Degree of Approximation of Functions ̃f ∈ H ω Class by the ( N p ⋅ E 1 ) Means in the Hölder Metric

The degree of approximation of a function f belonging to various classes using different summability method has been determined by several investigators like Khan [1, 2], V. N. Mishra and L. N. Mishra [3], Mishra et al. [4–6], and Mishra [7, 8]. Summability techniques were also applied on some engineering problems; for example, Chen and Jeng [9] 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 and Hong [10] used Cesàro sum regularization technique for hyper singularity of dual integral equation. Summability of Fourier series is useful for engineering analysis, for example, [11]. Recently, Mursaleen and Mohiuddine [12] discussed convergence methods for double sequences and their applications in various fields. In sequel, Alexits [13] studied the degree of approximation of the functions in H α class by the Cesàro means of their Fourier series in the sup-norm. Chandra ([14, 15]), Mohapatra and Chandra ([16, 17]), and Szal [18] have studied the approximation of functions in Hölder metric. Mishra et al. [6] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals. In 2008, Singh and Mahajan [19] studied error bound of periodic signals in the Hölder metric and generalized the result of Lal and Yadav [20] under much more general assumptions. Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon. The engineers and scientists use properties of Fourier approximation for designing digital filters. Especially, Psarakis andMoustakides [21] presented a new L 2 basedmethod for designing the finite impulse response (FIR) digital filters and got corresponding optimum approximations having improved performance.We also discuss an example when the Fourier series of the signal has Gibbs phenomenon. For a 2π-periodic signal f ∈ Lp := Lp[0, 2π], p ≥ 1, periodic integrable in the sense of Lebesgue.Then the Fourier series of f(x) is given by


Introduction
The degree of approximation of a function  belonging to various classes using different summability method has been determined by several investigators like Khan [1,2], V. N. Mishra and L. N. Mishra [3], Mishra et al. [4][5][6], and Mishra [7,8].Summability techniques were also applied on some engineering problems; for example, Chen and Jeng [9] implemented the Cesàro sum of order (, 1) and (, 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 and Hong [10] used Cesàro sum regularization technique for hyper singularity of dual integral equation.Summability of Fourier series is useful for engineering analysis, for example, [11].Recently, Mursaleen and Mohiuddine [12] discussed convergence methods for double sequences and their applications in various fields.In sequel, Alexits [13] studied the degree of approximation of the functions in   class by the Cesàro means of their Fourier series in the sup-norm.Chandra ( [14,15]), Mohapatra and Chandra ([16,17]), and Szal [18] have studied the approximation of functions in Hölder metric.Mishra et al. [6] used the technique of approximation of functions in measuring the errors in the input signals and the processed output signals.In 2008, Singh and Mahajan [19] studied error bound of periodic signals in the Hölder metric and generalized the result of Lal and Yadav [20] under much more general assumptions.Analysis of signals or time functions is of great importance, because it conveys information or attributes of some phenomenon.The engineers and scientists use properties of Fourier approximation for designing digital filters.Especially, Psarakis and Moustakides [21] presented a new  2 based method for designing the finite impulse response (FIR) digital filters and got corresponding optimum approximations having improved performance.We also discuss an example when the Fourier series of the signal has Gibbs phenomenon.
For a 2-periodic signal  ∈   :=   [0, 2],  ≥ 1, periodic integrable in the sense of Lebesgue.Then the Fourier series of () is given by (2) The conjugate series of Fourier series (1) is given by with th partial sum s (; ).

(4)
Let  2 denote the Banach space of all 2-periodic continuous functions defined on [−, ] under the sup-norm.The space   [0, 2] where  = ∞ includes the space  2 .For some positive constant , the function space   is defined by with norm ‖ ⋅ ‖  * defined by with the understanding that Δ 0 (, ) = 0.If there exist positive constants  and  such that (| − |) ≤ | − |  and  * (| − |) ≤  | − |  , 0 ≤  <  ≤ 1, then the space is Banach space [22] and the metric induced by the norm ‖ ⋅ ‖  on   is said to be Hölder metric.Clearly   is a Banach space which decreases as  increases; that is, Let ∑ ∞ =0   be a given infinite series with the sequence of th partial sums {  }.Let {  } be a nonnegative sequence of constants, real or complex, and let us write The sequence to sequence transformation defines the sequence {   } of Nörlund means of the sequence {  }, generated by the sequence of coefficients {  }.The series ∑ ∞ =0   is said to be summable   to the sum  if lim  → ∞    exists and is equal to .In the special case in which then Nörlund summability   reduces to the familiar (, ) summability.
An infinite series ∑ ∞ =0   is said to be (, 1) summable to  if The  1 transform is defined as the th partial sum of  1 summability and we denote it by Ẽ1  .If then the infinite series ∑ ∞ =0   is said to be (, 1) summable to .
The (, 1) transform of the (, 1) transform  1  defines the (, 1)(, 1) transform of the partial sums   of the series ∑ ∞ =0   ; that is, the product summability (, 1)(, 1) is obtained by superimposing (, 1) summability on (, 1) summability.Thus, if where  1  denotes the (, 1) transform of   , then the series ∑ ∞ =0   with the partial sums   is said to be summable (, 1)(, 1) to the definite number  and we can write The   transform of the  1 transform defines (  ⋅  1 ) product transform and denote it by t  ().If International Journal of Mathematics and Mathematical Sciences 3 then the infinite series ∑ ∞ =0   is said to be (  ⋅ 1 ) summable to .
We note that  1  , () 1  , and t  are also trigonometric polynomials of degree (or order) .
( The conjugate function f() is defined for almost every  by see [23, page 131].We write throughout the paper We note that the series, conjugate to a Fourier series, is not necessarily a Fourier series [23,24].Hence, a separate study of conjugate series is desirable and attracted the attention of researchers.

Known Results
In 2001, Lal and Yadav [20] established the following theorem to estimate the error between the input signal () and the signal obtained after passing through the (, 1)(, 1)transform.
Theorem 2 (see [20]).If a function  :  →  is 2-periodic function and belongs to class Lip , 0 <  ≤ 1, then degree of approximation by (, 1)(, 1) means of its Fourier series is given by Recently, Singh and Mahajan [19] generalized the above result under more general assumptions.They proved the following.

Main Theorem
In this paper, we prove a theorem on the degree of approximation of a function f(), conjugate to a 2-periodic function  belonging to f ∈   class by (  ⋅ Remark 6.The product transform (  ⋅  1 ) plays an important role in signal theory as a double digital filter and theory of Machines in Mechanical Engineering [4,5].

Lemmas
In order to prove our main Theorem 5, we require the following lemmas.
It is easy to verify the following.

Applications
The theory of approximation is a very extensive field and the study of theory of trigonometric approximation is of great mathematical interest and of great practical importance.As mentioned in [21], the   space in general and  2 and  ∞ in particular play an important role in the theory of signals and filters.From the point of view of the applications, sharper estimates of infinite matrices [25] are useful to get bounds for the lattice norms (which occur in solid state physics) of matrix valued functions and enable to investigate perturbations of matrix valued functions and compare them.

Example
In the example, we see how the sequence of averages (i.e.,  1  ()-means or (, 1) mean) and Nörlund mean   of partial sums of a Fourier series is better behaved than the sequence of partial sums   () itself. Let with ( + 2) = () for all real .Fourier series of () is given by Then th partial sum   () of Fourier series (68) and th Cesàro sum for  = 1, that is,  1  () for the series (68), are given by From Theorem 20 of Hardy's "Divergent Series, " if a Nörlund method   has increasing weights {  }, then it is stronger than (, 1).Now, take   to be the Nörlund matrix generated by   =  + 1, then Nörlund means   is given by  = 5, 10 the peaks become flatter (Figure 1).The Gibbs phenomenon is an overshoot a peculiarity of the Fourier series and other eigen function series at a simple discontinuity; that is, the convergence of Fourier series is very slow near the point of discontinuity.Thus, the product summability means of the Fourier series of () overshoot the Gibbs Phenomenon and show the smoothing effect of the method.Thus,  1  (),    (; ) is the better approximant than   () and   method is stronger than (, 1) method.

Conclusion
Several results concerning the degree of approximation of periodic signals (functions) by product summability means of Fourier series and conjugate Fourier series in generalized Hölder metric and Hölder metric have been reviewed.Using graphical representation  1  (),    (; ) is a better approximant to   (), but till now nothing has been done to show this.Some interesting application of the operator (  ⋅  1 ) used in this paper is pointed out in Remark 6.Also, the result of our theorem is more general rather than the results of any other previous proved theorems, which will enrich the literate of summability theory of infinite series.

2
International Journal of Mathematics and Mathematical Sciences with ( + 1)th partial sum   (; ) called trigonometric polynomial of degree (or order)  and given by [19]eans of conjugate series of its Fourier series.This work generalizes the results of Singh and Mahajan[19]on (  ⋅  1 ) () are increasing functions of .Let   be the Nörlund summability matrix generated by the nonnegative {  } such that ( + 1)   =  (  ) , ∀ ≥ 0.