PlanetMath (more info)
 Math for the people, by the people.
Encyclopedia | Requests | Forums | Docs | Wiki | Random | RSS  
Login
create new user
name:
pass:
forget your password?
Main Menu
sections
Encyclopædia
Papers
Books
Expositions

meta
Requests (231)
Orphanage
Unclass'd
Unproven (514)
Corrections (25)
Classification

talkback
Polls
Forums
Feedback
Bug Reports

downloads
Snapshots
PM Book

information
News
Docs
Wiki
ChangeLog
TODO List
Legalese
About
Owner confidence rating: High Entry average rating: No information on entry rating
quasisymmetric mapping (Definition)

A function $ \mu$ of the real line to itself is quasisymmetric (or $ M$-quasisymmetric) if it satisfies the following $ M$-condition.

There exists an $ M$, such that for all $ x, t$ (where $ t \not= x$)

$\displaystyle \frac{1}{M} \leq \frac{\mu(x+t) - \mu(x)}{\mu(x)-\mu(x-t)} \leq M .$    

Geometrically this means that the ratio of the length of the intervals $ \mu[(x-t,x)]$ and $ \mu[(x,x+t)]$ is bounded. This implies among other things that the function is one-to-one and continuous.

For example powers (as long as you make them one-to-one by for example using an odd power, or defining them as $ -\vert x\vert^p$ for negative $ x$ and $ \vert x\vert^p$ for positive $ x$ where $ p > 0$) are quasisymmetric. On the other hand functions like $ e^x - e^{-x}$, while one-to-one, onto and continuous, are not quasisymmetric. It would seem like a very strict condition, however it has been shown that there in fact exist functions that are quasisymmetric, but are not even absolutely continuous.

Quasisymmetric functions are an analogue of quasiconformal mappings.



"quasisymmetric mapping" is owned by jirka.
(view preamble | get metadata)

View style:

See Also: quasiconformal mapping, Beurling-Ahlfors quasiconformal extension

Also defines:  $M$-condition, quasisymmetric, $M$-quasisymmetric
Log in to rate this entry.
(view current ratings)

Cross-references: quasiconformal mappings, absolutely continuous, even, strict, onto, positive, negative, odd power, powers, continuous, one-to-one, implies, bounded, intervals, length, ratio, line, real, function
There are 4 references to this entry.

This is version 5 of quasisymmetric mapping, born on 2004-01-15, modified 2006-09-17.
Object id is 5516, canonical name is QuasisymmetricMapping.
Accessed 4675 times total.

Classification:
AMS MSC: 26A12 (Real functions :: Functions of one variable :: Rate of growth of functions, orders of infinity, slowly varying functions)
 26A15 (Real functions :: Functions of one variable :: Continuity and related questions )
 30C65 (Functions of a complex variable :: Geometric function theory :: Quasiconformal mappings in $R^n$, other generalizations)
Pending Errata and Addenda
None.
[ View all 1 ]
Discussion
Style: Expand: Order:
forum policy

No messages.

Interact
post | correct | update request | add derivation | add example | add (any)