Approximation of Signals by General Matrix Summability with Eﬀects of Gibbs Phenomenon

: In the proposed paper the degree of approximation of signals (functions) belonging to Lip ( α,p n ) class has been obtained using general sub-matrix summability and a new theorem is established that generalizes the results of Mittal and Singh [10] (see [M. L. Mittal and Mradul Veer Singh, Approximation of signals (functions) by trigonometric polynomials in L p -norm, Int. J. Math. Math. Sci. 2014 (2014), Article ID 267383, 1-6 ]). Furthermore, as regards to the convergence of Fourier series of the signals, the eﬀect of the Gibbs Phenomenon has been presented with a comparison among diﬀerent means that are generated from sub-matrix summability mean together with the partial sum of Fourier series of the signals.


Introduction, Definitions and Motivation
The study of the Theory of Approximation, is an exceptionally broad field and the investigation of the hypothesis of trigonometric estimation is of incredible scientific interest and of incredible functional significance.As mentioned in [14], the L p -space in general, and L 2 and L 1 specifically assume an essential part of the hypothesis of signals.It is believed that the Theory of Approximation which started from a surely understood hypothesis of Weierstrass, has turned into an exciting interdisciplinary field of study for the last 130 years.These approximations have expected imperative new measurements because of their wide applications in Signal Analysis (see [13]), in general and specifically in Digital Signal Processing.Investigation of signals or time capacities is of awesome significance, since it passes on data 142 B. B. Jena, Lakshmi Narayan Mishra, S. K. Paikray and U. K. Misra or characteristics of some phenomenon.The Engineers and Scientists use properties of Fourier approximation for outlining digital filters and signals.Particularly, Psarakis and Moustakides [14] exhibited another L 2 based technique for outlining the Finite Impulse Response digital filters and get comparing optimum approximations having enhanced execution.Recently, Diger et al. [4], and Mittal and Singh [10] have obtained numerous nice results on Theory of Approximation utilizing sub-Nörlund, sub-Riesz mean of summability techniques with monotonicity on the rows of the corresponding matrix T (a digital filter) by using sub-Cesàro mean of summability method presented earlier by Armitage and Maddox (see [1]).Till now, nothing appears to have been done for obtaining the degree of approximation of signals (functions) using general sub-matrix mean of summability method.The purpose of the present study is to establish certain new theorems in this direction that will generalize some existing results.
Let f (x) ∈ L p [0, 2π] (p ≥ 1) be a signal function with period 2π, then the Fourier series of f is given by Let s n (f ) be the n th partial sum of the Fourier series (1.1), then The integral modulus of continuity of f is defined by For, p → ∞, Lip(α, p) class reduces to the Lip(α) class.
In this paper throughout, .Lp will denote L p -norm and is defined by, and L ∞ -norm of a function over R is defined by, The degree of approximation of a function over R by trigonometric polynomial (t n ) of degree n under supremum norm .L∞ is defined by Zygmund (see [16]) and given us, and error E n of a function f ∈ L p is defined by The formula of Abel's Transformation is given by, A non-negative sequence (c n ) is known as almost monotone decreasing (increasing) if there exists a constant K = K(c), depending on the sequence c only, such that, A non-negative sequence (c n ) which is either almost increasing sequence or almost decreasing sequence is called an almost monotone sequence.
Let F ⊂ N be infinite and it be the range of strictly increasing sequence of positive integers of the form F = (λ(n)) ∞ n=1 .The method of sub-Cesàro summability (C λ ) is defined by where (x k ) is a real sequence.Therefore, C λ summability method is a subsequence of the Cesàro (C 1 ) summability method and hence it is regular for any λ.Also C λ is obtained by deleting a set of rows from Cesàro matrix.The reader will be known about the most fundamental properties of C λ method see (Armitage and Maddox [1], Osikiewicz [12]).In the present paper, to determine the degree of approximation of signals f ∈ Lip(α, p) by imposing n th degree trigonometric polynomial (T λ n (f )), we first set where .
Here, throughout the paper T = (a λ(n),k ) will denote a lower triangular infinite matrix of real numbers such that, We shall also use the notations, The result in equation (1.6) is the generalization of the following known results: [10]).In this case, we write where, [10]).In this case, we write where, [10]).In this case, we write (1.9) Next, we define the product of sub-Cesàro summability C λ n (f ) with a sub-Nörlund summability N λ n (f ) denoted by A λ(n),k (f )-summability and it has the mean given by, (1.10) Here, and the matrix A λ(n),k is said to be almost row monotone for each 0 ≤ k ≤ λ(n), whenever, T = (a λ(n),k ) is either almost increasing or almost decreasing in k and 0 ≤ k ≤ λ(n).
Remark 1.1.The product transforms A λ n (f ) of this form plays an important role as a double digital filter [5,6] in signal theory as well as the theory of Machines in Mechanical Engineering (see [5]).
Many researchers like, Quade [15], Mohapatra and Russell [11], Chandra [2] and a few others used different summability means to determine the degree of approximations of trigonometric polynomials.Further Mittal and Rhoades [7,8] estimate the error of trigonometric polynomials by Fourier series expansion.In 2002, Chandra [3] has established a result of the degree of approximation of the trigonometric polynomial using (N, p m ) matrix.After that Mittal et al. [9] proved a theorem on the degree approximation of the trigonometric polynomial using lower triangular infinite matrix.Very recently, Deger et al. [4] and Mittal and Singh [10] used a more general sub-Cesàro summability mean (C λ ) (see Armitage and Maddox [1]) to establish a result of the approximation of signals by trigonometric polynomials in L p -norm.In order to have some advance study in this direction, in the proposed paper, we have established a new theorem on the degree of approximation of signals (f ∈ Lip(α, p)) under some weaker conditions by using general sub-matrix mean T λ n (f ) (that is, weakening the conditions of the filter, we enhance the quality of digital filter) that generalizes some known theorems.

Known Results
Dealing with a degree of approximations, Deger et al. [4] and Mittal and Singh [10] in the year 2012 and 2014 respectively established the following theorems.
If the positive sequence (p n ) satisfies and (p n ) is a monotonic increasing sequence, then Theorem 2.3.(see [10]) If f ∈ Lip(α, p) and (p n ) is positive and if one the following conditions ) and p n is almost decreasing sequence; holds; (v) p = 1, δ ∈ (0, 1] and and the condition

Proof of the Theorem 3.1
Initially, we wish to prove cases (i) and (ii) together, we have Jena, Lakshmi Narayan Mishra, S. K. Paikray and U. K. Misra Next, under the condition (iii), we have By Abel's transformation, we have

Now by Triangle inequality,
Also, , we have (by condition (iii).
Finally, for the condition (iv), This completes the proof of Theorem 3.1.
Proof.We have, This completes the proof of Corollary 4.1.
Remark 4.3.In Theorem 3.1, as well as in Corollary 4.1 and 4.2, as (λ(n)) −α ≤ (n) −α (0 < α ≤ 1), so our result for sub matrix summability gives better estimates (that is, minimizes the error) in comparison to the earlier existing results for general matrix summability methods.
where A λ(n),k (f ) is the mean for the product (C λ m .N λ m ).
Proof.Using the conditions we have, This establishes Corollary 4.4.

Effects of Gibbs Phenomenon and Applications
As regards to the effect of the Gibbs Phenomenon in the following example, we will see how the sub-Cesàro mean C λ n (f ), sub-Nörlund mean N λ n (f ) that are generated for sub-matrix mean T λ n (f ) as mentioned by the authors and the product A λ(n),k (f ) mean of partial sums of Fourier series of 2π -periodic signal is better behaved than the sequence of partial sums s n (x) itself. Consider be periodic with period 2π.Clearly, it is an odd function.So its Fourier series is given by Thus the Fourier series of f (x) is, The n th partial sum s n (x) of Fourier series (5.1), is given by and the average sub-mean of Fourier series (5.1), is given by (λ(n) − k + 1)s k (f ), (5.4) to Gibbs Phenomenon we conclude the convergence of signals as follows: According to Gibbs Phenomenon, in the neighborhood of discontinuity, the convergence of Fourier series is not uniform and the sequence of partial sum is over estimated the signal by 18 percent, that is, in the neighborhood of discontinuity overshoots in the peaks of partial sum s n (x) are noticed closure of the line passing through a point of discontinuity as n-increases.
From the Figure 5(a) and 5(b), we observe that C λ n (f ), N λ m (f ) and A λ(n),k (f ) converges quickly to f (x) than the sequence of partial sum s λ(n) in the interval [−π, π].We further notice that in the neighborhood of discontinuity that is, in the neighborhood of −π, 0 and π, the graph of s 7 and s 14 show overshoots in peaks and move closer the line passing through points of discontinuity as λ(n) increases, but in the graph of C λ n (f ), N λ n (f ) and A λ(n),k (f ), λ(n) = 7, 14 the peaks become flatter.Clearly, the product summability means of the Fourier series of f (x) overshoot the Gibbs Phenomenon and show the smoothing effect of the method.Thus C λ n (f ), N λ n (f ) and A λ(n),k (f ) are the better approximates than s n (x) and product A λ(n),k (f ) summability is better behaved than the individual s λ(n) , C λ n and N λ n summability methods.
) 144 B. B. Jena, Lakshmi Narayan Mishra, S. K. Paikray and U. K. Misra Further, we have established a new result on the approximation of signals of (f ∈ Lip(α, p)) class by the product of sub-Cesàro mean and sub-Nörlund mean (A λ(n),k (f )).Next as regards to 146 B. B. Jena, Lakshmi Narayan Mishra, S. K. Paikray and U. K. Misra convergence, the signals for n th partial sum of Fourier series and signals or C λ n (f ) mean, (N λ n (f )) mean and (A λ(n),k (f )) means are plotted by using Matlab and are compared in a suitable example.