Trigonometric Approximation of Signals (Functions) Belonging to 𝑊(𝐿^𝑟,𝜉(𝑡)) Class by Matrix (𝐶¹⋅𝑁_𝑝) Operator

Abstract

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2π-periodic signals (functions) belonging to Lip(𝛼,𝑟) 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,𝑟)- 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 𝑊(𝐿^𝑟, 𝜉(𝑡)) classes by using Cesáro-Nörlund (𝐶¹⋅𝑁_𝑝) summability without monotonicity condition on {𝑝_𝑛}, 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) 𝑓∈𝐿^𝑟∶=𝐿^𝑟[0,2𝜋], 𝑟≥1, let 𝑠_𝑛(𝑓)=𝑠_𝑛(𝑓;𝑥)=𝑎⁰2+^𝑛_𝑘=1𝑎_𝑘cos𝑘𝑥+𝑏_𝑘sin𝑘𝑥=^𝑛_𝑘=0𝑢_𝑘(𝑓;𝑥)(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 {𝑝_𝑛} be a nonnegative sequence of real numbers such that 𝑃_𝑛(=∑^𝑛𝑘=0𝑝_𝑘≠0)→∞ as 𝑛→∞ and 𝑃₋₁=0=𝑝₋₁.

Define 𝑁_𝑛(𝑓)=𝑁_𝑛(𝑓;𝑥)=𝑃_𝑛𝑛−1_𝑘=0𝑝_𝑛−𝑘𝑠_𝑘(𝑓;𝑥),∀𝑛≥0,(1.2) the Nörlund (𝑁_𝑝) means of the sequence 𝑠_𝑛(𝑓) or Fourier series of 𝑓. The Fourier series of 𝑓 is said to be Nörlund (𝑁_𝑝) summable to 𝑠(𝑥) if 𝑁_𝑛(𝑓;𝑥)→𝑠(𝑥)as𝑛→∞. The Fourier series of 𝑓 is called Cesáro-Nörlund (𝐶¹⋅𝑁_𝑝) summable to 𝑆(𝑥) if 𝑡^𝑛𝐶𝑁(𝑓)=(𝑛+1)^𝑛−1_𝑘=0𝑃_𝑘𝑘−1_𝑖=0𝑝_𝑘−𝑖𝑠_𝑖(𝑓;𝑥)⟶𝑆(𝑥)as𝑛⟶∞.(1.3) We note that 𝑁_𝑛(𝑓) and 𝑡^𝑛𝐶𝑁(𝑓) are also trigonometric polynomials of degree (or order) 𝑛.

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

The 𝐿^𝑟-norm of signal 𝑓 is defined by ‖𝑓‖_𝑟=12𝜋^02𝜋||𝑓(𝑥)||^𝑟𝑑𝑥^1/𝑟(1≤𝑟<∞),‖𝑓‖_∞=sup_{[]𝑥∈0,2𝜋}||𝑓(𝑥)||.(1.4) A signal (function) 𝑓 is approximated by trigonometric polynomials 𝑇_𝑛(𝑓) of degree 𝑛, and the degree of approximation 𝐸_𝑛(𝑓) is given by 𝐸_𝑛(𝑓)=Min_𝑛‖‖𝑓(𝑥)−𝑇_𝑛(𝑓)‖‖_𝑟.(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(𝑥+𝑡)−𝑓(𝑥)|=𝑂(|𝑡|^𝛼),0<𝛼≤1, and 𝑓∈Lip(𝛼,𝑟) if ‖𝑓(𝑥+𝑡)−𝑓(𝑥)‖_𝑟=𝑂(|𝑡|^𝛼),0<𝛼≤1,𝑟≥1.

For a positive increasing function 𝜉(𝑡) and 𝑟≥1, 𝑓∈Lip(𝜉(𝑡),𝑟) if ‖𝑓(𝑥+𝑡)−𝑓(𝑥)‖_𝑟=𝑂(𝜉(𝑡)), and 𝑓∈𝑊(𝐿^𝑟,𝜉(𝑡)) if ‖[𝑓(𝑥+𝑡)−𝑓(𝑥)]sin^𝛽(𝑥/2)‖_𝑟=𝑂(𝜉(𝑡)),𝛽≥0.

If 𝛽=0, then 𝑊(𝐿^𝑟,𝜉(𝑡)) reduces to Lip(𝜉(𝑡),𝑟), and if 𝜉(𝑡)=𝑡^𝛼(0<𝛼≤1), then Lip(𝜉(𝑡),𝑟) class coincides with the class Lip(𝛼,𝑟). Lip(𝛼,𝑟)→Lip𝛼 for 𝑟→∞.

We also write 𝐾𝜙(𝑛(,𝑥𝑡),𝑡=)=𝜙(1𝑡)=𝑓(𝑥+𝑡)+𝑓(𝑥−𝑡)−2𝑓(𝑥),2𝜋(𝑛+1)^𝑛_𝑘=0𝑃_𝑘𝑘−1_𝑖=0𝑝_𝑖sin(𝑘−𝑖+1/2)𝑡sin(𝑡/2),(1.6)𝜏=[1/𝑡], the greatest integer contained in 1/𝑡, 𝑃_𝜏=𝑃[1/𝑡], Δ𝑝_𝑘≡𝑝_𝑘−𝑝_𝑘+1.

2. Known Results

Chandra [3] and Khan [4] have obtained the error estimates ‖𝑁_𝑛(𝑓;𝑥)−𝑓(𝑥)‖_𝑟=𝑂(𝑛^−𝛼) in Lip(𝛼,𝑟) class using monotonicity conditions on the means generating sequence {𝑝_𝑛}, which was generalized by Leindler [5] to almost monotone weights {𝑝_𝑛} 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 𝑊(𝐿^𝑟,𝜉(𝑡)) classes using Cesáro-Nörlund (𝐶¹⋅𝑁_𝑝) summability with nonincreasing weights {𝑝_𝑛}. He proved the following theorem.

Theorem 2.1. Let 𝑁_𝑝 be a regular Nörlund method defined by a sequence {𝑝_𝑛} such that 𝑃_𝜏𝑛_𝑘=𝜏𝑃_𝑘−1=𝑂(𝑛+1).(2.1)

Let 𝑓∈𝐿¹[0,2𝜋] be a 2π-periodic function belonging to Lip𝛼(0<𝛼≤1), then the degree of approximation of 𝑓 by 𝐶¹⋅𝑁_𝑝 means of its Fourier series is given by sup_{[]𝑥∈0,2𝜋}||𝑡^𝑛𝐶𝑁(𝑥)−𝑓(𝑥)||=‖‖𝑡^𝑛𝐶𝑁−𝑓‖‖_∞=⎧⎪⎨⎪⎩𝑂((𝑛+1)^−𝛼𝑂log(𝑛+)1,)𝜋𝑒0<𝛼<1,(𝑛+1),𝛼=1.(2.2)

Theorem 2.2. If 𝑓 is a 2π-periodic function and Lebesgue integrable on [0, 2π] and is belonging to 𝑊(𝐿^𝑟,𝜉(𝑡)) class, then its degree of approximation by 𝐶¹⋅𝑁_𝑝 means of its Fourier series is given by ‖‖𝑡^𝑛𝐶𝑁−𝑓‖‖_𝑟=𝑂(𝑛+1)^𝛽+1/𝑟𝜉(𝑛+1)⁻¹,(2.3) provided 𝜉(𝑡) satisfies the following conditions: 𝜉(𝑡𝑡)beadecreasingsequence,(2.4)_01/(𝑛+1)𝑡||𝜙(𝑡)||sin^𝛽𝑡𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟=𝑂(𝑛+1)⁻¹,(2.5)_{𝜋1/(𝑛+1)}𝑡^−𝛿||𝜙(𝑡)||𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟=𝑂(𝑛+1)^𝛿,(2.6) where 𝛿 is an arbitrary number such that 𝑠(1−𝛿)−1>0, 𝑟⁻¹+𝑠⁻¹=1, 1≤𝑟≤∞, conditions (2.5) and (2.6) hold uniformly in 𝑥.

Remark 2.3. In the proof of Theorem 2.1 of Lal [8, page 349], the estimate for 𝛼=1 is obtained as𝑂1𝑛+1+𝑂log𝑛(𝑛++11)𝜋=𝑂𝑛lo+g𝑒1+𝑂log𝑛(𝑛++11)𝜋=𝑂log(𝑛𝑛++11)𝜋𝑒.(2.7) Since 1/(𝑛+1)≤log(𝑛+1)𝜋/(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 {𝑝_𝑛} in the proof of Theorem 2.1 and Theorem 2.2, but not mentioned it in the statements. Further in condition (2.4), {𝜉(𝑡)/𝑡} is a function of 𝑡 not a sequence.
(ii) The condition (2.5) of Theorem 2.2 leads to the divergent integral ∫^{𝜀1/(𝑛+1)}𝑡^{−(𝛽+1)𝑠}𝑑𝑡 as 𝜀→0 and 𝛽≥0 [8, page 349]. Also in [8, pages 349-350], the author while writing the proof of Theorem 2.2 has used sin𝑡≥2𝑡/𝜋 in the interval [1/(𝑛+1),𝜋], which is not valid for 𝑡=𝜋.

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 {𝑝_𝑛}. More precisely, we prove the following theorem.

Theorem 3.1. Let 𝑁_𝑝 be the Nörlund summability matrix generated by the nonnegative sequence {𝑝_𝑛}, which satisfies (𝑛+1)𝑝_𝑛=𝑂𝑃_𝑛,∀𝑛≥0.(3.1) Then the degree of approximation of a 2π-periodic signal (function) 𝑓∈Lip𝛼 by 𝐶¹⋅𝑁_𝑝 means of its Fourier series is given by ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_∞=⎧⎪⎨⎪⎩𝑂(𝑛^−𝛼𝑂log),𝑛0<𝛼<1,𝑛,𝛼=1.(3.2)

Theorem 3.2. Let the condition (3.1) be satisfied. Then the degree of approximation of a 2π-periodic signal (function) 𝑓∈𝑊(𝐿^𝑟,𝜉(𝑡)) with 0≤𝛽≤1−1/𝑟 by 𝐶¹⋅𝑁_𝑝 means of its Fourier series is given by ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_𝑟=𝑂𝑛^𝛽+1/𝑟𝜉1𝑛,(3.3) provided positive increasing function 𝜉(𝑡) satisfies the condition (2.4) and ^0𝜋/𝑛||𝜙(𝑡)||sin^𝛽(𝑡/2)𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟=𝑂(1),(3.4)_{𝜋𝜋/𝑛}𝑡^−𝛿||𝜙(𝑡)||𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟=𝑂𝑛^𝛿,(3.5) where 𝛿 is an arbitrary number such that 𝑠(𝛽−𝛿)−1>0, 𝑟⁻¹+𝑠⁻¹=1, 𝑟≥1, conditions (3.4) and (3.5) hold uniformly in 𝑥.

Remark 3.3. For nonincreasing sequence {𝑝_𝑛}, we have 𝑃_𝑛=^𝑛_𝑘=0𝑝_𝑘≥𝑝_𝑛𝑛_𝑘=01=(𝑛+1)𝑝_𝑛,thatis,(𝑛+1)𝑝_𝑛=𝑂𝑃_𝑛.(3.6) Thus condition (3.1) holds for nonincreasing sequence {𝑝_𝑛}; 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)𝜉(𝜋/𝑛)≤𝑛𝜉(1/𝑛).

4. Lemmas

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

Lemma 4.1 (see [10, 5.11]). If {𝑝_𝑛} is nonnegative and nonincreasing sequence, then for 0≤𝑎<𝑏≤∞,0≤𝑡≤𝜋 and for any 𝑛|||||_𝑏_𝑘=𝑎𝑝_𝑘𝑒^{𝑖(𝑛−𝑘)𝑡}|||||=𝑂𝑃𝑡₋₁𝑂𝑡,forany𝑎,₋₁𝑝_𝑎,for𝑎≥𝑡⁻¹.(4.1)

Lemma 4.2 (see  [8, page 348]). For 0<𝑡≤𝜋/𝑛, 𝐾(𝑛,𝑡)=𝑂(𝑛).

Lemma 4.3. If {𝑝_𝑛} is nonnegative sequence satisfying (3.1), then for 𝜋𝑛⁻¹<𝑡≤𝜋, 𝐾(𝑛,𝑡)=𝑂𝑡⁻²(𝑛+1)+𝑂𝑡⁻¹.(4.2)

Proof. We have 𝐾(𝑛,𝑡)=12𝜋(𝑛+1)sin(𝑡/2)^𝑛_𝑘=0𝑃_𝑘𝑘−1_𝑟=0𝑝_𝑟sin𝑘−𝑟+1=12𝑡2𝜋(𝑛+1)sin(𝑡/2)^𝜏_𝑘=0+^𝑛_𝑘_=𝜏+1𝑃^𝑘𝑘−1_𝑟=0𝑝_𝑟sin𝑘−𝑟+1=𝐾₁2𝑡(𝑛,𝑡)+𝐾₂(𝑛,𝑡),say.(4.3) Now, using (sin𝑡/2)⁻¹≤𝜋/𝑡, for 0<𝑡≤𝜋, we get ||𝐾₁(𝑛,𝑡)||=𝑂(𝑛+1)⁻¹𝑡⁻¹^𝜏_𝑘=0𝑃^𝑘𝑘−1_𝑟=0𝑝_𝑟=𝑂(𝑛𝜏+𝑡⁻¹1)=𝑂(𝑛𝑡+⁻²1).(4.4) Using (sin𝑡/2)⁻¹≤𝜋/𝑡, for 0<𝑡≤𝜋 and changing the order of summation, we have ||𝐾₂(𝑛,𝑡)||=𝑂𝑡⁻¹(𝑛+1)⁻¹|||||^𝑛_𝑘_=𝜏+1𝑃_𝑘𝑘−1_𝑟=0𝑝_𝑟sin𝑘−𝑟+1=𝑂𝑡₋₁2𝑡|||||(𝑛+1)⁻¹|||||^𝜏+1_𝑟=0𝑝_𝑟𝑛𝑘_=𝜏+1𝑃_𝑘−1sin𝑘−𝑟+1+_𝑛2𝑡_𝑟=_𝜏+1𝑝_𝑟𝑛_𝑘=𝑟𝑃_𝑘−1sin𝑘−𝑟+12𝑡|||||.(4.5) Again using (sin𝑡/2)⁻¹≤𝜋/𝑡, for 0<𝑡≤𝜋, Lemma 4.1, (in view of 𝑃_𝑛 being positive and 𝑃_𝑛+1−1≤𝑃_𝑛−1 for all 𝑛≥0) and 𝑡⁻¹<𝜏+1, we get |||||_𝜏+1_𝑟=0𝑝_𝑟𝑛𝑘_=𝜏+1𝑃_𝑘−1sin𝑘−𝑟+12𝑡|||||≤⎛⎜⎜⎝^𝜏+1_𝑟=0𝑝_𝑟|||||_𝑛𝑘_=𝜏+1𝑃_𝑘−1𝑒^{𝑖(𝑘−𝑟)𝑡}|||||⎞⎟⎟⎠=𝑂𝑡₋₁𝑃_𝜏+1−1^𝜏+1_𝑟=0𝑝_𝑟=𝑂1𝑡.(4.6) Using Abel’s transformation, we get ^𝑛_𝑘=𝑟𝑃_𝑘−1sin𝑘−𝑟+12𝑡=^𝑛−1_𝑘=𝑟Δ𝑃_𝑘−1^𝑘_𝑗=0sin𝑘−𝑗+1+𝑃_𝑛𝑛−12𝑡_𝑗=0sin𝑘−𝑗+12𝑡−𝑃^{𝑟−1𝑟−1}_𝑗=0sin𝑘−𝑗+1=𝑂12𝑡𝑡^𝑛−1_𝑘=𝑟||Δ𝑃_𝑘−1||+𝑃_𝑛−1+𝑃_𝑟−1=𝑂1𝑡𝑃^𝑛−1+𝑃_𝑟−1,(4.7) in view of (sin𝑡/2)⁻¹≤𝜋/𝑡, for 0<𝑡≤𝜋 and 𝑃_𝑛≥𝑃_𝑛−1forall𝑛≥0.
Combining (4.5)–(4.7), we get ||𝐾₂(𝑛,𝑡)||=𝑂𝑡⁻²(𝑛+1)⁻¹1+^𝑛_𝑟=_𝜏+1=𝑂𝑡₋₂𝑝_𝑟𝑃_𝑛−1+𝑃_𝑟−1(𝑛+1)⁻¹1+𝑃_𝑛𝑛−1_𝑟=0𝑝_𝑟+^𝑛=𝑂𝑡₋₂_𝑟=𝜏+1𝑃𝑝^𝑟_𝑟(𝑛+1)⁻¹1+^𝑛_𝑟=_𝜏+1(𝑟+1)⁻¹=𝑂𝑡₋₂𝑛+11+𝑂𝜏𝑛−+𝜏1=𝑂𝑡⁻²(𝑛+1)⁻¹+𝑂𝑡⁻¹,(4.8) in view of (3.1) and 𝜏≤1/𝑡<(𝜏+1).
Finally collecting (4.3), (4.4) and (4.8), we get Lemma 4.3.

Proof of Theorem 3.1. We have 𝑠_𝑛(𝑓)−𝑓(𝑥)=12𝜋^𝜋0sins(in𝑛+(𝑡/12/)2)𝑡𝜙(𝑡)𝑑𝑡.(4.9) Denoting 𝐶¹⋅𝑁_𝑝 means of {𝑠_𝑛(𝑓)} by 𝑡^𝑛𝐶𝑁(𝑓), we write ||𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)||=12𝜋(𝑛+1)|||||^𝜋0𝜙(𝑡)^𝑛_𝑘=0𝑃_𝑘𝑘−1_𝑖=0𝑝_𝑖≤sin(𝑘sin−(𝑖𝑡/+2)1/2)𝑡𝑑𝑡_0𝜋/𝑛|||||||𝜙(𝑡)𝐾(𝑛,𝑡)||𝑑𝑡+^{𝜋𝜋/𝑛}||𝜙(𝑡)𝐾(𝑛,𝑡)||𝑑𝑡=𝐼₁+𝐼₂,say.(4.10) Now, using Lemma 4.2 and the fact that 𝑓∈Lip𝛼⇒𝜙(𝑡)∈Lip𝛼 [10], we have 𝐼₁=𝑂(𝑛)^0𝜋/𝑛𝑡^𝛼𝑑𝑡=𝑂(𝑛^−𝛼).(4.11) Using Lemma 4.3, we get 𝐼₂=𝑂^{𝜋𝜋/𝑛}𝑡^𝛼(𝑛𝑡+⁻²1)+𝑡⁻¹𝑑𝑡=𝑂𝐼²¹+𝑂𝐼₂₂,say,(4.12) where 𝐼₂₁=(𝑛+1)⁻¹^{𝜋𝜋/𝑛}𝑡^𝛼−2𝑑𝑡=⎧⎪⎨⎪⎩𝑂(𝑛^−𝛼𝑂log),𝑛0<𝛼<1,𝐼𝑛,𝛼=1,₂₂=^{𝜋𝜋/𝑛}𝑡^𝛼−1𝑑𝑡=𝑂^{𝜋𝜋/𝑛}𝜋𝑛𝑡𝑡^𝛼−1𝑑𝑡=⎧⎪⎨⎪⎩𝑂(𝑛^−𝛼𝑂log),𝑛0<𝛼<1,𝑛,𝛼=1,(4.13) Collecting (4.10)–(4.13) and writing 1/𝑛≤(log𝑛)/𝑛, for large values of n, we get ||𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)||=⎧⎪⎨⎪⎩𝑂(𝑛^−𝛼𝑂log),𝑛0<𝛼<1,𝑛,𝛼=1.(4.14) Hence, ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_∞=sup_{[]𝑥∈0,2𝜋}||𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)||=⎧⎪⎨⎪⎩𝑂(𝑛^−𝛼𝑂log),𝑛0<𝛼<1,𝑛,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 𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)=^0𝜋/𝑛𝜙(𝑡)𝐾(𝑛,𝑡)𝑑𝑡+^{𝜋𝜋/𝑛}𝜙(𝑡)𝐾(𝑛,𝑡)𝑑𝑡=𝐼₃+𝐼₄,say.(4.16) Using Hölder’s inequality, 𝜙(𝑡)∈𝑊(𝐿^𝑟,𝜉(𝑡)), condition (3.4), Lemma 4.2, and (sin𝑡/2)⁻¹≤𝜋/𝑡, for 0<𝑡≤𝜋, we have ||𝐼₃||=||||^0𝜋/𝑛𝜙(𝑡)sin^𝛽(𝑡/2)≤𝜉(𝑡)⋅𝜉s(i𝑡n)^𝛽𝐾(𝑡(/𝑛2,)𝑡)𝑑𝑡_0𝜋/𝑛||||||||𝜙(𝑡)sin^𝛽(𝑡/2)𝜉(𝑡)||||^𝑟𝑑𝑡^1/𝑟lim^𝜀→0^{𝜀𝜋/𝑛}||||𝜉(𝑡)𝐾(𝑛,𝑡)sin^𝛽(𝑡/2)||||^𝑠𝑑𝑡^1/𝑠=𝑂(1)lim_𝜀→0^{𝜀𝜋/𝑛}si𝜉n^𝛽(𝑡()𝑡𝑛/2)^𝑠𝑑𝑡^1/𝑠=𝑂𝑛𝜉𝜋𝑛lim^𝜀→0^{𝜀𝜋/𝑛}𝑡^−𝛽𝑠𝑑𝑡^1/𝑠=𝑂𝜉1𝑛𝑛^{𝛽+1−1/𝑠}=𝑂𝑛^𝛽+1/𝑟𝜉1𝑛,(4.17) in view of the mean value theorem for integrals, 𝑟⁻¹+𝑠⁻¹=1 and Note 1.
Similarly, using Hölder’s inequality, Lemma 4.3, |sin(𝑡/2)|≤1, (sin(𝑡/2))⁻¹≤𝜋/𝑡, condition (3.5), and the mean value theorem for integrals, we have ||𝐼₄||=𝑂^{𝜋𝜋/𝑛}𝜙(𝑡)𝑡⁻²(𝑛+1)⁻¹+𝑡⁻¹𝑑𝑡=𝑂𝐼₄₁+𝑂𝐼₄₂,say,(4.18) where 𝐼₄₁=^{𝜋𝜋/𝑛}𝜙(𝑡)𝑡⁻²(𝑛+1)⁻¹𝑑𝑡=𝑂(𝑛+1)⁻¹^{𝜋𝜋/𝑛}𝑡^−𝛿𝜙(𝑡)sin^𝛽(𝑡/2)=𝑂𝑛₋₁𝜉(𝑡)𝑡^2−𝛿s𝜉in(𝑡^𝛽)(𝑡/2)𝑑𝑡^{𝜋𝜋/𝑛}𝑡^−𝛿||𝜙(𝑡)||𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟^{𝜋𝜋/𝑛}𝑡𝜉(𝑡)^{2−𝛿+𝛽}^𝑠𝑑𝑡^1/𝑠=𝑂𝑛^𝛿−1^{𝑛/𝜋1/𝜋}𝜉𝑦(1/𝑦^{𝛿−2−𝛽})^𝑠𝑦⁻²𝑑𝑦^1/𝑠=𝑂𝑛^𝛿−1𝑛𝜋𝜉𝜋𝑛^{𝜀𝑛/𝜋}₁𝑦^{(1−𝛿+𝛽)𝑠−2}𝑑𝑦^1/𝑠=𝑂𝑛^𝛿𝜉1𝐼₄₂𝑛𝑛^{1−𝛿+𝛽−1/𝑠}=𝑂𝜉1𝑛𝑛^𝛽+1/𝑟,=^{𝜋𝜋/𝑛}𝜙(𝑡)𝑡⁻¹𝑑𝑡=^{𝜋𝜋/𝑛}𝑡^−𝛿𝜙(𝑡)sin^𝛽(𝑡/2)=𝑂_{𝜋𝜋/𝑛}𝜉(𝑡)𝑡^1−𝛿s𝜉in(𝑡^𝛽)(𝑡/2)𝑑𝑡||||𝑡^−𝛿𝜙(𝑡)sin^𝛽(𝑡/2)𝜉(𝑡)||||^𝑟𝑑𝑡^1/𝑟^{𝜋𝜋/𝑛}||||𝜉(𝑡)𝑡^1−𝛿sin^𝛽(𝑡/2)||||^𝑠𝑑𝑡^1/𝑠=𝑂^{𝜋𝜋/𝑛}||||𝑡^−𝛿𝜙(𝑡)𝜉(𝑡)||||^𝑟𝑑𝑡^1/𝑟^{𝜋𝜋/𝑛}|||𝜉(𝑡)𝑡^{1−𝛿+𝛽}|||^𝑠𝑑𝑡^1/𝑠=𝑂𝑛^𝛿^{𝑛/𝜋1/𝜋}𝜉𝑦(1/𝑦^{𝛿−1−𝛽})^𝑠𝑦⁻²𝑑𝑦^1/𝑠=𝑂𝑛^𝛿+1𝜋𝜉𝜋𝑛^{𝜀𝑛/𝜋}₂𝑦^{(𝛽−𝛿)𝑠−2}𝑑𝑦^1/𝑠=𝑂𝑛^𝛿+1𝜉1𝑛𝑛^{−𝛿+𝛽−1/𝑠}=𝑂𝜉1𝑛𝑛^𝛽+1/𝑟,(4.19) in view of increasing nature of 𝑦𝜉(1/𝑦), 𝑟⁻¹+𝑠⁻¹=1, where 𝜀₁,𝜀₂ lie in [𝜋⁻¹,𝑛𝜋⁻¹], and Note 1.
Collecting (4.16)–(4.19), we get ||𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)||=𝑂𝑛^𝛽+1/𝑟𝜉1𝑛.(4.20) Hence, ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_𝑟=12𝜋^02𝜋||𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)||^𝑟𝑑𝑥^1/𝑟=𝑂𝑛^𝛽+1/𝑟𝜉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 𝑓∈Lip(𝜉(𝑡),𝑟), ‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖_𝑟=𝑂(𝑛^1/𝑟𝜉(1/𝑛)).

Corollary 5.2. If 𝛽=0,𝜉(𝑡)=𝑡^𝛼(0<𝛼≤1), then for 𝑓∈Lip(𝛼,𝑟)(𝛼>1/𝑟), ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_𝑟=𝑂𝑛^{1/𝑟−𝛼}.(5.1)

Corollary 5.3. If 𝑟→∞ in Corollary 5.2, then for 𝑓∈Lip𝛼(0<𝛼<1), (5.1) gives ‖‖𝑡^𝑛𝐶𝑁(𝑓)−𝑓(𝑥)‖‖_∞=𝑂(𝑛^−𝛼).(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 {𝑝_𝑛} 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.