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International Journal of Mathematics and Mathematical Sciences
VolumeΒ 2012, Article IDΒ 298923, 13 pages
http://dx.doi.org/10.1155/2012/298923
Research Article

Product Summability Transform of Conjugate Series of Fourier Series

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

Received 22 February 2012; Revised 15 March 2012; Accepted 15 March 2012

Academic Editor: Ram U.Β Verma

Copyright Β© 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.

Abstract

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging to Lipπœ‰(𝑑)-class by (𝐸,1)(𝐢,1) product summability means of conjugate series of Fourier series has been generalized for the weighted π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)), (π‘Ÿβ‰₯1),(𝑑>0)-class, where πœ‰(𝑑) is nonnegative and increasing function of 𝑑, by 𝐸1𝑛𝐢1𝑛 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,π‘Ÿ) and π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) classes by NΓΆrlund and generalized NΓΆrlund 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 π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) 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(πœ‰(𝑑),π‘Ÿ) class and π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) by (𝐸,1)(𝐢,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 CesΓ ro sum of order (𝐢,1) and (𝐢,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 CesΓ ro sum technique for the derivative of the double layer potential. Similarly, Chen and Hong [13] used CesΓ ro sum regularization technique for hypersingularity of dual integral equation.

The generalized weighted π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)),(π‘Ÿβ‰₯1)-class is generalization of Lip𝛼,Lip(𝛼,π‘Ÿ) and Lip(πœ‰(𝑑),π‘Ÿ) classes. Therefore, in the present paper, a theorem on degree of approximation of conjugate of signals belonging to the generalized weighted π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)),π‘Ÿβ‰₯1 class by (𝐸,1)(𝐢,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(π‘₯) be a 2πœ‹-periodic function and integrable in the sense of Lebesgue. The Fourier series of 𝑓(π‘₯) at any point π‘₯ is given by π‘Žπ‘“(π‘₯)∼02+βˆžξ“π‘›=1ξ€·π‘Žπ‘›cos𝑛π‘₯+𝑏𝑛≑sin𝑛π‘₯βˆžξ“π‘›=0𝐴𝑛(π‘₯),(1.1) with 𝑛th partial sum 𝑠𝑛(𝑓;π‘₯).

The conjugate series of Fourier series (1.1) is given by βˆžξ“π‘›=1𝑏𝑛cos𝑛π‘₯βˆ’π‘Žπ‘›ξ€Έβ‰‘sin𝑛π‘₯βˆžξ“π‘›=1𝐡𝑛(π‘₯).(1.2) Let βˆ‘βˆžπ‘›=0𝑒𝑛 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)π‘›π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ π‘ π‘˜βŸΆπ‘ ,asπ‘›βŸΆβˆž,(1.3) then the infinite series βˆ‘βˆžπ‘›=0𝑒𝑛 is summable (𝐸,1) to a definite number 𝑠, Hardy [14].

If πœπ‘›=𝑠0+𝑠1+𝑠2+β‹―+𝑠𝑛=1𝑛+1𝑛+1π‘›ξ“π‘˜=0π‘ π‘˜βŸΆπ‘ ,asπ‘›βŸΆβˆž,(1.4) then the infinite series βˆ‘βˆžπ‘›=0𝑒𝑛 is summable to the definite number 𝑠 by (𝐢,1) method. The (𝐸,1) transform of the (𝐢,1) transform defines (𝐸,1)(𝐢,1) product transform and denotes it by 𝐸1𝑛𝐢1𝑛. Thus, if 𝐸1𝑛𝐢1𝑛=1(2)π‘›π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ πΆ1π‘˜βŸΆπ‘ ,asπ‘›βŸΆβˆž,(1.5) then the infinite series βˆ‘βˆžπ‘›=0𝑒𝑛 is said to be summable by (𝐸,1)(𝐢,1) method or summable (𝐸,1)(𝐢,1) to a definite number 𝑠. The (𝐸,1) is regular method of summability π‘ π‘›βŸΆπ‘ βŸΉπΆ1𝑛𝑠𝑛=πœπ‘›=1𝑛+1π‘›ξ“π‘˜=0π‘ π‘˜βŸΆπ‘ ,asπ‘›βŸΆβˆž,𝐢1𝑛methodisregular,⟹𝐸1𝑛𝐢1𝑛𝑠𝑛=𝐸1𝑛𝐢1π‘›βŸΆπ‘ ,asπ‘›βŸΆβˆž,𝐸1𝑛methodisregular,⟹𝐸1𝑛𝐢1𝑛methodisregular.(1.6) A function 𝑓(π‘₯)∈Lip𝛼, if 𝑓||𝑑(π‘₯+𝑑)βˆ’π‘“(π‘₯)=𝑂𝛼||ξ€Έfor0<𝛼≀1,𝑑>0,(1.7) and 𝑓(π‘₯)∈Lip(𝛼,π‘Ÿ), for 0≀π‘₯≀2πœ‹, if πœ”π‘Ÿξ‚΅ξ€œ(𝑑;𝑓)=02πœ‹||||𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿ=𝑂(|𝑑|𝛼),0<𝛼≀1,π‘Ÿβ‰₯1,𝑑>0.(1.8) Given a positive increasing function πœ‰(𝑑),𝑓(π‘₯)∈Lip(πœ‰(𝑑),π‘Ÿ), if πœ”π‘Ÿξ‚΅ξ€œ(𝑑;𝑓)=02πœ‹||||𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿ=𝑂(πœ‰(𝑑)),π‘Ÿβ‰₯1,𝑑>0.(1.9) Given positive increasing function πœ‰(𝑑), an integer π‘Ÿβ‰₯1,π‘“βˆˆπ‘Š(πΏπ‘Ÿ,πœ‰(𝑑)), ([2]), if πœ”π‘Ÿξ‚»ξ€œ(𝑑;𝑓)=02πœ‹||{𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)}sin𝛽π‘₯||π‘Ÿξ‚Όπ‘‘π‘₯1/π‘Ÿ=𝑂(πœ‰(𝑑)),(𝛽β‰₯0),𝑑>0.(1.10) For our convenience to evaluate 𝐼2 without error, we redefine the weighted class as follows: πœ”π‘Ÿξ‚΅ξ€œ(𝑑;𝑓)=02πœ‹||||𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯)π‘Ÿsinπ›½π‘Ÿξ‚€π‘₯2𝑑π‘₯1/π‘Ÿ[]=𝑂(πœ‰(𝑑)),𝛽β‰₯0,𝑑>0(16).(1.11) If 𝛽=0, then the weighted class π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)) coincides with the class Lip(πœ‰(𝑑),π‘Ÿ), we observe that π‘Šξ€·πΏπ‘Ÿξ€Έ,πœ‰(𝑑)𝛽=0βˆ’βˆ’βˆ’β†’Lip(πœ‰(𝑑),π‘Ÿ)πœ‰(𝑑)=π‘‘π›Όβˆ’βˆ’βˆ’βˆ’β†’Lip(𝛼,π‘Ÿ)π‘Ÿβ†’βˆžβˆ’βˆ’βˆ’βˆ’β†’Lip𝛼for0<𝛼≀1,π‘Ÿβ‰₯1,𝑑>0.(1.12)πΏπ‘Ÿ-norm of a function is defined by β€–π‘“β€–π‘Ÿ=ξ‚΅ξ€œ02πœ‹||||𝑓(π‘₯)π‘Ÿξ‚Άπ‘‘π‘₯1/π‘Ÿ,π‘Ÿβ‰₯1.(1.13) A signal 𝑓 is approximated by trigonometric polynomials πœπ‘› of order 𝑛, and the degree of approximation 𝐸𝑛(𝑓) is given by Rhoades [3] 𝐸𝑛(𝑓)=min𝑛‖‖𝑓(π‘₯)βˆ’πœπ‘›β€–β€–(𝑓;π‘₯)π‘Ÿ,(1.14) in terms of 𝑛, where πœπ‘›(𝑓;π‘₯) 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: ξƒ΄πΊπœ“(𝑑)=𝑓(π‘₯+𝑑)βˆ’π‘“(π‘₯βˆ’π‘‘),𝑛1(𝑑)=2𝑛+1πœ‹βŽ‘βŽ’βŽ’βŽ£π‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ 11+π‘˜π‘˜ξ“π‘£=0cos(𝑣+1/2)π‘‘βŽ€βŽ₯βŽ₯⎦.sin(𝑑/2)(1.15)

2. Previous Result

Nigam [8] has proved a theorem on the degree of approximation of a function 𝑓(π‘₯), conjugate to a periodic function 𝑓(π‘₯) with period 2πœ‹ and belonging to the class Lip(πœ‰(𝑑),π‘Ÿ)(π‘Ÿβ‰₯1), by (𝐸,1)(𝐢,1) product summability means of conjugate series of Fourier series. He has proved the following theorem.

Theorem 2.1 (see [8]). If 𝑓(π‘₯), conjugate to a 2πœ‹-periodic function 𝑓(π‘₯), belongs to Lip(πœ‰(𝑑),π‘Ÿ) class, then its degree of approximation by (𝐸,1)(𝐢,1) product summability means of conjugate series of Fourier series is given by ‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–π‘Ÿξ‚†=𝑂(𝑛+1)1/π‘Ÿπœ‰ξ‚€1,𝑛+1(2.1) provided πœ‰(𝑑) satisfies the following conditions: ξƒ©ξ€œ01/(𝑛+1)𝑑||||πœ“(𝑑)ξ‚Άπœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξ‚€1=𝑂,ξƒ©ξ€œπ‘›+1(2.2)πœ‹1/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώ||||πœ“(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿξ€½=𝑂(𝑛+1)𝛿,(2.3) 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, condition (2.2) and (2.3) hold uniformly in π‘₯ and ξ„ŸπΈ1𝑛𝐢1𝑛 is (𝐸,1)(𝐢,1) means of the series (1.2).

Remark 2.2. The proof proceeds by estimating |ξ„ŸπΈ1𝑛𝐢1π‘›βˆ’ξ‚π‘“|, which is represented in terms of an integral. The domain of integration is divided into two partsβ€”from [0,1/(𝑛+1)] and [1/(𝑛+1),πœ‹]. Referring to second integral as 𝐼2, and using HΓΆlder inequality, the author [8] obtains ||𝐼2||β‰€ξƒ―ξ€œπœ‹1/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώ||||πœ“(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©ξ€œπœ‹1/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)π‘‘βˆ’π›ΏβŽžβŽŸβŽŸβŽŸβŽ π‘ βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/𝑠=𝑂(𝑛+1)π›Ώξ€ΎβŽ§βŽͺ⎨βŽͺβŽ©ξ€œπœ‹1/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)π‘‘βˆ’π›ΏβŽžβŽŸβŽŸβŽŸβŽ π‘ βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/𝑠.(2.4) The author then makes the substitution 𝑦=1/𝑑 to obtain ξ€½=𝑂(𝑛+1)π›Ώξ€Ύξ‚Έξ€œπ‘›+11/πœ‹ξ‚΅πœ‰(1/𝑦)π‘¦π›Ώβˆ’1𝑠𝑑𝑦𝑦2ξ‚Ή1/𝑠.(2.5) In the next step πœ‰(1/𝑦) is removed from the integrand by replacing it with 𝑂(πœ‰(1/(𝑛+1))), while πœ‰(𝑑) is an increasing function, πœ‰(1/𝑦) is now a decreasing function. Therefore, this step is invalid.

Remark 2.3. The proof follows by obtaining |ξ„Ÿ(𝐸𝐢)1π‘›βˆ’ξ‚π‘“|, in Theorem 2 of Nigam and Sharma [9], which is expressed in terms of an integral. The domain of integration is divided into two partsβ€”from [0,1/(𝑛+1)] and [1/(𝑛+1),πœ‹]. Referring to second integral as 𝐼2.2, and using HΓΆlder inequality, the authors [9] obtain the following: ||𝐼2.2||β‰€ξƒ―ξ€œπœ‹1/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώ||||πœ“(𝑑)sin𝛽𝑑ξƒͺπœ‰(𝑑)π‘Ÿξƒ°π‘‘π‘‘1/π‘ŸβŽ§βŽͺ⎨βŽͺβŽ©ξ€œπœ‹1/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)π‘‘βˆ’π›Ώsinπ›½π‘‘βŽžβŽŸβŽŸβŽŸβŽ π‘ βŽ«βŽͺ⎬βŽͺβŽ­π‘‘π‘‘1/𝑠(2.6)=𝑂(𝑛+1)π›Ώξ€Ύξ‚»ξ€œπœ‹1/(𝑛+1)ξ‚΅πœ‰(𝑑)𝑑1βˆ’π›Ώ+𝛽𝑠𝑑𝑑1/𝑠.(2.7) The authors then make the substitution 𝑦=1/𝑑 to get ξ€½=𝑂(𝑛+1)π›Ώξ€Ύξ‚Έξ€œπ‘›+11/πœ‹ξ‚΅πœ‰(1/𝑦)π‘¦π›Ώβˆ’1βˆ’π›½ξ‚Άπ‘ π‘‘π‘¦π‘¦2ξ‚Ή1/𝑠.(2.8) In the next step, πœ‰(1/𝑦) is removed from the integrand by replacing it with 𝑂(πœ‰(1/(𝑛+1))), while πœ‰(𝑑) is an increasing function, πœ‰(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(𝑑)=𝑂(1/𝑑𝛽),1/(𝑛+1)β‰€π‘‘β‰€πœ‹ used by Nigam and Sharma [9] is not valid, since sin𝑑→0 as π‘‘β†’πœ‹.

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(π‘₯), conjugate to a 2πœ‹-periodic function 𝑓, belongs to the generalized weighted π‘Š(πΏπ‘Ÿ,πœ‰(𝑑))(π‘Ÿβ‰₯1)-class, then its degree of approximation by (𝐸,1)(𝐢,1) product summability means of conjugate series of Fourier series is given by ‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–π‘Ÿξ‚†=𝑂(𝑛+1)𝛽+1/π‘Ÿπœ‰ξ‚€1,𝑛+1(3.1) provided πœ‰(𝑑) satisfies the following conditions: ξƒ©ξ€œ0πœ‹/(𝑛+1)𝑑||||πœ“(𝑑)πœ‰ξ‚Ά(𝑑)π‘Ÿsinπ›½π‘Ÿξ‚€π‘‘2ξƒͺ𝑑𝑑1/π‘Ÿξ‚€1=𝑂,ξƒ©ξ€œπ‘›+1(3.2)πœ‹πœ‹/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώ||||πœ“(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒͺ𝑑𝑑1/π‘Ÿ=𝑂(𝑛+1)𝛿,ξ‚»πœ‰(3.3)(𝑑)𝑑isnonincreasingsequencein”𝑑”,(3.4) 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 (3.2) and (3.3) hold uniformly in π‘₯ and 𝐸1𝑛𝐢1𝑛 is ξ„Ÿ(𝐸,1)(𝐢,1) means of the series (1.2) and the conjugate function 𝑓(π‘₯) is defined for almost every π‘₯ by 1𝑓(π‘₯)=βˆ’ξ€œ2πœ‹πœ‹0ξ‚€π‘‘πœ“(𝑑)cot2𝑑𝑑=limβ„Žβ†’0ξ‚΅βˆ’1ξ€œ2πœ‹πœ‹β„Žξ‚€π‘‘πœ“(𝑑)cot2𝑑𝑑.(3.5)

Note 1. πœ‰(πœ‹/(𝑛+1))β‰€πœ‹πœ‰(1/(𝑛+1)), for (πœ‹/(𝑛+1))β‰₯(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)(𝐢,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 |ξ„ŸπΊπ‘›(𝑑)|=𝑂[1/𝑑] for 0<π‘‘β‰€πœ‹/(𝑛+1).

Proof. For 0<π‘‘β‰€πœ‹/(𝑛+1),sin(𝑑/2)β‰₯(𝑑/πœ‹) and |cos𝑛𝑑|≀1. |||𝐺𝑛(|||=1𝑑)2πœ‹(2)𝑛||||||π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜π‘˜ξ“π‘£=0cos(𝑣+1/2)π‘‘βŽ€βŽ₯βŽ₯⎦||||||≀1sin(𝑑/2)2πœ‹(2)π‘›π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜π‘˜ξ“π‘£=0||||cos(𝑣+1/2)𝑑||||⎀βŽ₯βŽ₯⎦=1sin(𝑑/2)2𝑑(2)π‘›π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ 1(π‘˜+1)π‘˜ξ“π‘£=0⎀βŽ₯βŽ₯⎦=1(1)2𝑑(2)π‘›π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦=12𝑑(2)𝑛(2)𝑛1=𝑂𝑑,sinceπ‘›ξ“π‘˜=0βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ =(2)𝑛.(4.1) This completes the proof of Lemma 4.1.

Lemma 4.2. Consider |𝐺𝑛(𝑑)|=𝑂[1/𝑑], for 0<π‘‘β‰€πœ‹ and any 𝑛.

Proof. For 0<πœ‹/(𝑛+1)β‰€π‘‘β‰€πœ‹,sin(𝑑/2)β‰₯(𝑑/πœ‹). |||𝐺𝑛(|||=1𝑑)2πœ‹(2)𝑛||||||π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜π‘˜ξ“π‘£=0cos(𝑣+1/2)π‘‘βŽ€βŽ₯βŽ₯⎦||||||≀1sin(𝑑/2)2𝑑(2)𝑛||||||π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0𝑒𝑖(𝑣+1/2)π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||≀12𝑑(2)𝑛||||||π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||||𝑒𝑖(𝑑/2)||≀12𝑑(2)𝑛||||||π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||≀12𝑑(2)𝑛||||||πœβˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||+12𝑑(2)𝑛||||||π‘›ξ“π‘˜=πœβŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||.(4.2) Now considering first term of (4.2) 12𝑑(2)𝑛||||||πœβˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||≀12𝑑(2)𝑛||||||πœβˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜π‘˜ξ“π‘£=01ξƒ°βŽ€βŽ₯βŽ₯⎦||||||||𝑒𝑖𝑣𝑑||≀12𝑑(2)𝑛||||||πœβˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦||||||.(4.3) Now considering second term of (4.2) and using Abel’s lemma 12𝑑(2)𝑛||||||π‘›ξ“π‘˜=πœβŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜Reπ‘˜ξ“π‘£=0π‘’π‘–π‘£π‘‘ξƒ°βŽ€βŽ₯βŽ₯⎦||||||≀12𝑑(2)π‘›π‘›ξ“π‘˜=πœβŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€11+π‘˜max0β‰€π‘šβ‰€π‘˜|||||π‘šξ“π‘£=0𝑒𝑖𝑣𝑑|||||≀12𝑑(2)π‘›π‘›ξ“π‘˜=πœβŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ‚€1ξ‚βŽ€βŽ₯βŽ₯⎦=11+π‘˜(1+π‘˜)2𝑑(2)π‘›π‘›ξ“π‘˜=πœβŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦.(4.4) On combining (4.2), (4.3), and (4.4) |||𝐺𝑛|||≀1(𝑑)2𝑑(2)π‘›πœβˆ’1ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ βŽ€βŽ₯βŽ₯⎦+12𝑑(2)π‘›π‘›ξ“π‘˜=πœβŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ,|||𝐺𝑛|||1(𝑑)=𝑂𝑑.(4.5) This completes the proof of Lemma 4.2.

5. Proof of Theorem

Let 𝑠𝑛(π‘₯) denotes the 𝑛th partial sum of series (1.2). Then following Nigam [8], we have ξ„Ÿπ‘ π‘›ξ„Ÿ1(π‘₯)βˆ’π‘“(π‘₯)=ξ€œ2πœ‹πœ‹0πœ“(𝑑)cos(𝑛+1/2)𝑑sin(𝑑/2)𝑑𝑑.(5.1) Therefore, using (1.2), the (𝐢,1) transform 𝐢1𝑛 of 𝑠𝑛 is given by 𝐢1π‘›βˆ’ξƒ΄1𝑓(π‘₯)=ξ€œ2πœ‹(𝑛+1)πœ‹0πœ“(𝑑)π‘›ξ“π‘˜=0cos(𝑛+1/2)𝑑sin(𝑑/2)𝑑𝑑.(5.2) Now denoting ξ„Ÿ(𝐸,1)(𝐢,1) transform of 𝑠𝑛 as ξ„Ÿ(𝐸1𝑛,𝐢1𝑛), we write ξ„Ÿξ€·πΈ1𝑛,𝐢1π‘›ξ€Έβˆ’ξƒ΄1𝑓(π‘₯)=2πœ‹(2)π‘›π‘›ξ“π‘˜=0βŽ‘βŽ’βŽ’βŽ£βŽ›βŽœβŽœβŽπ‘›π‘˜βŽžβŽŸβŽŸβŽ ξ€œπœ‹0πœ“(𝑑)ξ‚€1sin(𝑑/2)1+π‘˜π‘˜ξ“π‘£=0ξƒ°βŽ€βŽ₯βŽ₯⎦=ξ€œcos(𝑣+1/2)π‘‘π‘‘π‘‘πœ‹0ξƒ΄πΊπœ“(𝑑)𝑛=ξ‚Έξ€œ(𝑑)𝑑𝑑0πœ‹/(𝑛+1)+ξ€œπœ‹πœ‹/(𝑛+1)ξ‚Ήξƒ΄πΊπœ“(𝑑)𝑛(𝑑)𝑑𝑑=𝐼1+𝐼2(say).(5.3) We consider, |𝐼1∫|≀0πœ‹/(𝑛+1)𝐺|πœ“(𝑑)||𝑛(𝑑)|𝑑𝑑.

Using HΓΆlder’s inequality ||𝐼1||β‰€ξƒ¬ξ€œ0πœ‹/(𝑛+1)𝑑||||πœ“(𝑑)ξ‚Άπœ‰(𝑑)π‘Ÿsinπ›½π‘Ÿξ‚€π‘‘2𝑑𝑑1/π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ£ξ€œ0πœ‹/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)𝑑sinπ›½βŽžβŽŸβŽŸβŽŸβŽ (𝑑/2)π‘ βŽ€βŽ₯βŽ₯βŽ₯βŽ¦π‘‘π‘‘1/𝑠1=π‘‚ξ‚βŽ‘βŽ’βŽ’βŽ’βŽ£ξ€œπ‘›+10πœ‹/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)𝑑sinπ›½βŽžβŽŸβŽŸβŽŸβŽ (𝑑/2)π‘ βŽ€βŽ₯βŽ₯βŽ₯βŽ¦π‘‘π‘‘1/𝑠1by(3.2)=π‘‚ξ‚ξƒ¬ξ€œπ‘›+10πœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)𝑑2sin𝛽(𝑑/2)𝑠𝑑𝑑1/𝑠1byLemma4.1=π‘‚ξ‚ξƒ¬ξ€œπ‘›+10πœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)(𝑑/2)𝛽𝑑2sin𝛽(𝑑/2)β‹…(𝑑/2)𝛽𝑠𝑑𝑑1/𝑠1=𝑂𝑛+12(πœ‹/2(𝑛+1))ξ‚Άsin(πœ‹/2(𝑛+1))π›½π‘ ξ€œβ„Žπœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)𝑑2+𝛽𝑠𝑑𝑑1/𝑠1,asβ„ŽβŸΆ0=π‘‚ξ‚ξ‚Έξ€œπ‘›+1β„Žπœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)𝑑2+𝛽𝑠𝑑𝑑1/𝑠.(5.4) Since πœ‰(𝑑) is a positive increasing function and by using second mean value theorem for integrals, we have 𝐼11=π‘‚ξ‚†ξ‚€ξ‚πœ‰ξ‚€πœ‹π‘›+1ξƒ¬ξ€œπ‘›+1ξ‚ξ‚‡βˆˆπœ‹/(𝑛+1)ξ‚€1𝑑2+𝛽𝑠𝑑𝑑1/π‘ πœ‹,forsome0<∈<1𝑛+1=𝑂1𝑛+1πœ‹πœ‰ξ‚Έξ€œπ‘›+1ξ‚ξ‚‡βˆˆπœ‹/(𝑛+1)π‘‘βˆ’π›½π‘ βˆ’2𝑠𝑑𝑑1/𝑠.(5.5) Note that πœ‰(πœ‹/(𝑛+1))β‰€πœ‹πœ‰(1/(𝑛+1)), 𝐼11=π‘‚ξ‚†ξ‚€ξ‚πœ‰ξ‚€1𝑛+1𝑑𝑛+1ξ‚ξ‚‡βˆ’π›½π‘ βˆ’2𝑠+1ξ‚Όβˆ’π›½π‘ βˆ’2𝑠+1βˆˆπœ‹/(𝑛+1)ξƒ­1/𝑠1=π‘‚ξ‚ƒξ‚€ξ‚πœ‰ξ‚€1𝑛+1𝑛+1(𝑛+1)𝛽+2βˆ’1/π‘ ξ‚„ξ‚ƒπœ‰ξ‚€1=𝑂𝑛+1(𝑛+1)𝛽+1βˆ’1/π‘ ξ‚„ξ‚ƒπœ‰ξ‚€1=𝑂𝑛+1(𝑛+1)𝛽+1/π‘Ÿξ‚„βˆ΅π‘Ÿβˆ’1+π‘ βˆ’1=1,1β‰€π‘Ÿβ‰€βˆž.(5.6) Now, we consider ||𝐼2||β‰€ξ€œπœ‹πœ‹/(𝑛+1)|||||||ξƒ΄πΊπœ“(𝑑)𝑛|||(𝑑)𝑑𝑑.(5.7) Using HΓΆlder’s inequality ||𝐼2||β‰€ξƒ¬ξ€œπœ‹πœ‹/(𝑛+1)ξƒ©π‘‘βˆ’π›Ώsin𝛽||||(𝑑/2)πœ“(𝑑)ξƒͺπœ‰(𝑑)π‘Ÿξƒ­π‘‘π‘‘1/π‘ŸβŽ‘βŽ’βŽ’βŽ’βŽ£ξ€œπœ‹πœ‹/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)π‘‘βˆ’π›Ώsin𝛽(βŽžβŽŸβŽŸβŽŸβŽ π‘‘/2)π‘ βŽ€βŽ₯βŽ₯βŽ₯βŽ¦π‘‘π‘‘1/𝑠=𝑂(𝑛+1)π›Ώξ€ΎβŽ‘βŽ’βŽ’βŽ’βŽ£ξ€œπœ‹πœ‹/(𝑛+1)βŽ›βŽœβŽœβŽœβŽ|||ξƒ΄πΊπœ‰(𝑑)𝑛|||(𝑑)π‘‘βˆ’π›Ώsinπ›½βŽžβŽŸβŽŸβŽŸβŽ (𝑑/2)π‘ βŽ€βŽ₯βŽ₯βŽ₯βŽ¦π‘‘π‘‘1/𝑠(by(3.3)=𝑂𝑛+1)π›Ώξ€Ύξ‚Έξ€œπœ‹πœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)π‘‘βˆ’π›Ώπ‘‘sin𝛽(𝑑/2)𝑠𝑑𝑑1/𝑠byLemma4.2=𝑂(𝑛+1)π›Ώξ€Ύξ‚Έξ€œπœ‹πœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)π‘‘βˆ’π›Ώ+1sin𝛽(𝑑/2)𝑠𝑑𝑑1/𝑠=𝑂(𝑛+1)π›Ώξ€Ύξ‚Έξ€œπœ‹πœ‹/(𝑛+1)ξ‚΅πœ‰(𝑑)π‘‘βˆ’π›Ώ+𝛽+1𝑠𝑑𝑑1/𝑠.(5.8) Now putting 𝑑=1/𝑦, we have 𝐼2ξ€½=𝑂(𝑛+1)π›Ώξ€Ύξ‚Έξ€œ(𝑛+1)/πœ‹1/πœ‹ξ‚΅πœ‰(1/𝑦)π‘¦π›Ώβˆ’π›½βˆ’1𝑠𝑑𝑦𝑦2ξ‚Ή1/𝑠.(5.9) Since πœ‰(𝑑) is a positive increasing function, so πœ‰(1/𝑦)/(1/𝑦) is also a positive increasing function and using second mean value theorem for integrals, we have ξ‚»=𝑂(𝑛+1)π›Ώπœ‰(πœ‹/(𝑛+1))ξ€œπœ‹/(𝑛+1)ξ‚Όξ‚Έπœ‚(𝑛+1)/πœ‹ξ‚΅π‘‘π‘¦π‘¦βˆ’π›½π‘ +𝛿𝑠+2ξ‚Άξ‚Ή1/𝑠1,forsomeπœ‹β‰€πœ‚β‰€π‘›+1πœ‹ξ‚†=𝑂(𝑛+1)𝛿+1πœ‰ξ‚€1𝑦𝑛+1ξ‚ξ‚‡βˆ’π›Ώπ‘ βˆ’2+𝛽𝑠+1ξ‚Ήβˆ’π›Ώπ‘ βˆ’2+𝛽𝑠+11(𝑛+1)/πœ‹ξƒ°1/𝑠1,forsomeπœ‹β‰€1≀𝑛+1πœ‹ξ‚†=𝑂(𝑛+1)𝛿+1πœ‰ξ‚€1𝑦𝑛+1ξ‚ξ‚‡ξ‚†βˆ’π›Ώπ‘ βˆ’1+𝛽𝑠1𝑛+1/πœ‹ξ‚‡1/𝑠(=𝑂𝑛+1)𝛿+1πœ‰ξ‚€1(𝑛+1𝑛+1)βˆ’π›Ώβˆ’1/𝑠+π›½ξ‚†πœ‰ξ‚€1=𝑂𝑛+1(𝑛+1)𝛿+1βˆ’π›Ώβˆ’1/𝑠+π›½ξ‚‡ξ‚†πœ‰ξ‚€1=𝑂𝑛+1(𝑛+1)𝛽+1/π‘Ÿξ‚‡βˆ΅π‘Ÿβˆ’1+π‘ βˆ’1=1,1β‰€π‘Ÿβ‰€βˆž.(5.10) Combining 𝐼1 and 𝐼2 yields |||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||=𝑂(𝑛+1)1/π‘Ÿ+π›½πœ‰ξ‚€1.𝑛+1(5.11) Now, using the πΏπ‘Ÿ-norm of a function, we get ‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–π‘Ÿ=ξ‚»ξ€œ02πœ‹|||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||π‘Ÿξ‚Όπ‘‘π‘₯1/π‘Ÿξ‚»ξ€œ=𝑂02πœ‹ξ‚€(𝑛+1)𝛽+1/π‘Ÿπœ‰ξ‚€1𝑛+1ξ‚ξ‚π‘Ÿξ‚Όπ‘‘π‘₯1/π‘Ÿξƒ―=𝑂(𝑛+1)𝛽+1/π‘Ÿπœ‰ξ‚€1ξ‚ξ‚΅ξ€œπ‘›+102πœ‹ξ‚Άπ‘‘π‘₯1/π‘Ÿξƒ°ξ‚€=𝑂(𝑛+1)𝛽+1/π‘Ÿπœ‰ξ‚€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 πœ‰(𝑑)=𝑑𝛼,0<𝛼≀1, then the weighted class π‘Š(πΏπ‘Ÿ,πœ‰(𝑑)),π‘Ÿβ‰₯1 reduces to the class Lip(𝛼,π‘Ÿ) and the degree of approximation of a function 𝑓(π‘₯) conjugate to a 2πœ‹-periodic function 𝑓 belonging to the class Lip(𝛼,π‘Ÿ), is given by |||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||ξ‚΅1=𝑂(𝑛+1)π›Όβˆ’1/π‘Ÿξ‚Ά.(6.1)

Proof. The result follows by setting 𝛽=0 in (3.1), we have ‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–π‘Ÿ=ξ‚»ξ€œ02πœ‹|||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||π‘Ÿξ‚Όπ‘‘π‘₯1/π‘Ÿξ‚€=𝑂(𝑛+1)1/π‘Ÿπœ‰ξ‚€1ξ‚΅1𝑛+1=𝑂(𝑛+1)π›Όβˆ’1/π‘Ÿξ‚Ά,π‘Ÿβ‰₯1.(6.2) Thus, we get |||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||β‰€ξ‚»ξ€œ02πœ‹|||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||π‘Ÿξ‚Όπ‘‘π‘₯1/π‘Ÿξ‚΅1=𝑂(𝑛+1)π›Όβˆ’1/π‘Ÿξ‚Ά,π‘Ÿβ‰₯1.(6.3) This completes the proof of Corollary 6.1.

Corollary 6.2. If πœ‰(𝑑)=𝑑𝛼 for 0<𝛼<1 and π‘Ÿβ†’βˆž in Corollary 6.1, then π‘“βˆˆLip𝛼 and |||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||ξ‚΅1=𝑂(𝑛+1)𝛼.(6.4)

Proof. For π‘Ÿ=∞ in Corollary 6.1, we get ‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–βˆž=sup0≀π‘₯≀2πœ‹|||𝐸1𝑛𝐢1𝑛|||ξ‚΅1(π‘₯)βˆ’π‘“(π‘₯)=𝑂(𝑛+1)𝛼.(6.5) Thus, we get |||𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“|||≀‖‖𝐸1𝑛𝐢1π‘›βˆ’ξ‚π‘“β€–β€–βˆž=sup0≀π‘₯≀2πœ‹|||𝐸1𝑛𝐢1𝑛|||ξ‚΅1(π‘₯)βˆ’π‘“(π‘₯)=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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