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Favard Operator
In functional analysis, a branch of mathematics, the Favard operators are defined by: : mathcal_n(f)x) = \frac \sum_^\infty where x\in\mathbb, n\in\mathbb. They are named after Jean Favard Jean Favard (28 August 190221 January 1965) was a French mathematician who worked on analysis. Favard was born in Peyrat-la-Nonière. During World War II he was a prisoner of war in Germany. He also was a President of the French Mathematical So .... Generalizations A common generalization is: : mathcal_n(f)x) = \frac \sum_^\infty where (\gamma_n)_^\infty is a positive sequence that converges to 0. This reduces to the classical Favard operators when \gamma_n^2=1/(2n). References * This paper also discussed Szász–Mirakyan operators, which is why Favard is sometimes credited with their development (e.g. Favard–Szász operators Footnotes Approximation theory {{mathanalysis-stub ... [...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] |
Jean Favard
Jean Favard (28 August 190221 January 1965) was a French mathematician who worked on analysis. Favard was born in Peyrat-la-Nonière. During World War II he was a prisoner of war in Germany. He also was a President of the French Mathematical Society in 1946. He died in La Tronche, aged 62. See also * Favard measure (se * Bohr–Favard inequality (se * Favard inequality (se * Favard constant * Favard–Akhiezer–Krein theorem * Favard interpolation * Favard theorem * Favard problem (se * Favard operators External linksCOMITE DES AMIS DE JEAN-FAVARD*ThLycée Jean Favardis named after him. Favard is mentioned as a prisoner of war. * {{DEFAULTSORT:Favard, Jean 1902 births 1965 deaths Mathematical analysts 20th-century French mathematicians ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Of A Sequence
As the positive integer n becomes larger and larger, the value n\cdot \sin\left(\tfrac1\right) becomes arbitrarily close to 1. We say that "the limit of the sequence n\cdot \sin\left(\tfrac1\right) equals 1." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the \lim symbol (e.g., \lim_a_n).Courant (1961), p. 29. If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests. Limits can be defined in any metric or topological space, but are usually first encountered in the real numbers. History The Greek philosopher Zeno of Elea is famous for formulating paradoxes that involve limiting processes. Leucippus, Democritus, Antiphon, Eudoxus, and Archimedes developed the method of exhaustion, which uses an infinite sequence of ... [...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] |
