Function series

A function sequence is a sequence whose individual terms are functions. Sequences of functions and their convergence properties are available for all branches of analysis is of great importance. Above all, this examines the sense in which the sequence converges if the limit function if limit formations can be interchanged with functional consequences inherits properties of the sequence or. Many important examples are sets of features, such as power series or Fourier series.

• 3.1 boundedness
• 3.2 Local uniform boundedness

• 4.1 Classical concepts of convergence 4.1.1 Pointwise convergence
• 4.1.2 Uniform convergence
• 4.1.3 Local uniform convergence
• 4.1.4 Compact Convergence
• 4.1.5 Normal convergence

• 4.2.1 Pointwise convergence almost everywhere
• 4.2.2 Convergence in measure
• 4.2.3 Lp- convergence and convergence in Sobolev spaces
• 4.2.4 Almost uniform convergence
• 4.2.5 Weak convergence

Definition

A (real) function is a sequence of functions. General can also be other quantities, such as intervals definition and target amount; However, they must be the same for all functions.

Abstract can be a sequence of functions as picture

Be defined for a set of definitions and a target amount.

Examples

Interchanging limit and integral sign

For the following, with

Holds for every fixed

It converges pointwise to the zero function. However, for all

So

Pointwise convergence is therefore not sufficient, so limit and integral sign may be reversed; so this substitution is permitted, is a more stringent convergence behavior, namely the so-called uniform convergence, sufficiently.

Power series

In the Analysis function consequences often occur as sums of functions, ie, as a series, especially as a power series, or more generally as a Laurent series.

Fourier analysis and approximation theory

In the approximation theory is investigated how well can represent functions as the limit of sequences of functions, in particular, the quantitative estimate of the error is of interest. The functional consequences usually occur in this case as a function of rows, ie as a sum. For example, Fourier series converge in the sense against the function to be plotted. Better approximations in the sense of uniform convergence is obtained often with a series of Chebyshev polynomials.

Stochastics

In the stochastic random variable is defined as a measurable function of a Maßraums with a probability measure. Sequences of random variables are therefore special functional consequences, as are statistics such as the sample mean function consequences. Important convergence properties of these function sequences are, for example, the laws of large numbers or the central limit theorems.

Numerical Mathematics

In numerical mathematics sequences of functions appear for example in solving partial differential equations, which is a (not necessarily linear ) differential operator and the unknown function. In the numerical solution of the finite element method with about one obtains functions as a solution of the discretized version of the equation, where the fineness of the discretization called. In the analysis of the numerical algorithm is now the properties of the discrete solutions, which form a functional, examines; in particular, it makes sense that the sequence of discrete solutions converges to the solution of the original problem in refining the discretization.

Properties

Narrowness

A functional sequence is bounded in an amount if there exists a constant such that.

Local uniform boundedness

A sequence of functions is locally uniformly bounded in an open area, if one exists for every open area, with a constant, so that.

Concepts of convergence

The limit f of a function sequence is called the limit function. Since the sequences of functions occurring in the applications can have very different behavior with increasing index, it is necessary to introduce many different concepts of convergence for sequences of functions. From a more abstract point of view it is usually around the convergence with respect to certain standards or general topologies on the corresponding function spaces; few show but also other concepts of convergence.

The different concepts of convergence differ mainly by the implicit properties of the limit function. The most important are:

Classical concepts of convergence

Pointwise convergence

The pointwise limit exists

At each point of its domain, the function sequence is pointwise convergent called. For example, applies

The limit function is therefore discontinuous.

Uniform convergence

A sequence of functions is uniformly convergent to a function when the maximum differences between and converge to zero. This notion of convergence is convergence in terms of the supremum norm.

Uniform convergence implies some properties of the limit function, if the followers they have:

• The uniform limit of continuous functions is continuous.
• The uniform limit of a sequence ( Riemann and Lebesgue ) integrable functions on a compact interval is ( Riemann and Lebesgue ) integrable, and the integral of the limit function is the limit of the integrals of the followers: Is uniformly convergent to, the following applies

• Converges a sequence of differentiable functions pointwise to a function and is the sequence of derivatives is uniformly convergent, so is differentiable and it holds

Local uniform convergence

Many series in the theory of functions, in particular power series are not uniformly convergent, because the convergence of increasing arguments is getting worse. If one demands the uniform convergence only locally, ie in a neighborhood of each point, one comes to the notion of locally uniform convergence, which is sufficient for many applications in the analysis. As with the uniform convergence of the continuity of the sequence elements carries over in the locally uniform convergence to the limit function.

Compact convergence

A similarly good convergence concept is that of compact convergence, uniform convergence on compact subsets calls only. From the locally uniform convergence follows the compact convergence, for locally compact spaces, which often occur in applications that converse is also true.

Normal convergence

In mathematics, the concept of normal convergence of the characterization of infinite series of functions is used. Introduced the concept of the French mathematician René Louis Baire.

Measure-theoretical concepts of convergence

In the measure theoretic convergence terms, the limit function is usually not unique, but only almost everywhere uniquely defined. Alternatively, this convergence as convergence of equivalence classes of functions that match almost everywhere understand. As such an equivalence class then the limit is uniquely determined.

Pointwise convergence almost everywhere

If a measure space and a sequence of measurable functions on given with definition set, then the sequence of functions is pointwise convergent almost everywhere with respect to called when the pointwise limit

Almost everywhere with respect exists, so if a set of measure zero () exists, so that limited converges pointwise to the complement.

The convergence almost everywhere with respect to a probability measure is called the Stochastic almost sure convergence.

For example, applies

Another example is the sequence of functions, wherein for,

This sequence converges to zero, since they have the values ​​0 and 1 infinitely often takes for each fixed. For every subsequence, however, a partial subsequence can be specified, so that

Were it a topology of pointwise convergence almost everywhere, so would the fact that every subsequence contains a part of the subsequence that converges to 0, follow that must converge to 0. However, since not converge, it can therefore be no topology of convergence almost everywhere. The pointwise convergence almost everywhere so an example of a convergence concept which, although the Fréchet axioms is sufficient, but can not be generated by a topology.

Convergence in measure

In a measure space on a sequence of measurable functions is called convergent dimensions after to a function, if for every

Applies.

In a finite measure space, so if true, the convergence is in measure weaker than the convergence almost everywhere: a sequence of measurable functions converges almost everywhere to a function, it also converges in measure against.

In the stochastic convergence is called Stochastic convergence or convergence in probability in measure.

Lp- convergence and convergence in Sobolev spaces

A functional sequence is said to converge to, if it converges in the sense of the corresponding Lp- space, ie, if

Is a finite measure, so true, it follows from the inequality for the generalized means that there exists a constant, so that; particular, therefore, follows from the convergence of towards the convergence of against.

In the stochastic convergence is called convergence in pth mean.

From the convergence convergence follows according to the measure, as is evident from the Chebyshev 's inequality in the form

Looks.

A generalization of the Lp- convergence is the convergence in Sobolev spaces, which takes into account not only the convergence of the function values ​​, but also the convergence of certain derivatives. The Sobolewschen embedding theorem describes the dependencies of the concepts of convergence in the various Sobolev spaces.

Almost uniform convergence

In a measure space is a sequence of measurable convergent called it real - or complex-valued functions almost uniformly to a function if exists for each a lot, so converges uniformly on the complement against.

From the almost uniform convergence of the pointwise convergence follows almost everywhere; follows from the set of Yegorov that in a finite measure space and vice versa from the pointwise convergence almost everywhere follows the almost uniform convergence. In a finite measure space, ie, in particular for real-valued random variables, convergence almost everywhere and almost uniform convergence of real-valued sequences of functions are equivalent.

From the almost uniform convergence also the convergence follows according to the measure. Conversely, to the extent that after convergent sequence contains a subsequence, which almost uniformly (and hence almost everywhere) converges to the same limit result.

Weak convergence

Hierarchical order concepts of convergence in spaces with finite measure

In measure spaces with finite measure, ie, if true, it is mostly possible to arrange the different concepts of convergence according to their strength. This is especially true in probability spaces because there is true yes.

From the uniform convergence, the convergence follows dimensions according to the two different ways, which leads via the pointwise convergence of a:

• Uniformly locally uniformly (ie uniformly on a neighborhood of every point ).
• Locally uniformly compact (ie, uniformly on every compact subset ).
• Compact point-wise ( each point is indeed a compact subset ).
• Pointwise pointwise almost everywhere (or almost certain ).
• Pointwise almost everywhere almost evenly.
• Almost uniformly in measure (or stochastically or in probability).

The other way of the uniform convergence on the convergence degree after the results of the convergence:

• Uniformly in.
• Real in at all.
• Real in at all.
• In for the dimensions to (or stochastically or in probability).

From the convergence degree after you get to the weak convergence:

• In measure weakly (or in distribution).

Important theorems about sequences of functions

• Set of Arzelà - Ascoli
• Set of Dini
• Set of Yegorov

Swell