DSpace Collection: Volume 28 Number 4
http://hdl.handle.net/10525/465
Serdica Mathematical Journal Volume 28, Number 4, 2002The Collection's search engineSearch the Channelsearch
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Approximation Classes for Adaptive Methods
http://hdl.handle.net/10525/513
Title: Approximation Classes for Adaptive Methods<br/><br/>Authors: Binev, Peter; Dahmen, Wolfgang; DeVore, Ronald; Petrushev, Pencho<br/><br/>Abstract: Adaptive Finite Element Methods (AFEM) are numerical procedures that approximate the solution to a partial differential equation (PDE)by piecewise polynomials on adaptively generated triangulations. Only recently has any analysis of the convergence of these methods [10, 13] or theirrates of convergence [2] become available. In the latter paper it is shownthat a certain AFEM for solving Laplace’s equation on a polygonal domainΩ ⊂ R^2 based on newest vertex bisection has an optimal rate of convergencein the following sense. If, for some s > 0 and for each n = 1, 2, . . ., the solution u can be approximated in the energy norm to order O(n^(−s )) by piecewiselinear functions on a partition P obtained from n newest vertex bisections,then the adaptively generated solution will also use O(n) subdivisions (andfloating point computations) and have the same rate of convergence. Thequestion arises whether the class of functions A^s with this approximationrate can be described by classical measures of smoothness. The purpose ofthe present paper is to describe such approximation classes A^s by Besovsmoothness.<br/><br/>Description: * This work has been supported by the Office of Naval Research Contract Nr. N0014-91-J1343,the Army Research Office Contract Nr. DAAD 19-02-1-0028, the National Science Foundationgrants DMS-0221642 and DMS-0200665, the Deutsche Forschungsgemeinschaft grant SFB 401,the IHP Network “Breaking Complexity” funded by the European Commission and the Alexan-der von Humboldt Foundation.On Representations of Algebraic Polynomials by Superpositions of Plane Waves
http://hdl.handle.net/10525/512
Title: On Representations of Algebraic Polynomials by Superpositions of Plane Waves<br/><br/>Authors: Oskolkov, K.<br/><br/>Abstract: Let P be a bi-variate algebraic polynomial of degree n with thereal senior part, and Y = {yj }1,n an n-element collection of pairwise noncolinear unit vectors on the real plane. It is proved that there exists a rigidrotation Y^φ of Y by an angle φ = φ(P, Y ) ∈ [0, π/n] such that P equals thesum of n plane wave polynomials, that propagate in the directions ∈ Y^φ .<br/><br/>Description: * The author was supported by NSF Grant No. DMS 9706883.Nearly Coconvex Approximation
http://hdl.handle.net/10525/511
Title: Nearly Coconvex Approximation<br/><br/>Authors: Leviatan, D.; Shevchuk, I.<br/><br/>Abstract: Let f ∈ C[−1, 1] change its convexity finitely many times, in theinterval. We are interested in estimating the degree of approximation of f bypolynomials, and by piecewise polynomials, which are nearly coconvex withit, namely, polynomials and piecewise polynomials that preserve the convexityof f except perhaps in some small neighborhoods of the points where fchanges its convexity. We obtain Jackson type estimates and summarize thepositive and negative results in a truth-table as we have previously done fornearly comonotone approximation.<br/><br/>Description: * Part of this work was done while the second author was on a visit at Tel Aviv University in March 2001Spline Subdivision Schemes for Compact Sets. A Survey
http://hdl.handle.net/10525/510
Title: Spline Subdivision Schemes for Compact Sets. A Survey<br/><br/>Authors: Dyn, Nira; Farkhi, Elza<br/><br/>Abstract: Attempts at extending spline subdivision schemes to operateon compact sets are reviewed. The aim is to develop a procedure for approximating a set-valued function with compact images from a finite set ofits samples. This is motivated by the problem of reconstructing a 3D objectfrom a finite set of its parallel cross sections. The first attempt is limited tothe case of convex sets, where the Minkowski sum of sets is successfully applied to replace addition of scalars. Since for nonconvex sets the Minkowskisum is too big and there is no approximation result as in the case of convexsets, a binary operation, called metric average, is used instead. With themetric average, spline subdivision schemes constitute approximating operators for set-valued functions which are Lipschitz continuous in the Hausdorffmetric. Yet this result is not completely satisfactory, since 3D objects arenot continuous in the Hausdorff metric near points of change of topology,and a special treatment near such points has yet to be designed.<br/><br/>Description: Dedicated to the memory of our colleague Vasil Popov January 14, 1942 – May 31, 1990* Partially supported by ISF-Center of Excellence, and by The Hermann Minkowski Center for Geometry at Tel Aviv University, IsraelPenalized Least Squares Fitting
http://hdl.handle.net/10525/509
Title: Penalized Least Squares Fitting<br/><br/>Authors: von Golitschek, Manfred; Schumaker, Larry<br/><br/>Abstract: Bounds on the error of certain penalized least squares datafitting methods are derived. In addition to general results in a fairly abstractsetting, more detailed results are included for several particularly interestingspecial cases, including splines in both one and several variables.<br/><br/>Description: * Supported by the Army Research Office under grant DAAD-19-02-10059.Greedy Approximation with Regard to Bases and General Minimal Systems
http://hdl.handle.net/10525/508
Title: Greedy Approximation with Regard to Bases and General Minimal Systems<br/><br/>Authors: Konyagin, S.; Temlyakov, V.<br/><br/>Abstract: This paper is a survey which also contains some new results onthe nonlinear approximation with regard to a basis or, more generally, withregard to a minimal system. Approximation takes place in a Banach or ina quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important in applications because of its increasedefficiency. Two types of nonlinear approximation are employed frequentlyin applications. Adaptive methods are used in PDE solvers. The m-termapproximation considered here is used in image and signal processing as wellas the design of neural networks. The basic idea behind nonlinear approximation is that the elements used in the approximation do not come froma fixed linear space but are allowed to depend on the function being approximated. The fundamental question of nonlinear approximation is howto construct good methods (algorithms) of nonlinear approximation. In thispaper we discuss greedy type and thresholding type algorithms.<br/><br/>Description: *This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003Generalization of a Conjecture in the Geometry of Polynomials
http://hdl.handle.net/10525/507
Title: Generalization of a Conjecture in the Geometry of Polynomials<br/><br/>Authors: Sendov, Bl.<br/><br/>Abstract: In this paper we survey work on and around the followingconjecture, which was first stated about 45 years ago: If all the zeros of analgebraic polynomial p (of degree n ≥ 2) lie in a disk with radius r, then,for each zero z1 of p, the disk with center z1 and radius r contains at leastone zero of the derivative p′ . Until now, this conjecture has been proved forn ≤ 8 only. We also put the conjecture in a more general framework involvinghigher order derivatives and sets defined by the zeros of the polynomials.Bibliography of the works of Prof. V. Popov
http://hdl.handle.net/10525/506
Title: Bibliography of the works of Prof. V. Popov<br/><br/>Authors: Popov, Vasil<br/><br/>Abstract: Bibliography of the research publications of Vasil Atanasov PopovCurriculum vita of Prof. Vasil Atanasov Popov
http://hdl.handle.net/10525/505
Title: Curriculum vita of Prof. Vasil Atanasov Popov<br/><br/>Authors: Ivanov, Kamen; Petrushev, Pencho<br/><br/>Abstract: Our primary goal in this preamble is to highlight the best of Vasil Popov’smathematical achievements and ideas. V. Popov showed his extraordinary talentfor mathematics in his early papers in the (typically Bulgarian) area of approximationin the Hausdorff metric. His results in this area are very well presentedin the monograph of his advisor Bl. Sendov, “Hausdorff Approximation”.