|
In the mathematical field of analysis, uniform convergence is a type of convergence stronger than pointwise convergence. A sequence { fn } of functions converges uniformly to a limiting function f if the speed of convergence of fn(x) to f(x) does not depend on x. For other meanings of mathematics or math, see mathematics (disambiguation). ...
Analysis is the generic name given to any branch of mathematics that depends upon the concepts of limits and convergence. ...
In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ...
Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...
In mathematics, a sequence is a list of objects (or events) arranged in a linear fashion, such that the order of the members is well defined and significant. ...
Partial plot of a function f. ...
The concept is important because several properties of the functions fn, such as continuity, differentiability and Riemann integrability, are only transferred to the limit f if the convergence is uniform. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...
In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...
If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...
In mathematics, the concept of a limit is used to describe the behavior of a function as its argument either gets close to some point, or as it becomes arbitrarily large; or the behavior of a sequences elements, as their index increases indefinitely. ...
[edit]
History
Augustin Louis Cauchy in 1821 published a faulty proof of the false statement that the pointwise limit of a sequence of continuous functions is always continuous. Joseph Fourier and Niels Henrik Abel found counter examples in the context of Fourier series. Dirichlet then analyzed Cauchy's proof and found the mistake: the notion of pointwise convergence had to be replaced by uniform convergence. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ...
Jean Baptiste Joseph Fourier (March 21, 1768 - May 16, 1830) was a French mathematician and physicist who is best known for initiating the investigation of Fourier series and their application to problems of heat flow. ...
Niels Henrik Abel (August 5, 1802–April 6, 1829), Norwegian mathematician, was born in Finnøy. ...
The Fourier series is a mathematical tool used for analyzing an arbitrary periodic function by decomposing it into a weighted sum of much simpler sinusoidal component functions sometimes referred to as normal Fourier modes, or simply modes for short. ...
Johann Peter Gustav Lejeune Dirichlet (February 13, 1805 - May 5, 1859) was a German mathematician credited with the modern formal definition of a function. ...
The concept of uniform convergence was probably first used by Christoph Gudermann. Later his pupil Karl Weierstrass coined the term gleichmäßig konvergent (German: uniform convergence) which he used in his 1841 paper Zur Theorie der Potenzreihen, published in 1894. Independently a similar concept was used by Philipp Ludwig von Seidel and George Gabriel Stokes but without having any major impact on further development. G. H. Hardy compares the three definitions in his paper Sir George Stokes and the concept of uniform convergence and remarks: Weierstrass's discovery was the earliest, and he alone fully realized its far-reaching importance as one of the fundamental ideas of analysis Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany. ...
Karl Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // Biography Karl Weierstrass was born in Ostenfelde, Westphalia (today Germany). ...
Philipp Ludwig von Seidel (1821–1896) was a German mathematician. ...
George Gabriel Stokes Sir George Gabriel Stokes, 1st Baronet (13 August 1819–1 February 1903) was an Anglo-Irish mathematician and physicist. ...
G. H. Hardy Professor Godfrey Harold Hardy FRS (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
Under the influence of Weierstrass and Bernhard Riemann this concept and related questions were intensely studied at the end of the 19th century by Hermann Hankel, Paul du Bois-Reymond, Ulisse Dini, Cesare Arzelà and others. Bernhard Riemann. ...
Hermann Hankel (February 14, 1839 - August 29, 1873) was a German mathematician who was born in Halle, Germany and died in Schramberg (near Tübingen), Germany. ...
Paul David Gustav du Bois-Reymond (December 2, 1831 - April 7, 1889) was a mathematician who was born in Berlin, Germany and died in Freiburg, Germany. ...
Ulisse Dini (Born November 14, 1845 in Pisa, Italy-Died October 28, 1918 in Pisa, Italy) was a mathematician and politician. ...
Cesare Arzelà (1847-1912) was an Italian mathematician who taught at Bologna and is recognized for contributions in sequences of functions. ...
[edit]
Definition Suppose S is a set and fn : S → R are real-valued functions for every natural number n. We say that the sequence (fn) is uniformly convergent with limit f : S → R if In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory and computer science. ...
- for every ε > 0, there exists a natural number N such that for all x in S and all n ≥ N, |fn(x) − f(x)| < ε.
The sequence (fn) is said to be locally uniformly convergent with limit f if for every x in S, if there exists an r > 0 such that (fn) converges uniformly on B(x,r) ∩ S. [edit]
Notes Compare uniform convergence to the concept of pointwise convergence: The sequence (fn) converges pointwise with limit f : S → R if and only if Suppose { fn } is a sequence of functions sharing the same domain in common (for the moment, we defer making precise the nature of the values of these functions, but the reader may take them to be real numbers if that makes anyone feel good). ...
- for every x in S and every ε > 0, there exists a natural number N such that for all n ≥ N, |fn(x) − f(x)| < ε.
In the case of uniform convergence, N can only depend on ε, while in the case of pointwise convergence N may depend on ε and x. It is therefore plain that uniform convergence implies pointwise convergence. The converse is not true, as the following example shows: take S to be the unit interval [0,1] and define fn(x) = xn for every natural number n. Then (fn) converges pointwise to the function f defined by f(x) = 0 if x < 1 and f(1) = 1. This convergence is not uniform: for instance for ε = 1/4, there exists no N as required by the definition. In mathematics, the unit interval is the interval [0,1], that is the set of all real numbers x such that zero is less than or equal to x and x is less than or equal to one. ...
[edit]
Generalizations One may straightforwardly extend the concept to functions S → M, where (M, d) is a metric space, by replacing |fn(x) - f(x)| with d(fn(x), f(x)). In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...
The most general setting is the uniform convergence of nets of functions S → X, where X is a uniform space. We say that the net (fα) converges uniformly with limit f : S → X iff In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces. ...
In the mathematical field of topology, a uniform space is a set with a uniform structure. ...
IFF, Iff or iff can stand for: Interchange File Format - a computer file format introduced by Electronic Arts Identification, friend or foe - a radio based identification system utilizing transponders iff - the mathematics concept if and only if International Flavors and Fragrances - a company producing flavors and fragrances International Freedom Foundation...
- for every entourage V in X, there exists an α0, such that for every x in I and every α≥α0: (fα(x), f(x)) is in V.
The above mentioned theorem, stating that the uniform limit of continuous functions is continuous, remains correct in these settings. In topology, one defines uniform spaces in order to study concepts such as uniform continuity, completeness and uniform convergence. ...
[edit]
Examples Given a topological space X, we can equip the space of bounded real or complex-valued functions over X with the uniform norm topology. Then uniform convergence simply means convergence in the uniform norm topology. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In mathematics, a function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. ...
In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. ...
In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...
In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...
In the absence of a more specific context, convergence denotes the approach toward a definite value, as time goes on; or to a definite point, a common view or opinion, or toward a fixed or equilibrium state. ...
In mathematical analysis, the uniform norm assigns to real- or complex-valued functions f the nonnegative number This norm is also called the supremum norm or the Chebyshev norm. ...
[edit]
Properties - Every uniformly convergent sequence is locally uniformly convergent
- Every locally uniformly convergent sequence is compactly convergent
- For locally compact spaces local uniform convergence and compact convergence coincide
[edit] In mathematics compact convergence is a type of convergence which generalizes the idea of uniform convergence. ...
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. ...
Applications If S is a real interval (or indeed any topological space), we can talk about the continuity of the functions fn and f. The following is the more important result about uniform continuity: In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...
Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...
In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...
- Uniform convergence theorem. If (fn) is a sequence of continuous functions which converges uniformly towards the function f, then f is continuous as well.
 Counterexample to a strengthening of the uniform convergence theorem, in which pointwise convergence, rather than uniform convergence, is assumed. The continuous green functions sin n( x) converge to the non-continuous red function because convergence is not uniform The former theorem is important, since pointwise convergence of continuous functions is not enough to guarantee continuity of the limit function as the image illustrates. Image File history File links Counterexample to the converse of the uniform convergence theorem. ...
Image File history File links Counterexample to the converse of the uniform convergence theorem. ...
If S is an interval and all the functions fn are differentiable and converge to a limit f, it is often desirable to differentiate the limit function f by taking the limit of the derivatives of fn. This is however in general not possible: even if the convergence is uniform, the limit function need not be differentiable, and even if it is differentiable, the derivative of the limit function need not be equal to the limit of the derivatives. Consider for instance fn(x) = 1/n sin(nx) with uniform limit 0, but the derivatives do not approach 0. The precise statement covering this situation is as follows: In mathematics, the derivative is defined as the instantaneous rate of change of a function. ...
- If fn converges uniformly to f, and if all the fn are differentiable, and if the derivatives f'n converge uniformly to g, then f is differentiable and its derivative is g.
Similarly, one often wants to exchange integrals and limit processes. For the Riemann integral, one needs to require uniform convergence: If you are having difficulty understanding this article, you might wish to learn more about algebra, functions, and mathematical limits. ...
- If (fn) is a sequence of Riemann integrable functions which uniformly converge with limit f, then f is Riemann integrable and its integral can be computed as the limit of the integrals of the fn.
Much stronger theorems in this respect, which require not much more than pointwise convergence, can be obtained if one abandons the Riemann integral and uses the Lebesgue integral instead. The integral can be interpreted as the area under a curve. ...
- If S is a compact interval (or in general a compact topological space), and (fn) is a monotone increasing sequence (meaning fn(x) ≤ fn+1(x) for all n and x) of continuous functions with a pointwise limit f which is also continuous, then the convergence is necessarily uniform (Dini's theorem). Uniform convergence is also guaranteed if S is a compact interval and (fn) is an equicontinuous sequence that converges pointwise.
[edit] In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ...
In mathematics, functions between ordered sets are monotonic (or monotone) if they preserve the given order. ...
In mathematical analysis, a sequence of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood (a precise definition appears below). ...
References [edit] Konrad Hermann Theodor Knopp (22 July 1882, Berlin, Germany – 20 April 1957, Annecy, France) was a mathematician. ...
G. H. Hardy Godfrey Harold Hardy (February 7, 1877 – December 1, 1947) was a prominent British mathematician, known for his achievements in number theory and mathematical analysis. ...
The Cambridge Philosophical Society (CPS) is a scientific society at University of Cambridge. ...
Nicolas Bourbaki is the pseudonym under which a group of mainly French 20th-century mathematicians wrote a series of books of exposition of modern advanced mathematics, beginning in 1935. ...
External links - Uniform convergence on PlanetMath
- Limit point of function on PlanetMath
- Converges uniformly on PlanetMath
- Convergent series on PlanetMath
- Graphic examples of uniform convergence of Fourier series from the University of Colorado
|
Feeds |
Contact