This monograph examines in detail two aspects in the field of interpolation of functions -the Global Smoothness Preservation Property (GSPP) and the Shape Preservation Property (SPP). By considering well-known classical interpolation operators such as Lagrange, Grunwald, Hermite-Fejer and Shepard type, the study is mainly developed for the univariate and bivariate cases. One of the first books on the subject, it presents to the reader, recent work featuring many new interesting results in this field, including an excellent survey of past research. Accompanied by numerous open problems, an...
This monograph examines in detail two aspects in the field of interpolation of functions -the Global Smoothness Preservation Property (GSPP) and th...
An introduction to a method of research which can be used in various fields of mathematics. It examines, in a systematic way, the quantitative characterizations of the "deviation from a (given) property", called the "defect of a property", in: set theory; topology; measure theory; real, complex and functional analysis; algebra; geometry; number theory; and fuzzy mathematics. Besides well-known "defects", the text introduces and studies new ones, such as: measures of noncompactness for fuzzy sets; fuzzy and intuitionistic entropies; the defect of (sub, super)additivity; complementarity;...
An introduction to a method of research which can be used in various fields of mathematics. It examines, in a systematic way, the quantitative charact...
In many problems arising in engineering and science one requires approxi- tion methods to reproduce physical reality as well as possible. Very schema- cally, if the input data represents a complicated discrete/continuous quantity of information, of shape S (S could mean, for example, that we have a monotone/convex collection of data), then one desires to represent it by the less-complicated output information, that approximates well the input data and, in addition, has the same shape S. This kind of approximation is called shape-preserving approximation and arises in computer-aided geometric...
In many problems arising in engineering and science one requires approxi- tion methods to reproduce physical reality as well as possible. Very schema-...
The monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several complex variables, that is, to extend their quantitative convergence properties to larger sets in the complex plane rather than the real intervals. The operators studied are of the following types: Bernstein, Bernstein—Faber, Bernstein-Butzer, q-Bernstein, Bernstein-Stancu, Bernstein-Kantorovich, Favard-Szász-Mirakjan, Baskakov and Balázs-Szabados.The second main objective is to provide a study of the approximation and...
The monograph, as its first main goal, aims to study the overconvergence phenomenon of important classes of Bernstein-type operators of one or several...
This monograph deals with the quantitative overconvergence phenomenon in complex approximation by various operators. The book is divided into three chapters. First, the results for the Schurer-Faber operator, Beta operators of first kind, Bernstein-Durrmeyer-type operators and Lorentz operator are presented. The main focus is on results for several q-Bernstein kind of operators with q > 1, when the geometric order of approximation 1/qn is obtained not only in complex compact disks but also in quaternion compact disks and in other compact subsets of the complex plane. The focus...
This monograph deals with the quantitative overconvergence phenomenon in complex approximation by various operators. The book is divided into three...
We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of their convexity properties. To our knowledge it is the first time the entire calculus of moduli of smoothness has been included in a book. We then present numerous applications of Approximation Theory, giving exact val- ues of errors in explicit forms. The K-functional method is systematically avoided since it produces nonexplicit constants. All other related books so far have allocated very little space to the computational aspect of moduli of...
We study in Part I of this monograph the computational aspect of almost all moduli of continuity over wide classes of functions exploiting some of the...
This book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Laplace, wave, telegraph, Burgers, Black-Merton-Scholes, Schrödinger and Korteweg-de Vries equations.The complexification of the spatial variable is done by two different methods. The first method is that of complexifying the spatial variable in the corresponding semigroups of operators. In this case, the solutions are studied within the context of the theory of semigroups of linear operators. It is also interesting to observe that these...
This book investigates several classes of partial differential equations of real time variable and complex spatial variables, including the heat, Lapl...
This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operators. The exposition highlights the max-product operators as those which allow one to obtain, in many cases, more valuable estimates than those obtained by classical approaches. The text considers a wide variety of operators which are studied for a number of interesting problems such as quantitative estimates, convergence, saturation results, localization, to name several.Additionally, the book discusses the perfect analogies between the...
This monograph presents a broad treatment of developments in an area of constructive approximation involving the so-called "max-product" type operator...
