Search results
Results From The WOW.Com Content Network
A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...
Both tests are the best of their kind. For the Dini-Lipschitz test, it is possible to construct a function f with its modulus of continuity satisfying the test with O instead of o, i.e. = (). and the Fourier series of f diverges. For the Dini test, the statement of precision is slightly longer: it says that for any function Ω such that
For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone. In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions.
In mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. ... The modulus of continuity of ...
This concept is very useful for iterated function systems where contraction mappings are often used. Banach's fixed-point theorem is also applied in proving the existence of solutions of ordinary differential equations , and is used in one proof of the inverse function theorem .
In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines.
In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity.
Modulus, the absolute value of a real or complex number ( | a |) Moduli space, in mathematics a geometric space whose points represent algebro-geometric objects; Conformal modulus, a measure of the size of a curve family; Modulus of continuity, a function gauging the uniform continuity of a function; Similarly, the modulus of a Dirichlet character