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International Journal of Mathematics and Mathematical Sciences

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International Journal of Mathematics and Mathematical Sciences / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 298923 | https://doi.org/10.1155/2012/298923

Vishnu Narayan Mishra, Kejal Khatri, Lakshmi Narayan Mishra, "Product Summability Transform of Conjugate Series of Fourier Series", International Journal of Mathematics and Mathematical Sciences, vol. 2012, Article ID 298923, 13 pages, 2012. https://doi.org/10.1155/2012/298923

Product Summability Transform of Conjugate Series of Fourier Series

Vishnu Narayan Mishra,1 Kejal Khatri,1 and Lakshmi Narayan Mishra2

1Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Ichchhanath Mahadev Road, Surat 395 007, India

2Dr. Ram Manohar Lohia Avadh University, Hawai Patti Allahabad Road, Faizabad 224 001, India

Academic Editor: Ram U. Verma
Received22 Feb 2012
Revised15 Mar 2012
Accepted15 Mar 2012
Published29 Jul 2012

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 𝑎𝑓(𝑥)∼02+∞𝑛=1𝑎𝑛cos𝑛𝑥+𝑏𝑛≡sin𝑛𝑥∞𝑛=0𝐴𝑛(𝑥),(1.1) with 𝑛th partial sum 𝑠𝑛(𝑓;𝑥).

The conjugate series of Fourier series (1.1) is given by ∞𝑛=1𝑏𝑛cos𝑛𝑥−𝑎𝑛≡sin𝑛𝑥∞𝑛=1𝐵𝑛(𝑥).(1.2) Let ∑∞𝑛=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)𝑛𝑛𝑘=0⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠𝑠𝑘⟶𝑠,as𝑛⟶∞,(1.3) then the infinite series ∑∞𝑛=0𝑢𝑛 is summable (𝐸,1) to a definite number 𝑠, Hardy [14].

If 𝜏𝑛=𝑠0+𝑠1+𝑠2+⋯+𝑠𝑛=1𝑛+1𝑛+1𝑛𝑘=0𝑠𝑘⟶𝑠,as𝑛⟶∞,(1.4) then the infinite series ∑∞𝑛=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)𝑛𝑛𝑘=0⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠𝐶1𝑘⟶𝑠,as𝑛⟶∞,(1.5) then the infinite series ∑∞𝑛=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⟹𝐶1𝑛𝑠𝑛=𝜏𝑛=1𝑛+1𝑛𝑘=0𝑠𝑘⟶𝑠,as𝑛⟶∞,𝐶1𝑛methodisregular,⟹𝐸1𝑛𝐶1𝑛𝑠𝑛=𝐸1𝑛𝐶1𝑛⟶𝑠,as𝑛⟶∞,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)𝑟𝑑𝑥1/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)𝑟𝑑𝑥1/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||𝑟𝑑𝑥1/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)𝑟sin𝛽𝑟𝑥2𝑑𝑥1/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 𝑊𝐿𝑟,𝜉(𝑡)𝛽=0−−−→Lip(𝜉(𝑡),𝑟)𝜉(𝑡)=𝑡𝛼−−−−→Lip(𝛼,𝑟)𝑟→∞−−−−→Lip𝛼for0<𝛼≤1,𝑟≥1,𝑡>0.(1.12)𝐿𝑟-norm of a function is defined by ‖𝑓‖𝑟=02𝜋||||𝑓(𝑥)𝑟𝑑𝑥1/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+1𝜋⎡⎢⎢⎣𝑛𝑘=0⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘𝑘𝑣=0cos(𝑣+1/2)𝑡⎤⎥⎥⎦.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/𝑟𝜉1,𝑛+1(2.1) provided 𝜉(𝑡) satisfies the following conditions: 01/(𝑛+1)𝑡||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟1=𝑂,𝑛+1(2.2)𝜋1/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟=𝑂(𝑛+1)𝛿,(2.3) where 𝛿 is an arbitrary number such that 𝑠(1−𝛿)−1>0,𝑟−1+𝑠−1=1,1≤𝑟≤∞, 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)𝑟𝑑𝑡1/𝑟⎧⎪⎨⎪⎩𝜋1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿⎞⎟⎟⎟⎠𝑠⎫⎪⎬⎪⎭𝑑𝑡1/𝑠=𝑂(𝑛+1)𝛿⎧⎪⎨⎪⎩𝜋1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿⎞⎟⎟⎟⎠𝑠⎫⎪⎬⎪⎭𝑑𝑡1/𝑠.(2.4) The author then makes the substitution 𝑦=1/𝑡 to obtain =𝑂(𝑛+1)𝛿𝑛+11/𝜋𝜉(1/𝑦)𝑦𝛿−1𝑠𝑑𝑦𝑦21/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)sin𝛽𝑡𝜉(𝑡)𝑟𝑑𝑡1/𝑟⎧⎪⎨⎪⎩𝜋1/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sin𝛽𝑡⎞⎟⎟⎟⎠𝑠⎫⎪⎬⎪⎭𝑑𝑡1/𝑠(2.6)=𝑂(𝑛+1)𝛿𝜋1/(𝑛+1)𝜉(𝑡)𝑡1−𝛿+𝛽𝑠𝑑𝑡1/𝑠.(2.7) The authors then make the substitution 𝑦=1/𝑡 to get =𝑂(𝑛+1)𝛿𝑛+11/𝜋𝜉(1/𝑦)𝑦𝛿−1−𝛽𝑠𝑑𝑦𝑦21/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/𝑟𝜉1,𝑛+1(3.1) provided 𝜉(𝑡) satisfies the following conditions: 0𝜋/(𝑛+1)𝑡||||𝜓(𝑡)𝜉(𝑡)𝑟sin𝛽𝑟𝑡2𝑑𝑡1/𝑟1=𝑂,𝑛+1(3.2)𝜋𝜋/(𝑛+1)𝑡−𝛿||||𝜓(𝑡)𝜉(𝑡)𝑟𝑑𝑡1/𝑟=𝑂(𝑛+1)𝛿,𝜉(3.3)(𝑡)𝑡isnonincreasingsequencein”𝑡”,(3.4) where 𝛿 is an arbitrary number such that 𝑠(1−𝛿)−1>0,𝑟−1+𝑠−1=1,1≤𝑟≤∞, 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)=−2𝜋𝜋0𝑡𝜓(𝑡)cot2𝑑𝑡=limℎ→0−12𝜋𝜋ℎ𝑡𝜓(𝑡)cot2𝑑𝑡.(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||||||𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘𝑘𝑣=0cos(𝑣+1/2)𝑡⎤⎥⎥⎦||||||≤1sin(𝑡/2)2𝜋(2)𝑛𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘𝑘𝑣=0||||cos(𝑣+1/2)𝑡||||⎤⎥⎥⎦=1sin(𝑡/2)2𝑡(2)𝑛𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠1(𝑘+1)𝑘𝑣=0⎤⎥⎥⎦=1(1)2𝑡(2)𝑛𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠⎤⎥⎥⎦=12𝑡(2)𝑛(2)𝑛1=𝑂𝑡,since𝑛𝑘=0⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠=(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||||||𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘𝑘𝑣=0cos(𝑣+1/2)𝑡⎤⎥⎥⎦||||||≤1sin(𝑡/2)2𝑡(2)𝑛||||||𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖(𝑣+1/2)𝑡⎤⎥⎥⎦||||||≤12𝑡(2)𝑛||||||𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||||𝑒𝑖(𝑡/2)||≤12𝑡(2)𝑛||||||𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||+12𝑡(2)𝑛||||||𝑛𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||.(4.2) Now considering first term of (4.2) 12𝑡(2)𝑛||||||𝜏−1𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘𝑘𝑣=01⎤⎥⎥⎦||||||||𝑒𝑖𝑣𝑡||≤12𝑡(2)𝑛||||||𝜏−1𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠⎤⎥⎥⎦||||||.(4.3) Now considering second term of (4.2) and using Abel’s lemma 12𝑡(2)𝑛||||||𝑛𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘Re𝑘𝑣=0𝑒𝑖𝑣𝑡⎤⎥⎥⎦||||||≤12𝑡(2)𝑛𝑛𝑘=𝜏⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠11+𝑘max0≤𝑚≤𝑘|||||𝑚𝑣=0𝑒𝑖𝑣𝑡|||||≤12𝑡(2)𝑛𝑛𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠1⎤⎥⎥⎦=11+𝑘(1+𝑘)2𝑡(2)𝑛𝑛𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠⎤⎥⎥⎦.(4.4) On combining (4.2), (4.3), and (4.4) |||𝐺𝑛|||≤1(𝑡)2𝑡(2)𝑛𝜏−1𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠⎤⎥⎥⎦+12𝑡(2)𝑛𝑛𝑘=𝜏⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠,|||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 𝑠𝑛1(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)𝑛𝑘=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)𝑛𝑛𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑛𝑘⎞⎟⎟⎠𝜋0𝜓(𝑡)1sin(𝑡/2)1+𝑘𝑘𝑣=0⎤⎥⎥⎦=cos(𝑣+1/2)𝑡𝑑𝑡𝜋0𝐺𝜓(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)𝑟sin𝛽𝑟𝑡2𝑑𝑡1/𝑟⎡⎢⎢⎢⎣0𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡sin𝛽⎞⎟⎟⎟⎠(𝑡/2)𝑠⎤⎥⎥⎥⎦𝑑𝑡1/𝑠1=𝑂⎡⎢⎢⎢⎣𝑛+10𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡sin𝛽⎞⎟⎟⎟⎠(𝑡/2)𝑠⎤⎥⎥⎥⎦𝑑𝑡1/𝑠1by(3.2)=𝑂𝑛+10𝜋/(𝑛+1)𝜉(𝑡)𝑡2sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠1byLemma4.1=𝑂𝑛+10𝜋/(𝑛+1)𝜉(𝑡)(𝑡/2)𝛽𝑡2sin𝛽(𝑡/2)⋅(𝑡/2)𝛽𝑠𝑑𝑡1/𝑠1=𝑂𝑛+12(𝜋/2(𝑛+1))sin(𝜋/2(𝑛+1))𝛽𝑠ℎ𝜋/(𝑛+1)𝜉(𝑡)𝑡2+𝛽𝑠𝑑𝑡1/𝑠1,asℎ⟶0=𝑂𝑛+1ℎ𝜋/(𝑛+1)𝜉(𝑡)𝑡2+𝛽𝑠𝑑𝑡1/𝑠.(5.4) Since 𝜉(𝑡) is a positive increasing function and by using second mean value theorem for integrals, we have 𝐼11=𝑂𝜉𝜋𝑛+1𝑛+1∈𝜋/(𝑛+1)1𝑡2+𝛽𝑠𝑑𝑡1/𝑠𝜋,forsome0<∈<1𝑛+1=𝑂1𝑛+1𝜋𝜉𝑛+1∈𝜋/(𝑛+1)𝑡−𝛽𝑠−2𝑠𝑑𝑡1/𝑠.(5.5) Note that 𝜉(𝜋/(𝑛+1))≤𝜋𝜉(1/(𝑛+1)), 𝐼11=𝑂𝜉1𝑛+1𝑡𝑛+1−𝛽𝑠−2𝑠+1−𝛽𝑠−2𝑠+1∈𝜋/(𝑛+1)1/𝑠1=𝑂𝜉1𝑛+1𝑛+1(𝑛+1)𝛽+2−1/𝑠𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1−1/𝑠𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1/𝑟∵𝑟−1+𝑠−1=1,1≤𝑟≤∞.(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)𝑟𝑑𝑡1/𝑟⎡⎢⎢⎢⎣𝜋𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sin𝛽(⎞⎟⎟⎟⎠𝑡/2)𝑠⎤⎥⎥⎥⎦𝑑𝑡1/𝑠=𝑂(𝑛+1)𝛿⎡⎢⎢⎢⎣𝜋𝜋/(𝑛+1)⎛⎜⎜⎜⎝|||𝐺𝜉(𝑡)𝑛|||(𝑡)𝑡−𝛿sin𝛽⎞⎟⎟⎟⎠(𝑡/2)𝑠⎤⎥⎥⎥⎦𝑑𝑡1/𝑠(by(3.3)=𝑂𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿𝑡sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠byLemma4.2=𝑂(𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿+1sin𝛽(𝑡/2)𝑠𝑑𝑡1/𝑠=𝑂(𝑛+1)𝛿𝜋𝜋/(𝑛+1)𝜉(𝑡)𝑡−𝛿+𝛽+1𝑠𝑑𝑡1/𝑠.(5.8) Now putting 𝑡=1/𝑦, we have 𝐼2=𝑂(𝑛+1)𝛿(𝑛+1)/𝜋1/𝜋𝜉(1/𝑦)𝑦𝛿−𝛽−1𝑠𝑑𝑦𝑦21/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)/𝜋𝑑𝑦𝑦−𝛽𝑠+𝛿𝑠+21/𝑠1,forsome𝜋≤𝜂≤𝑛+1𝜋=𝑂(𝑛+1)𝛿+1𝜉1𝑦𝑛+1−𝛿𝑠−2+𝛽𝑠+1−𝛿𝑠−2+𝛽𝑠+11(𝑛+1)/𝜋1/𝑠1,forsome𝜋≤1≤𝑛+1𝜋=𝑂(𝑛+1)𝛿+1𝜉1𝑦𝑛+1−𝛿𝑠−1+𝛽𝑠1𝑛+1/𝜋1/𝑠(=𝑂𝑛+1)𝛿+1𝜉1(𝑛+1𝑛+1)−𝛿−1/𝑠+𝛽𝜉1=𝑂𝑛+1(𝑛+1)𝛿+1−𝛿−1/𝑠+𝛽𝜉1=𝑂𝑛+1(𝑛+1)𝛽+1/𝑟∵𝑟−1+𝑠−1=1,1≤𝑟≤∞.(5.10) Combining 𝐼1 and 𝐼2 yields |||𝐸1𝑛𝐶1𝑛−𝑓|||=𝑂(𝑛+1)1/𝑟+𝛽𝜉1.𝑛+1(5.11) Now, using the 𝐿𝑟-norm of a function, we get ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖𝑟=02𝜋|||𝐸1𝑛𝐶1𝑛−𝑓|||𝑟𝑑𝑥1/𝑟=𝑂02𝜋(𝑛+1)𝛽+1/𝑟𝜉1𝑛+1𝑟𝑑𝑥1/𝑟=𝑂(𝑛+1)𝛽+1/𝑟𝜉1𝑛+102𝜋𝑑𝑥1/𝑟=𝑂(𝑛+1)𝛽+1/𝑟𝜉1.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|||𝑟𝑑𝑥1/𝑟=𝑂(𝑛+1)1/𝑟𝜉11𝑛+1=𝑂(𝑛+1)𝛼−1/𝑟,𝑟≥1.(6.2) Thus, we get |||𝐸1𝑛𝐶1𝑛−𝑓|||≤02𝜋|||𝐸1𝑛𝐶1𝑛−𝑓|||𝑟𝑑𝑥1/𝑟1=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 𝑟→∞ 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 ‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖∞=sup0≤𝑥≤2𝜋|||𝐸1𝑛𝐶1𝑛|||1(𝑥)−𝑓(𝑥)=𝑂(𝑛+1)𝛼.(6.5) Thus, we get |||𝐸1𝑛𝐶1𝑛−𝑓|||≤‖‖𝐸1𝑛𝐶1𝑛−𝑓‖‖∞=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.

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Copyright © 2012 Vishnu Narayan Mishra 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.

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