Approximation of signals belonging to generalized Lipschitz class using ( N , pn , qn ) ( E , s )-summability mean of Fourier series

Abstract: Degree of approximation of functions of different classes has been studied by several researchers by different summability methods. In the proposed paper, we have established a new theorem for the approximation of a signal (function) belonging to the W(L r , (t))-class by (N, p n , q n )(E, s)-product summability means of a Fourier series. The result obtained here, generalizes several known theorems.


Introduction
The theory of summability arose from the process of summation of series and the significance of the concept of summability has been rightly demonstrated in varying contexts, e.g. in Fourier analysis, approximation theory and fixed point theory and many other fields. The theory of approximation of functions has been originated from a well-known theorem of Weierstrass, it has become an exciting

PUBLIC INTEREST STATEMENT
The theory of summability is a wide field of mathematics as regards to the study of Analysis and Functional Analysis. It has many applications, for instance, in numerical analysis (to speed of the rate of convergence), complex analysis (for analytic continuums), operator theory, the theory of orthogonal series, and approximation theory, etc.; while the classical summability theory deals with the generalization of the convergence of sequences or series of real or complex numbers. Further, in classical summability theory, the use of infinite matrices has been a significant research area in mathematical analysis as regards to the study of summability of divergent sequences and series. Recently, various summability methods have interesting applications in approximation theory. The approximation of functions by positive linear operators is a significant research area in mathematical analysis with key relevance to studies of computer-aided geometric design and solution of differential equations.
interdisciplinary field of study for last 130 years. The approximation of functions by generalized Fourier series, based on trigonometric polynomial is a closely related topic in the recent development of engineering and mathematics. The almost summability method and statistical summability method are now an active area of research in summability theory. The error approximation of periodic functions belonging to various Lipschitz classes through summability method is also an active area of research in the last decades. The engineers and scientist use the properties of approximation of functions for designing digital filters. Psarakis and Moustakides (1997) presented a new L 2 -based method for designing Finite Impulse Response digital filters for getting optimum approximation. In similar manner, L p -space, L 2 -space, and L ∞ -space play an important role for designing digital filters.
The approximation of functions belonging to various Lipschitz classes, through trigonometric Fourier approximation using different summability means has been proved by various investigators, like Nigam (2011), Lal (2000, Paikray, Jati, Misra, and Sahoo (2012) and many others. Recently, by generalized Hölders inequality and Minkowski's inequality, Mishra, Sonavane, and Mishra (2013) have proved L r approximation of signals belonging to W(L r , (t))-class by C 1 .N p -summability means of conjugate series of Fourier series. Mishra and Sonavane (2015) has proved approximation of functions belonging to the Lipschitz class by product mean (N, p n )(E, 1) of Fourier series. In an attempt to make an advance study in this direction, in this paper, we obtain a theorem on the approximation of functions belonging to W(L r , (t)) by (N, p n , q n )(E, s)-summability means of Fourier series which generalizes several known and unknown results.

Definition and notations
Let ∑ u n be an infinite series with the sequence of partial sum {s n }. Let {p n } and {q n } be sequences of positive real number such that, and let R n = p 0 q n + p 1 q n−1 + ⋯ + p n q 0 (≠ 0), p −1 = q −1 = R −1 = 0.
The sequence to sequence transformation (Mishra, Palo, Padhy, Samanta, & Misra, 2014), defines the sequence {t N n } of the (N, p n , q n ) mean of the sequence {s n } generated by the sequence of coefficients p n and q n .
Similarly, we define the extended Riesz mean, Analogous to regularity conditions of Riesz summability (Hardy, 1949), we have where, C is any positive integer independent of n.
The sequence to sequence transformation (Hardy, 1949), defines the sequence {E s n } of (E, s) mean of the sequence {s n }.
If E s n → s as n → ∞, then ∑ u n summable to s with respect to (E, s) summability and (E, s) method is regular (Hardy, 1949).
Further as (N, p n ) and (E, s) means are regular, so (N, p n , q n )(E, s) mean is also regular.
Remark 1 If we put q n = 1 in Equation (2.1) then (N, p n , q n )-summability method reduces to (N, p n )summability and for p n = 1 it reduces to (N, q n )-summability.
Let f is a 2 periodic function belonging to L r [0, 2 ], r ≥ 1, with the partial sum s n (f ), then Here, as regards to Lipschitz classes we may recall that, a signal (function) f ∈ Lip( ), if Further as regards to the norm in L ∞ and L r -spaces, we may recall that L ∞ -norm of a function f :R → R is defined by and L r -norm of a function f :R → R is defined by Next, the degree of approximation of a function f :R → R by a trigonometric polynomial t n of order n under ‖ ⋅ ‖ ∞ is defined by and the degree of approximation of E n (f ) of a function f ∈ L r is given by We use the following notations throughout this paper: Remark 2 If we take = 0, then W(L r , (t))-class coincides with the class Lip( (t), r); if (t) = t then the class Lip( (t), r) coincides with Lip( , r)-class and if r → ∞ then Lip( , r)-class reduces to the Lip( ).

Known theorems
Dealing with the product (C, 1)(E, q) mean, in Nigam (2011) proved the following theorem.
Theorem 1 If f is a 2 periodic function, Lebesgue integrable on [0, 2 ] and belongs to W(L r , (t)) class, then its degree of approximation is given by provided (t) satisfies the following conditions: and where is any arbitrary number such that s(1 − ) − 1 > 0, 1 r + 1 s = 1, conditions (3.3) and (3.4) hold uniformly in x and C 1 n E q n is (C, 1)(E, q) means of the Fourier series (2.4).
Next, dealing with degree of approximation, in Mishra et. al. (2014) proved the following theorem.
Theorem 2 For a positive increasing function (t) and an integer l > 1, if f is a 2 -periodic function on the class Lip( (t), l), then the degree of approximation by product (E, s)(N, p n , q n )-summability mean of Fourier series (2.4) is given by . (3.1) be a decreasing sequence, where n is (E, s)(N, p n , q n )-summability mean.

Main theorem
Here, just by replacing Nörlund summability by extended Riesz summability and taking the reverse composition, we have proved the following theorem.
Theorem 3 Let f be a 2 periodic function which is integrable in Lebesgue sense in [0, 2 ]. If f ∈ W(L r , (t)) class, then its degree of approximation is given by where T NE n is the (N, p n , q n )(E, s) transform of s n , provided (t) satisfies the following conditions; and To prove the theorem, we need the following lemmas.
Proof For 0 ≤ t ≤ 1 n+1 , as sin nt ≤ n sin t; so we have Proof For 1 n+1 < t ≤ , as sin t 2 ≥ t (Jordan's lemma) and sin nt ≤ 1; so be a decreasing sequence,

Proof of main theorem
Using Riemann-Lebesgue theorem, so, by using Minkowski's inequality, Further f ∈ W(L r , (t)) implies ∈ W(L r , (t)), thus Now by Hölder's inequality and Lemma 1, we have sin(n + 1 2 t) sin t Also, by 2nd mean value theorem, we have Now by Hölder inequality and Lemma 2, we have Again by using 2nd mean value theorem