scholarly journals Approximation of signals (functions) belonging to the weightedW(Lp,ξ(t))-class by linear operators

2006 ◽  
Vol 2006 ◽  
pp. 1-10 ◽  
Author(s):  
M. L. Mittal ◽  
B. E. Rhoades ◽  
Vishnu Narayan Mishra
Keyword(s):  
Degree Of Approximation ◽  
Linear Operators ◽  
Conjugate Series ◽  
General Matrix ◽  
Matrix Summability ◽  
Approximation Of Signals ◽  
Summability Matrix ◽  
Matrix Operators ◽  
Increasing Function ◽  
Approximation Of A Function

Mittal and Rhoades (1999–2001) and Mittal et al. (2006) have initiated the studies of error estimatesEn(f)through trigonometric Fourier approximations (TFA) for the situations in which the summability matrixTdoes not have monotone rows. In this paper, we determine the degree of approximation of a functionf˜, conjugate to a periodic functionfbelonging to the weightedW(Lp,ξ(t))-class(p≥1), whereξ(t)is nonnegative and increasing function oftby matrix operatorsT(without monotone rows) on a conjugate series of Fourier series associated withf. Our theorem extends a recent result of Mittal et al. (2005) and a theorem of Lal and Nigam (2001) on general matrix summability. Our theorem also generalizes the results of Mittal, Singh, and Mishra (2005) and Qureshi (1981-1982) for Nörlund(Np)-matrices.

Degree of approximation of signals (functions) in Besov space using linear operators

2016 ◽  
Vol 09 (01) ◽  
pp. 1650009 ◽  
Author(s):  
M. L. Mittal ◽  
Mradul Veer Singh
Keyword(s):  
Besov Space ◽  
Error Estimates ◽  
Degree Of Approximation ◽  
Linear Operators ◽  
Approximation Of Functions ◽  
Fourier Approximation ◽  
General Matrix ◽  
Matrix Formula ◽  
Approximation Of Signals ◽  
Summability Matrix

Mittal, Rhoades (1999–2001), Mittal et al. (2005, 2006, 2011) have initiated a study of error estimates through trigonometric Fourier approximation (tfa) for the situation in which the summability matrix [Formula: see text] does not have monotone rows. Recently Mohanty et al. (2011) have obtained a theorem on the degree of approximation of functions in Besov space [Formula: see text] by choosing [Formula: see text] to be a Nörlund ([Formula: see text])-matrix with non-increasing weights [Formula: see text]. In this paper, we continue the work of Mittal et al. and extend the result of Mohanty et al. (2011) to the general matrix [Formula: see text].

scholarly journals Approximation of signals by general matrix summability with effects of Gibbs Phenomenon

2019 ◽  
Vol 38 (6) ◽  
pp. 141-158 ◽  
Author(s):  
B. B. Jena ◽  
Lakshmi Narayan Mishra ◽  
S. K. Paikray ◽  
U. K. Misra
Keyword(s):  
Fourier Series ◽  
Gibbs Phenomenon ◽  
Degree Of Approximation ◽  
Trigonometric Polynomials ◽  
General Matrix ◽  
Matrix Summability ◽  
Partial Sum ◽  
Approximation Of Signals

In the proposed paper the degree of approximation of signals (functions) belonging to $Lip(\alpha,p_{n})$ 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, \textit{Int. J. Math. Math. Sci.,} \textbf{2014} (2014), ArticleID 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.

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.

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.

scholarly journals Product Summability Transform of Conjugate Series of Fourier Series

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Vishnu Narayan Mishra ◽  
Kejal Khatri ◽  
Lakshmi Narayan Mishra
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Conjugate Series ◽  
Increasing Function

A known theorem, Nigam (2010) dealing with the degree of approximation of conjugate of a signal belonging toLipξ(t)-class by(E,1)(C,1)product summability means of conjugate series of Fourier series has been generalized for the weightedW(Lr,ξ(t)),(r≥1),(t>0)-class, whereξ(t)is nonnegative and increasing function oft, byEn1Cn1~which is in more general form of Theorem 2 of Nigam and Sharma (2011).

scholarly journals Degree of approximation of conjugate of a function belonging toLip(ξ(t),p)class by matrix summability means of conjugate Fourier series

2001 ◽  
Vol 27 (9) ◽  
pp. 555-563 ◽  
Author(s):  
Shyam Lal ◽  
Hare Krishna Nigam
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Conjugate Series ◽  
Conjugate Fourier Series ◽  
Matrix Summability

We determine the degree of approximation of conjugate of a function belonging toLip(ξ(t),p)class by matrix summability means of a conjugate series of a Fourier series.

scholarly journals Trigonometric Approximation of Signals (Functions) Belonging toW(Lr, ξ(t))Class by Matrix(C1·Np)Operator

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Uaday Singh ◽  
M. L. Mittal ◽  
Smita Sonker
Keyword(s):  
Degree Of Approximation ◽  
Trigonometric Approximation ◽  
Monotonicity Condition ◽  
Approximation Of Functions ◽  
Periodic Signals ◽  
Summability Matrices ◽  
Approximation Of Signals ◽  
Summability Matrix ◽  
Class Of Functions ◽  
General Summability

Various investigators such as Khan (1974), Chandra (2002), and Liendler (2005) have determined the degree of approximation of 2π-periodic signals (functions) belonging to Lip(α,r)class of functions through trigonometric Fourier approximation using different summability matrices with monotone rows. Recently, Mittal et al. (2007 and 2011) have obtained the degree of approximation of signals belonging to Lip(α,r)- class by general summability matrix, which generalize some of the results of Chandra (2002) and results of Leindler (2005), respectively. In this paper, we determine the degree of approximation of functions belonging to Lip αandW(Lr,ξ(t)) classes by using Cesáro-Nörlund(C1·Np)summability without monotonicity condition on{pn}, which in turn generalizes the results of Lal (2009). We also note some errors appearing in the paper of Lal (2009) and rectify them in the light of observations of Rhoades et al. (2011).

scholarly journals The degree of approximation by Hausdorff means of a conjugate Fourier series

2016 ◽  
Vol 70 (2) ◽  
pp. 63
Author(s):  
Sergiusz Kęska
Keyword(s):  
Fourier Series ◽  
Degree Of Approximation ◽  
Lipschitz Class ◽  
Conjugate Series ◽  
Conjugate Fourier Series ◽  
Hausdorff Means ◽  
Approximation Of A Function

The purpose of this paper is to analyze the degree of approximation of a function \(\overline f\) that is a conjugate of a function \(f\) belonging to the Lipschitz class by Hausdorff means of a conjugate series of the Fourier series.

scholarly journals Trigonometric approximation of functions $f(x,y)$ of generalized Lipschitz class by double Hausdorff matrix summability method

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Abhishek Mishra ◽  
Vishnu Narayan Mishra ◽  
M. Mursaleen
Keyword(s):  
Double Fourier Series ◽  
Degree Of Approximation ◽  
Trigonometric Approximation ◽  
Lipschitz Class ◽  
Gamma Delta ◽  
Approximation Of Functions ◽  
Matrix Summability ◽  
Alpha Beta ◽  
Generalized Lipschitz Class ◽  
Hausdorff Matrix

AbstractIn this paper, we establish a new estimate for the degree of approximation of functions $f(x,y)$ f ( x , y ) belonging to the generalized Lipschitz class $Lip ((\xi _{1}, \xi _{2} );r )$ L i p ( ( ξ 1 , ξ 2 ) ; r ) , $r \geq 1$ r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from $Lip ((\alpha ,\beta );r )$ L i p ( ( α , β ) ; r ) and $Lip(\alpha ,\beta )$ L i p ( α , β ) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and $(C, \gamma , \delta )$ ( C , γ , δ ) means.

Some negative results for the interpolation monotone approximation of functions having a fractional derivative

2020 ◽  
pp. 122-127
Author(s):  
T. O. Petrova ◽  
I. P. Chulakov
Keyword(s):  
Fractional Derivative ◽  
Convex Function ◽  
Nondecreasing Function ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Monotone Approximation ◽  
Convex Approximation ◽  
Negative Results ◽  
Positive Functions ◽  
Approximation Of A Function

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

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