A numerical comparative study of generalized Bernstein-Kantorovich operators

<p style='text-indent:20px;'>In this paper, a new generalization of the Bernstein-Kantorovich type operators involving multiple shape parameters is introduced. Certain Voronovskaja and Grüss-Voronovskaya type approximation results, statistical convergence and statistical rate of convergence of proposed operators are obtained by means of a regular summability matrix. Some illustrative graphics that demonstrate the convergence behavior, accuracy and consistency of the operators are given via Maple algorithms. The proposed operators are comprehensively compared with classical Bernstein, Bernstein-Kantorovich and other new modifications of Bernstein operators such as <inline-formula><tex-math id="M1">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein, <inline-formula><tex-math id="M2">\begin{document}$ \lambda $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich, <inline-formula><tex-math id="M3">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein and <inline-formula><tex-math id="M4">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula>-Bernstein-Kantorovich operators.</p>

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

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].

On matrix methods of convergence of order α in (ℓ)-groups

We introduce a concept of convergence of order ?, with 0 < ? ? 1, with respect to a summability matrix method A for sequences (which generalizes the notion of statistical convergence of order ?), taking values in (?)-groups. Some main properties and differences with the classical A-convergence are investigated. A Cauchy-type criterion and a closedness result for the space of convergent sequences according our notion is proved.

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

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).

Multidimensional matrix characterization of equivalent double sequences

In 1936 Hamilton presented a Silverman-Toeplitz type characterization of c″0 (i.e. the space of bounded double Pringsheim null sequences). In this paper we begin with the presentation of a notion of asymptotically statistical regular. Using this definition and the concept of maximum remaining difference for double sequence, we present the following Silverman-Toeplitz type characterization of double statistical rate of convergence: let A be a nonnegative c″0−c″0 summability matrix and let [x] and [y] be member of l″ such that with [x] ∈ P0, and [y] ∈ Pδ for some δ > 0 then µ(Ax) µ(Ay). In addition other implications and variations shall also be presented.

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

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).

A Voronovskaya-type formula for SMK operators via statistical convergence

In this paper, we obtain a statistical Voronovskaya-type theorem for the Szász-Mirakjan-Kantorovich (SMK) operators by using the notion of A-statistical convergence, where A is a non-negative regular summability matrix.

Uniformly summable double sequences

In 1945 Brudno presented the following important theorem: If A and B are regular summability matrix methods such that every bounded sequence summed by A is also summed by B , then it is summed by B to the same value. In 1960 Petersen extended Brudno’s theorem by using uniformly summable methods. The goal of this paper is to extend Petersen’s theorem to double sequences by using four dimensional matrix transformations and notion of uniformly summable methods for double sequences. In addition to this extension we shall also present an accessible analogue of this theorem.

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

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.