Borislav Draganov

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• DSc
• Professor (Full) at Sofia University "St. Kliment Ohridski"

In this study, we begin by deriving a two-term strong converse inequality that characterizes the rate of approximation for generalized exponential sampling operators, utilizing the logarithmic moduli of smoothness. Subsequently, we combine the direct and converse estimates to establish the saturation property and identify the class of these approxi...

We establish a direct and a matching two-term converse estimate by a K-functional and moduli of smoothness for the rate of approximation by generalised Kantorovich sampling operators in weighted variable exponent Lebesgue spaces. They yield the saturation property and class of these operators. The weight is power-type with nonpositive exponents at...

We establish a strong converse inequality by a K-functional of the rate of approximation by the Kantorovich operators in variable exponent Lebesgue spaces. It shows that a recently obtained direct inequality is optimal. A Voronovskaya inequality is also proved. The approach applied heavily relies on the boundedness of the Hardy-Littlewood maximal o...

We establish a direct estimate of the rate of the weighted simultaneous approximation by generalized Kantorovich sampling operators in Lp by a modulus of smoothness. The weights are power-type with nonpositive exponents at infinity. The unweighted case is also covered. We include an example of a sampling operator of that type whose kernel is suppor...

We establish two direct estimates by K-functionals of the rate of approximation by the Kantorovich operators in variable exponent Lebesgue spaces. They extend known results in the non-variable exponent Lebesgue spaces. The approach applied heavily relies on the boundedness of the Hardy–Littlewood maximal operator.

We establish a direct and a matching two-term converse estimate by a K-functional and a modulus of smoothness for the rate of approximation by generalized Kantorovich sampling operators in variable exponent Lebesgue spaces. They yield the saturation property and class of these operators. We also prove a Voronovskaya-type estimate. A full-text acce...

We establish a direct and a matching two-term strong converse inequality by moduli of smoothness for the rate of the simultaneous approximation by generalized sampling operators and their Kantorovich modification in the Lp-norm, in particular, the uniform norm on R. They yield the saturation property and class for the simultaneous approximation by...

We present a general method for establishing quantitative Voronovskaya-type estimates of convolution operators on homogeneous Banach spaces of periodic functions of one real variable or of functions on the real line. The method is based on properties of the Fourier transform of the kernel of the operator. We illustrate the elegance and the efficien...

We construct a sampling operator with the property that the smoother a function is, the faster its approximation is. We establish a direct estimate and a weak converse estimate of its rate of approximation in the uniform norm by means of a modulus of smoothness and a $K$-functional. The case of weighted approximation is also considered. The weights...

We establish a two-term strong converse inequality for the rate of approximation of generalized sampling operators by means of the classical moduli of smoothness. It matches an already known direct estimate. We combine the direct and the converse estimates to derive the saturation property and class of this approximation operator. We demonstrate th...

"We establish a two-term strong converse estimate of the rate of approximation by the iterated Boolean sums of the Bernstein operator. The characterization is stated in terms of appropriate moduli of smoothness or K-functionals."

We establish direct estimates of the rate of weighted simultaneous approximation by the Baskakov operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals.

We establish direct estimates of the rate of weighted simultaneous approximation by the Szász–Mirakjan operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals. A read-only version of the paper is availab...

The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving. Published in Ann. Sofia Univ., Fac. Math and Inf. 109 (2019), 79–100. Available at https://www.fmi.uni-sofia.bg/sites/defaul...

We establish two-term strong converse estimates of the rate of weighted simultaneous approximation by the Szasz-Mirakjan operator for smooth functions in the supremum norm on the non-negative semi-axis. We consider Jacobi-type weights. The estimates are stated in terms of appropriate moduli of smoothness or K-functionals.

The Bernstein polynomials with integer coefficients do not generally preserve monotonicity and convexity. We establish sufficient conditions under which they do. We also observe that they are asymptotically shape preserving.

We correct a mistake in the statements of Corollaries 4.11 and 4.12. The mistake has no bearing to their proofs or other results in the paper.

It is demonstrated that multiplier methods naturally yield better constants in strong converse inequalities for the Bernstein-Durrmeyer operator. The absolute constants obtained in some of the inequalities are independent of the weight and the dimension. The estimates are stated in terms of the K-functional that is naturally associated to the opera...

We prove a weak converse estimate for the simultaneous approximation by several forms of the Bernstein polynomials with integer coefficients. It is stated in terms of moduli of smoothness. In particular, it yields a big $O$-characterization of the rate of that approximation. We also show that the approximation process generated by these Bernstein p...

We characterize the approximation of functions in the Lp-norm by the Szász-Mirakjan-Kantorovich operator. We prove a direct and a strong converse inequality of type B in terms of an appropriate K-functional.

It is demonstrated that multiplier methods naturally yield better constants in strong converse inequalities for the Bernstein-Durrmeyer operator. The absolute constants obtained in some of the inequalities are independent of the weight and the dimension. The estimates are stated in terms of the K-functional that is naturally associated to the opera...

We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the...

We prove that several forms of the Bernstein polynomials with integer coefficients possess the property of simultaneous approximation, that is, they approximate not only the function but also its derivatives. We establish direct estimates of the error of that approximation in uniform norm by means of moduli of smoothness. Moreover, we show that the...

As is known, if f∈C2[0,1]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in C^2[0,1]$$\end{document}, then, for the Bernstein operator Bn\documentclass[12pt]{minimal}...

We characterize Peetre $K$-functionals with weights of power-type asymptotics at the ends of the interval by means of the classical moduli of smoothness taken on a proper linear transforms of the function. Negative exponents at finite ends of the interval are included. We also point out applications with regard to weighted approximation by Bernstei...

Recently, it has been shown that the rate of simultaneous approximation by means of the Bernstein operator $B_n$ satisfies the direct estimate $$ \|w(B_n f - f)^{(s)}\|\le c\,\inf_{g\in C^{s+2}[0,1]}\left\{\|w(f^{(s)}-g^{(s)})\| + n^{-1}\|w(Dg)^{(s)}\|\right\}, $$ where $\|\circ\|$ is the supremum norm on the interval $[0,1]$, $w$ is a Jacobi weig...

We establish matching direct and two-term strong converse estimates of the rate of weighted simultaneous approximation by the Bernstein operator and its iterated Boolean sums for smooth functions in -norm, . We consider Jacobi weights. The characterization is stated in terms of appropriate moduli of smoothness or -functionals. Also, analogous resul...

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. A read-only version of the paper is available at the following link, provided by...

We construct moduli of smoothness that generalize the well-known classical moduli and possess similar properties. They are related to a linear differential operator L just as the classical moduli are related to the ordinary derivative. The generalized moduli are used to characterize the approximation error of the corresponding L-splines in Lp[a, b]...

The paper presents an improved Jackson inequality and the corresponding inverse inequality for the best trigonometric approximation in terms of the moduli of smoothness equivalent to zero on the trigonometric polynomials whose degree does not exceed a certain number. The deduced inequalities are analogous to Timan’s inequalities. The relations betw...

The paper presents a description of the optimal rate of approximation as well as of a broad class of functions that possess it for convolution operators acting in the so-called homogeneous Banach spaces of functions on $\R^d$. The description is the same in any such space and uses the Fourier transform. Simple criteria for establishing upper estima...

We establish a sharp characterization of the error of the Szasz-Mirakjan operator in uniform norm with power-type weights in terms of a K-functional. The weight exponents are optimal. We also state a sharp characterization of the $K$-functional by means of the classical unweighted fixed-step modulus of smoothness. Published in Proc. intern. conf....

We present upper estimates of the approximation rate of combinations $B_{r,n}$ of iterates of the Bernstein operator B_n, defined by $I-B_{r,n}=(I-B_n)^r,\ r\in\N$. The treatment is based on (weighted) simultaneous approximation by the Bernstein operator. We give a sufficient condition on the smoothness of the function that implies approximation ra...

Best trigonometric approximation in homogeneous Banach spaces of periodic functions is characterized by two moduli of smoothness, which are equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is just similar to the one given by the classical modulus of smoothness. The moduli possesses properties...

Best trigonometric approximation in L p , 1≦p≦∞, is characterized by a modulus of smoothness, which is equivalent to zero if the function is a trigonometric polynomial of a given degree. The characterization is similar to the one given by the classical modulus of smoothness. The modulus possesses properties similar to those of the classical one.

The paper is concerned with establishing direct estimates for convolution operators on homogeneous Banach spaces of periodic functions by means of an appropriately defined K-functional. The differential operator in the K-functional is defined by means of a strong limit and described explicitly in terms of its Fourier coefficients. The description i...

We present a characterization of the approximation errors of the Post–Widder and the Gamma operators in Lp(0,∞),1≤p≤∞, with a weight xγ0(1+x)γ∞−γ0 with arbitrary real γ0,γ∞. Characteristics of two types are used — weighted K-functionals of the approximated function itself and the classical fixed-step moduli of smoothness taken on simple modificatio...

The paper presents a method for establishing direct and strong converse inequalities in terms of K-functionals for convolution operators acting in homogeneous Banach spaces of multivariate functions. The method is based on the behaviour of the Fourier transform of the kernel of the convolution operator.

The purpose of the paper is to give an upper estimate of the rate of simultaneous approximation of Fejér-type operators in the L p -norm and in a generalized Hölder L p -norm. The estimates involve moduli of smoothness of second order. A sufficient condition for the optimal order of approximation is established. Published in East J. Approx. 14 (20...

The weighted approximation errors of the Post-Widder and the Gamma operators are characterized for functions in Lp(0,∞),1⩽p⩽∞, with a weight xγ, γ∈R. Direct and strong converse theorems are proved. Two types of characteristics are used—weighted K-functionals of the approximated function itself and the classical fixed step moduli of smoothness taken...

The paper presents a method of relating two $K$-functionals by means of a continuous linear transform of the function. In particular, a characterization of various weighted $K$-functionals by unweighted fixed-step moduli of smoothness is derived. This is applied in estimating the rate of convergence of several approximation processes. Available at...

Certain types of weighted Peetre K-functionals are characterized by means of the classical moduli of smoothness taken on a proper linear transforms of the function. The weights with power-type asymptotic at the ends of the interval with arbitrary real exponents are considered. Available at http://www.math.bas.bg/~serdica/2007/2007-059-124.pdf

We present a characterization of a large class of weighted K-functionals in terms of the classical fixed step moduli of smoothness and proper modifications of the underlying function. It gives new estimates of the error of various approximation processes. Published in Proc. intern. conf. "Constructive Theory of Functions, Varna 2005'', Ed. B. Boja...

The purpose of this paper is to present a characterization of certain types of generalized weighted Peetre K-functionals by means of a modulus of smoothness. This new modulus is based on the classical one taken on a certain linear transform of the function. A new modulus of smoothness which describes the best algebraic approximation is introduced.

A new approach to establishing generalized Taylor's expansions is used to prove the trigonometric analogue of Taylor's formula. We derive point-wise estimates of the error in the trigonometric interpolation and approximation by convolutional linear operators. Published in Annuaire Univ. Sofia Fac. Math. Inform. 96 (2004), 141-154. Available at htt...

The rate of convergence of best trigonometric approximation in L p and C-norm is characterized by a new modulus of smoothness. This modulus is equivalent to 0 on the trigonometric polynomials up to a given degree. Published in East J. Approx. 8 (2002), 4, 465-499. Available at https://www.fmi.uni-sofia.bg/sites/default/files/users/u847/bta1_4.pdf