Hölder Continuous

	Hölder Continuous Hölder: * ''Hölder, Hoelder'' as surname * Hölder condition * Hölder's inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces. The numbers an ... * Hölder mean * Jordan–Hölder theorem {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hölder (surname) Hölder is a German surname Personal names in German-speaking Europe consist of one or several given names (''Vorname'', plural ''Vornamen'') and a surname (''Nachname, Familienname''). The ''Vorname'' is usually gender-specific. A name is usually cited in the "Name order, .... Notable people with the surname include: * Otto Hölder (1859–1937), German mathematician * Ernst Hölder (1901−1990), German mathematician, son of Otto See also * Holder (surname) {{DEFAULTSORT:Holder German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hölder Condition In mathematics, a real or complex-valued function on -dimensional Euclidean space satisfies a Hölder condition, or is Hölder continuous, when there are real constants , , such that , f(x) - f(y) , \leq C\, x - y\, ^ for all and in the domain of . More generally, the condition can be formulated for functions between any two metric spaces. The number \alpha is called the ''exponent'' of the Hölder condition. A function on an interval satisfying the condition with is constant (see proof below). If , then the function satisfies a Lipschitz condition. For any , the condition implies the function is uniformly continuous. The condition is named after Otto Hölder. If \alpha = 0, the function is simply bounded (any two values f takes are at most C apart). We have the following chain of inclusions for functions defined on a closed and bounded interval of the real line with : where . Hölder spaces Hölder spaces consisting of functions satisfying a Hölder conditio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hölder's Inequality In mathematical analysis, Hölder's inequality, named after Otto Hölder, is a fundamental inequality (mathematics), inequality between Lebesgue integration, integrals and an indispensable tool for the study of Lp space, spaces. The numbers and above are said to be Hölder conjugates of each other. The special case gives a form of the Cauchy–Schwarz inequality. Hölder's inequality holds even if is Infinity, infinite, the right-hand side also being infinite in that case. Conversely, if is in and is in , then the pointwise product is in . Hölder's inequality is used to prove the Minkowski inequality, which is the triangle inequality in the space , and also to establish that is the dual space of for . Hölder's inequality (in a slightly different form) was first found by . Inspired by Rogers' work, gave another proof as part of a work developing the concept of convex function, convex and concave functions and introducing Jensen's inequality, which was in turn named ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hölder Mean In mathematics, generalised means (or power mean or Hölder mean from Otto Hölder) are a family of functions for aggregating sets of numbers. These include as special cases the Pythagorean means (arithmetic, geometric, and harmonic means). Definition If is a non-zero real number, and x_1, \dots, x_n are positive real numbers, then the generalized mean or power mean with exponent of these positive real numbers is M_p(x_1,\dots,x_n) = \left( \frac \sum_^n x_i^p \right)^ . (See -norm). For we set it equal to the geometric mean (which is the limit of means with exponents approaching zero, as proved below): M_0(x_1, \dots, x_n) = \left(\prod_^n x_i\right)^ . Furthermore, for a sequence of positive weights we define the weighted power mean as M_p(x_1,\dots,x_n) = \left(\frac \right)^ and when , it is equal to the weighted geometric mean: M_0(x_1,\dots,x_n) = \left(\prod_^n x_i^\right)^ . The unweighted means correspond to setting all . Special cases A few particula ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]