|
Fatou's Theorem
In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk. Motivation and statement of theorem If we have a holomorphic function f defined on the open unit disk \mathbb=\, it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius r. This defines a new function: :\begin f_r:S^1 \to \Complex \\ f_(e^)=f(re^) \end where :S^1:=\=\, is the unit circle. Then it would be expected that the values of the extension of f onto the circle should be the limit of these functions, and so the question reduces to determining when f_r converges, and in what sense, as r\to 1, and how well defined is this limit. In particular, if the L^p norms of these f_r are well beh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear engineering, nuclear, aerospace engineering, aerospace, mechanical engineering, mechanical and electrical engineering. As a differentiable function of a complex variable is equal to its Taylor series (that is, it is Analyticity of holomorphic functions, analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions). History Complex analysis is one of the classical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Fatou
Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography Pierre Fatou's parents were Prosper Ernest Fatou (1832-1891) and Louise Eulalie Courbet (1844-1911), both of whom were in the military. Pierre's family would have liked for him to enter the military as well, but his health was not sufficiently good for him to pursue a military career. Fatou entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an intern (''stagiaire'') in the Paris Observatory. Fatou was promoted to assistant astronomer in 1904 and to astronomer (''astronome titulaire'') in 1928. He worked in this observatory until his death. Fatou was awarded the Becquerel prize in 1918; he was a knight of the Legion of Honour (1923). He was the president of the French ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Functions
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (''analytic''). Holomorphic functions are the central objects of study in complex analysis. Though the term ''analytic function'' is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis. Holomorphic functions are also sometimes referred to as ''regular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-p-space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. Pointwise operations Formal definition A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by Commonly, ''o'' and ''O'' are denoted by the same symbol. A similar definition is used for unary operations ''o'', and for operations of other arity. Examples \begin (f+g)(x) & = f(x)+g(x) & \text \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text \\ (\lambda \cdot f)(x) & = \lambda \cdot f(x) & \text \end where f, g : X \to R. See also pointwise product, and scalar. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poisson Kernel
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disk. The kernel can be understood as the derivative of the Green's function for the Laplace equation. It is named for Siméon Poisson. Poisson kernels commonly find applications in control theory and two-dimensional problems in electrostatics. In practice, the definition of Poisson kernels are often extended to ''n''-dimensional problems. Two-dimensional Poisson kernels On the unit disc In the complex plane, the Poisson kernel for the unit disc is given by P_r(\theta) = \sum_^\infty r^e^ = \frac = \operatorname\left(\frac\right), \ \ \ 0 \le r < 1. This can be thought of in two ways: either as a function of ''r'' and ''θ'', or as a family of functions of ''θ'' indexed by ''r''. If is the open |
Hardy–Littlewood Maximal Function
In mathematics, the Hardy–Littlewood maximal operator ''M'' is a significant non-linear operator used in real analysis and harmonic analysis. Definition The operator takes a locally integrable function ''f'' : R''d'' → C and returns another function ''Mf''. For any point ''x'' ∈ R''d'', the function ''Mf'' returns the maximum of a set of reals, namely the set of average values of ''f'' for all the balls ''B''(''x'', ''r'') of any radius ''r'' at ''x''. Formally, : Mf(x)=\sup_ \frac\int_ , f(y), \, dy where , ''E'', denotes the ''d''-dimensional Lebesgue measure of a subset ''E'' ⊂ R''d''. The averages are jointly continuous in ''x'' and ''r'', therefore the maximal function ''Mf'', being the supremum over ''r'' > 0, is measurable. It is not obvious that ''Mf'' is finite almost everywhere. This is a corollary of the Hardy–Littlewood maximal inequality. Hardy–Littlewood maximal inequality This theorem of G. H. Hardy and J. E. Littlewood states that ''M' ... [...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 |
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] |
