scholarly journals On the degree of approximation of the conjugate of a function belonging to the weighted $ W(L^p, \xi(t))$ class by matrix means of the conjugate series of a Fourier series

2002 ◽  
Vol 33 (4) ◽  
pp. 365-370 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fourier Series ◽  
Series Representation ◽  
Triangular Matrix ◽  
Degree Of Approximation ◽  
Conjugate Series ◽  
Matrix Means ◽  
The Matrix

In a recent paper Lal [1] obtained a theorem on the degree of approximation of the conjugate of a function belonging to the weighted $ W(L^p, \xi(t))$ class using a triangular matrix transform of the conjugate series of the Fourier series representation of the function. The matrix involved was assumed to have monotone increasing rows. We establish the same result by removing the monotonicity conditon.

scholarly journals On the degree of approximation of functions belonging to a Lipschitz class by Hausdorff means of its Fourier series

2003 ◽  
Vol 34 (3) ◽  
pp. 245-247 ◽  
Author(s):  
B. E. Rhoades
Keyword(s):  
Fourier Series ◽  
Series Representation ◽  
Triangular Matrix ◽  
Degree Of Approximation ◽  
Lipschitz Class ◽  
The Matrix ◽  
Hausdorff Matrices ◽  
Cesàro Matrix ◽  
Hausdorff Means ◽  
Euler Matrix

In a recent paper Lal and Yadav [1] obtained a theorem on the degree of approximation for a function belonging to a Lipschitz class using a triangular matrix transform of the Fourier series representation of the function. The matrix involved was the product of $ (C, 1) $, the Cesaro matrix of order one, with $ (E, 1) $, the Euler matrix of order one. In this paper we extend this result to a much wider class of Hausdorff matrices.

scholarly journals On the degree of approximation of conjugate of a function belonging to weighted $ W(L^p,\xi(t))$ class by matrix summability means of conjugate series of a Fourier series

2000 ◽  
Vol 31 (4) ◽  
pp. 279-288 ◽  
Author(s):  
Shyam Lal
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Conjugate Series ◽  
Matrix Summability

In this paper a new theorem on the degree of approximation of conjugate of a function belonging to weighted $ W(L^p,\xi(t))$ class by Matrix summability means of conjugate series of a Fourier series has been established. The main theorem is a generalization of serveral known and unknown results.

On the Strong Summability by Triangular Means of the Derived Fourier Series and its Conjugate Series

1966 ◽  
Vol 9 (05) ◽  
pp. 647-654
Author(s):  
Narendra K. Govil
Keyword(s):  
Fourier Series ◽  
Triangular Matrix ◽  
Regular Sequence ◽  
Strong Summability ◽  
List Type ◽  
Conjugate Series

The triangular matrix (A) = (X ), where n = 0, 1, 2,…; k = 0, 1, 2, …; and λn, k = 0 for k > n is regular (in the sense of defining a regular sequence to sequence transform) if for every fixed k ; independently of n;

scholarly journals Degree of approximation of conjugate of $ {Lip(\alpha,p)}$ function by ${(C,1)}$ ${(E,1)}$ means of conjugate of a Fourier series

2002 ◽  
Vol 33 (3) ◽  
pp. 269-274 ◽  
Author(s):  
Shyam Lal ◽  
Prem Narain Singh
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Conjugate Series

An estimate of degree of approximation of conjugates of Lip$ (\alpha, p)$ functions by ($ C$,1) ($ E$,1) product means of conjugate series of a Fourier Series is obtained.

Trigonometric approximation of functions belonging to certain Lipschitz classes by C1 ⋅ T operator

2014 ◽  
Vol 07 (04) ◽  
pp. 1450064 ◽  
Author(s):  
Uaday Singh ◽  
Shailesh Kumar Srivastava
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Degree Of Approximation ◽  
Approximation Properties ◽  
Periodic Functions ◽  
Lipschitz Classes ◽  
Matrix Means ◽  
Fir Digital Filters ◽  
Approximation Of Periodic Functions

The study of approximation properties of the periodic functions in Lp (p ≥ 1)-spaces, in general and in Lipschitz classes Lip α, Lip (α, p), Lip (ξ(t), p) and weighted Lipschitz class W(Lp, ω(t), β), in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals [E. Z. Psarakis and G. V. Moustakides, An L2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Systems I Fundam. Theory Appl., 44(7) (1997) 551–601]. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x) - Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f. In this paper, we discuss the approximation properties of the periodic functions in the Lipschitz classes Lip α and W(Lp , ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix means of the Fourier series associated with the function. The degree of approximation obtained in our theorems of this paper is free from p and sharper than earlier results.

scholarly journals Approximation by fourier stieltjes series

1988 ◽  
Vol 38 (1) ◽  
pp. 87-92
Author(s):  
S.M. Mazhar
Keyword(s):  
Rate Of Convergence ◽  
Triangular Matrix ◽  
Conjugate Series ◽  
Matrix Means ◽  
Stieltjes Series

In this paper certain estimates of the rate of convergence of triangular matrix means of the Fourier Stieltjes series and its conjugate series are obtained.

scholarly journals On Degree of Approximation by Product Means of Conjugate Series of a Fourier Series

2012 ◽  
Vol 4 ◽  
pp. 5-11 ◽  
Author(s):  
S.K. Paikray ◽  
R.K. Jati ◽  
N.C. Sahoo ◽  
U.K. Misra
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Conjugate Series ◽  
Approximation Of A Function

In this paper a theorem on degree of approximation of a function f ∈ Lip(α, r) by product summability (E, q)(N, pn) of conjugate series of Fourier series associated with f has been established.

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