MATRIX TRANSFORMATIONS AND WALSH’S EQUICONVERGENCE THEOREM

In 1977, Jacob defines Gα, for any 0≤α<∞, as the set of all complex sequences x such that |xk|1/k≤α. In this paper, we apply Gu−Gv matrix transformation on the sequences of operators given in the famous Walsh's equiconvergence theorem, where we have that the difference of two sequences of operators converges to zero in a disk. We show that the Gu−Gv matrix transformation of the difference converges to zero in an arbitrarily large disk. Also, we give examples of such matrices.


Introduction
If x = (x k ) is a complex number sequence and A = [a nk ] is an infinite matrix, then Ax is the sequence whose nth term is given by (Ax) n = ∞ k=0 a nk x k . (1.1) The matrix A is called X − Y matrix if Ax is in the set Y whenever x is in X.For 0 ≤ α < ∞, let G α = {x : limsup|x k | 1/k ≤ α}.For various values of α, this sequence space has been studied extensively by many authors (see [3,8,9]).In particular, Jacob [5, page 186] proves the following result.
Theorem 1.1.An infinite matrix A is a G u − G v matrix if and only if for each number w such that 0 < w < 1/v, there exist numbers B and s such that 0 < s < 1/u and for all n and k.

Preliminaries
Let f be an analytic function in the disk D R = {z ∈ C : |z| < R} for some R > 1.If f (z) has the Taylor series expansion f (z) = ∞ k=0 a k z k , then for each positive integer n, let 2648 Matrix transformations and Walsh's equiconvergence theorem be the nth partial sum of f (z).Also, let L n (z; f ) denote the unique Lagrange interpolation polynomial of degree at most n which interpolates f (z) in the (n + 1)st roots of unity, that is, where ω = e 2πi/(n+1) .Then the well-known Walsh's equiconvergence theorem [10] states that the convergence being uniform and geometric on any closed subdisk of D R 2 .This theorem has been extended in various ways by several authors.In [7], Price used certain arithmetical means and in [6], Lou used commutators of interpolation operators to enlarge the disk D R 2 of equiconvergence.In [1], Brück applied certain summability methods to the difference L n − S n in order to enlarge the disk D R 2 .Also, in [2], the authors extended the disk of convergence by substituting the nth partial sum S n (z; f ) by polynomials where l is a fixed positive integer.
Our aim is to apply a certain class of matrices to L n and S n and enlarge the disk D R 2 of Walsh's equiconvergence to D ρ for any ρ > R 2 .
Throughout this paper, we let Γ be any circle |t| = r with 1 < r < R. For any function f analytic in D R , we have by Cauchy integral formula (2.5) Interchanging the summation and the integral, we see that (2.7) C. R. Selvaraj and S. Selvaraj 2649 Similarly, we can express S n (z; f ) as follows: Therefore, (2.9) For simplicity, we will denote L n (z; f ) by L n (z) and S n (z; f ) by S n (z).

Main result
Therefore, by Theorem 1.1, for any w such that 1 < w < 1/v, there exist numbers B and s such that 0 < s < 1/u and Consequently, the matrix A is a summability matrix which transforms null sequences into null sequences.This is because We define λ n (z) = ∞ k=0 a nk L k (z) and σ n (z) = ∞ k=0 a nk S k (z).Then, for |z| < ρ, we obtain that The interchange of the integral and the summation is justified by showing that the series k a nk and k a nk (z/t) k converge absolutely as follows.Using (3.1), we get that the series 2650 Matrix transformations and Walsh's equiconvergence theorem which converges for each n since s < 1/u < 1 and that the series which also converges for each n, since |z|s/r < |z|/ru < |z|/ρ < 1.Also, The interchange of the integral and the summation is justified as follows.Using (3.1), we see that for each n and each j, because s/r j < 1/ur j < 1/ρr j−1 < 1, and similarly (3.9) Proof.Using the expressions obtained for λ n (z) and σ n (z), we get that It can be easily proved that the two series on the right-hand side of the above inequality converge by using the ratio test.Therefore, w > 1 implies that for each |z| < ρ.

Examples
First, we give below an obvious example for such a matrix A. Choose u > ρ/r and v such that 0 < v < 1. Define the matrix A by For each w so that 0 < w < 1/v, we have where 1/t < 1/u.Hence by Theorem 1.1, A is a G u − G v matrix.Our next example is the Sonnenschein matrix A(g) = [a nk ] which is defined by [4, page 257] where g is analytic at z = 0 and a 00 = 1, and a 0k = 0 for k ≥ 1.Clearly, for each n ≥ 1, As we easily see that the first (n − 1) derivatives of [g(z)] n contains g(z) as its factor.So, if g(0) = 0, then the first (n − 1) terms of the series ∞ k=0 a nk z k vanish and the matrix A(g) = [a nk ] reduces to an upper triangular matrix.Now, for u > ρ/r and 0 < v < 1, choose Let g(z) = 1/(z − 2l) + 1/2l so that g(0) = 0. Therefore, the Sonnenschein matrix A(g) = [a nk ] is an upper triangular matrix.Since g(z) is analytic at z = 0 and on 2652 Matrix transformations and Walsh's equiconvergence theorem Therefore by Cauchy integral formula, Then for any w such that 0 < w < 1/v, we have where

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: