Abstract

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging to Lip𝜉(𝑡)-class by (𝐸,1)(𝐶,1) product summability means of conjugate series of Fourier series has been generalized for the weighted 𝑊(𝐿𝑟,𝜉(𝑡)), (𝑟≥1),(𝑡>0)-class, where 𝜉(𝑡) is nonnegative and increasing function of 𝑡, by 𝐸1𝑛𝐶1𝑛 which is in more general form of Theorem 2 of Nigam and Sharma (2011).

1. Introduction

Khan [1, 2] has studied the degree of approximation of a function belonging to Lip(𝛼,𝑟) and 𝑊(𝐿𝑟,𝜉(𝑡)) classes by Nörlund and generalized Nörlund means. Working in the same direction Rhoades [3], Mittal et al. [4], Mittal and Mishra [5], and Mishra [6, 7] have studied the degree of approximation of a function belonging to 𝑊(𝐿𝑟,𝜉(𝑡)) class by linear operators. Thereafter, Nigam [8] and Nigam and Sharma [9] discussed the degree of approximation of conjugate of a function belonging to Lip(𝜉(𝑡),𝑟) class and 𝑊(𝐿𝑟,𝜉(𝑡)) by (𝐸,1)(𝐶,1) product summability means, respectively. Recently, Rhoades et al. [10] have determined very interesting result on the degree of approximation of a function belonging to Lip𝛼 class by Hausdorff means. Summability techniques were also applied on some engineering problems like Chen and Jeng [11] who implemented the Cesàro sum of order (𝐶,1) and (𝐶,2), in order to accelerate the convergence rate to deal with the Gibbs phenomenon, for the dynamic response of a finite elastic body subjected to boundary traction. Chen et al. [12] applied regularization with Cesàro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [13] used Cesàro sum regularization technique for hypersingularity of dual integral equation.

The generalized weighted 𝑊(𝐿𝑟,𝜉(𝑡)),(𝑟≥1)-class is generalization of Lip𝛼,Lip(𝛼,𝑟) and Lip(𝜉(𝑡),𝑟) classes. Therefore, in the present paper, a theorem on degree of approximation of conjugate of signals belonging to the generalized weighted 𝑊(𝐿𝑟,𝜉(𝑡)),𝑟≥1 class by (𝐸,1)(𝐶,1) product summability means of conjugate series of Fourier series has been established which is in more general form than that of Nigam and Sharma [9]. We also note some errors appearing in the paper of Nigam [8], Nigam and Sharma [9] and rectify the errors pointed out in Remarks 2.2, 2.3 and 2.4.

Let 𝑓(𝑥) be a 2𝜋-periodic function and integrable in the sense of Lebesgue. The Fourier series of 𝑓(𝑥) at any point 𝑥 is given by ğ‘Žğ‘“(𝑥)∼02+âˆžî“ğ‘›=1î€·ğ‘Žğ‘›cos𝑛𝑥+𝑏𝑛≡sinğ‘›ğ‘¥âˆžî“ğ‘›=0𝐴𝑛(𝑥),(1.1) with 𝑛th partial sum 𝑠𝑛(𝑓;𝑥).

The conjugate series of Fourier series (1.1) is given by âˆžî“ğ‘›=1𝑏𝑛cosğ‘›ğ‘¥âˆ’ğ‘Žğ‘›î€¸â‰¡sinğ‘›ğ‘¥âˆžî“ğ‘›=1𝐵𝑛(𝑥).(1.2) Let âˆ‘âˆžğ‘›=0𝑢𝑛 be a given infinite series with sequence of its 𝑛th partial sums {𝑠𝑛}. The (𝐸,1) transform is defined as the 𝑛th partial sum of (𝐸,1) summability, and we denote it by 𝐸1𝑛.

If 𝐸1𝑛=1(2)𝑛𝑛𝑘=0âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ ğ‘ ğ‘˜âŸ¶ğ‘ ,asğ‘›âŸ¶âˆž,(1.3) then the infinite series âˆ‘âˆžğ‘›=0𝑢𝑛 is summable (𝐸,1) to a definite number 𝑠, Hardy [14].

If 𝜏𝑛=𝑠0+𝑠1+𝑠2+⋯+𝑠𝑛=1𝑛+1𝑛+1𝑛𝑘=0𝑠𝑘⟶𝑠,asğ‘›âŸ¶âˆž,(1.4) then the infinite series âˆ‘âˆžğ‘›=0𝑢𝑛 is summable to the definite number 𝑠 by (𝐶,1) method. The (𝐸,1) transform of the (𝐶,1) transform defines (𝐸,1)(𝐶,1) product transform and denotes it by 𝐸1𝑛𝐶1𝑛. Thus, if 𝐸1𝑛𝐶1𝑛=1(2)𝑛𝑛𝑘=0âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ ğ¶1𝑘⟶𝑠,asğ‘›âŸ¶âˆž,(1.5) then the infinite series âˆ‘âˆžğ‘›=0𝑢𝑛 is said to be summable by (𝐸,1)(𝐶,1) method or summable (𝐸,1)(𝐶,1) to a definite number 𝑠. The (𝐸,1) is regular method of summability 𝑠𝑛⟶𝑠⟹𝐶1𝑛𝑠𝑛=𝜏𝑛=1𝑛+1𝑛𝑘=0𝑠𝑘⟶𝑠,asğ‘›âŸ¶âˆž,𝐶1𝑛methodisregular,⟹𝐸1𝑛𝐶1𝑛𝑠𝑛=𝐸1𝑛𝐶1𝑛⟶𝑠,asğ‘›âŸ¶âˆž,𝐸1𝑛methodisregular,⟹𝐸1𝑛𝐶1𝑛methodisregular.(1.6) A function 𝑓(𝑥)∈Lip𝛼, if 𝑓||𝑡(𝑥+𝑡)−𝑓(𝑥)=𝑂𝛼||for0<𝛼≤1,𝑡>0,(1.7) and 𝑓(𝑥)∈Lip(𝛼,𝑟), for 0≤𝑥≤2𝜋, if 𝜔𝑟(𝑡;𝑓)=02𝜋||||𝑓(𝑥+𝑡)−𝑓(𝑥)𝑟𝑑𝑥1/𝑟=𝑂(|𝑡|𝛼),0<𝛼≤1,𝑟≥1,𝑡>0.(1.8) Given a positive increasing function 𝜉(𝑡),𝑓(𝑥)∈Lip(𝜉(𝑡),𝑟), if 𝜔𝑟(𝑡;𝑓)=02𝜋||||𝑓(𝑥+𝑡)−𝑓(𝑥)𝑟𝑑𝑥1/𝑟=𝑂(𝜉(𝑡)),𝑟≥1,𝑡>0.(1.9) Given positive increasing function 𝜉(𝑡), an integer 𝑟≥1,𝑓∈𝑊(𝐿𝑟,𝜉(𝑡)), ([2]), if 𝜔𝑟(𝑡;𝑓)=02𝜋||{𝑓(𝑥+𝑡)−𝑓(𝑥)}sin𝛽𝑥||𝑟𝑑𝑥1/𝑟=𝑂(𝜉(𝑡)),(𝛽≥0),𝑡>0.(1.10) For our convenience to evaluate 𝐼2 without error, we redefine the weighted class as follows: 𝜔𝑟(𝑡;𝑓)=02𝜋||||𝑓(𝑥+𝑡)−𝑓(𝑥)𝑟sin𝛽𝑟𝑥2𝑑𝑥1/𝑟[]=𝑂(𝜉(𝑡)),𝛽≥0,𝑡>0(16).(1.11) If 𝛽=0, then the weighted class 𝑊(𝐿𝑟,𝜉(𝑡)) coincides with the class Lip(𝜉(𝑡),𝑟), we observe that 𝑊𝐿𝑟,𝜉(𝑡)𝛽=0−−−→Lip(𝜉(𝑡),𝑟)𝜉(𝑡)=𝑡𝛼−−−−→Lip(𝛼,𝑟)ğ‘Ÿâ†’âˆžâˆ’âˆ’âˆ’âˆ’â†’Lip𝛼for0<𝛼≤1,𝑟≥1,𝑡>0.(1.12)𝐿𝑟-norm of a function is defined by ‖𝑓‖𝑟=02𝜋||||𝑓(𝑥)𝑟𝑑𝑥1/𝑟,𝑟≥1.(1.13) A signal 𝑓 is approximated by trigonometric polynomials 𝜏𝑛 of order 𝑛, and the degree of approximation 𝐸𝑛(𝑓) is given by Rhoades [3] 𝐸𝑛(𝑓)=min𝑛‖‖𝑓(𝑥)−𝜏𝑛‖‖(𝑓;𝑥)𝑟,(1.14) in terms of 𝑛, where 𝜏𝑛(𝑓;𝑥) is a trigonometric polynomial of degree 𝑛. This method of approximation is called trigonometric Fourier approximation (TFA) [4].

We use the following notations throughout this paper: 𝐺𝜓(𝑡)=𝑓(𝑥+𝑡)−𝑓(𝑥−𝑡),𝑛1(𝑡)=2𝑛+1ğœ‹âŽ¡âŽ¢âŽ¢âŽ£ğ‘›î“ğ‘˜=0âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ 11+𝑘𝑘𝑣=0cos(𝑣+1/2)ğ‘¡âŽ¤âŽ¥âŽ¥âŽ¦.sin(𝑡/2)(1.15)

2. Previous Result

Nigam [8] has proved a theorem on the degree of approximation of a function 𝑓(𝑥), conjugate to a periodic function 𝑓(𝑥) with period 2𝜋 and belonging to the class Lip(𝜉(𝑡),𝑟)(𝑟≥1), by (𝐸,1)(𝐶,1) product summability means of conjugate series of Fourier series. He has proved the following theorem.

Theorem 2.1 (see [8]). If 𝑓(𝑥), conjugate to a 2𝜋-periodic function 𝑓(𝑥), belongs to Lip(𝜉(𝑡),𝑟) class, then its degree of approximation by (𝐸,1)(𝐶,1) product summability means of conjugate series of Fourier series is given by ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖𝑟=𝑂(𝑛+1)1/𝑟𝜉1,𝑛+1(2.1) provided 𝜉(𝑡) satisfies the following conditions: 01/(𝑛+1)𝑡||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟1=𝑂,𝑛+1(2.2)𝜋1/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟=𝑂(𝑛+1)𝛿,(2.3) where 𝛿 is an arbitrary number such that 𝑠(1−𝛿)−1>0,𝑟−1+𝑠−1=1,1â‰¤ğ‘Ÿâ‰¤âˆž, condition (2.2) and (2.3) hold uniformly in 𝑥 and 𝐸1𝑛𝐶1𝑛 is (𝐸,1)(𝐶,1) means of the series (1.2).

Remark 2.2. The proof proceeds by estimating |𝐸1𝑛𝐶1𝑛−𝑓|, which is represented in terms of an integral. The domain of integration is divided into two parts—from [0,1/(𝑛+1)] and [1/(𝑛+1),𝜋]. Referring to second integral as 𝐼2, and using Hölder inequality, the author [8] obtains ||𝐼2||≤𝜋1/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/ğ‘ŸâŽ§âŽªâŽ¨âŽªâŽ©î€œğœ‹1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)ğ‘¡âˆ’ğ›¿âŽžâŽŸâŽŸâŽŸâŽ ğ‘ âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¡1/𝑠=𝑂(𝑛+1)ğ›¿î€¾âŽ§âŽªâŽ¨âŽªâŽ©î€œğœ‹1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)ğ‘¡âˆ’ğ›¿âŽžâŽŸâŽŸâŽŸâŽ ğ‘ âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¡1/𝑠.(2.4) The author then makes the substitution 𝑦=1/𝑡 to obtain =𝑂(𝑛+1)𝛿𝑛+11/𝜋𝜉(1/𝑦)𝑦𝛿−1𝑠𝑑𝑦𝑦21/𝑠.(2.5) In the next step 𝜉(1/𝑦) is removed from the integrand by replacing it with 𝑂(𝜉(1/(𝑛+1))), while 𝜉(𝑡) is an increasing function, 𝜉(1/𝑦) is now a decreasing function. Therefore, this step is invalid.

Remark 2.3. The proof follows by obtaining |(𝐸𝐶)1𝑛−𝑓|, in Theorem 2 of Nigam and Sharma [9], which is expressed in terms of an integral. The domain of integration is divided into two parts—from [0,1/(𝑛+1)] and [1/(𝑛+1),𝜋]. Referring to second integral as 𝐼2.2, and using Hölder inequality, the authors [9] obtain the following: ||𝐼2.2||≤𝜋1/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)sin𝛽𝑡𝜉(𝑡)𝑟𝑑𝑡1/ğ‘ŸâŽ§âŽªâŽ¨âŽªâŽ©î€œğœ‹1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sinğ›½ğ‘¡âŽžâŽŸâŽŸâŽŸâŽ ğ‘ âŽ«âŽªâŽ¬âŽªâŽ­ğ‘‘ğ‘¡1/𝑠(2.6)=𝑂(𝑛+1)𝛿𝜋1/(𝑛+1)𝜉(𝑡)𝑡1−𝛿+𝛽𝑠𝑑𝑡1/𝑠.(2.7) The authors then make the substitution 𝑦=1/𝑡 to get =𝑂(𝑛+1)𝛿𝑛+11/𝜋𝜉(1/𝑦)𝑦𝛿−1−𝛽𝑠𝑑𝑦𝑦21/𝑠.(2.8) In the next step, 𝜉(1/𝑦) is removed from the integrand by replacing it with 𝑂(𝜉(1/(𝑛+1))), while 𝜉(𝑡) is an increasing function, 𝜉(1/𝑦) is now a decreasing function. Therefore, in view of second mean value theorem of integral, this step is invalid.

Remark 2.4. The condition 1/sin𝛽(𝑡)=𝑂(1/𝑡𝛽),1/(𝑛+1)≤𝑡≤𝜋 used by Nigam and Sharma [9] is not valid, since sin𝑡→0 as 𝑡→𝜋.

3. Main Result

It is well known that the theory of approximation, that is, TFA, which originated from a theorem of Weierstrass, has become an exciting interdisciplinary field of study for the last 130 years. These approximations have assumed important new dimensions due to their wide applications in signal analysis [15], in general, and in digital signal processing [16] in particular, in view of the classical Shannon sampling theorem. Broadly speaking, signals are treated as function of one variable and images are represented by functions of two variables.

This has motivated Mittal and Rhoades [17–20] and Mittal et al. [4, 21] to obtain many results on TFA using summability methods without rows of the matrix. In this paper, we prove the following theorem.

Theorem 3.1. If 𝑓(𝑥), conjugate to a 2𝜋-periodic function 𝑓, belongs to the generalized weighted 𝑊(𝐿𝑟,𝜉(𝑡))(𝑟≥1)-class, then its degree of approximation by (𝐸,1)(𝐶,1) product summability means of conjugate series of Fourier series is given by ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖𝑟=𝑂(𝑛+1)𝛽+1/𝑟𝜉1,𝑛+1(3.1) provided 𝜉(𝑡) satisfies the following conditions: 0𝜋/(𝑛+1)𝑡||||𝜓(𝑡)𝜉(𝑡)𝑟sin𝛽𝑟𝑡2𝑑𝑡1/𝑟1=𝑂,𝑛+1(3.2)𝜋𝜋/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟=𝑂(𝑛+1)𝛿,𝜉(3.3)(𝑡)𝑡isnonincreasingsequencein”𝑡”,(3.4) where 𝛿 is an arbitrary number such that 𝑠(1−𝛿)−1>0,𝑟−1+𝑠−1=1,1â‰¤ğ‘Ÿâ‰¤âˆž, conditions (3.2) and (3.3) hold uniformly in 𝑥 and 𝐸1𝑛𝐶1𝑛 is (𝐸,1)(𝐶,1) means of the series (1.2) and the conjugate function 𝑓(𝑥) is defined for almost every 𝑥 by 1𝑓(𝑥)=−2𝜋𝜋0𝑡𝜓(𝑡)cot2𝑑𝑡=limℎ→0−12ğœ‹ğœ‹â„Žî‚€ğ‘¡ğœ“(𝑡)cot2𝑑𝑡.(3.5)

Note 1. 𝜉(𝜋/(𝑛+1))≤𝜋𝜉(1/(𝑛+1)), for (𝜋/(𝑛+1))≥(1/(𝑛+1)).

Note 2. Also for 𝛽=0, Theorem 3.1 reduces to Theorem 2.1, and thus generalizes the theorem of Nigam [8]. Also our Theorem 3.1 in the modified form of Theorem 2 of Nigam and Sharma [9].

Note 3. The product transform (𝐸,1)(𝐶,1) plays an important role in signal theory as a double digital filter [6] and the theory of machines in mechanical engineering.

4. Lemmas

For the proof of our theorem, the following lemmas are required.

Lemma 4.1. Consider |𝐺𝑛(𝑡)|=𝑂[1/𝑡] for 0<𝑡≤𝜋/(𝑛+1).

Proof. For 0<𝑡≤𝜋/(𝑛+1),sin(𝑡/2)≥(𝑡/𝜋) and |cos𝑛𝑡|≤1. |||𝐺𝑛(|||=1𝑡)2𝜋(2)𝑛||||||𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘𝑘𝑣=0cos(𝑣+1/2)ğ‘¡âŽ¤âŽ¥âŽ¥âŽ¦||||||≤1sin(𝑡/2)2𝜋(2)𝑛𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘𝑘𝑣=0||||cos(𝑣+1/2)𝑡||||⎤⎥⎥⎦=1sin(𝑡/2)2𝑡(2)𝑛𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ 1(𝑘+1)𝑘𝑣=0⎤⎥⎥⎦=1(1)2𝑡(2)𝑛𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦=12𝑡(2)𝑛(2)𝑛1=𝑂𝑡,since𝑛𝑘=0âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ =(2)𝑛.(4.1) This completes the proof of Lemma 4.1.

Lemma 4.2. Consider |𝐺𝑛(𝑡)|=𝑂[1/𝑡], for 0<𝑡≤𝜋 and any 𝑛.

Proof. For 0<𝜋/(𝑛+1)≤𝑡≤𝜋,sin(𝑡/2)≥(𝑡/𝜋). |||𝐺𝑛(|||=1𝑡)2𝜋(2)𝑛||||||𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘𝑘𝑣=0cos(𝑣+1/2)ğ‘¡âŽ¤âŽ¥âŽ¥âŽ¦||||||≤1sin(𝑡/2)2𝑡(2)𝑛||||||𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0𝑒𝑖(𝑣+1/2)ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||≤12𝑡(2)𝑛||||||𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||||𝑒𝑖(𝑡/2)||≤12𝑡(2)𝑛||||||𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||+12𝑡(2)𝑛||||||𝑛𝑘=ğœâŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||.(4.2) Now considering first term of (4.2) 12𝑡(2)𝑛||||||𝜏−1𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘𝑘𝑣=01⎤⎥⎥⎦||||||||𝑒𝑖𝑣𝑡||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦||||||.(4.3) Now considering second term of (4.2) and using Abel’s lemma 12𝑡(2)𝑛||||||𝑛𝑘=ğœâŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘Re𝑘𝑣=0ğ‘’ğ‘–ğ‘£ğ‘¡îƒ°âŽ¤âŽ¥âŽ¥âŽ¦||||||≤12𝑡(2)𝑛𝑛𝑘=ğœâŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€11+𝑘max0≤𝑚≤𝑘|||||𝑚𝑣=0𝑒𝑖𝑣𝑡|||||≤12𝑡(2)𝑛𝑛𝑘=ğœâŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î‚€1⎤⎥⎥⎦=11+𝑘(1+𝑘)2𝑡(2)𝑛𝑛𝑘=ğœâŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦.(4.4) On combining (4.2), (4.3), and (4.4) |||𝐺𝑛|||≤1(𝑡)2𝑡(2)𝑛𝜏−1𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ âŽ¤âŽ¥âŽ¥âŽ¦+12𝑡(2)𝑛𝑛𝑘=ğœâŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ ,|||𝐺𝑛|||1(𝑡)=𝑂𝑡.(4.5) This completes the proof of Lemma 4.2.

5. Proof of Theorem

Let 𝑠𝑛(𝑥) denotes the 𝑛th partial sum of series (1.2). Then following Nigam [8], we have 𝑠𝑛1(𝑥)−𝑓(𝑥)=2𝜋𝜋0𝜓(𝑡)cos(𝑛+1/2)𝑡sin(𝑡/2)𝑑𝑡.(5.1) Therefore, using (1.2), the (𝐶,1) transform 𝐶1𝑛 of 𝑠𝑛 is given by 𝐶1𝑛−1𝑓(𝑥)=2𝜋(𝑛+1)𝜋0𝜓(𝑡)𝑛𝑘=0cos(𝑛+1/2)𝑡sin(𝑡/2)𝑑𝑡.(5.2) Now denoting (𝐸,1)(𝐶,1) transform of 𝑠𝑛 as (𝐸1𝑛,𝐶1𝑛), we write 𝐸1𝑛,𝐶1𝑛−1𝑓(𝑥)=2𝜋(2)𝑛𝑛𝑘=0âŽ¡âŽ¢âŽ¢âŽ£âŽ›âŽœâŽœâŽğ‘›ğ‘˜âŽžâŽŸâŽŸâŽ î€œğœ‹0𝜓(𝑡)1sin(𝑡/2)1+𝑘𝑘𝑣=0⎤⎥⎥⎦=cos(𝑣+1/2)𝑡𝑑𝑡𝜋0𝐺𝜓(𝑡)𝑛=(𝑡)𝑑𝑡0𝜋/(𝑛+1)+𝜋𝜋/(𝑛+1)𝐺𝜓(𝑡)𝑛(𝑡)𝑑𝑡=𝐼1+𝐼2(say).(5.3) We consider, |𝐼1∫|≤0𝜋/(𝑛+1)𝐺|𝜓(𝑡)||𝑛(𝑡)|𝑑𝑡.

Using Hölder’s inequality ||𝐼1||≤0𝜋/(𝑛+1)𝑡||||𝜓(𝑡)𝜉(𝑡)𝑟sin𝛽𝑟𝑡2𝑑𝑡1/ğ‘ŸâŽ¡âŽ¢âŽ¢âŽ¢âŽ£î€œ0𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡sinğ›½âŽžâŽŸâŽŸâŽŸâŽ (𝑡/2)ğ‘ âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦ğ‘‘ğ‘¡1/𝑠1=ğ‘‚î‚âŽ¡âŽ¢âŽ¢âŽ¢âŽ£î€œğ‘›+10𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡sinğ›½âŽžâŽŸâŽŸâŽŸâŽ (𝑡/2)ğ‘ âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦ğ‘‘ğ‘¡1/𝑠1by(3.2)=𝑂𝑛+10𝜋/(𝑛+1)𝜉(𝑡)𝑡2sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠1byLemma4.1=𝑂𝑛+10𝜋/(𝑛+1)𝜉(𝑡)(𝑡/2)𝛽𝑡2sin𝛽(𝑡/2)⋅(𝑡/2)𝛽𝑠𝑑𝑡1/𝑠1=𝑂𝑛+12(𝜋/2(𝑛+1))sin(𝜋/2(𝑛+1))ğ›½ğ‘ î€œâ„Žğœ‹/(𝑛+1)𝜉(𝑡)𝑡2+𝛽𝑠𝑑𝑡1/𝑠1,asℎ⟶0=𝑂𝑛+1â„Žğœ‹/(𝑛+1)𝜉(𝑡)𝑡2+𝛽𝑠𝑑𝑡1/𝑠.(5.4) Since 𝜉(𝑡) is a positive increasing function and by using second mean value theorem for integrals, we have 𝐼11=𝑂𝜉𝜋𝑛+1𝑛+1∈𝜋/(𝑛+1)1𝑡2+𝛽𝑠𝑑𝑡1/𝑠𝜋,forsome0<∈<1𝑛+1=𝑂1𝑛+1𝜋𝜉𝑛+1∈𝜋/(𝑛+1)𝑡−𝛽𝑠−2𝑠𝑑𝑡1/𝑠.(5.5) Note that 𝜉(𝜋/(𝑛+1))≤𝜋𝜉(1/(𝑛+1)), 𝐼11=𝑂𝜉1𝑛+1𝑡𝑛+1−𝛽𝑠−2𝑠+1−𝛽𝑠−2𝑠+1∈𝜋/(𝑛+1)1/𝑠1=𝑂𝜉1𝑛+1𝑛+1(𝑛+1)𝛽+2−1/𝑠𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1−1/𝑠𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1/𝑟∵𝑟−1+𝑠−1=1,1â‰¤ğ‘Ÿâ‰¤âˆž.(5.6) Now, we consider ||𝐼2||≤𝜋𝜋/(𝑛+1)|||||||𝐺𝜓(𝑡)𝑛|||(𝑡)𝑑𝑡.(5.7) Using Hölder’s inequality ||𝐼2||≤𝜋𝜋/(𝑛+1)𝑡−𝛿sin𝛽||||(𝑡/2)𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/ğ‘ŸâŽ¡âŽ¢âŽ¢âŽ¢âŽ£î€œğœ‹ğœ‹/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sin𝛽(âŽžâŽŸâŽŸâŽŸâŽ ğ‘¡/2)ğ‘ âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦ğ‘‘ğ‘¡1/𝑠=𝑂(𝑛+1)ğ›¿î€¾âŽ¡âŽ¢âŽ¢âŽ¢âŽ£î€œğœ‹ğœ‹/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sinğ›½âŽžâŽŸâŽŸâŽŸâŽ (𝑡/2)ğ‘ âŽ¤âŽ¥âŽ¥âŽ¥âŽ¦ğ‘‘ğ‘¡1/𝑠(by(3.3)=𝑂𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿𝑡sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠byLemma4.2=𝑂(𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿+1sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠=𝑂(𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿+𝛽+1𝑠𝑑𝑡1/𝑠.(5.8) Now putting 𝑡=1/𝑦, we have 𝐼2=𝑂(𝑛+1)𝛿(𝑛+1)/𝜋1/𝜋𝜉(1/𝑦)𝑦𝛿−𝛽−1𝑠𝑑𝑦𝑦21/𝑠.(5.9) Since 𝜉(𝑡) is a positive increasing function, so 𝜉(1/𝑦)/(1/𝑦) is also a positive increasing function and using second mean value theorem for integrals, we have =𝑂(𝑛+1)𝛿𝜉(𝜋/(𝑛+1))𝜋/(𝑛+1)𝜂(𝑛+1)/𝜋𝑑𝑦𝑦−𝛽𝑠+𝛿𝑠+21/𝑠1,forsome𝜋≤𝜂≤𝑛+1𝜋=𝑂(𝑛+1)𝛿+1𝜉1𝑦𝑛+1−𝛿𝑠−2+𝛽𝑠+1−𝛿𝑠−2+𝛽𝑠+11(𝑛+1)/𝜋1/𝑠1,forsome𝜋≤1≤𝑛+1𝜋=𝑂(𝑛+1)𝛿+1𝜉1𝑦𝑛+1−𝛿𝑠−1+𝛽𝑠1𝑛+1/𝜋1/𝑠(=𝑂𝑛+1)𝛿+1𝜉1(𝑛+1𝑛+1)−𝛿−1/𝑠+𝛽𝜉1=𝑂𝑛+1(𝑛+1)𝛿+1−𝛿−1/𝑠+𝛽𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1/𝑟∵𝑟−1+𝑠−1=1,1â‰¤ğ‘Ÿâ‰¤âˆž.(5.10) Combining 𝐼1 and 𝐼2 yields |||𝐸1𝑛𝐶1𝑛−𝑓|||=𝑂(𝑛+1)1/𝑟+𝛽𝜉1.𝑛+1(5.11) Now, using the 𝐿𝑟-norm of a function, we get ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖𝑟=02𝜋|||𝐸1𝑛𝐶1𝑛−𝑓|||𝑟𝑑𝑥1/𝑟=𝑂02𝜋(𝑛+1)𝛽+1/𝑟𝜉1𝑛+1𝑟𝑑𝑥1/𝑟=𝑂(𝑛+1)𝛽+1/𝑟𝜉1𝑛+102𝜋𝑑𝑥1/𝑟=𝑂(𝑛+1)𝛽+1/𝑟𝜉1.𝑛+1(5.12) This completes the proof of Theorem 3.1.

6. Applications

The theory of approximation is a very extensive field, which has various applications, and the study of the theory of trigonometric Fourier approximation is of great mathematical interest and of great practical importance. From the point of view of the applications, Sharper estimates of infinite matrices [22] are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enables to investigate perturbations of matrix valued functions and compare them.

The following corollaries may be derived from Theorem 3.1.

Corollary 6.1. If 𝜉(𝑡)=𝑡𝛼,0<𝛼≤1, then the weighted class 𝑊(𝐿𝑟,𝜉(𝑡)),𝑟≥1 reduces to the class Lip(𝛼,𝑟) and the degree of approximation of a function 𝑓(𝑥) conjugate to a 2𝜋-periodic function 𝑓 belonging to the class Lip(𝛼,𝑟), is given by |||𝐸1𝑛𝐶1𝑛−𝑓|||1=𝑂(𝑛+1)𝛼−1/𝑟.(6.1)

Proof. The result follows by setting 𝛽=0 in (3.1), we have ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖𝑟=02𝜋|||𝐸1𝑛𝐶1𝑛−𝑓|||𝑟𝑑𝑥1/𝑟=𝑂(𝑛+1)1/𝑟𝜉11𝑛+1=𝑂(𝑛+1)𝛼−1/𝑟,𝑟≥1.(6.2) Thus, we get |||𝐸1𝑛𝐶1𝑛−𝑓|||≤02𝜋|||𝐸1𝑛𝐶1𝑛−𝑓|||𝑟𝑑𝑥1/𝑟1=𝑂(𝑛+1)𝛼−1/𝑟,𝑟≥1.(6.3) This completes the proof of Corollary 6.1.

Corollary 6.2. If 𝜉(𝑡)=𝑡𝛼 for 0<𝛼<1 and ğ‘Ÿâ†’âˆž in Corollary 6.1, then 𝑓∈Lip𝛼 and |||𝐸1𝑛𝐶1𝑛−𝑓|||1=𝑂(𝑛+1)𝛼.(6.4)

Proof. For 𝑟=∞ in Corollary 6.1, we get ‖‖𝐸1𝑛𝐶1ğ‘›âˆ’î‚ğ‘“â€–â€–âˆž=sup0≤𝑥≤2𝜋|||𝐸1𝑛𝐶1𝑛|||1(𝑥)−𝑓(𝑥)=𝑂(𝑛+1)𝛼.(6.5) Thus, we get |||𝐸1𝑛𝐶1𝑛−𝑓|||≤‖‖𝐸1𝑛𝐶1ğ‘›âˆ’î‚ğ‘“â€–â€–âˆž=sup0≤𝑥≤2𝜋|||𝐸1𝑛𝐶1𝑛|||1(𝑥)−𝑓(𝑥)=𝑂(𝑛+1)𝛼.(6.6) This completes the proof of Corollary 6.2.

Acknowledgments

The authors are highly thankful to the anonymous referees for the careful reading, their critical remarks, valuable comments, and several useful suggestions which helped greatly for the overall improvements and the better presentation of this paper. The authors are also grateful to all the members of editorial board of IJMMS, especially Professor Noran El-Zoheary, Editorial Staff H. P. C. and Professor Ram U. Verma, Texas A&M University, USA, the Editor of IJMMS, Omnia Kamel Editorial office, HPC, Riham Taha Accounts Receivable specialist, HPC for their kind cooperation during communication. The authors are also thankful to the Cumulative Professional Development Allowance (CPDA) SVNIT, Surat (Gujarat), India, for their financial assistance.