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Baskakov Operator
In functional analysis, a branch of mathematics, the Baskakov operators are generalizations of Bernstein polynomials, Szász–Mirakyan operator In functional analysis, a discipline within mathematics, the Szász–Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mira ...s, and Lupas operators. They are defined by : mathcal_n(f)x) = \sum_^\infty where x\in ,b)\subset\mathbb (b can be \infty), n\in\mathbb, and (\phi_n)_ is a sequence of functions defined on [0,b/math> that have the following properties for all n,k\in\mathbb: #\phi_n\in\mathcal^\infty[0,b]. Alternatively, \phi_n has a Taylor series on [0,b). #\phi_n(0) = 1 #\phi_n is completely monotone, i.e. (-1)^k\phi_n^\geq 0. #There is an integer c such that \phi_n^ = -n\phi_^ whenever n>\max\ They are named after V. A. Baskakov, who studied their convergence to bounded, continuous functions. Basic result ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous function, continuous, unitary operator, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of variati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein Polynomials
In the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial that is a linear combination of Bernstein basis polynomials. The idea is named after Sergei Natanovich Bernstein. A numerically stable way to evaluate polynomials in Bernstein form is de Casteljau's algorithm. Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Weierstrass approximation theorem. With the advent of computer graphics, Bernstein polynomials, restricted to the interval , 1 became important in the form of Bézier curves. Definition The ''n''+1 Bernstein basis polynomials of degree ''n'' are defined as : b_(x) = \binom x^ \left( 1 - x \right)^, \quad \nu = 0, \ldots, n, where \tbinom is a binomial coefficient. So, for example, b_(x) = \tbinomx^2(1-x)^3 = 10x^2(1-x)^3. The first few Bernstein basis polynomials for blending 1, 2, 3 or 4 values together are: : \begin b_(x) & = 1, \\ b_(x) & = 1 - x, & b_(x) & = x \\ b_(x) & = (1 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Szász–Mirakyan Operator
In functional analysis, a discipline within mathematics, the Szász–Mirakyan operators (also spelled "Mirakjan" and "Mirakian") are generalizations of Bernstein polynomials to infinite intervals, introduced by Otto Szász in 1950 and G. M. Mirakjan in 1941. They are defined by :\left mathcal_n(f)\rightx) := e^\sum_^\infty where x\ina theorem stating that Bernstein polynomials approximate continuous functions on ,1 Generalizations A Kantorovich Leonid Vitalyevich Kantorovich ( rus, Леони́д Вита́льевич Канторо́вич, , p=lʲɪɐˈnʲit vʲɪˈtalʲjɪvʲɪtɕ kəntɐˈrovʲɪtɕ, a=Ru-Leonid_Vitaliyevich_Kantorovich.ogg; 19 January 19127 April 1986) was a Soviet ...-type generalization is sometimes discussed in the literature. These generalizations are also called the Szász–Mirakjan–Kantorovich operators. In 1976, C. P. May showed that the Baskakov operators can reduce to the Szász–Mirakyan operators. References * * (See also: Favard oper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michiel Hazewinkel
Michiel Hazewinkel (born 22 June 1943) is a Dutch mathematician, and Emeritus Professor of Mathematics at the Centre for Mathematics and Computer Science and the University of Amsterdam, particularly known for his 1978 book ''Formal groups and applications'' and as editor of the ''Encyclopedia of Mathematics''. Biography Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam. He received his BA in mathematics and physics in 1963, his MA in mathematics with a minor in philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields".Michiel Hazewinkel, Curriculum vitae at michhaz.home.xs4all.nl. Accessed September 10, 2013 After graduation Hazewinkel started hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Doklady Akademii Nauk SSSR
The ''Proceedings of the USSR Academy of Sciences'' (russian: Доклады Академии Наук СССР, ''Doklady Akademii Nauk SSSR'' (''DAN SSSR''), french: Comptes Rendus de l'Académie des Sciences de l'URSS) was a Soviet journal that was dedicated to publishing original, academic research papers in physics, mathematics, chemistry, geology, and biology. It was first published in 1933 and ended in 1992 with volume 322, issue 3. Today, it is continued by ''Doklady Akademii Nauk'' (russian: Доклады Академии Наук), which began publication in 1992. The journal is also known as the ''Proceedings of the Russian Academy of Sciences (RAS)''. ''Doklady'' has had a complicated publication and translation history. A number of translation journals exist which publish selected articles from the original by subject section; these are listed below. History The Russian Academy of Sciences dates from 1724, with a continuous series of variously named publications dat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
