Product Summability Transform of Conjugate Series of Fourier Series

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(𝐿_𝑟,𝜉(𝑡)), (𝑟≥1),(𝑡>0)-class, where 𝜉(𝑡) is nonnegative and increasing function of 𝑡, by 𝐸_1𝑛𝐶^1𝑛 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(𝐿_𝑟,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(𝐿_𝑟,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(𝐿_𝑟,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(𝐿_𝑟,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(𝐿_𝑟,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 𝑓(𝑥)∼𝑎⁰2+^∞_𝑛=1𝑎_𝑛cos𝑛𝑥+𝑏_𝑛sin𝑛𝑥≡^∞_𝑛=0𝐴_𝑛(𝑥),(1.1) with 𝑛th partial sum 𝑠_𝑛(𝑓;𝑥).

The conjugate series of Fourier series (1.1) is given by ^∞_𝑛=1𝑏_𝑛cos𝑛𝑥−𝑎_𝑛sin𝑛𝑥≡^∞_𝑛=1𝐵_𝑛(𝑥).(1.2) Let ∑^∞𝑛=0𝑢_𝑛 be a given infinite series with sequence of its 𝑛th partial sums {𝑠_𝑛}. The (𝐸,1) transform is defined as the 𝑛th partial sum of (𝐸,1) summability, and we denote it by 𝐸^1𝑛.

If 𝐸^1𝑛=1(2)^𝑛𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝑠_𝑘⟶𝑠,as𝑛⟶∞,(1.3) then the infinite series ∑^∞𝑛=0𝑢_𝑛 is summable (𝐸,1) to a definite number 𝑠, Hardy [14].

If 𝜏_𝑛=𝑠⁰+𝑠₁+𝑠₂+⋯+𝑠_𝑛𝑛+1=𝑛+11^𝑛_𝑘=0𝑠_𝑘⟶𝑠,as𝑛⟶∞,(1.4) then the infinite series ∑^∞𝑛=0𝑢_𝑛 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 𝐸^1𝑛𝐶^1𝑛. Thus, if 𝐸^1𝑛𝐶^1𝑛=1(2)^𝑛𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠𝐶_1𝑘⟶𝑠,as𝑛⟶∞,(1.5) then the infinite series ∑^∞𝑛=0𝑢_𝑛 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 𝑠_𝑛⟶𝑠⟹𝐶^1𝑛𝑠_𝑛=𝜏_𝑛=1𝑛+1^𝑛_𝑘=0⟹𝐸_1𝑛𝑠_𝑘⟶𝑠,as𝑛⟶∞,𝐶^1𝑛methodisregular,⟹𝐸_1𝑛𝐶^1𝑛𝑠_𝑛=𝐸^1𝑛𝐶^1𝑛⟶𝑠,as𝑛⟶∞,𝐸^1𝑛methodisregular,𝐶^1𝑛methodisregular.(1.6) A function 𝑓(𝑥)∈Lip𝛼, if 𝑓(𝑥+𝑡)−𝑓(𝑥)=𝑂||𝑡^𝛼||for0<𝛼≤1,𝑡>0,(1.7) and 𝑓(𝑥)∈Lip(𝛼,𝑟), for 0≤𝑥≤2𝜋, if 𝜔_𝑟(𝑡;𝑓)=^02𝜋||𝑓(𝑥+𝑡)−𝑓(𝑥)||^𝑟𝑑𝑥^1/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 𝜔_𝑟(𝑡;𝑓)=^02𝜋||𝑓(𝑥+𝑡)−𝑓(𝑥)||^𝑟𝑑𝑥^1/𝑟=𝑂(𝜉(𝑡)),𝑟≥1,𝑡>0.(1.9) Given positive increasing function 𝜉(𝑡), an integer 𝑟≥1,𝑓∈𝑊(𝐿_𝑟,𝜉(𝑡)), ([2]), if 𝜔_𝑟(𝑡;𝑓)=^02𝜋||{𝑓(𝑥+𝑡)−𝑓(𝑥)}sin^𝛽𝑥||^𝑟𝑑𝑥^1/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: 𝜔_𝑟(𝑡;𝑓)=^02𝜋||𝑓(𝑥+𝑡)−𝑓(𝑥)||^𝑟sin^𝛽𝑟𝑥2𝑑𝑥^1/𝑟=𝑂(𝜉(𝑡)),𝛽≥0,𝑡>0([16]).(1.11) If 𝛽=0, then the weighted class 𝑊(𝐿_𝑟,𝜉(𝑡)) coincides with the class Lip(𝜉(𝑡),𝑟), we observe that 𝑊𝐿_𝑟,𝜉(𝑡)−−−→^𝛽=0Lip(𝜉(𝑡),𝑟)−−−−→^{𝜉(𝑡)=𝑡𝛼}Lip(𝛼,𝑟)−−−−^𝑟→∞→Lip𝛼for0<𝛼≤1,𝑟≥1,𝑡>0.(1.12)𝐿_𝑟-norm of a function is defined by ‖𝑓‖_𝑟=^02𝜋||𝑓(𝑥)||^𝑟𝑑𝑥^1/𝑟,𝑟≥1.(1.13) A signal 𝑓 is approximated by trigonometric polynomials 𝜏_𝑛 of order 𝑛, and the degree of approximation 𝐸_𝑛(𝑓) is given by Rhoades [3] 𝐸_𝑛(𝑓)=min_𝑛‖‖𝑓(𝑥)−𝜏_𝑛(𝑓;𝑥)‖‖_𝑟,(1.14) in terms of 𝑛, where 𝜏_𝑛(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: 𝐺𝜓(𝑡)=𝑓(𝑥+𝑡)−𝑓(𝑥−𝑡),_𝑛(𝑡)=12^𝑛+1𝜋⎡⎢⎢⎣^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘^𝑘_𝑣=0cos(𝑣+1/2)𝑡sin(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 ‖‖𝐸_1𝑛𝐶^1𝑛−𝑓‖‖_𝑟=𝑂(𝑛+1)^1/𝑟𝜉1𝑛+1,(2.1) provided 𝜉(𝑡) satisfies the following conditions: ^01/(𝑛+1)𝑡||𝜉𝜓(𝑡()𝑡)||^𝑟𝑑𝑡^1/𝑟=𝑂1𝑛+1,(2.2)_{𝜋1/(𝑛+1)}𝑡^−𝛿||𝜓(𝑡)||𝜉(𝑡)^𝑟𝑑𝑡^1/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 𝐸_1𝑛𝐶^1𝑛 is (𝐸,1)(𝐶,1) means of the series (1.2).

Remark 2.2. The proof proceeds by estimating |𝐸^1𝑛𝐶^1𝑛−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)^𝑟𝑑𝑡^1/𝑟⎧⎪⎨⎪⎩_{𝜋1/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡^−𝛿⎞⎟⎟⎟⎠^𝑠𝑑𝑡⎫⎪⎬⎪⎭^1/𝑠=𝑂(𝑛+1)^𝛿⎧⎪⎨⎪⎩^{𝜋1/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡^−𝛿⎞⎟⎟⎟⎠^𝑠𝑑𝑡⎫⎪⎬⎪⎭^1/𝑠.(2.4) The author then makes the substitution 𝑦=1/𝑡 to obtain =𝑂(𝑛+1)^𝛿^𝑛+11/𝜋𝜉𝑦(1/𝑦^𝛿−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 𝐼_2.2, and using Hölder inequality, the authors [9] obtain the following: ||𝐼_2.2||≤^{𝜋1/(𝑛+1)}𝑡^−𝛿||𝜓(𝑡)||sin^𝛽𝑡𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟⎧⎪⎨⎪⎩_{𝜋1/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡^−𝛿sin^𝛽𝑡⎞⎟⎟⎟⎠^𝑠=𝑂(𝑛+1)_𝛿𝑑𝑡⎫⎪⎬⎪⎭^1/𝑠(2.6)^{𝜋1/(𝑛+1)}𝜉(𝑡)𝑡^{1−𝛿+𝛽}^𝑠𝑑𝑡^1/𝑠.(2.7) The authors then make the substitution 𝑦=1/𝑡 to get =𝑂(𝑛+1)^𝛿^𝑛+11/𝜋𝑦𝜉(1/𝑦)^{𝛿−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/sin^𝛽(𝑡)=𝑂(1/𝑡^𝛽),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(𝐿_𝑟,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 ‖‖𝐸_1𝑛𝐶^1𝑛−𝑓‖‖_𝑟=𝑂(𝑛+1)^𝛽+1/𝑟𝜉1𝑛+1,(3.1) provided 𝜉(𝑡) satisfies the following conditions: ^{0𝜋/(𝑛+1)}𝑡||𝜉𝜓(𝑡()𝑡)||^𝑟sin^𝛽𝑟𝑡2𝑑𝑡^1/𝑟=𝑂1𝑛+1,(3.2)_{𝜋𝜋/(𝑛+1)}𝑡^−𝛿||𝜓(𝑡)||𝜉(𝑡)^𝑟𝜉(𝑡)𝑑𝑡^1/𝑟=𝑂(𝑛+1)^𝛿,(3.3)𝑡isnonincreasingsequencein”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 𝐸_1𝑛𝐶^1𝑛 is (𝐸,1)(𝐶,1) means of the series (1.2) and the conjugate function 𝑓(𝑥) is defined for almost every 𝑥 by 𝑓(𝑥)=−12𝜋^𝜋0𝜓(𝑡)cot𝑡2𝑑𝑡=lim^ℎ→0−21𝜋^𝜋ℎ𝜓(𝑡)cot𝑡2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 |𝐺_𝑛(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. |||𝐺_𝑛(𝑡)|||=12𝜋(2)^𝑛||||||_𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘^𝑘_𝑣=0cos(𝑣+1/2)𝑡≤1sin(𝑡/2)⎤⎥⎥⎦||||||2𝜋(2)^𝑛𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘^𝑘_𝑣=0||cos(𝑣+1/2)𝑡||=1||sin(𝑡/2)||⎤⎥⎥⎦2𝑡(2)^𝑛𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠1(𝑘+1)^𝑘_𝑣=0=1(1)⎤⎥⎥⎦2𝑡(2)^𝑛𝑛_𝑘=0=1⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠⎤⎥⎥⎦2𝑡(2)^𝑛(2)^𝑛=𝑂1𝑡,since^𝑛_𝑘=0⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠=(2)_𝑛.(4.1) This completes the proof of Lemma 4.1.

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

Proof. For 0<𝜋/(𝑛+1)≤𝑡≤𝜋,sin(𝑡/2)≥(𝑡/𝜋). |||𝐺_𝑛(𝑡)|||=12𝜋(2)^𝑛||||||_𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘^𝑘_𝑣=0cos(𝑣+1/2)𝑡≤1sin(𝑡/2)⎤⎥⎥⎦||||||2𝑡(2)^𝑛||||||_𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^{𝑖(𝑣+1/2)𝑡}≤1⎤⎥⎥⎦||||||2𝑡(2)^𝑛||||||_𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡⎤⎥⎥⎦||||||||𝑒^𝑖(𝑡/2)≤1||2𝑡(2)^𝑛||||||_𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡≤1⎤⎥⎥⎦||||||2𝑡(2)^𝑛||||||_𝜏−1_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡+1⎤⎥⎥⎦||||||2𝑡(2)^𝑛||||||_𝑛_𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡⎤⎥⎥⎦||||||.(4.2) Now considering first term of (4.2) 12𝑡(2)^𝑛||||||_𝜏−1_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡⎤⎥⎥⎦||||||≤2𝑡(12)^𝑛||||||_𝜏−1_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘^𝑘_𝑣=0≤11⎤⎥⎥⎦||||||||𝑒^𝑖𝑣𝑡||2𝑡(2)^𝑛||||||_𝜏−1_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠⎤⎥⎥⎦||||||.(4.3) Now considering second term of (4.2) and using Abel’s lemma 12𝑡(2)^𝑛||||||_𝑛_𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘Re^𝑘_𝑣=0𝑒^𝑖𝑣𝑡⎤⎥⎥⎦||||||≤2𝑡(12)^𝑛𝑛_𝑘=𝜏⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠11+𝑘max^{0≤𝑚≤𝑘}|||||_𝑚_𝑣=0𝑒^𝑖𝑣𝑡≤1|||||2𝑡(2)^𝑛𝑛_𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠1=11+𝑘(1+𝑘)⎤⎥⎥⎦2𝑡(2)^𝑛𝑛_𝑘=𝜏⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠⎤⎥⎥⎦.(4.4) On combining (4.2), (4.3), and (4.4) |||𝐺_𝑛(𝑡)|||≤12𝑡(2)^𝑛𝜏−1_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠⎤⎥⎥⎦+12𝑡(2)^𝑛𝑛_𝑘=𝜏|||𝐺_𝑛⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠,(𝑡)|||=𝑂1𝑡.(4.5) This completes the proof of Lemma 4.2.

5. Proof of Theorem

Let 𝑠_𝑛(𝑥) denotes the 𝑛th partial sum of series (1.2). Then following Nigam [8], we have 𝑠_𝑛(𝑥)−𝑓(𝑥)=12𝜋^𝜋0𝜓(𝑡)cos(𝑛+1/2)𝑡sin(𝑡/2)𝑑𝑡.(5.1) Therefore, using (1.2), the (𝐶,1) transform 𝐶^1𝑛 of 𝑠_𝑛 is given by 𝐶_1𝑛−𝑓(𝑥)=12𝜋(𝑛+1)^𝜋0𝜓(𝑡)^𝑛_𝑘=0cos(𝑛+1/2)𝑡sin(𝑡/2)𝑑𝑡.(5.2) Now denoting (𝐸,1)(𝐶,1) transform of 𝑠_𝑛 as (𝐸_1𝑛,𝐶^1𝑛), we write 𝐸_1𝑛,𝐶^1𝑛−𝑓(𝑥)=12𝜋(2)^𝑛𝑛_𝑘=0⎡⎢⎢⎣⎛⎜⎜⎝𝑘𝑛⎞⎟⎟⎠^𝜋0𝜓(𝑡)sin(𝑡/2)1+1𝑘^𝑘_𝑣=0=cos(𝑣+1/2)𝑡𝑑𝑡_𝜋0⎤⎥⎥⎦=𝜓(𝑡)𝐺_𝑛(𝑡)𝑑𝑡_{0𝜋/(𝑛+1)}+^{𝜋𝜋/(𝑛+1)}𝜓(𝑡)𝐺^𝑛=𝐼₁(𝑡)𝑑𝑡+𝐼₂(say).(5.3) We consider, |𝐼₁|≤∫^{0𝜋/(𝑛+1)}|𝜓(𝑡)||𝐺_𝑛(𝑡)|𝑑𝑡.

Using Hölder’s inequality ||𝐼₁||≤^{0𝜋/(𝑛+1)}𝑡||𝜉𝜓(𝑡()𝑡)||^𝑟sin^𝛽𝑟𝑡2𝑑𝑡^1/𝑟⎡⎢⎢⎢⎣_{0𝜋/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡sin^𝛽(𝑡/2)⎞⎟⎟⎟⎠^𝑠𝑑𝑡⎤⎥⎥⎥⎦^1/𝑠=𝑂1𝑛+1⎡⎢⎢⎢⎣^{0𝜋/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡sin^𝛽(𝑡/2)⎞⎟⎟⎟⎠^𝑠=𝑂1𝑑𝑡⎤⎥⎥⎥⎦^1/𝑠by(3.2)𝑛+1^{0𝜋/(𝑛+1)}𝑡²sin𝜉(^𝛽𝑡()𝑡/2)^𝑠=𝑂1𝑑𝑡^1/𝑠byLemma4.1𝑛+1^{0𝜋/(𝑛+1)}𝑡²sin𝜉_𝛽((𝑡𝑡)/(2𝑡/)2⋅)(^𝛽𝑡/2)^𝛽^𝑠𝑑𝑡^1/𝑠=𝑂1𝑛+1si2n(𝜋(𝜋/2/2(𝑛(𝑛++11))))^𝛽𝑠^{ℎ𝜋/(𝑛+1)}𝜉𝑡(𝑡^2+𝛽)^𝑠=𝑂1𝑑𝑡^1/𝑠,asℎ⟶0𝑛+1^{ℎ𝜋/(𝑛+1)}𝜉𝑡(𝑡^2+𝛽)^𝑠𝑑𝑡^1/𝑠.(5.4) Since 𝜉(𝑡) is a positive increasing function and by using second mean value theorem for integrals, we have 𝐼₁=𝑂1𝑛+1𝜉𝑛+𝜋1^{∈𝜋/(𝑛+1)}𝑡1_2+𝛽^𝑠𝑑𝑡^1/𝑠,forsome0<∈<𝜋=𝑂1𝑛+1𝑛+1𝜋𝜉𝑛+11^{∈𝜋/(𝑛+1)}𝑡^{−𝛽𝑠−2𝑠}𝑑𝑡^1/𝑠.(5.5) Note that 𝜉(𝜋/(𝑛+1))≤𝜋𝜉(1/(𝑛+1)), 𝐼₁=𝑂1𝑛+1𝜉𝑛+11−𝛽𝑡𝑠−2𝑠+^{−𝛽𝑠−2𝑠+1}1^{∈𝜋/(𝑛+1)}^1/𝑠=𝑂1=𝑂𝜉𝑛+11𝜉𝑛+11(𝑛+1)^{𝛽+2−1/𝑠}=𝑂𝜉𝑛+11(𝑛+1)^{𝛽+1−1/𝑠}𝑛+1(𝑛+1)^𝛽+1/𝑟∵𝑟⁻¹+𝑠⁻¹=1,1≤𝑟≤∞.(5.6) Now, we consider ||𝐼₂||≤^{𝜋𝜋/(𝑛+1)}||𝜓(𝑡)|||||𝐺_𝑛(𝑡)|||𝑑𝑡.(5.7) Using Hölder’s inequality ||𝐼₂||≤^{𝜋𝜋/(𝑛+1)}𝑡^−𝛿sin^𝛽(𝑡/2)||𝜓(𝑡)||𝜉(𝑡)^𝑟𝑑𝑡^1/𝑟⎡⎢⎢⎢⎣_{𝜋𝜋/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡^−𝛿sin^𝛽(𝑡/2)⎞⎟⎟⎟⎠^𝑠𝑑𝑡⎤⎥⎥⎥⎦^1/𝑠=𝑂(𝑛+1)^𝛿⎡⎢⎢⎢⎣^{𝜋𝜋/(𝑛+1)}⎛⎜⎜⎜⎝𝜉(𝑡)|||𝐺_𝑛(𝑡)|||𝑡^−𝛿sin^𝛽(𝑡/2)⎞⎟⎟⎟⎠^𝑠=𝑂(𝑛+1)_𝛿𝑑𝑡⎤⎥⎥⎥⎦^1/𝑠by(3.3)^{𝜋𝜋/(𝑛+1)}𝜉(𝑡)𝑡^−𝛿𝑡sin^𝛽(𝑡/2)^𝑠=𝑂(𝑛+1)_𝛿𝑑𝑡^1/𝑠byLemma4.2^{𝜋𝜋/(𝑛+1)}𝜉(𝑡)𝑡^−𝛿+1sin^𝛽(𝑡/2)^𝑠𝑑𝑡^1/𝑠=𝑂(𝑛+1)^𝛿^{𝜋𝜋/(𝑛+1)}𝑡𝜉(𝑡)^{−𝛿+𝛽+1}^𝑠𝑑𝑡^1/𝑠.(5.8) Now putting 𝑡=1/𝑦, we have 𝐼₂=𝑂(𝑛+1)^𝛿^{(𝑛+1)/𝜋1/𝜋}𝜉𝑦(1/𝑦^{𝛿−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}𝑦𝑑𝑦^{−𝛽𝑠+𝛿𝑠+2}^1/𝑠,forsome1=𝑂(𝑛+1)_𝛿+1𝜋≤𝜂≤𝑛+𝜋1𝜉1𝑛+1−𝛿𝑠𝑦−2+𝛽𝑠^{−𝛿𝑠−2+𝛽𝑠+1}+1^{1(𝑛+1)/𝜋}^1/𝑠,forsome1=𝑂(𝑛+1)_𝛿+1𝜋≤1≤𝑛+𝜋1𝜉1𝑛+1𝑦^{−𝛿𝑠−1+𝛽𝑠}^1𝑛+1/𝜋^1/𝑠=𝑂(𝑛+1)^𝛿+1𝜉1𝑛+1(𝑛+1)^{−𝛿−1/𝑠+𝛽}=𝑂𝜉1=𝑂𝜉𝑛+11(𝑛+1)^{𝛿+1−𝛿−1/𝑠+𝛽}𝑛+1(𝑛+1)^𝛽+1/𝑟∵𝑟⁻¹+𝑠⁻¹=1,1≤𝑟≤∞.(5.10) Combining 𝐼₁ and 𝐼₂ yields |||𝐸_1𝑛𝐶^1𝑛−𝑓|||=𝑂(𝑛+1)^1/𝑟+𝛽𝜉1𝑛+1.(5.11) Now, using the 𝐿_𝑟-norm of a function, we get ‖‖𝐸_1𝑛𝐶^1𝑛−𝑓‖‖_𝑟=^02𝜋|||𝐸_1𝑛𝐶^1𝑛−𝑓|||^𝑟𝑑𝑥^1/𝑟=𝑂^02𝜋(𝑛+1)^𝛽+1/𝑟𝜉1𝑛+1^𝑟𝑑𝑥^1/𝑟=𝑂(𝑛+1)^𝛽+1/𝑟𝜉1𝑛+1^02𝜋𝑑𝑥^1/𝑟=𝑂(𝑛+1)_𝛽+1/𝑟𝜉1𝑛+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)=𝑡^𝛼,0<𝛼≤1, then the weighted class 𝑊(𝐿_𝑟,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 |||𝐸_1𝑛𝐶^1𝑛−𝑓|||=𝑂1(𝑛+1)^{𝛼−1/𝑟}.(6.1)

Proof. The result follows by setting 𝛽=0 in (3.1), we have ‖‖𝐸_1𝑛𝐶^1𝑛−𝑓‖‖_𝑟=^02𝜋|||𝐸_1𝑛𝐶^1𝑛−𝑓|||^𝑟𝑑𝑥^1/𝑟=𝑂(𝑛+1)^1/𝑟𝜉1=𝑂1𝑛+1(𝑛+1)^{𝛼−1/𝑟},𝑟≥1.(6.2) Thus, we get |||𝐸_1𝑛𝐶^1𝑛−𝑓|||≤^02𝜋|||𝐸_1𝑛𝐶^1𝑛−𝑓|||^𝑟𝑑𝑥^1/𝑟=𝑂1(𝑛+1)^{𝛼−1/𝑟},𝑟≥1.(6.3) This completes the proof of Corollary 6.1.

Corollary 6.2. If 𝜉(𝑡)=𝑡^𝛼 for 0<𝛼<1 and 𝑟→∞ in Corollary 6.1, then 𝑓∈Lip𝛼 and |||𝐸_1𝑛𝐶^1𝑛−𝑓|||=𝑂1(𝑛+1)^𝛼.(6.4)

Proof. For 𝑟=∞ in Corollary 6.1, we get ‖‖𝐸_1𝑛𝐶^1𝑛−𝑓‖‖_∞==𝑂sup_{0≤𝑥≤2𝜋}|||𝐸1^1𝑛𝐶^1𝑛(𝑥)−𝑓(𝑥)|||(𝑛+1)^𝛼.(6.5) Thus, we get |||𝐸_1𝑛𝐶^1𝑛−𝑓|||≤‖‖𝐸^1𝑛𝐶^1𝑛−𝑓‖‖_∞=sup_{0≤𝑥≤2𝜋}|||𝐸_1𝑛=𝑂1𝐶^1𝑛(𝑥)−𝑓(𝑥)|||(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.