Skip to main content

Gancho Tachev

Followers

5

Following

1

Public Views

Oxford Brookes University

IIT Kanpur

Universidad Nacional de Colombia

Florentin Smarandache

University of New Mexico

Angelamaria Cardone

University of Salerno

khursheed jamal Ansari

King Khalid University

Abant Izzet Baysal University, Bolu, Turkey

Abant Izzet Baysal University, Bolu, Turkey

Yildiz Technical University

Charles University, Prague

Interests

Uploads

Papers by Gancho Tachev

Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Approximation with Positive Linear Operators and Linear Combinations, 2017

Let E be a subspace of C[0, ∞) which contains the polynomials and L n : E → C[0, ∞) be a sequence... more Let E be a subspace of C[0, ∞) which contains the polynomials and L n : E → C[0, ∞) be a sequence of linear positive operators. The weighted modulus of continuity, considered by Acar–Aral–Rasa in [7] is denoted by \(\Omega (f;\delta )\) and given by $$\displaystyle{\Omega (f;\delta ) =\sup _{0\leq h<\delta,x\in [0,\infty )}\frac{\vert f(x + h) - f(x)\vert } {(1 + h^{2})(1 + x^{2})} }$$

Direct Estimates for Some New Operators

Approximation with Positive Linear Operators and Linear Combinations, 2017

Inverse Estimates and Saturation Results for Linear Combinations

Let the Ditzian–Totik moduli of smoothness ω φ r (f, t) p be given as ( 2.1.2) and the equivalent... more Let the Ditzian–Totik moduli of smoothness ω φ r (f, t) p be given as ( 2.1.2) and the equivalent K-functional K r, φ (f, t r ) p be given as ( 2.1.4). As in Section 2.1, we suppose the same definitions for the weight function φ(x) and for the domain D of the operators \(B_{n},S_{n},V _{n},\widehat{B}_{n},\widehat{S}_{n},\widehat{V }_{n}.\) To establish the inverse results for approximation by L n, r we need two Bernstein type inequalities and the Berens–Lorentz lemma , which results we formulate as follows:

Direct Estimates for Approximation by Linear Combinations

The aim of this chapter is to collect the known results for the error of approximation by linear ... more The aim of this chapter is to collect the known results for the error of approximation by linear combinations L n, r , measured in different norms L p (B) and usually in terms of Ditzian–Totik moduli of smoothness ω φ r (f, t) p , or the ordinary moduli of continuity ω r (f, t) p . We will see the importance of the information about central moments of the p.l.o. (respectively of L n, r ).

Approximation with Positive Linear Operators and Linear Combinations

Developments in Mathematics, 2017

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Moments and Combinations of Positive Linear Operators

Approximation with Positive Linear Operators and Linear Combinations, 2017

The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the importa... more The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the important areas of researchers related to approximation theory. Apart from the earlier known examples several new sequences of p.l.o. were introduced and their approximation properties have been discussed in the last few decades. There are several books in approximation theory, which deal with the linear and nonlinear operators of different kind. We mention here some of the books available in the related area, which are due to DeVore [42], DeVore–Lorentz [43], Ditzian–Totik [50] and Pǎltǎnea [153].

Quantitative Voronovskaya type theorems for a general sequence of linear positive operators

Filomat, 2019

The present paper deal with the obtaining quantitative form of the results presented Butzer & Kar... more The present paper deal with the obtaining quantitative form of the results presented Butzer & Karsli [1]. That is, we prove quantitative simultaneous results by general sequence of positive linear operators which are valid for unbounded functions with polynomial growth. We present some applications of the general results by considering particular sequences of positive linear operators.

On two modified Phillips operators

Studia Universitatis Babes-Bolyai Matematica, 2019

In this note we introduce two new modified Phillips operators G 1 n and G 2 n. We obtain direct e... more In this note we introduce two new modified Phillips operators G 1 n and G 2 n. We obtain direct estimates for approximation of bounded continuous functions, defined on [0, ∞) by G 1 n , as well as for approximation of unbounded continuous functions by G 2 n. We improve some previous results on this topic.

Better Numerical Approximation by Durrmeyer Type Operators

Results in Mathematics, 2019

The main object of this paper is to construct new Durrmeyer type operators which have better feat... more

Improved approximation on Durrmeyer-type operators

Boletín de la Sociedad Matemática Mexicana, 2018

In the present paper, we consider the Durrmeyer-type operators, some of which reproduce the linea... more In the present paper, we consider the Durrmeyer-type operators, some of which reproduce the linear functions, while some the constant functions only. We observe that the order of approximation for these operators is O(n −1); here, we consider the linear combinations of such operators and study some approximation properties for these operators. Our results essentially improve the order of approximation.

On Approximation Properties of Phillips Operators Preserving Exponential Functions

Mediterranean Journal of Mathematics, 2017

In the present paper, we study a modification of the Phillips operators, which reproduces constan... more In the present paper, we study a modification of the Phillips operators, which reproduces constant and the exponential functions. We obtain the moments using the concept of moment-generating function for the Phillips operators. Here we discuss a uniform convergence estimate for this modified forms. Also some direct estimates, which also involve the asymptotic-type result are established.

General Form of Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Results in Mathematics, 2015

In the present paper we use weighted modulus ωϕ(f ; h) introduced by Pǎltǎnea in (Bull Transilvan... more In the present paper we use weighted modulus ωϕ(f ; h) introduced by Pǎltǎnea in (Bull Transilvania Univ Brasov 4(53):67-74, 2011), and defined as ωϕ(f ; h) = sup |f (x) − f (y)| : x ≥ 0, y ≥ 0, |x − y| ≤ hϕ x + y 2 , where h ≥ 0ϕ(x) = √ x 1+x m , x ∈ [0, ∞), m ∈ N, m ≥ 2. We establish the general form of Voronovskaja's theorem in terms of the modulus ωϕ(f ; h). In Sect. 3 applications are given for the Szász-Mirakyan, Baskakov operators and the Phillips operators.

On the composition and decomposition of positive linear operators (II)

Studia Scientiarum Mathematicarum Hungarica, 2010

This note discusses the indecomposability and decomposability of certain operators occurring freq... more This note discusses the indecomposability and decomposability of certain operators occurring frequently in approximation theory: piecewise linear interpolation and Bernsteintype operators. The second topic includes the central (absolute) moments of the composition of two operators and their asymptotic behavior.

From Bernstein Polynomials to Lagrange Interpolation

For a given continuous function f (x) on [0, 1] we construct sequence of algebraic polynomials ba... more

Linear Combinations of Genuine-Szasz-Mirakyan-Durrmeyer Operators

On the second moments of variation-diminishing splines

The paper is devoted to establish new properties of Schoenberg’s variation-diminishing splines an... more The paper is devoted to establish new properties of Schoenberg’s variation-diminishing splines and it delopes results recently obtained by L. Beutel, H. H. Gonska, D. Kacso and G. Tachev [“On variation-diminishing Schoenberg operators”. in: Multivariate Approximation and Interpolation with Applications, Monogr. Academia Ciencias Zaragoza 20, 9–58 (2002; Zbl 1113.41014)]. The main result of the paper is Theorem 3,where a representation of second moments of Schoenberg’s operators is established.This way,the authors give a correction to an earlier result due to M. J. Marsden [Canad. J. Math. 38, No. 5, 1081–1093 (1986; Zbl 0595.41013)], arriving at a clean representation of the second moments. Applying the main result and additional informations,upper and lower bounds for the second moments are established.

A lower bound for the second moment of Schoenberg operator

In this paper we represent a new lower bound for the second moment for Schoenberg variation-dimin... more

Refined Estimates for the Equivalence Between Ditzian-Totik Moduli of Smoothness and K-Functionals

The aim of this note is to study the magnitude of the constants in the equivalence between the fi... more

On the Conjecture of Cao, Gonska and Kacsó

We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus ar... more We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus are possible, when we measure the pointwise approximation of continuous function by Bernstein operator. In this case we confirm the conjecture made by Cao, Gonska and Kacsó. To prove this we first establish sharp upper and lower bounds for pointwise approximation of the function g(x) = x ln(x) + (1 − x) ln(1 − x), x ∈ [0, 1] by Bernstein operator.

Generalized Inverse Theorem for Schoenberg Operator

For Schoenberg splines of degree -fixed and , we establish inverse result in terms of the second ... more For Schoenberg splines of degree -fixed and , we establish inverse result in terms of the second order classical moduli. As a consequence, we generalize the earlier inverse estimates of Berens-Lorenz and of Beutel, Gonska etc. for (k = 1, 2, 3). Mathematics Subject Classification (2000)- 41A10, 41A15, 41A17, 41A25, 4

Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Approximation with Positive Linear Operators and Linear Combinations, 2017

Let E be a subspace of C[0, ∞) which contains the polynomials and L n : E → C[0, ∞) be a sequence... more Let E be a subspace of C[0, ∞) which contains the polynomials and L n : E → C[0, ∞) be a sequence of linear positive operators. The weighted modulus of continuity, considered by Acar–Aral–Rasa in [7] is denoted by \(\Omega (f;\delta )\) and given by $$\displaystyle{\Omega (f;\delta ) =\sup _{0\leq h<\delta,x\in [0,\infty )}\frac{\vert f(x + h) - f(x)\vert } {(1 + h^{2})(1 + x^{2})} }$$

Direct Estimates for Some New Operators

Approximation with Positive Linear Operators and Linear Combinations, 2017

Inverse Estimates and Saturation Results for Linear Combinations

Let the Ditzian–Totik moduli of smoothness ω φ r (f, t) p be given as ( 2.1.2) and the equivalent... more Let the Ditzian–Totik moduli of smoothness ω φ r (f, t) p be given as ( 2.1.2) and the equivalent K-functional K r, φ (f, t r ) p be given as ( 2.1.4). As in Section 2.1, we suppose the same definitions for the weight function φ(x) and for the domain D of the operators \(B_{n},S_{n},V _{n},\widehat{B}_{n},\widehat{S}_{n},\widehat{V }_{n}.\) To establish the inverse results for approximation by L n, r we need two Bernstein type inequalities and the Berens–Lorentz lemma , which results we formulate as follows:

Direct Estimates for Approximation by Linear Combinations

The aim of this chapter is to collect the known results for the error of approximation by linear ... more The aim of this chapter is to collect the known results for the error of approximation by linear combinations L n, r , measured in different norms L p (B) and usually in terms of Ditzian–Totik moduli of smoothness ω φ r (f, t) p , or the ordinary moduli of continuity ω r (f, t) p . We will see the importance of the information about central moments of the p.l.o. (respectively of L n, r ).

Approximation with Positive Linear Operators and Linear Combinations

Developments in Mathematics, 2017

The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Moments and Combinations of Positive Linear Operators

Approximation with Positive Linear Operators and Linear Combinations, 2017

The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the importa... more The convergence of a sequence of positive linear operators (abbrev. p.l.o.) is one of the important areas of researchers related to approximation theory. Apart from the earlier known examples several new sequences of p.l.o. were introduced and their approximation properties have been discussed in the last few decades. There are several books in approximation theory, which deal with the linear and nonlinear operators of different kind. We mention here some of the books available in the related area, which are due to DeVore [42], DeVore–Lorentz [43], Ditzian–Totik [50] and Pǎltǎnea [153].

Quantitative Voronovskaya type theorems for a general sequence of linear positive operators

Filomat, 2019

The present paper deal with the obtaining quantitative form of the results presented Butzer & Kar... more The present paper deal with the obtaining quantitative form of the results presented Butzer & Karsli [1]. That is, we prove quantitative simultaneous results by general sequence of positive linear operators which are valid for unbounded functions with polynomial growth. We present some applications of the general results by considering particular sequences of positive linear operators.

On two modified Phillips operators

Studia Universitatis Babes-Bolyai Matematica, 2019

In this note we introduce two new modified Phillips operators G 1 n and G 2 n. We obtain direct e... more In this note we introduce two new modified Phillips operators G 1 n and G 2 n. We obtain direct estimates for approximation of bounded continuous functions, defined on [0, ∞) by G 1 n , as well as for approximation of unbounded continuous functions by G 2 n. We improve some previous results on this topic.

Better Numerical Approximation by Durrmeyer Type Operators

Results in Mathematics, 2019

The main object of this paper is to construct new Durrmeyer type operators which have better feat... more

Improved approximation on Durrmeyer-type operators

Boletín de la Sociedad Matemática Mexicana, 2018

In the present paper, we consider the Durrmeyer-type operators, some of which reproduce the linea... more In the present paper, we consider the Durrmeyer-type operators, some of which reproduce the linear functions, while some the constant functions only. We observe that the order of approximation for these operators is O(n −1); here, we consider the linear combinations of such operators and study some approximation properties for these operators. Our results essentially improve the order of approximation.

On Approximation Properties of Phillips Operators Preserving Exponential Functions

Mediterranean Journal of Mathematics, 2017

In the present paper, we study a modification of the Phillips operators, which reproduces constan... more In the present paper, we study a modification of the Phillips operators, which reproduces constant and the exponential functions. We obtain the moments using the concept of moment-generating function for the Phillips operators. Here we discuss a uniform convergence estimate for this modified forms. Also some direct estimates, which also involve the asymptotic-type result are established.

General Form of Voronovskaja’s Theorem in Terms of Weighted Modulus of Continuity

Results in Mathematics, 2015

In the present paper we use weighted modulus ωϕ(f ; h) introduced by Pǎltǎnea in (Bull Transilvan... more In the present paper we use weighted modulus ωϕ(f ; h) introduced by Pǎltǎnea in (Bull Transilvania Univ Brasov 4(53):67-74, 2011), and defined as ωϕ(f ; h) = sup |f (x) − f (y)| : x ≥ 0, y ≥ 0, |x − y| ≤ hϕ x + y 2 , where h ≥ 0ϕ(x) = √ x 1+x m , x ∈ [0, ∞), m ∈ N, m ≥ 2. We establish the general form of Voronovskaja's theorem in terms of the modulus ωϕ(f ; h). In Sect. 3 applications are given for the Szász-Mirakyan, Baskakov operators and the Phillips operators.

On the composition and decomposition of positive linear operators (II)

Studia Scientiarum Mathematicarum Hungarica, 2010

This note discusses the indecomposability and decomposability of certain operators occurring freq... more This note discusses the indecomposability and decomposability of certain operators occurring frequently in approximation theory: piecewise linear interpolation and Bernsteintype operators. The second topic includes the central (absolute) moments of the composition of two operators and their asymptotic behavior.

From Bernstein Polynomials to Lagrange Interpolation

For a given continuous function f (x) on [0, 1] we construct sequence of algebraic polynomials ba... more

Linear Combinations of Genuine-Szasz-Mirakyan-Durrmeyer Operators

On the second moments of variation-diminishing splines

The paper is devoted to establish new properties of Schoenberg’s variation-diminishing splines an... more The paper is devoted to establish new properties of Schoenberg’s variation-diminishing splines and it delopes results recently obtained by L. Beutel, H. H. Gonska, D. Kacso and G. Tachev [“On variation-diminishing Schoenberg operators”. in: Multivariate Approximation and Interpolation with Applications, Monogr. Academia Ciencias Zaragoza 20, 9–58 (2002; Zbl 1113.41014)]. The main result of the paper is Theorem 3,where a representation of second moments of Schoenberg’s operators is established.This way,the authors give a correction to an earlier result due to M. J. Marsden [Canad. J. Math. 38, No. 5, 1081–1093 (1986; Zbl 0595.41013)], arriving at a clean representation of the second moments. Applying the main result and additional informations,upper and lower bounds for the second moments are established.

A lower bound for the second moment of Schoenberg operator

In this paper we represent a new lower bound for the second moment for Schoenberg variation-dimin... more

Refined Estimates for the Equivalence Between Ditzian-Totik Moduli of Smoothness and K-Functionals

The aim of this note is to study the magnitude of the constants in the equivalence between the fi... more

On the Conjecture of Cao, Gonska and Kacsó

We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus ar... more We consider the question if lower estimates in terms of the second order Ditzian-Totik modulus are possible, when we measure the pointwise approximation of continuous function by Bernstein operator. In this case we confirm the conjecture made by Cao, Gonska and Kacsó. To prove this we first establish sharp upper and lower bounds for pointwise approximation of the function g(x) = x ln(x) + (1 − x) ln(1 − x), x ∈ [0, 1] by Bernstein operator.

Generalized Inverse Theorem for Schoenberg Operator

For Schoenberg splines of degree -fixed and , we establish inverse result in terms of the second ... more For Schoenberg splines of degree -fixed and , we establish inverse result in terms of the second order classical moduli. As a consequence, we generalize the earlier inverse estimates of Berens-Lorenz and of Beutel, Gonska etc. for (k = 1, 2, 3). Mathematics Subject Classification (2000)- 41A10, 41A15, 41A17, 41A25, 4