Vishnu Narayan Mishra

Statistical approximation by Kantorovich-type discrete q-Beta operators

The aim of the present paper is to introduce a Kantorovich-type modification ofthe q-discrete beta operators and to investigate their statistical andweighted statistical approximation properties. Rates of statistical convergenceby means of the modulus of continuity and the Lipschitz-type function are alsoestablished for operators. Finally, we construct a bivariate generalization ofthe operator and also obtain the statistical approximation properties.MSC: 41A25, 41A36.

A SHORT NOTE ON APPROXIMATION PROPERTIES OF STANCU GENERALIZATION OF Q-DURRMEYER OPERATORS

In the present paper, we introduce a simple Stancu generalization of q-analogue of well-known Durrmeyer operators. We first estimate moments of q-Durrmeyer-Stancu operators. We also establish the rate of convergence as well as Voronovskaja type asymptotic formula for q-Durrmeyer-Stancu operators.

Simultaneous Approximation for Generalized Srivastava-Gupta Operators

We introduce a new Stancu type generalization of Srivastava-Gupta operators to approximate integrable functions on the interval and estimate the rate of convergence for functions having derivatives of bounded variation. Also we present simultenaous approximation by new operators in the end of the paper.

Approximation by the Durrmeyer-Baskakov-Stancu operators

In the present paper we propose the Durrmeyer-Baskakov-Stancu operators. We establish some direct results in the polynomial weighted space of continuous functions defined on the interval [0,∞). Also, Voronovskaja type theorem is studied.

Graph convergence for the H(⋅,⋅)-mixed mapping with an application for solving the system of generalized variational inclusions

In this paper, we investigate a class of accretive mappings called theH(⋅,⋅)-mixed mappingsin Banach spaces. We prove that the proximal-point mapping associated with theH(⋅,⋅)-mixed mapping issingle-valued and Lipschitz continuous. Some examples are given to justify thedefinition of H(⋅,⋅)-mixed mapping.Further, a concept of graph convergence concerned with theH(⋅,⋅)-mixed mapping isintroduced in Banach spaces and some equivalence theorems betweengraph-convergence and proximal-point mapping convergence for theH(⋅,⋅)-mixed mappingssequence are proved. As an application, we consider a system of generalizedvariational inclusions involving H(⋅,⋅)-mixed mappingsin real q-uniformly smooth Banach spaces. Using the proximal-pointmapping method, we prove the existence and uniqueness of solution and suggest aniterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithmunder some suitable conditions.MSC: 47J19, 49J40, 49J53.

Product (N, pn) (C, 1) summability of a sequence of Fourier coefficients

PurposeThe purpose of the present paper is to study the product (N,u2009pn) (C,u20091) summability of a sequence of Fourier coefficients which extends a theorem of Prasad.MethodsWe use Np.u2009Cu20091 summability methods with dropping monotonicity on the generating sequence {pnu2009−u2009k} (that is, by weakening the conditions on the filter, we improve the quality of digital filter).ResultsLet Bn(x) denote the nth term of conjugate series of a Fourier series. Mohanty and Nanda were the first to establish a result for C1 summability of the sequence {nu2009Bn(x)}. Varshney improved the result for H1.u2009C1 summability which was generalized by various investigators using different summability methods with different sets of conditions. In this paper, we extend a result of Prasad by dropping the monotonicity on the sequence {pnu2009−u2009k}.ConclusionsVarious results pertaining to the C1 and H1.u2009C1 summabilities of the sequence {nu2009Bn(x)} have been reviewed and the condition of monotonicity on the means generating the sequence {pnu2009−u2009k} has been relaxed. Moreover, a proper set of conditions have been discussed to rectify the errors pointed out in Remark 3.2 (1) and (2).

The Durrmeyer type modification of the q-Baskakov type operators with two parameter α and β

In this paper, we are dealing with q analogue of Durrmeyer type modified the Baskakov operators with two parameter α and β, which introduces a new sequence of positive linear q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous function defined on the interval [0, ∞). We study moments, weighted approximation properties, the rate of convergence using a weighted modulus of smoothness, asymptotic formula and better error estimation for these operators.

Approximation properties of q-Baskakov-Durrmeyer-Stancu operators

PurposeThe purpose of this paper is to introduce and study the Baskakov- Durrmeyer- Stancu operators based on q-integers.MethodsFirst we use property of q-calculus to estimate moments of these operators. Also study some approximation properties, asymptotic formula including q-derivative and point-wise estimation for the operatorsℒn,q(α,β).ResultsWe studied better error estimations for these operators using King type approach.ConclusionsThe results proposed here are new and have a better rate of convergence.MSC 2000Primary 41A 25,41A 35.

Trigonometric approximation of functions belonging to Lipschitz class by matrix ( C 1 ⋅ N p ) operator of conjugate series of Fourier series

In the present paper, a new theorem on the degree of approximation of a function f˜, conjugate to a 2π periodic function f belonging to the Lipα (0<α≤1) class without the monotonicity condition on the generating sequence {pn} has been established, which in turn generalizes the results of Lal (Appl. Math. Comput. 209: 346-350, 2009) on a Fourier series.MSC:40G05, 41A10, 42B05, 42B08.

An iterative algorithm for a system of generalized implicit variational inclusions

In this paper, we introduce a system of generalized implicit variational inclusions which consists of three variational inclusions. We design an iterative algorithm with error terms based on relaxed resolvent operator due to Ahmad et al. (Stat Optim Inf Comput 4:183–193, 2016) for approximating the solution of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. An example is given which satisfy all the conditions of our main result.