DSpace Collection: Volume 33, Number 4
http://hdl.handle.net/10525/2551
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Approximation of Univariate Set-Valued Functions - an Overview
http://hdl.handle.net/10525/2573
Title: Approximation of Univariate Set-Valued Functions - an Overview<br/><br/>Authors: Dyn, Nira; Farkhi, Elza; Mokhov, Alona<br/><br/>Abstract: The paper is an updated survey of our work on the approximation of univariate set-valued functions by samples-based linear approximation operators, beyond the results reported in our previous overview. Ourapproach is to adapt operators for real-valued functions to set-valued functions, by replacing operations between numbers by operations between sets.For set-valued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metricaverages and metric linear combinations of sets are used. We obtain generalapproximation results and apply them to Bernstein polynomial operators,Schoenberg spline operators and polynomial interpolation operators.<br/><br/>Description: 2000 Mathematics Subject Classification: 26E25, 41A35, 41A36, 47H04, 54C65.Equivalence Between K-functionals Based on Continuous Linear Transforms
http://hdl.handle.net/10525/2572
Title: Equivalence Between K-functionals Based on Continuous Linear Transforms<br/><br/>Authors: Draganov, Borislav; Ivanov, Kamen<br/><br/>Abstract: The paper presents a method of relating two K-functionals bymeans of a continuous linear transform of the function. In particular, acharacterization of various weighted K-functionals by unweighted fixed-stepmoduli of smoothness is derived. This is applied in estimating the rate ofconvergence of several approximation processes.<br/><br/>Description: 2000 Mathematics Subject Classification: 46B70, 41A10, 41A25, 41A27, 41A35, 41A36,42A10.Some Coefficient Estimates for Polynomials on the Unit Interval
http://hdl.handle.net/10525/2571
Title: Some Coefficient Estimates for Polynomials on the Unit Interval<br/><br/>Authors: Qazi, M. A.; Rahman, Q. I.<br/><br/>Abstract: In this paper we present some inequalities about the moduli of the coefficients of polynomials of the form f (x) : = еn = 0nan xn, where a0, ј, an О C. They can be seen as generalizations, refinements or analogues of the famous inequality of P. L. Chebyshev, according to which |an| Ј 2n-1 if | еn = 0n an xn | Ј 1 for -1 Ј x Ј 1.<br/><br/>Description: 2000 Mathematics Subject Classification: 26C05, 26C10, 30A12, 30D15, 42A05, 42C05.A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations
http://hdl.handle.net/10525/2570
Title: A general Approach to Methods with a Sparse Jacobian for Solving Nonlinear Systems of Equations<br/><br/>Authors: Kyurkchiev, Nikolay; Iliev, Anton<br/><br/>Abstract: Here we give methodological survey of contemporary methodsfor solving nonlinear systems of equations in Rn. The reason of this reviewis that many authors in present days rediscovered such classical methods.In particular, we consider Newton’s-type algorithms with sparse Jacobian.Method for which the inverse matrix of the Jacobian is replaced by theinverse matrix of the Vandermondian is proposed. A number of illustrativenumerical examples are displayed. We demonstrate Herzberger’s model withfixed-point relations to the some discrete versions of Halley’s and Euler-Chebyshev’s methods for solving such kind of systems.<br/><br/>Description: 2000 Mathematics Subject Classification: 65H10.Sobolev Type Decomposition of Paley-Wiener-Schwartz Space with Application to Sampling Theory
http://hdl.handle.net/10525/2569
Title: Sobolev Type Decomposition of Paley-Wiener-Schwartz Space with Application to Sampling Theory<br/><br/>Authors: Dryanov, Dimiter<br/><br/>Abstract: We characterize Paley-Wiener-Schwartz space of entire functions as a union of three-parametric linear normed subspaces determined bythe type of the entire functions, their polynomial asymptotic on the real line,and the index p ≥ 1 of a Sobolev type Lp-summability on the real line withan appropriate weight function. An entire function belonging to a sub-spaceof the decomposition is exactly recovered by a sampling series, locally uniformly convergent on the complex plane. The sampling formulas obtainedextend the Shannon sampling theorem, certain representation formulas dueto Bernstein, and a transcendental interpolating theory due to Levin.<br/><br/>Description: 2000 Mathematics Subject Classification: 94A12, 94A20, 30D20, 41A05.Smale's Conjecture on Mean Values of Polynomials and Electrostatics
http://hdl.handle.net/10525/2568
Title: Smale's Conjecture on Mean Values of Polynomials and Electrostatics<br/><br/>Authors: Dimitrov, Dimitar<br/><br/>Abstract: A challenging conjecture of Stephen Smale on geometry ofpolynomials is under discussion. We consider an interpretation which turnsout to be an interesting problem on equilibrium of an electrostatic field thatobeys the law of the logarithmic potential. This interplay allows us to studythe quantities that appear in Smale’s conjecture for polynomials whose zerosbelong to certain specific regions. A conjecture concerning the electrostaticequilibrium related to polynomials with zeros in a ring domain is formulatedand discussed.<br/><br/>Description: 2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.Complex Analogues of the Rolle's Theorem
http://hdl.handle.net/10525/2567
Title: Complex Analogues of the Rolle's Theorem<br/><br/>Authors: Sendov, Blagovest<br/><br/>Abstract: Classical Rolle’s theorem and its analogues for complex algebraicpolynomials are discussed. A complex Rolle’s theorem is conjectured.<br/><br/>Description: 2000 Mathematics Subject Classification: 30C10.