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Riesz–Thorin Theorem
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about ''interpolation of operators''. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to which is a Hilbert space, or to and . Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps. Motivation First we need the following definition: :Definition. Let be two numbers such that . Then for define by: . By splitting up the function in as the product and applying Hölder's inequality to its ... [...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] |
Olof Thorin
Olov (or Olof) is a Swedish form of Olav/Olaf, meaning "ancestor's descendant". A common short form of the name is ''Olle''. The name may refer to: *Per-Olov Ahrén (1926–2004), Swedish clergyman, bishop of Lund from 1980 to 1992 *Per-Olov Brasar (born 1950), retired professional ice hockey forward * Olov Englund (born 1983), Swedish bandy player *Per Olov Enquist (1934–2020), one of Sweden's internationally best known authors * Olle Hagnell (1924–2011), Swedish psychiatrist *Karl Olov Hedberg (1923–2007), botanist, taxonomist, author, professor at Uppsala University *Olle Hellbom (1925–1982), Swedish film director *Per Olov Jansson (1920–2019), Finnish photographer *Olof Johansson (born 1937), Swedish politician *Per-Olov Kindgren (born 1956), Swedish musician, composer, guitarist and music teacher *Olov Lambatunga, Archbishop of Uppsala, Sweden, 1198–1206 * Sven-Olov Lawesson (1926–1988), Swedish chemist known for his popularization of Lawesson's reagent within t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Locally Compact Abelian Group
In several mathematical areas, including harmonic analysis, topology, and number theory, locally compact abelian groups are abelian groups which have a particularly convenient topology on them. For example, the group of integers (equipped with the discrete topology), or the real numbers or the circle (both with their usual topology) are locally compact abelian groups. Definition and examples A topological group is called ''locally compact'' if the underlying topological space is locally compact and Hausdorff; the topological group is called ''abelian'' if the underlying group is abelian. Examples of locally compact abelian groups include: * \R^n for ''n'' a positive integer, with vector addition as group operation. * The positive real numbers \R^+ with multiplication as operation. This group is isomorphic to (\R, +) by the exponential map. * Any finite abelian group, with the discrete topology. By the structure theorem for finite abelian groups, all such groups are produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hausdorff–Young Inequality
The Hausdorff−Young inequality is a foundational result in the mathematical field of Fourier analysis. As a statement about Fourier series, it was discovered by and extended by . It is now typically understood as a rather direct corollary of the Plancherel theorem, found in 1910, in combination with the Riesz-Thorin theorem, originally discovered by Marcel Riesz in 1927. With this machinery, it readily admits several generalizations, including to multidimensional Fourier series and to the Fourier transform on the real line, Euclidean spaces, as well as more general spaces. With these extensions, it is one of the best-known results of Fourier analysis, appearing in nearly every introductory graduate-level textbook on the subject. The nature of the Hausdorff-Young inequality can be understood with only Riemann integration and infinite series as prerequisite. Given a continuous function , define its "Fourier coefficients" by :c_n=\int_0^1 e^f(x)\,dx for each integer . The Hausdorf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''period''), the number of components, and their amplitudes and phase parameters. With appropriate choices, one cycle (or ''period'') of the summation can be made to approximate an arbitrary function in that interval (or the entire function if it too is periodic). The number of components is theoretically infinite, in which case the other parameters can be chosen to cause the series to converge to almost any ''well behaved'' periodic function (see Pathological and Dirichlet–Jordan test). The components of a particular function are determined by ''analysis'' techniques described in this article. Sometimes the components are known first, and the unknown function is ''synthesized'' by a Fourier series. Such is the case of a discrete-ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivation
Motivation is the reason for which humans and other animals initiate, continue, or terminate a behavior at a given time. Motivational states are commonly understood as forces acting within the agent that create a disposition to engage in goal-directed behavior. It is often held that different mental states compete with each other and that only the strongest state determines behavior. This means that we can be motivated to do something without actually doing it. The paradigmatic mental state providing motivation is desire. But various other states, such as beliefs about what one ought to do or intentions, may also provide motivation. Motivation is derived from the word 'motive', which denotes a person's needs, desires, wants, or urges. It is the process of motivating individuals to take action in order to achieve a goal. The psychological elements fueling people's behavior in the context of job goals might include a desire for money. Various competing theories have been proposed co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Mean Oscillation
In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces ''Hp'' that the space ''L''∞ of essentially bounded functions plays in the theory of ''Lp''-spaces: it is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time. Historical note According to , the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set belonging to R''n'' into R''n'' and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by , where several properties of this function spaces were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
In complex analysis, the Hardy spaces (or Hardy classes) ''Hp'' are certain spaces of holomorphic functions on the unit disk or upper half plane. They were introduced by Frigyes Riesz , who named them after G. H. Hardy, because of the paper . In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (in the sense of distributions) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the ''Lp'' spaces of functional analysis. For 1 ≤ ''p'' < ∞ these real Hardy spaces ''Hp'' are certain s of ''Lp'', while for ''p'' < 1 the ''Lp'' spaces have some undesirable properties, and the Hardy spaces are much better behaved. There are also higher-dimensional generalizations, consisting of certain holomorphic functions on |
Isidore Isaac Hirschman, Jr
Isidore ( ; also spelled Isador, Isadore and Isidor) is an English and French masculine given name. The name is derived from the Greek name ''Isídōros'' (Ἰσίδωρος) and can literally be translated to "gift of Isis." The name has survived in various forms throughout the centuries. Although it has never been a common name, it has historically been popular due to its association with Catholic figures and among the Jewish diaspora. Isidora is the feminine form of the name. Pre-modern era :''Ordered chronologically'' Religious figures * Isidore of Alexandria (died 403), Egyptian priest, saint * Isidore of Chios (died 251), Roman Christian martyr * Isidore of Scété (died c. 390), 4th-century A.D. Egyptian Christian priest and desert ascetic * Isidore of Pelusium (died c. 449), Egyptian monk, saint and prolific letter writer * Isidore of Seville (c. 560–636), Catholic saint and scholar, last of the Fathers of the Church and Archbishop of Seville * Isidore the Laborer (c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Fefferman
Charles Louis Fefferman (born April 18, 1949) is an American mathematician at Princeton University, where he is currently the Herbert E. Jones, Jr. '43 University Professor of Mathematics. He was awarded the Fields Medal in 1978 for his contributions to mathematical analysis. Early life and education Fefferman was born to a Jewish family, in Washington, DC. Fefferman was a child prodigy. Fefferman entered the University of Maryland at age 14, and had written his first scientific paper by the age of 15. He graduated with degrees in math and physics at 17, and earned his PhD in mathematics three years later from Princeton University, under Elias Stein. His doctoral dissertation was titled "Inequalities for strongly singular convolution operators". Fefferman achieved a full professorship at the University of Chicago at the age of 22, making him the youngest full professor ever appointed in the United States. Career At the age of 25, he returned to Princeton as a full professor, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elias Stein
Elias Menachem Stein (January 13, 1931 – December 23, 2018) was an American mathematician who was a leading figure in the field of harmonic analysis. He was the Albert Baldwin Dod Professor of Mathematics, Emeritus, at Princeton University, where he was a faculty member from 1963 until his death in 2018. Biography Stein was born in Antwerp Belgium, to Elkan Stein and Chana Goldman, Ashkenazi Jews from Belgium.University of St Andrews, Scotland - School of Mathematics and Statistics: "Elias Menachem Stein" by J.J. O'Connor and E F Robertson February 2010 After the German invasion in 1940, the Stein family fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fatou’s Lemma
In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou. Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem. Standard statement In what follows, \operatorname_ denotes the \sigma-algebra of Borel sets on ,+\infty/math>. Fatou's lemma remains true if its assumptions hold \mu-almost everywhere. In other words, it is enough that there is a null set N such that the values \ are non-negative for every . To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on N. Proof Fatou's lemma does ''not'' require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below. In each case, the proof begins by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
