Product Summability Transform of Conjugate Series of Fourier Series

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 ̃ E1 nC n which is in more general form of Theorem 2 of Nigam and Sharma 2011 .


Introduction
, 2 has studied the degree of approximation of a function belonging to Lip α, r and W L r , ξ t classes by N örlund and generalized N örlund 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 International Journal of Mathematics and Mathematical Sciences 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 with nth partial sums n f; x .The conjugate series of Fourier series 1.1 is given by Let ∞ n 0 u 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 then the infinite series ∞ n 0 u n is summable E, 1 to a definite number s, Hardy 14 .If then the infinite series ∞ n 0 u 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 then the infinite series ∞ n 0 u 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 International Journal of Mathematics and Mathematical Sciences 3 Given a positive increasing function ξ t , f x ∈ Lip ξ t , r , if 1.9 Given positive increasing function ξ t , an integer r ≥ 1, f ∈ W L r , ξ t , 2 , if For our convenience to evaluate I 2 without error, we redefine the weighted class as follows: 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 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 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 .

International Journal of Mathematics and Mathematical Sciences
We use the following notations throughout this paper: 1.15

Previous Result
provided ξ t satisfies the following conditions: where δ is an arbitrary number such that s Remark 2.2.The proof proceeds by estimating | E 1 n C 1 n − 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 I 2 , and using Hölder inequality, the author 8 obtains

2.4
The author then makes the substitution y 1/t to obtain 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 1 n − 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 H ölder inequality, the authors 9 obtain the following: The authors then make the substitution y 1/t to get 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.

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.

International Journal of Mathematics and Mathematical Sciences
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 provided ξ t satisfies the following conditions: where δ is an arbitrary number such that s 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.

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

4.1
This completes the proof of Lemma 4.1.

4.2
Now considering first term of 4.2

International Journal of Mathematics and Mathematical Sciences
Now considering second term of 4.2 and using Abel's lemma

4.5
This completes the proof of Lemma 4.2.

Proof of Theorem
Let s n x denotes the nth partial sum of series 1.2 .Then following Nigam 8 , we have Therefore, using 1.2 , the C, 1 transform C 1 n of s n is given by Using H ölder's inequality

5.4
Since ξ t is a positive increasing function and by using second mean value theorem for integrals, we have

5.6
Now, we consider ψ t G n t dt.

International Journal of Mathematics and Mathematical Sciences
Using H ölder's inequality

5.8
Now putting t 1/y, we have 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

5.10
Combining I 1 and I 2 yields Now, using the L r -norm of a function, we get

5.12
This completes the proof of Theorem 3.1.

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 Proof.The result follows by setting β 0 in 3.1 , we have

6.2
Thus, we get This completes the proof of Corollary 6.1.

International Journal of Mathematics and Mathematical Sciences
Corollary 6.2.If ξ t t α for 0 < α < 1 and r → ∞ in Corollary 6.1, then f ∈ Lipα and 6.4 Proof.For r ∞ in Corollary 6.1, we get

6.5
Thus, we get

6.6
This completes the proof of Corollary 6.2.
Nigam 8has 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