βˆ’1+π‘ƒπ‘Ÿβˆ’1ξ€Έ,(4.7) in view of (sin𝑑/2)βˆ’1β‰€πœ‹/𝑑, for 0<π‘‘β‰€πœ‹ and 𝑃𝑛β‰₯π‘ƒπ‘›βˆ’1forall𝑛β‰₯0.
Combining (4.5)–(4.7), we get ||𝐾2||𝑑(𝑛,𝑑)=π‘‚βˆ’2(𝑛+1)βˆ’11+π‘›ξ“π‘Ÿ=𝜏+1π‘π‘Ÿξ€·π‘ƒπ‘›βˆ’1+π‘ƒπ‘Ÿβˆ’1ξ€Έξƒͺ𝑑=π‘‚βˆ’2(𝑛+1)βˆ’11+π‘ƒπ‘›π‘›βˆ’1ξ“π‘Ÿ=0π‘π‘Ÿ+π‘›ξ“π‘Ÿ=𝜏+1ξ‚΅π‘π‘Ÿπ‘ƒπ‘Ÿξ‚Άξƒͺ𝑑=π‘‚βˆ’2(𝑛+1)βˆ’11+π‘›ξ“π‘Ÿ=𝜏+1(π‘Ÿ+1)βˆ’1ξƒͺ𝑑=π‘‚βˆ’2𝑛+11+π‘‚π‘›βˆ’πœξ€·π‘‘πœ+1=π‘‚βˆ’2(𝑛+1)βˆ’1𝑑+π‘‚βˆ’1ξ€Έ,(4.8) in view of (3.1) and πœβ‰€1/𝑑<(𝜏+1).
Finally collecting (4.3), (4.4) and (4.8), we get Lemma 4.3.

Proof of Theorem 3.1. We have 𝑠𝑛1(𝑓)βˆ’π‘“(π‘₯)=ξ€œ2πœ‹πœ‹0ξ‚΅sin(𝑛+1/2)𝑑sin(𝑑/2)πœ™(𝑑)𝑑𝑑.(4.9) Denoting 𝐢1⋅𝑁𝑝 means of {𝑠𝑛(𝑓)} by 𝑑𝑛𝐢𝑁(𝑓), we write ||𝑑𝑛𝐢𝑁||=ξ‚΅1(𝑓)βˆ’π‘“(π‘₯)ξ‚Ά|||||ξ€œ2πœ‹(𝑛+1)πœ‹0πœ™(𝑑)π‘›ξ“π‘˜=0π‘ƒπ‘˜π‘˜βˆ’1𝑖=0𝑝𝑖sin(π‘˜βˆ’π‘–+1/2)𝑑|||||β‰€ξ€œsin(𝑑/2)𝑑𝑑0πœ‹/𝑛||πœ™||ξ€œ(𝑑)𝐾(𝑛,𝑑)𝑑𝑑+πœ‹πœ‹/𝑛||||πœ™(𝑑)𝐾(𝑛,𝑑)𝑑𝑑=𝐼1+𝐼2,say.(4.10) Now, using Lemma 4.2 and the fact that π‘“βˆˆLipπ›Όβ‡’πœ™(𝑑)∈Lip𝛼 [10], we have 𝐼1ξ€œ=𝑂(𝑛)0πœ‹/𝑛𝑑𝛼𝑑𝑑=𝑂(π‘›βˆ’π›Ό).(4.11) Using Lemma 4.3, we get 𝐼2ξ‚»ξ€œ=π‘‚πœ‹πœ‹/π‘›π‘‘π›Όξ‚΅π‘‘βˆ’2(𝑛+1)+π‘‘βˆ’1𝐼𝑑𝑑=𝑂21𝐼+𝑂22ξ€Έ,say,(4.12) where 𝐼21=(𝑛+1)βˆ’1ξ€œπœ‹πœ‹/π‘›π‘‘π›Όβˆ’2⎧βŽͺ⎨βŽͺβŽ©π‘‘π‘‘=𝑂(π‘›βˆ’π›Όπ‘‚ξ‚΅),0<𝛼<1,log𝑛𝑛𝐼,𝛼=1,22=ξ€œπœ‹πœ‹/π‘›π‘‘π›Όβˆ’1ξ‚΅ξ€œπ‘‘π‘‘=π‘‚πœ‹πœ‹/π‘›πœ‹π‘‘π‘›π‘‘π›Όβˆ’1ξ‚Ά=⎧βŽͺ⎨βŽͺβŽ©π‘‘π‘‘π‘‚(π‘›βˆ’π›Ό)𝑂,0<𝛼<1,log𝑛𝑛,𝛼=1,(4.13) Collecting (4.10)–(4.13) and writing 1/𝑛≀(log𝑛)/𝑛, for large values of n, we get ||𝑑𝑛𝐢𝑁||=⎧βŽͺ⎨βŽͺ⎩(𝑓)βˆ’π‘“(π‘₯)𝑂(π‘›βˆ’π›Όπ‘‚ξ‚΅),0<𝛼<1,log𝑛𝑛,𝛼=1.(4.14) Hence, ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)∞=sup[]π‘₯∈0,2πœ‹||𝑑𝑛𝐢𝑁||=⎧βŽͺ⎨βŽͺ⎩(𝑓)βˆ’π‘“(π‘₯)𝑂(π‘›βˆ’π›Όπ‘‚ξ‚΅),0<𝛼<1,log𝑛𝑛,𝛼=1.(4.15) This completes the proof of Theorem 3.1.

Proof of Theorem 3.2. Following the proof of Theorem 3.1, we have π‘‘π‘›πΆπ‘ξ€œ(𝑓)βˆ’π‘“(π‘₯)=0πœ‹/π‘›ξ€œπœ™(𝑑)𝐾(𝑛,𝑑)𝑑𝑑+πœ‹πœ‹/π‘›πœ™(𝑑)𝐾(𝑛,𝑑)𝑑𝑑=𝐼3+𝐼4,say.(4.16) Using HΓΆlder’s inequality, πœ™(𝑑)βˆˆπ‘Š(πΏπ‘Ÿ,πœ‰(𝑑)), condition (3.4), Lemma 4.2, and (sin𝑑/2)βˆ’1β‰€πœ‹/𝑑, for 0<π‘‘β‰€πœ‹, we have ||𝐼3||=||||ξ€œ0πœ‹/π‘›πœ™(𝑑)sin𝛽(𝑑/2)β‹…πœ‰(𝑑)πœ‰(𝑑)𝐾(𝑛,𝑑)sin𝛽||||β‰€ξƒ©ξ€œ(𝑑/2)𝑑𝑑0πœ‹/𝑛||||πœ™(𝑑)sin𝛽(𝑑/2)||||πœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξƒ©limπœ€β†’0ξ€œπœ€πœ‹/𝑛||||πœ‰(𝑑)𝐾(𝑛,𝑑)sin𝛽||||(𝑑/2)𝑠ξƒͺ𝑑𝑑1/𝑠=𝑂(1)limπœ€β†’0ξ€œπœ€πœ‹/π‘›ξ‚΅πœ‰(𝑑)𝑛sin𝛽(𝑑/2)𝑠𝑑𝑑1/π‘ ξ‚€ξ‚€πœ‹=π‘‚π‘›πœ‰π‘›ξ‚΅ξ‚ξ‚limπœ€β†’0ξ€œπœ€πœ‹/π‘›π‘‘βˆ’π›½π‘ ξ‚Άπ‘‘π‘‘1/π‘ ξ‚€πœ‰ξ‚€1=𝑂𝑛𝑛𝛽+1βˆ’1/𝑠𝑛=𝑂𝛽+1/π‘Ÿπœ‰ξ‚€1𝑛,(4.17) in view of the mean value theorem for integrals, π‘Ÿβˆ’1+π‘ βˆ’1=1 and Note 1.
Similarly, using HΓΆlder’s inequality, Lemma 4.3, |sin(𝑑/2)|≀1, (sin(𝑑/2))βˆ’1β‰€πœ‹/𝑑, condition (3.5), and the mean value theorem for integrals, we have ||𝐼4||ξ‚΅ξ€œ=π‘‚πœ‹πœ‹/π‘›ξ€·π‘‘πœ™(𝑑)βˆ’2(𝑛+1)βˆ’1+π‘‘βˆ’1𝐼𝑑𝑑=𝑂41𝐼+𝑂42ξ€Έ,say,(4.18) where 𝐼41=ξ‚΅ξ€œπœ‹πœ‹/π‘›πœ™(𝑑)π‘‘βˆ’2(𝑛+1)βˆ’1𝑑𝑑=𝑂(𝑛+1)βˆ’1ξ€Έξ‚΅ξ€œπœ‹πœ‹/π‘›π‘‘βˆ’π›Ώπœ™(𝑑)sin𝛽(𝑑/2)πœ‰(𝑑)πœ‰(𝑑)𝑑2βˆ’π›Ώsin𝛽𝑛(𝑑/2)𝑑𝑑=π‘‚βˆ’1ξ€Έξƒ©ξ€œπœ‹πœ‹/π‘›ξƒ©π‘‘βˆ’π›Ώ||||πœ™(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξ‚΅ξ€œπœ‹πœ‹/π‘›ξ‚΅πœ‰(𝑑)𝑑2βˆ’π›Ώ+𝛽𝑠𝑑𝑑1/𝑠𝑛=π‘‚π›Ώβˆ’1ξ€Έξ‚΅ξ€œπ‘›/πœ‹1/πœ‹ξ‚΅πœ‰(1/𝑦)π‘¦π›Ώβˆ’2βˆ’π›½ξ‚Άπ‘ π‘¦βˆ’2𝑑𝑦1/𝑠𝑛=π‘‚π›Ώβˆ’1ξ‚€π‘›πœ‹ξ‚πœ‰ξ‚€πœ‹π‘›ξ‚΅ξ€œξ‚ξ‚πœ€π‘›/πœ‹1𝑦(1βˆ’π›Ώ+𝛽)π‘ βˆ’2𝑑𝑦1/𝑠𝑛=π‘‚π›Ώπœ‰ξ‚€1𝑛𝑛1βˆ’π›Ώ+π›½βˆ’1/π‘ ξ‚ξ‚€πœ‰ξ‚€1=𝑂𝑛𝑛𝛽+1/π‘Ÿξ‚,𝐼42=ξ€œπœ‹πœ‹/π‘›πœ™(𝑑)π‘‘βˆ’1ξ€œπ‘‘π‘‘=πœ‹πœ‹/π‘›π‘‘βˆ’π›Ώπœ™(𝑑)sin𝛽(𝑑/2)πœ‰(𝑑)πœ‰(𝑑)𝑑1βˆ’π›Ώsinπ›½ξƒ©ξ€œ(𝑑/2)𝑑𝑑=π‘‚πœ‹πœ‹/𝑛||||π‘‘βˆ’π›Ώπœ™(𝑑)sin𝛽(𝑑/2)||||πœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξƒ©ξ€œπœ‹πœ‹/𝑛||||πœ‰(𝑑)𝑑1βˆ’π›Ώsin𝛽||||(𝑑/2)𝑠ξƒͺ𝑑𝑑1/π‘ ξƒ©ξ€œ=π‘‚πœ‹πœ‹/𝑛||||π‘‘βˆ’π›Ώπœ™(𝑑)||||πœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξ‚΅ξ€œπœ‹πœ‹/𝑛|||πœ‰(𝑑)𝑑1βˆ’π›Ώ+𝛽|||𝑠𝑑𝑑1/𝑠𝑛=π‘‚π›Ώξ€Έξ‚΅ξ€œπ‘›/πœ‹1/πœ‹ξ‚΅πœ‰(1/𝑦)π‘¦π›Ώβˆ’1βˆ’π›½ξ‚Άπ‘ π‘¦βˆ’2𝑑𝑦1/𝑠𝑛=𝑂𝛿+1πœ‹ξ‚Άπœ‰ξ‚€πœ‹π‘›ξ‚ξ€œξ‚Άξ‚΅πœ€π‘›/πœ‹2𝑦(π›½βˆ’π›Ώ)π‘ βˆ’2𝑑𝑦1/𝑠𝑛=𝑂𝛿+1πœ‰ξ‚€1π‘›ξ‚π‘›βˆ’π›Ώ+π›½βˆ’1/π‘ ξ‚ξ‚€πœ‰ξ‚€1=𝑂𝑛𝑛𝛽+1/π‘Ÿξ‚,(4.19) in view of increasing nature of π‘¦πœ‰(1/𝑦), π‘Ÿβˆ’1+π‘ βˆ’1=1, where πœ€1,πœ€2 lie in [πœ‹βˆ’1,π‘›πœ‹βˆ’1], and Note 1.
Collecting (4.16)–(4.19), we get ||𝑑𝑛𝐢𝑁||𝑛(𝑓)βˆ’π‘“(π‘₯)=𝑂𝛽+1/π‘Ÿπœ‰ξ‚€1𝑛.(4.20) Hence, ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)π‘Ÿ=ξ‚΅1ξ€œ2πœ‹02πœ‹||𝑑𝑛𝐢𝑁||(𝑓)βˆ’π‘“(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿξ‚€π‘›=𝑂𝛽+1/π‘Ÿπœ‰ξ‚€1𝑛.(4.21) This completes the proof of Theorem 3.2.

5. Corollaries

The following corollaries can be derived from Theorem 3.2.

Corollary 5.1. If 𝛽=0, then for π‘“βˆˆLip(πœ‰(𝑑),π‘Ÿ), ‖𝑑𝑛𝐢𝑁(𝑓)βˆ’π‘“(π‘₯)β€–π‘Ÿ=𝑂(𝑛1/π‘Ÿπœ‰(1/𝑛)).

Corollary 5.2. If 𝛽=0,πœ‰(𝑑)=𝑑𝛼(0<𝛼≀1), then for π‘“βˆˆLip(𝛼,π‘Ÿ)(𝛼>1/π‘Ÿ), ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)π‘Ÿξ€·π‘›=𝑂1/π‘Ÿβˆ’π›Όξ€Έ.(5.1)

Corollary 5.3. If π‘Ÿβ†’βˆž in Corollary 5.2, then for π‘“βˆˆLip𝛼(0<𝛼<1), (5.1) gives ‖‖𝑑𝑛𝐢𝑁‖‖(𝑓)βˆ’π‘“(π‘₯)∞=𝑂(π‘›βˆ’π›Ό).(5.2)

6. Conclusion

Various results pertaining to the degree of approximation of periodic functions (signals) belonging to the Lipchitz classes have been reviewed and the condition of monotonocity on the means generating sequence {𝑝𝑛} has been relaxed. Further, a proper set of conditions have been discussed to rectify the errors pointed out in Remarks 2.3 and 2.4.

Acknowledgment

The authors are very grateful to the reviewer for his kind suggestions for the improvement of this paper.

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