scholarly journals Approximation of Signals (Functions) by Trigonometric Polynomials inLp-Norm

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
M. L. Mittal ◽  
Mradul Veer Singh
Keyword(s):  
Error Estimates ◽  
Trigonometric Polynomials ◽  
Fourier Approximation ◽  
Approximation Of Signals ◽  
Summability Matrix ◽  
Cesàro Matrix

Mittal and Rhoades (1999, 2000) and Mittal et al. (2011) have initiated a study of error estimatesEn(f)through trigonometric-Fourier approximation (tfa) for the situations in which the summability matrixTdoes not have monotone rows. In this paper, the first author continues the work in the direction forTto be aNp-matrix. We extend two theorems on summability matrixNpof Deger et al. (2012) where they have extended two theorems of Chandra (2002) usingCλ-method obtained by deleting a set of rows from Cesàro matrixC1. Our theorems also generalize two theorems of Leindler (2005) toNp-matrix which in turn generalize the result of Chandra (2002) and Quade (1937).

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 Deterministic and random approximation by the combination of algebraic polynomials and trigonometric polynomials

2021 ◽  
Vol 19 (1) ◽  
pp. 1047-1055
Author(s):  
Zhihua Zhang
Keyword(s):  
Fourier Series ◽  
Trigonometric Polynomial ◽  
Algebraic Polynomial ◽  
Fourier Coefficients ◽  
Random Processes ◽  
Qualitative Theory ◽  
Trigonometric Polynomials ◽  
Algebraic Polynomials ◽  
Fourier Approximation ◽  
Random Differential Equations

Abstract Fourier approximation plays a key role in qualitative theory of deterministic and random differential equations. In this paper, we will develop a new approximation tool. For an m m -order differentiable function f f on [ 0 , 1 0,1 ], we will construct an m m -degree algebraic polynomial P m {P}_{m} depending on values of f f and its derivatives at ends of [ 0 , 1 0,1 ] such that the Fourier coefficients of R m = f − P m {R}_{m}=f-{P}_{m} decay fast. Since the partial sum of Fourier series R m {R}_{m} is a trigonometric polynomial, we can reconstruct the function f f well by the combination of a polynomial and a trigonometric polynomial. Moreover, we will extend these results to the case of random processes.

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.

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 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.

Systematic effects in observed parallaxes

1978 ◽  
Vol 48 ◽  
pp. 31-35
Author(s):  
R. B. Hanson
Keyword(s):  
Error Estimates ◽  
Zero Point ◽  
Absolute Zero ◽  
Apparent Magnitude ◽  
Coherent System ◽  
Preliminary Results ◽  
General Catalogue ◽  
Systematic Effects ◽  
New Evidence ◽  
Right Ascension

Several outstanding problems affecting the existing parallaxes should be resolved to form a coherent system for the new General Catalogue proposed by van Altena, as well as to improve luminosity calibrations and other parallax applications. Lutz has reviewed several of these problems, such as: (A) systematic differences between observatories, (B) external error estimates, (C) the absolute zero point, and (D) systematic observational effects (in right ascension, declination, apparent magnitude, etc.). Here we explore the use of cluster and spectroscopic parallaxes, and the distributions of observed parallaxes, to bring new evidence to bear on these classic problems. Several preliminary results have been obtained.

Approximation of Unbounded Functions by Trigonometric Polynomials

2017 ◽  
Vol 13 (4) ◽  
pp. 106-116
Author(s):  
Alaa A. Auad ◽  
◽  
Mousa M. Khrajan
Keyword(s):  
Trigonometric Polynomials
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