Ditzian-Totik Modulus Research Articles

Bèzier variant of the generalized Baskakov Kantorovich operators

In this paper, we introduce the Bezier variant of the generalized Baskakov Kantorovich operators. We establish a direct approximation theorem with the aid of the Ditzian–Totik modulus of smoothness and also study the rate of convergence for the functions having a derivative of bounded variation for these operators.

Approximation of a kind of new type Bézier operators

In this paper, a kind of new type Bezier operators is introduced. The Korovkin type approximation theorem of these operators is investigated. The rates of convergence of these operators are studied by means of modulus of continuity. Then, by using the Ditzian-Totik modulus of smoothness, a direct theorem concerned with an approximation for these operators is also obtained.

A Global Inverse Theorem for Combinations of Phillips Operators

In the present paper, we prove a global inverse theorem for approximation of bounded continuous functions on the interval [0, ∞) by linear combinations of Phillips operators. Our estimate is in terms of Ditzian–Totik modulus of smoothness and the combinations are in the general form of linear combinations, considered in the book of Ditzian–Totik. Our main theorem extends and generalizes the previous results, obtained by C. P. May, V. Gupta, R. P. Agrawal, etc.

Direct and Inverse Estimates For Combinations of Bernstein Polynomials with Endpoint Singularities

We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein polynomials by the rth Ditzian-Totik modulus of smoothness ω r φ (f, t)w where φ is an admissible step-weight function.

SOME APPROXIMATION PROPERTIES OF MODIFIED BASKAKOV STANCU OPERATORS

In the present paper, we prove a global direct theorem for the modified Baskakovstancu operators in terms of Ditzian-Totik modulus of smoothness. Here, we have modified our operators by taking weight function of Beta operators and then generalizing it as stancu type generalized operators. We will also see that taking weight function of Beta operators will give better approximation. We study a global direct theorem using simultaneous approximation for ourstancu type generalized operator in �� �� 0, ∞ . Here, first we estimate recurrence relation for moments and then develop some global direct results by making our stancu type generalized operators positive using differential and integral operators. In this paper, our effort is to give better global approximation for our stancu type generalized operator than the earlier integral modifications of Baskakov operators studied by various authors. Here, we will extend our results for the whole interval 0, ∞ . In this paper, we will also make use of the fact that second modulus of smoothness introduced by Ditzian-Totik is equivalent to modified k-functional and �� �� [0, ∞) is not contained in �� 1 [0, ∞) for obtaining results. Here, Riesz-Thorin theorem and Leibnitz theorem is used extensively for doing simultaneous approximation. We have also used Fubini’s theorem for obtaining results.

Approximation for genuine summation-integral type link operators

In the present article, we extend the studies on recently introduced sequence of the genuine summation-integral type operators [7]. Here, we establish a link between the genuine operators and the discrete operators. We also establish a quantitative asymptotic formula, a direct estimate in terms of Ditzian–Totik modulus of smoothness and finally, we present the rate of convergence for functions having derivatives of bounded variation.

Modified gamma operators in L p spaces

In this paper, we consider certain positive linear operators connected with the classical gamma operators in Lp (1 ≤ p ≤ ∞) spaces. We study approximation properties of these operators, including theorems on the degree of approximation. The results are given in terms of some Ditzian–Totik modulus of smoothness.

On Simultaneous Approximation by Iterated Boolean Sums of Bernstein Operators

The paper presents upper estimates of the error of weighted and unweighted simultaneous approximation by the Bernstein operators and their iterated Boolean sums. The estimates are stated in terms of the Ditzian–Totik modulus of smoothness or appropriate K-functionals.

Lp-Approximation (p≥1) by q-Kantorovich Operators

For a new q-Kantorovich operator we establish direct approximation theorems in the space Lp[0,1],1≤p≤∞, via Ditzian-Totik modulus of smoothness of second order.

A Korovkin Type Approximation Theorem and Its Applications

We present a Korovkin type approximation theorem for a sequence of positive linear operators defined on the space of all real valued continuous and periodic functions viaA-statistical approximation, for the rate of the third order Ditzian-Totik modulus of smoothness. Finally, we obtain an interleave between Riesz's representation theory and Lebesgue-Stieltjes integral-i, for Riesz's functional supremum formula via statistical limit.

Global smoothness preservation with second order modulus of smoothness

Abstract We establish the global smoothness preservation of a function f by the sequence of linear positive operators. Our estimate is in terms of the second order Ditzian-Totik modulus of smoothness. Application is given to the Bernstein operator.

Strong type of Steckin inequality for the linear combination of Bernstein operators

In this paper we obtain a new strong type of Steckin inequality for the linear combinations of Bernstein operators, which gives the optimal approximation rate. Moreover, a method to prove lower estimates for linear operators is introduced. As a result the lower estimate for the linear combinations of Bernstein operators is obtained by using the Ditzian–Totik modulus of smoothness.

Approximation by limit q-Bernstein operator

Abstract We establish quantitative estimates for the limit q-Bernstein operator introduced in [3], via the second order Ditzian-Totik modulus of smoothness.

Simultaneous approximation by combinations of Bernstein–Kantorovich operators

We prove some new direct and converse results on simultaneous approximation by the combinations of Bernstein–Kantorovich operators using the Ditzian–Totik modulus of smoothness.

I-approximation properties of certain class of linear positive operators

In this paper we studyI-approximation properties of certain class of linear positive operators. The two main tools used in this paper areI-convergence and Ditzian-Totik modulus of smoothness. Furthermore, we defineq-Lupaş-Durrmeyer operators and give local and global approximation results for such operators.

On discrete q-beta operators

The aim of the present paper is to introduce and study the q-analogue of discrete beta operators. First, we show some approximation properties of these operators. Then, we establish some global direct error estimates for the above operators using the second order Ditzian–Totik modulus of smoothness. Finally, we define and study the limit discrete q-beta operator.

Quantitative estimates for the Lupaş q-analogue of the Bernstein operator

Abstract We establish quantitative results for the approximation properties of the q-analogue of the Bernstein operator defined by Lupaş in 1987 and for the approximation properties of the limit Lupaş operator introduced by Ostrovska in 2006, via Ditzian-Totik modulus of smoothness. Our results are local and global approximation theorems.

Inverse Estimates for Lupas-Baskakov Operators

We establish Stechkin-Marchaud-type inequal-ities for some Feller operators by using some modified Ditzian-Totik modulus of smooth -ness. Then we derive some inverse results for this family of operators. Moreover combining the inverse results with the direct estimates we obtain a approximation equivalent theorem.

Direct and Inverse Estimates For Combinations of Bernstein Polynomials with Endpoint Singularities

We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein polynomials by the $r$th Ditzian-Totik modulus of smoothness $\omega_\phi^{r}(f,t)_w$ where $\phi$ is an admissible step-weight function.

Approximation theorems for certain positive linear operators

In this work we prove approximation theorems for certain positive linear operators via Ditzian–Totik moduli ω 2 , ϕ ( f , ⋅ ) of second order where the step-weights are functions whose squares are concave. The results obtained are applied to the q -Lupaş–Bernstein operators, the ω , q -Bernstein operators and the convergence of the iterates of the q -Bernstein polynomials.