|
Riesz–Thorin Theorem
In mathematical analysis, 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 inequa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm \, T\, of a linear map T : X \to Y is the maximum factor by which it "lengthens" vectors. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \text v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also know ... [...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 In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fo .... 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. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motivation
Motivation is an mental state, internal state that propels individuals to engage in goal-directed behavior. It is often understood as a force that explains why people or animals initiate, continue, or terminate a certain behavior at a particular time. It is a complex phenomenon and its precise definition is disputed. It contrasts with #Amotivation and akrasia, amotivation, which is a state of apathy or listlessness. Motivation is studied in fields like psychology, neuroscience, motivation science, and philosophy. Motivational states are characterized by their direction, Motivational intensity, intensity, and persistence. The direction of a motivational state is shaped by the goal it aims to achieve. Intensity is the strength of the state and affects whether the state is translated into action and how much effort is employed. Persistence refers to how long an individual is willing to engage in an activity. Motivation is often divided into two phases: in the first phase, the indi ... [...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 \Omega belonging to \mathbb^n into \mathbb^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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hardy Space
In complex analysis, the Hardy spaces (or Hardy classes) H^p are 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 spaces of distributions on the real -space \mathbb^n, defined (in the sense of distributions) as boundary values of the holomorphic functions. Hardy spaces are related to the ''Lp'' spaces. For 1 \leq p < \infty these Hardy spaces are s of spaces, while for |
Isidore Isaac Hirschman, Jr
Isidore ( ; also spelled Isador, Isadore and Isidor) is a masculine given name. The name is derived from the Greek name ''Isídōros'' (Ἰσίδωρος, latinized ''Isidorus'') 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 * Isidorus (2nd century AD), pagan Egyptian priest * Isidore, son of Basilides, the Egyptian Christian Gnostic (2nd century AD) * Isidore of Chios (died 251), Roman Christian martyr * Isidore of Scetes (died c. 390), 4th-century Egyptian Christian priest and desert ascetic * Isidore of Alexandria (died 403), Egyptian Christian priest, saint * Isidore of Pelusium (died c. 449), Egyptian monk, saint and prolific letter writer * Isidore of ... [...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. He was a child prodigy, 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, becoming the young ... [...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 ... [...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 and natural proof. A proof directly from the definitions of integrals is given further below. Via the Monotone Conver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
