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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 |
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] |
Edmund Landau
Edmund Georg Hermann Landau (14 February 1877 – 19 February 1938) was a German mathematician who worked in the fields of number theory and complex analysis. Biography Edmund Landau was born to a Jewish family in Berlin. His father was Leopold Landau, a gynecologist and his mother was Johanna Jacoby. Landau studied mathematics at the University of Berlin, receiving his doctorate in 1899 and his habilitation (the post-doctoral qualification required to teach in German universities) in 1901. His doctoral thesis was 14 pages long. In 1895, his paper on scoring chess tournaments is the earliest use of eigenvector centrality. Landau taught at the University of Berlin from 1899 to 1909, after which he held a chair at the University of Göttingen. He married Marianne Ehrlich, the daughter of the Nobel Prize-winning biologist Paul Ehrlich, in 1905. At the 1912 International Congress of Mathematicians Landau listed four problems in number theory about primes that he said were parti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet mathematician who contributed to the mathematics of probability theory, topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and computational complexity. Biography Early life Andrey Kolmogorov was born in Tambov, about 500 kilometers south-southeast of Moscow, in 1903. His unmarried mother, Maria Y. Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do nobleman. Little is known about Andrey's father. He was supposedly named Nikolai Matveevich Kataev and had been an agronomist. Kataev had been exiled from St. Petersburg to the Yaroslavl province after his participation in the revolutionary movem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Inequality
In the field of mathematical analysis, an interpolation inequality is an inequality of the form : \, u_ \, _ \leq C \, u_ \, _^ \, u_ \, _^ \dots \, u_ \, _^, \quad n \geq 2, where for 0\leq k \leq n, u_k is an element of some particular vector space X_k equipped with norm \, \cdot\, _k and \alpha_k is some real exponent, and C is some constant independent of u_0,..,u_n. The vector spaces concerned are usually function spaces, and many interpolation inequalities assume u_0 = u_1 = \cdots = u_n and so bound the norm of an element in one space with a combination norms in other spaces, such as Ladyzhenskaya's inequality and the Gagliardo-Nirenberg interpolation inequality, both given below. Nonetheless, some important interpolation inequalities involve distinct elements u_0,..,u_n, including Hölder's Inequality and Young's inequality for convolutions which are also presented below. Applications The main applications of interpolation inequalities lie in fields of study, s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacques Hadamard
Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teacher, Amédée Hadamard, of Jewish descent, and Claire Marie Jeanne Picard, Hadamard was born in Versailles, France and attended the Lycée Charlemagne and Lycée Louis-le-Grand, where his father taught. In 1884 Hadamard entered the École Normale Supérieure, having placed first in the entrance examinations both there and at the École Polytechnique. His teachers included Tannery, Hermite, Darboux, Appell, Goursat and Picard. He obtained his doctorate in 1892 and in the same year was awarded the for his essay on the Riemann zeta function. In 1892 Hadamard married Louise-Anna Trénel, also of Jewish descent, with whom he had three sons and two daughters. The following year he took up a lectureship in the University of Bordeaux, where he ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Georgiy Shilov
Georgi Evgen'evich Shilov (russian: Гео́ргий Евге́ньевич Ши́лов; 3 February 1917 – 17 January 1975) was a Soviet mathematician and expert in the field of functional analysis, who contributed to the theory of normed rings and generalized functions. He was born in Ivanovo-Voznesensk. After graduating from Moscow State University in 1938, he served in the army during World War II. He earned a doctorate in physical-mathematical sciences in 1951, also at MSU, and briefly taught at Kyiv University until returning as a professor at MSU in 1954. There, he supervised over 40 graduate students, including Mikhail Agranovich, Valentina Borok, Gregory Eskin, and Arkadi Nemirovski. Shilov often collaborated with colleague Israel Gelfand on research that included generalized functions and partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a mul ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Isaac Jacob Schoenberg
Isaac Jacob Schoenberg (April 21, 1903 – February 21, 1990) was a Romanian-American mathematician, known for his invention of splines. Life and career Schoenberg was born in Galați. He studied at the University of Iași, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur. He presented his thesis to the University of Iași, obtaining his Ph.D. in 1926. In Göttingen, he met Edmund Landau, who arranged a visit for Schoenberg to the Hebrew University of Jerusalem in 1928. During this visit, Schoenberg began his work on total positivity and variation-diminishing linear transformations. In 1930, he returned from Jerusalem, and married Landau's daughter Charlotte in Berlin. In 1930, he was awarded a Rockefeller Fellowship, which enabled him to go to the United States, visiting the University of Chicago, Harvard, and the Institute for Advanced Study in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kallman–Rota Inequality
In mathematics, the Kallman–Rota inequality, introduced by , is a generalization of the Landau–Kolmogorov inequality to Banach spaces. It states that if ''A'' is the infinitesimal generator of a one-parameter contraction semigroup In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', ... then : \, Af\, ^2 \le 4\, f\, \, A^2f\, . References *. {{DEFAULTSORT:Kallman-Rota inequality Inequalities ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contraction (operator Theory)
In operator theory, a bounded operator ''T'': ''X'' → ''Y'' between normed vector spaces ''X'' and ''Y'' is said to be a contraction if its operator norm , , ''T'' , , ≤ 1. This notion is a special case of the concept of a contraction mapping, but every bounded operator becomes a contraction after suitable scaling. The analysis of contractions provides insight into the structure of operators, or a family of operators. The theory of contractions on Hilbert space is largely due to Béla Szőkefalvi-Nagy and Ciprian Foias. Contractions on a Hilbert space If ''T'' is a contraction acting on a Hilbert space \mathcal, the following basic objects associated with ''T'' can be defined. The defect operators of ''T'' are the operators ''DT'' = (1 − ''T*T'')½ and ''DT*'' = (1 − ''TT*'')½. The square root is the square root of a matrix, positive semidefinite one given by the spectral theorem. The defect spaces \mathcal_T a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
