|
Rademacher System
In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form: : \. The Rademacher system is stochastically independent, and is closely related to the Walsh system. Specifically, the Walsh system can be constructed as a product of Rademacher functions. To see that the Rademacher system is an incomplete orthogonal system and not an orthonormal basis, consider the function on the unit interval defined by the following equation: f(t) = 4 \left, x-\frac 12\ - 1 This function is orthogonal to all the functions in the Rademacher system, yet is nonzero. References * * * * External links Rademacher systemin the ''Encyclopedia of Mathematics The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries cover ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Rademacher
Hans Adolph Rademacher (; 3 April 1892 – 7 February 1969) was a German-born American mathematician, known for work in mathematical analysis and number theory. Biography Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göttingen; Constantin Carathéodory supervised his dissertation. In 1919, he became '' privatdozent'' under Constantin Carathéodory at University of Berlin The Humboldt University of Berlin (, abbreviated HU Berlin) is a public research university in the central borough of Mitte in Berlin, Germany. The university was established by Frederick William III on the initiative of Wilhelm von Humbol .... In 1922, he became an assistant professor at the University of Hamburg, where he supervised budding mathematicians like Theodor Estermann. He was dismissed from his position at the University of Breslau by the Nazis in 1933 due to his public support of the Weimar Republic, and emigrated from Europe in 1934. After leaving Germany, he moved ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Orthogonal System
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization. The choice of an origin and an orthonormal basis forms a coordinate frame known as an ''orthonormal frame''. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Interval
In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology. In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: , , and . However, the notation ' is most commonly reserved for the closed interval . Properties The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space, it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval. In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Walsh Transform
The Hadamard transform (also known as the Walsh–Hadamard transform, Hadamard–Rademacher–Walsh transform, Walsh transform, or Walsh–Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutive, linear operation on real numbers (or complex, or hypercomplex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size-2 discrete Fourier transforms (DFTs), and is in fact equivalent to a multidimensional DFT of size . It decomposes an arbitrary input vector into a superposition of Walsh functions. The transform is named for the French mathematician Jacques Hadamard (), the German-American mathematician Hans Rademacher, and the American mathematician Joseph L. Walsh. Definition The Hadamard transform ''H''''m'' is a 2''m'' × 2''m'' matrix, the Hadamard matrix (scaled by a normalization factor), that transforms 2''m'' r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Encyclopedia Of Mathematics
The ''Encyclopedia of Mathematics'' (also ''EOM'' and formerly ''Encyclopaedia of Mathematics'') is a large reference work in mathematics. Overview The 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, and the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer. The CD-ROM contains animations and three-dimensional objects. The encyclopedia has been translated from the Soviet ''Matematicheskaya entsiklopediya'' (1977) originally edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles. Until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge. This URL now redirects to the new wiki A wiki ( ) is a form of hypertext publication on the internet which is collaboratively edited and manage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
