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Favard Constant
In mathematics, the Favard constant, also called the Akhiezer–Krein–Favard constant, of order ''r'' is defined as :K_r = \frac \sum\limits_^ \left \frac \right. This constant is named after the French mathematician Jean Favard, and after the Soviet mathematicians Naum Akhiezer and Mark Krein. Particular values :K_0 = 1. :K_1 = \frac. Uses This constant is used in solutions of several extremal problems, for example * Favard's constant is the sharp constant in Jackson's inequality for trigonometric polynomials * the sharp constants in the Landau–Kolmogorov inequality are expressed via Favard's constants * Norms of periodic perfect spline Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 f ...s. References * Mathematical constants {{mathanalysis-stub ... [...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] |
Naum Akhiezer
Naum Ilyich Akhiezer ( uk, Нау́м Іллі́ч Ахіє́зер; russian: link=no, Нау́м Ильи́ч Ахие́зер; 6 March 1901 – 3 June 1980) was a Soviet and Ukrainian mathematician of Jewish origin, known for his works in approximation theory and the theory of differential and integral operators.NAUM IL’ICH AKHIEZER (ON THE 100TH ANNIVERSARY OF HIS BIRTH), by V. A. Marchenko, Yu. A. Mitropol’skii, A. V. Pogorelov, A. M. Samoilenko, I. V. Skrypnik, and E. Ya. Khruslov (restricted access) He is also known as the author of classical books on various subjects in |
Mark Krein
Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of functional analysis. He is known for works in operator theory (in close connection with concrete problems coming from mathematical physics), the problem of moments, classical analysis and representation theory. He was born in Kyiv, leaving home at age 17 to go to Odessa. He had a difficult academic career, not completing his first degree and constantly being troubled by anti-Semitic discrimination. His supervisor was Nikolai Chebotaryov. He was awarded the Wolf Prize in Mathematics in 1982 (jointly with Hassler Whitney), but was not allowed to attend the ceremony. David Milman, Mark Naimark, Israel Gohberg, Vadym Adamyan, Mikhail Livsic and other known mathematicians were his students. He died in Odessa. On 14 January 2008, the memo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jackson's Inequality
In approximation theory, Jackson's inequality is an inequality bounding the value of function's best approximation by algebraic or trigonometric polynomials in terms of the modulus of continuity or modulus of smoothness of the function or of its derivatives. Informally speaking, the smoother the function is, the better it can be approximated by polynomials. Statement: trigonometric polynomials For trigonometric polynomials, the following was proved by Dunham Jackson: :Theorem 1: If f: ,2\pito \C is an r times differentiable periodic function such that :: \left , f^(x) \right , \leq 1, \qquad x\in ,2\pi :then, for every positive integer n, there exists a trigonometric polynomial T_ of degree at most n-1 such that ::\left , f(x) - T_(x) \right , \leq \frac, \qquad x\in ,2\pi :where C(r) depends only on r. The Akhiezer– Krein– Favard theorem gives the sharp value of C(r) (called the Akhiezer–Krein–Favard constant): : C(r) = \frac \sum_^\infty \frac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Polynomial
In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The coefficients may be taken as real numbers, for real-valued functions. For complex coefficients, there is no difference between such a function and a finite Fourier series. Trigonometric polynomials are widely used, for example in trigonometric interpolation applied to the interpolation of periodic functions. They are used also in the discrete Fourier transform. The term ''trigonometric polynomial'' for the real-valued case can be seen as using the analogy: the functions sin(''nx'') and cos(''nx'') are similar to the monomial basis for polynomials. In the complex case the trigonometric polynomials are spanned by the positive and negative powers of ''e''''ix'', Laurent polynomials in ''z'' under the change of variables ''z'' = ''e''''ix' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Landau–Kolmogorov Inequality
In mathematics, the Landau–Kolmogorov inequality, named after Edmund Landau and Andrey Kolmogorov, is the following family of interpolation inequalities between different derivatives of a function ''f'' defined on a subset ''T'' of the real numbers: : \, f^\, _ \le C(n, k, T) ^ ^ \text 1\le k < n. On the real line For ''k'' = 1, ''n'' = 2 and ''T'' = [''c'',∞) or ''T'' = R, the inequality was first proved by Edmund Landau with the sharp constants ''C''(2, 1, [''c'',∞)) = 2 and ''C''(2, 1, R) = √2. Following contributions by Jacques Hadamard and Georgiy Shilov, Andrey Kolmogorov found the sharp constants and arbitrary ''n'', ''k'': : where ''a''''n'' are the Favard constants.On the half-line Following work by Matorin and others, the extremising functions were found by |
Perfect Spline
Perfect commonly refers to: * Perfection, completeness, excellence * Perfect (grammar), a grammatical category in some languages Perfect may also refer to: Film * Perfect (1985 film), ''Perfect'' (1985 film), a romantic drama * Perfect (2018 film), ''Perfect'' (2018 film), a science fiction thriller Literature * Perfect (Friend novel), ''Perfect'' (Friend novel), a 2004 novel by Natasha Friend * Perfect (Hopkins novel), ''Perfect'' (Hopkins novel), a young adult novel by Ellen Hopkins * Perfect (Joyce novel), ''Perfect'' (Joyce novel), a 2013 novel by Rachel Joyce * Perfect (Shepard novel), ''Perfect'' (Shepard novel), a Pretty Little Liars novel by Sara Shepard * ''Perfect'', a young adult science fiction novel by Dyan Sheldon Music * Perfect interval, in music theory * Perfect Records, a record label Artists * Perfect (musician) (born 1980), reggae singer * Perfect (Polish band) * Perfect (American band), an American alternative rock group Albums * Perfect (Intwine album ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
