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).

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 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.

scholarly journals Applications of Cesàro Submethod to Trigonometric Approximation of Signals (Functions) Belonging to ClassLip(α,p)inLp-Norm

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
M. L. Mittal ◽  
Mradul Veer Singh
Keyword(s):  
Trigonometric Approximation ◽  
Lipschitz Class ◽  
Approximation Of Functions ◽  
Approximation Of Signals

We prove two Theorems on approximation of functions belonging to Lipschitz classLip(α,p)inLp-norm using Cesàro submethod. Further we discuss few corollaries of our Theorems and compare them with the existing results. We also note that our results give sharper estimates than the estimates in some of the known results.

scholarly journals On trigonometric approximation of functions in the Lq norm

2018 ◽  
Vol 51 (1) ◽  
pp. 17-26 ◽  
Author(s):  
Ram N. Mohapatra ◽  
Bogdan Szal
Keyword(s):  
Fourier Series ◽  
Modulus Of Continuity ◽  
Degree Of Approximation ◽  
Trigonometric Approximation ◽  
Infinite Matrix ◽  
Approximation Of Functions ◽  
Integral Modulus

Abstract In this paper we obtain a degree of approximation of functions in Lq by operators associated with their Fourier series using integral modulus of continuity. These results generalize many known results and are proved under less stringent conditions on the infinite matrix.

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.

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 $r є (3,4)$.

scholarly journals Error bounds in the approximation of functions

1972 ◽  
Vol 6 (1) ◽  
pp. 11-18 ◽  
Author(s):  
Badri N. Sahney ◽  
V. Venu Gopal Rao
Keyword(s):  
Trigonometric Polynomial ◽  
Error Bounds ◽  
Degree Of Approximation ◽  
Approximation Of Functions ◽  
Image Position

Let f(x) ε Lipα, 0 < α < 1, in the range (-π, π), and periodic with period 2π, outside this range. Also let.We define the norm asand let the degree of approximation be given bywhere Tn (x) is some n–th trigonometric polynomial.

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