Baskakov Operators Research Articles

A note on the convergence of Baskakov type operators

In the present paper, we estimate the rate of convergence on functions of polynomial type for some variant of the Baskakov operators. The form of the operator discussed in the present paper makes results helpful from the computational point of view.

Simultaneous Approximation for Generalized Baskakov-Durrmeyer-Type Operators

The present paper deals with the study of a Durrmeyer-type integral modification of certain modified Baskakov operators. Here we study simultaneous approximation properties for these operators by using the iterative combinations. We obtain an asymptotic formula and an error estimation in terms of higher order modulus of continuity for these operators.

The complete asymptotic expansion for Baskakov operators

In this paper, we derive the complete asymptotic expansion of classical Baskakov operators Vn(f;x) in the form of all coefficients of n−k,k = 0, 1 ... being calculated explicitly in terms of Stirling number of the first and second kind and another number G(i, p). As a corollary, we also get the Voronovskaja-type result for the operators.

Direct estimate on pointwise simultaneous approximation by combinations of Baskakov operators

In this paper we obtain a direct estimate on the pointwise simultanous approximation by the combinations of Baskakov operators.

Direct estimates in simultaneous approximation for Durrmeyer type operators

In the present paper, we study a Durrmeyer type integral modification of the wellknown Baskakov operators with the weight function of Beta basis function. Some approximation properties of these operators were recently studied by Finta [2]. Here we study simultaneous approximation properties for these operators. We estimate local direct result in terms of modulus of continuity. The operators considered in this paper reproduce not only the constant functions but also the linear ones, due to this property we can improve the order of approximation for these operators by applying the iterative combinations, which were first studied by Micchelli [7]. We establish an asymptotic formula and error estimation in terms of higher order modulus of continuity in simultaneous approximation for the Micchelli combinations of these operators. Mathematics subject classification (2000): 41A30, 41A36.

Rate of convergence of generalized rational type Baskakov operators

We consider the operators introduced in [V. Gupta, N. Ispir, On the Bezier variant of generalized Kantorovich type Balazs operators, Appl. Math. Lett. 18 (9) (2005) 1053–1061] and estimate their rate of convergence for absolutely continuous functions having a derivative coinciding a.e. with a function of bounded variation.

Pointwise Simultaneous Approximation by Combinations of Baskakov Operators

In this paper, we investigate the relation between the rate of convergence for the derivatives of the combinations of Baskakov operators and the smoothness for the derivatives of the functions approximated. We give some direct and inverse results on pointwise simultaneous approximation by the combinations of Baskakov operators. We also give a new equivalent result on pointwise approximation by these operators.

An identity for a general class of approximation operators

We prove an identity for basis functions of a general family of positive linear operators. It covers as special cases the Bernstein, Szász–Mirakjan and Baskakov operators. A corollary of our result can be considered a pointwise orthogonality relation. The Bernstein case is the univariate case of a remarkable identity which recently was presented by Jetter and Stöckler. As an application we give a representation of a restricted dual basis and define a class of quasi-interpolants.

Statistically-Oriented Iterative Algorithm for Improved Approximation by Modified Baskakov Operator

Statistically-Oriented Iterative Algorithm for Improved Approximation by Modified Baskakov Operator

On the degree of approximation of functions of two variables by some operators

We introduce operators of Szász-Mirakyan and Baskakov type in polynomial weighted spaces of functions of two variables. We prove a theorem on the degree of approximation and a Voronovskaya type theorem for these operators. Similar results for Bernstein, Szász-Mirakyan and Baskakov operators of functions of one variable were given in [3-6].

On converse approximation theorems

We establish sufficient conditions for obtaining a strong converse inequality of type B in terms of a unified K-functional for a sequence of linear positive operators ( L n ) n ⩾ 1 , L n : C [ 0 , ∞ ) → C [ 0 , ∞ ) . This K-functional, introduced by Guo et al. (see, e.g., [S. Guo, Q. Qi, Strong converse inequalities for Baskakov operators, J. Approx. Theory 124 (2003) 219–231]), will be considered for more general weight functions. As applications we investigate the situation for Baskakov type operators and Szász–Mirakjan type operators.

An estimate on the convergence of Baskakov–Bézier operators

In the present paper we estimate the rate of convergence on functions of bounded variation for the Bézier variant of the Baskakov operators B n , α ( f , x ) . Here we have studied the rate of convergence of B n , α ( f , x ) for the case 0 < α < 1 .

On multivariate Baskakov operator

In this paper, we study multivariate Baskakov operator B n , d ( f , x ) . We first show that the operator can retain some properties of the original function f, such as monotony, semi-additivity and Lipschitz condition, etc. Secondly, we discuss the monotony on the sequence of multivariate Baskakov operator B n , d ( f , x ) for n when the function f is convex. Then, we propose, for estimating the rate of approximation, a new modulus of smoothness and prove the modulus to be equivalent to certain K-functional. Finally, with the modulus of smoothness as metric, we establish a strong direct theorem by using a decomposition technique for the operator.

Rate of convergence of bounded variation functions by a Bézier‐Durrmeyer variant of the Baskakov operators

We consider a Bézier‐Durrmeyer integral variant of the Baskakov operators and study the rate of convergence for functions of bounded variation.

On the Rate of Convergence for Modified Baskakov Operators

We consider some linear operators in certain weighted spaces of functions of one variable and study approximation properties of these operators. The method was inspired by Kirov [4].

Approximation by Kantorovich type operators

We study approxiamtion properties of the Kantorovich type operators associated with the Szasz-Mirakyan and Baskakov operators. We prove that the approximation order of smoothness functions by considered operators is better than for the classical Szasz-Mirakyan and Baskakov operators given in [2]. Our paper is motivated by results obtained in [2], [6], [9] and [12].

Strong converse inequalities for Baskakov operators

We give a strong converse inequality of type B in terms of unified K-functional K λ α( f,t 2)(0⩽λ⩽1, 0<α<2) for Baskakov operators.

On convergence rate of approximation for two-dimensional Baskakov operators

In this paper, we study the convergence rate of two-dimensional Baskakov operators with Jacobi-weights making use of multivariate decompose skills and results of one-dimensional Baskakov operators, and obtain the direct approximationtheorem.

On uniform approximation by some classical Bernstein-type operators

We investigate the functions for which certain classical families of operators of probabilistic type over noncompact intervals provide uniform approximation on the whole interval. The discussed examples include the Szász operators, the Szász–Durrmeyer operators, the gamma operators, the Baskakov operators, and the Meyer–König and Zeller operators. We show that some results of Totik remain valid for unbounded functions, at the same time that we give simple rates of convergence in terms of the usual modulus of continuity. We also show by a counterexample that the result for Meyer–König and Zeller operators does not extend to Cheney and Sharma operators.

Lp‐inverse theorem for modified beta operators

We obtain a converse theorem for the linear combinations of modified beta operators whose weight function is the Baskakov operators. To prove our inverse theorem, we use the technique of linear approximating method, namely, Steklov mean.