Denjoy Theorem (other)
In mathematics, Denjoy's theorem may refer to several theorems proved by Arnaud Denjoy, including * Denjoy–Carleman theorem * Denjoy–Koksma inequality * Denjoy–Luzin theorem * Denjoy–Luzin–Saks theorem * Denjoy–Riesz theorem * Denjoy–Wolff theorem * Denjoy–Young–Saks theorem * Denjoy's theorem on rotation number In mathematics, the Denjoy theorem gives a sufficient condition for a diffeomorphism of the circle to be topologically conjugate to a diffeomorphism of a special kind, namely an irrational rotation. proved the theorem in the course of his topolog ...

Arnaud Denjoy
Arnaud Denjoy (; 5 January 1884 – 21 January 1974) was a French mathematician. Biography Denjoy was born in Auch, Gers. His contributions include work in harmonic analysis and differential equations. His integral was the first to be able to integrate all derivatives. Among his students is Gustave Choquet. He is also known for the more general broad Denjoy integral, or Khinchin integral. Denjoy was an Invited Speaker of the ICM with talk ''Sur une classe d'ensembles parfaits en relation avec les fonctions admettant une dérivée seconde généralisée'' in 1920 at Strasbourg and with talk ''Les equations differentielles periodiques'' in 1950 at Cambridge, Massachusetts. In 1931 he was the president of the Société Mathématique de France. In 1942 he was elected a member of the Académie des sciences and was its president in 1962. Denjoy married in 1923 and was the father of three sons. He died in Paris in 1974. He was an atheist with a strong interest in philosophy, psy ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Carleman Theorem
In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If ''f'' is an analytic function on an interval 'a'',''b''nbsp;⊂ R, and at some point ''f'' and all of its derivatives are zero, then ''f'' is identically zero on all of 'a'',''b'' Quasi-analytic classes are broader classes of functions for which this statement still holds true. Definitions Let M=\_^\infty be a sequence of positive real numbers. Then the Denjoy-Carleman class of functions ''C''''M''( 'a'',''b'' is defined to be those ''f'' ∈ ''C''∞( 'a'',''b'' which satisfy :\left , \frac(x) \right , \leq A^ k! M_k for all ''x'' ∈  'a'',''b'' some constant ''A'', and all non-negative integers ''k''. If ''M''''k'' = 1 this is exactly the class of real analytic functions on 'a'',''b'' The class ''C''''M''( 'a'',''b'' is said to be ''quasi-analytic'' if whenever ''f'' ∈& ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Koksma Inequality
In mathematics, the Denjoy–Koksma inequality, introduced by as a combination of work of Arnaud Denjoy and the Koksma–Hlawka inequality of Jurjen Ferdinand Koksma, is a bound for Weyl sums \sum_^f(x+k\omega) of functions ''f'' of bounded variation. Statement Suppose that a map ''f'' from the circle ''T'' to itself has irrational rotation number ''α'', and ''p''/''q'' is a rational approximation to ''α'' with ''p'' and ''q'' coprime, , ''α'' â€“ ''p''/''q'',  < 1/''q''². Suppose that ''φ'' is a function of bounded variation, and ''μ'' a

Denjoy–Luzin Theorem
In mathematics, the Denjoy–Luzin theorem, introduced independently by and states that if a trigonometric series converges absolutely on a set of positive measure, then the sum of its coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ... converges absolutely, and in particular the trigonometric series converges absolutely everywhere. References * * * Fourier series Theorems in analysis {{mathanalysis-stub ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Luzin–Saks Theorem
In mathematics, the Denjoy–Luzin–Saks theorem states that a function of generalized bounded variation in the restricted sense has a derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ... almost everywhere, and gives further conditions of the set of values of the function where the derivative does not exist. N. N. Luzin and A. Denjoy proved a weaker form of the theorem, and later strengthened their theorem. References * Theorems in analysis {{mathanalysis-stub ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Riesz Theorem
In topology, the Denjoy–Riesz theorem states that every compact set of totally disconnected points in the Euclidean plane can be covered by a continuous image of the unit interval, without self-intersections (a Jordan arc). Definitions and statement A topological space is zero-dimensional according to the Lebesgue covering dimension if every finite open cover has a refinement that is also an open cover by disjoint sets. A topological space is totally disconnected if it has no nontrivial connected subsets; for points in the plane, being totally disconnected is equivalent to being zero-dimensional. The Denjoy–Riesz theorem states that every compact totally disconnected subset of the plane is a subset of a Jordan arc. History credits the result to publications by Frigyes Riesz in 1906, and Arnaud Denjoy in 1910, both in '' Comptes rendus de l'Académie des sciences''. As describe,. Riesz actually gave an incorrect argument that every totally disconnected set in the plane is a ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Wolff Theorem
In mathematics, the Denjoy–Wolff theorem is a theorem in complex analysis and dynamical systems concerning fixed points and iterations of holomorphic mappings of the unit disc in the complex numbers into itself. The result was proved independently in 1926 by the French mathematician Arnaud Denjoy and the Dutch mathematician Julius Wolff. Statement Theorem. Let ''D'' be the open unit disk in C and let ''f'' be a holomorphic function mapping ''D'' into ''D'' which is not an automorphism of ''D'' (i.e. a Möbius transformation). Then there is a unique point ''z'' in the closure of ''D'' such that the iterates of ''f'' tend to ''z'' uniformly on compact subsets of ''D''. If ''z'' lies in ''D'', it is the unique fixed point of ''f''. The mapping ''f'' leaves invariant hyperbolic disks centered on ''z'', if ''z'' lies in ''D'', and disks tangent to the unit circle at ''z'', if ''z'' lies on the boundary of ''D''. When the fixed point is at ''z'' = 0, the hyperbolic disks ...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  

Denjoy–Young–Saks Theorem
In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere. proved the theorem for continuous functions, extended it to measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in di ...s, and extended it to arbitrary functions. and give historical accounts of the theorem. Statement If ''f'' is a real valued function defined on an interval, then with the possible exception of a set of measure 0 on the interval, the Dini derivatives of ''f'' satisfy one of the following four conditions at each point: *''f'' has a finite derivative *''D''+''f'' = ''D''–''f'' is finite, ''D''−''f'' = ∞, ''D''+''f'' = –∞. *''D''−''f'' = ''D''+''f'' is finite, ''D''+''f'' = ∞, ''D''–''f'' = –∠...
[...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]