Uniform convergence

In the Analysis describes uniform convergence the property of a sequence of functions , with someone outside the function argument "speed" to a limit function to converge . In contrast to point-by-point convergence , the concept of uniform convergence allows important properties of the functions such as continuity and Riemann integrability to be transferred to the limit function . ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$

history

The term is usually ascribed to Karl Weierstrass in the 1840s (first in a paper in 1841, but which was not published until 1894), who in turn found it hinted at by his teacher Christoph Gudermann (1838), and was still missing in the original structure of the analysis Augustin-Louis Cauchy . This led to some errors in Cauchy's Cours d'Analyse of 1821, particularly in the so-called Cauchy's sum theorem. Cauchy claimed to have proven that a convergent series of continuous functions is continuous, but Niels Henrik Abel soon afterwards gave a counterexample in 1826 . Philipp Ludwig Seidel ( infinitely slow convergence ) and George Gabriel Stokes in 1847 ( infinitely slow convergence , points with non uniform convergence ) independently proved that the theorem applies when point-by-point convergence is replaced by uniform convergence (according to today's understanding ). Seidel tied directly to Cauchy and Peter Gustav Lejeune Dirichlet , who had given examples of Fourier series that converge to discontinuous functions. Stokes, on the other hand, did not refer to Cauchy, but to an essay on power series by John Radford Young from 1846. After Ivor Grattan-Guinness , the Swede Emanuel G. Björling (1846/47) may have joined the two as the originator of the concept. There was also a discussion ( Pierre Dugac 2003) whether Cauchy knew and implicitly used the term (and the related one of uniform continuity) in another textbook a little later in 1823. A group of math historians and mathematicians such as Detlef Laugwitz and Abraham Robinson later tried to save Cauchy's proof by pursuing the idea that Cauchy, who himself explicitly introduced infinitely small quantities in his textbook, had used a form of non-standard analysis , which was however not the case with the most Cauchy researchers failed and was taken as an example of a forced interpretation of the history of mathematics from a modern point of view. In his book, Klaus Viertel came to a more nuanced picture of an only gradual development of the concepts of continuity and convergence in today's sense, even in the context of the Weierstrass School, where the concept was also subject to change over time. At the beginning of the 20th century there were already various further developments of the term (quasi-convergence in Godfrey Harold Hardy 1918, William Henry Young 1903, 1907).

definition

A sequence of functions is given

${\displaystyle \left(f_{n}\colon D_{f}\subseteq \mathbb {R} \to \mathbb {R} \right)_{n\in \mathbb {N} }}$,

which assigns a real-valued function to every natural number , and a function . Let all and be defined on the same definition set . The sequence converges uniformly to if and only if ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle D_{f}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

${\displaystyle \lim _{n\rightarrow \infty }\,\sup _{x\in D_{f}}\left|f_{n}(x)-f(x)\right|=0.}$

One considers here the absolute difference from and for all from the domain of definition . The set of these differences is either unbounded or has a smallest upper bound, a supremum . Uniform convergence of against means that this supremum exists for almost everyone and tends to zero when it approaches infinity. ${\displaystyle f_{n}\left(x\right)}$ ${\displaystyle f\left(x\right)}$ ${\displaystyle x}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle n}$

This fact can also be defined differently: All terms are as above. Then converges uniformly to if and only if there exists for all one such that for all and for all applies: ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle n\geq N}$ ${\displaystyle x\in D_{f}}$

${\displaystyle \left|f_{n}(x)-f(x)\right|<\varepsilon .}$

example

Let it be a real number. The function sequence converges to the null function for uniform . It has to be shown that ${\displaystyle 0<q<1}$ ${\displaystyle \left(f_{n}\colon \left[0,q\right]\to \mathbb {R} ;\,x\mapsto x^{n}\right)_{n\in \mathbb {N} }}$ ${\displaystyle n\to \infty }$ ${\displaystyle f\colon \left[0,q\right]\to \mathbb {R} ;\,x\mapsto 0}$

${\displaystyle \lim _{n\to \infty }\,\sup _{x\in [0,q]}|f_{n}(x)|=0}$.

Each of the is increasing to non-negative and monotonous , so and because of this goes against . ${\displaystyle f_{n}}$ ${\displaystyle [0,q]}$ ${\displaystyle \textstyle \sup _{x\in [0,q]}|f_{n}(x)|=q^{n}}$ ${\displaystyle q<1}$ ${\displaystyle 0}$

The specification of the convergence range is essential here: the sequence still converges point by point to the null function on the unit interval open to the right , but no longer uniformly. It is now true , so in particular is ${\displaystyle f_{n}(x)=x^{n}}$ ${\displaystyle [0,1)}$ ${\displaystyle \textstyle \sup _{x\in [0,1)}|f_{n}(x)|=1}$

${\displaystyle \lim _{n\to \infty }\sup _{x\in [0,1)}|f_{n}(x)|=1\neq 0}$.

Comparison between regular and point convergence

The choice of with uniform convergence depends only on. In contrast, point-wise convergence depends both on and on. Formulated to both concepts of convergence using quantifiers, one sees that they are in the order of "introduction" of and different from each other and thus the function of two variables (see underlined): ${\displaystyle N}$ ${\displaystyle \varepsilon }$ ${\displaystyle N}$ ${\displaystyle \varepsilon }$ ${\displaystyle x}$ ${\displaystyle x}$ ${\displaystyle N}$

point-wise convergence:	${\displaystyle \forall \varepsilon >0\ {\underline {\forall x\in D_{f}\ \exists N\in \mathbb {N} }}\ \forall n\geq N:\quad \left|f_{n}(x)-f(x)\right|<\varepsilon ,}$ and
uniform convergence:	${\displaystyle \forall \varepsilon >0\ {\underline {\exists N\in \mathbb {N} \ \forall x\in D_{f}}}\ \forall n\geq N:\quad \left|f_{n}(x)-f(x)\right|<\varepsilon ,}$

d. . h, for pointwise convergence must for each and every natural number type, such that for all the following applies: . ${\displaystyle x}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N}$ ${\displaystyle n\geq N}$ ${\displaystyle \left|f_{n}(x)-f(x)\right|<\varepsilon }$

Point-by-point convergence follows from uniform convergence, but not vice versa. For example, the sequence of functions converges defined by ${\displaystyle F=(f_{n})_{n\in \mathbb {N} }}$

${\displaystyle f_{n}(x)={\begin{cases}0,&x\leq n\\1,&x>n\end{cases}}}$

pointwise against the null function for each , but is not a uniformly convergent sequence. ${\displaystyle f\equiv 0}$ ${\displaystyle x\in \mathbb {R} }$

designation

One of the following terms is mostly used for the uniform convergence of a sequence of functions that strives against ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$

${\displaystyle f_{n}{\underset {n\ }{\Rightarrow }}f,}$

or

${\displaystyle f_{n}{\underset {n\ }{\rightrightarrows }}f,}$

or

${\displaystyle \lim _{n\to \infty }f_{n}=f.}$

Uniform convergence in one point

A sequence of functions is said to converge uniformly towards if ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle \xi }$ ${\displaystyle f}$

${\displaystyle \forall \varepsilon >0\ \exists N\in \mathbb {N} \ \exists \delta >0\ \forall x\in (D_{f}\cap \{y\mid |y-\xi |<\delta \})\ \forall n\geq N:\left|f_{n}(x)-f(x)\right|<\varepsilon .}$

If the validity of the inequality for at least one is required instead of for all , then the convergence is called uniform. Uniformly convergent sequences are also uniformly convergent. The uniform convergence does not imply point-wise convergence. ${\displaystyle n}$ ${\displaystyle |f_{n}(x)-f(x)|<\varepsilon }$ ${\displaystyle n}$

Be

• ${\displaystyle {\mathfrak {G}}\,}$the class of uniformly convergent function sequences,
• ${\displaystyle {\mathfrak {J}}\,}$the class of function sequences that converge uniformly at every point and
• ${\displaystyle {\mathfrak {P}}\,}$the class of function sequences converging pointwise at each point.

Thus applies: . ${\displaystyle {\mathfrak {G}}\varsubsetneq {\mathfrak {J}}\varsubsetneq {\mathfrak {P}}}$

The sequence of functions mentioned above is in , so it converges equally at every point, but not globally. ${\displaystyle F}$ ${\displaystyle {\mathfrak {J}}\setminus {\mathfrak {G}}}$

An example of a sequence of functions from is defined by ${\displaystyle {\mathfrak {P}}\setminus {\mathfrak {J}}}$ ${\displaystyle (h_{n})_{n\in \mathbb {N} }}$

${\displaystyle h_{n}(x)={\begin{cases}0,&x\in \textstyle A_{n}:=(\mathbb {R} \setminus \mathbb {Q} )\cup \{y\in \mathbb {Q} \mid y={\tfrac {p}{q}},p\in \mathbb {Z} ,q\in \mathbb {N} ,0<q\leq n\}\\1,&x\notin A_{n}\end{cases}}}$

The function sequence converges point by point to the null function. Because every rational number lies in all those whose is equal to or greater than the denominator in the completely abbreviated representation of the fraction . On the other hand, there are always only finitely many rational numbers in the intersection of an arbitrary interval . Therefore there are always (infinitely many rational) numbers for each and every number whose distance is too small and which are not in . So the sequence does not converge evenly at any point. ${\displaystyle (h_{n})_{n\in \mathbb {N} }}$ ${\displaystyle y}$ ${\displaystyle A_{n}}$ ${\displaystyle n}$ ${\displaystyle y}$ ${\displaystyle A_{n}}$ ${\displaystyle n}$ ${\displaystyle z\in A_{n}}$ ${\displaystyle z}$ ${\displaystyle A_{n}}$ ${\displaystyle \textstyle (h_{n})_{n\in \mathbb {N} }}$

Inferences

As already mentioned, the concept of uniform convergence enables statements about the limit function based on properties of the sequence, which is not possible with point-wise convergence. In the following, the terms are the same as in the definition above, let us be a real interval . The following sentences result : ${\displaystyle I}$

continuity

• Let it be a sequence of continuous functions. If converges uniformly to , then is continuous. Instead of demanding uniform convergence, it is also sufficient to assume simple-uniform convergence . ${\displaystyle F=(f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle f}$
• Let be a sequence of functions that converges pointwise. All are also in steady. is continuous if and only if is uniformly convergent at the point . ${\displaystyle F=(f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$ ${\displaystyle \xi }$ ${\displaystyle f}$ ${\displaystyle \xi }$ ${\displaystyle F}$ ${\displaystyle \xi }$
• The set of points of uniform convergence and the set of points of uniform convergence of a function sequence that converges point by point are each G _δ sets .
• The uniformly convergent function sequences with a compact domain of definition are all equally continuous .
• Let be a compact interval and an equally continuous sequence. If it converges to point by point , then it also converges uniformly. ${\displaystyle I}$ ${\displaystyle F=(f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle I}$ ${\displaystyle F}$ ${\displaystyle f}$
• Let be a sequence of functions with a compact domain . has a uniformly convergent subsequence if and only if is uniformly continuous and is bounded in every point of ( Arzelà-Ascoli's theorem ). ${\displaystyle F=(f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle D}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle D}$

Differentiability

There is no such strong result for the differentiability of the limit function as for the continuity. Let them be differentiable to and uniformly convergent to . In general, the limit function need not even be differentiable, and if it is, its derivative need by no means be equal to the limit of the derivatives of the sequence. For example, B. the sequence of functions defined by uniformly towards 0, but not the sequence of derivatives . ${\displaystyle f_{n}}$ ${\displaystyle I}$ ${\displaystyle f}$ ${\displaystyle \textstyle f_{n}(x)={\frac {\sin(nx)}{n}}}$ ${\displaystyle (f'_{n})_{n\in \mathbb {N} }}$

In general one can say: they are all differentiable. If converges at a point and the sequence of derivatives converges uniformly against , then converges point-wise (even locally uniformly) against a and is differentiable with the derivative . ${\displaystyle f_{n}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle (f_{n}')_{n\in \mathbb {N} }}$ ${\displaystyle g}$ ${\displaystyle f_{n}}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle g}$

Integrability

For the Riemann integral on intervals, integration and limit value formation can be swapped with uniform convergence:

All are (Riemann-) integratable. If converges uniformly to , then Riemann is integrable, and the integral of is the limit of the integrals of . ${\displaystyle f_{n}}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle f_{n}}$

An example of a point-by-point, but not uniformly convergent sequence of functions, in which the integral cannot be interchanged with the limit value, is provided by this sequence of functions: For each the function is defined by ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle f_{n}\colon [0,2]\to \mathbb {R} }$

${\displaystyle f_{n}(x)={\begin{cases}n^{2}x&0\leq x\leq 1/n\\2n-n^{2}x&1/n\leq x\leq 2/n\\0&x\geq 2/n\end{cases}}}$

continuous and therefore Riemann integrable. The following applies to the integral

${\displaystyle \int _{0}^{2}f_{n}(x)\,\mathrm {d} x=1}$.

The sequence of functions converges point by point to the null function for all . So is ${\displaystyle (f_{n})_{n\in \mathbb {N} }\;}$ ${\displaystyle f(x)=0}$ ${\displaystyle x\in [0,2]}$

${\displaystyle 1=\lim _{n\to \infty }\int _{0}^{2}f_{n}(x)\,\mathrm {d} x\neq \int _{0}^{2}\lim _{n\to \infty }f_{n}(x)\,\mathrm {d} x=0.}$

Point-wise convergence is therefore not sufficient for limit value and integral sign to be interchanged.

Dini's theorem

→ Main article : Dini's theorem

If there is a compact interval and a monotonic sequence of continuous functions (i.e. ≥ or ≤ for each and every one ) that converges point-wise to an equally continuous function , then it also converges uniformly. ${\displaystyle I}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$ ${\displaystyle f_{n+1}(x)}$ ${\displaystyle f_{n}(x)}$ ${\displaystyle f_{n+1}(x)}$ ${\displaystyle f_{n}(x)}$ ${\displaystyle n}$ ${\displaystyle x}$ ${\displaystyle f}$ ${\displaystyle (f_{n})_{n\in \mathbb {N} }}$

Generalizations

Uniform convergence of complex sequences of functions

definition

The uniform convergence for complex function sequences is defined in the same way as in the case of real function sequences. A sequence of functions

${\displaystyle F=(f_{n}\colon D_{f}\subseteq \mathbb {C} \to \mathbb {C} )_{n\in \mathbb {N} }}$

means against

${\displaystyle f\colon D_{f}\subseteq \mathbb {C} \to \mathbb {C} }$

uniformly convergent if

${\displaystyle \forall \varepsilon \in \mathbb {R} _{+}\ \exists N\in \mathbb {N} \ \forall z\in D_{f}\ \forall n\geq N:\left|f_{n}(z)-f(z)\right|<\varepsilon .}$

Chordal uniform convergence

${\displaystyle F}$is called chordal uniformly convergent if

${\displaystyle \forall \varepsilon \in \mathbb {R} _{+}\ \exists N\in \mathbb {N} \ \forall z\in D_{f}\ \forall n\geq N:\chi (f_{n}(z),f(z))<\varepsilon ,}$

in which

${\displaystyle \chi (w,z)={\frac {|w-z|}{\sqrt {(1+|w|^{2})(1+|z|^{2})}}}}$

is the term for chordal distance.

Be

• ${\displaystyle {\mathfrak {K}}(D)\,}$ the class of function sequences that are uniformly convergent, ${\displaystyle D}$
• ${\displaystyle {\mathfrak {H}}(D)\,}$ the class of function sequences uniformly convergent on chordal and ${\displaystyle D}$
• ${\displaystyle {\mathfrak {B}}(D)\,}$ the class of the function sequences pointwise convergent to an in bounded function. ${\displaystyle D}$ ${\displaystyle D}$

It applies

${\displaystyle {\mathfrak {H}}(D)\cap {\mathfrak {B}}(D)\subset {\mathfrak {K}}(D)\varsubsetneq {\mathfrak {H}}(D)\,}$

properties

Similar to the uniform convergence of real function sequences, the uniform limit value can also be interchanged with the differential or the curve integral in complexes .

Uniform convergence μ-almost everywhere

→ Main article : Uniform convergence μ-almost everywhere

The uniform convergence μ-almost everywhere is a dimensional theoretical modification of the uniform convergence. It only demands uniform convergence on almost all points. On a zero set there does not have to be any uniform convergence or even no convergence at all. The uniform convergence corresponds to the convergence in the p-th mean for the borderline case and can thus be embedded in the theory of Lp spaces via the corresponding integral norms by means of the essential supremum . One then speaks of the convergence in . ${\displaystyle p\to \infty }$ ${\displaystyle {\mathcal {L}}^{\infty }}$

Almost uniform convergence

→ Main article : Almost uniform convergence

Like the uniform convergence μ-almost everywhere, the almost uniform convergence is a measure-theoretical variant of the uniform convergence. It demands that there is uniform convergence on the complement of a set of arbitrarily small measure. This is a real tightening of the uniform convergence μ-almost everywhere.

Uniform convergence in metric spaces

Be a quantity, a metric space and a sequence of functions. This sequence of functions is uniformly convergent to , if all one does so ${\displaystyle S}$ ${\displaystyle (M,d)}$ ${\displaystyle (f_{n}\colon S\to M)_{n\in \mathbb {N} }}$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle N\in \mathbb {N} }$ ${\displaystyle \forall n\geq N}$

${\displaystyle \sup _{x\in S}\ d(f_{n}(x),f(x))<\varepsilon }$

applies.

Uniform convergence in uniform spaces

Entirely analogously, uniform convergence of functions can in a uniform space with a system of neighborhoods define: a filter (or more generally a filter base ) on the set of functions for a quantity converges to a function , if for each neighborhood a exists, so ${\displaystyle Y}$ ${\displaystyle \Phi }$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle X\to Y}$ ${\displaystyle X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle E\in \Phi }$ ${\displaystyle F\in {\mathcal {F}}}$

${\displaystyle \left\{\left(f(x),g(x)\right)\mid x\in X,\ g\in F\right\}\subseteq E}$.

See also

• The concept of uniform convergence for the special case of bounded functions is the same as that of convergence with respect to the supremum norm .
• Compact convergence
• Locally uniform convergence
• Abelian criterion for uniform convergence

literature

• Klaus Viertel: History of Uniform Convergence . Springer 2014

Web links

• Eric W. Weisstein : uniform convergence . In: MathWorld (English).
• uniform convergence (lecture Saarbrücken University)

Individual evidence

1. ↑ St. Goebbels, St. Ritter: Understanding and applying mathematics - from the basics to Fourier series and Laplace transformation. Spektrum, Heidelberg 2011. ISBN 978-3-8274-2761-8 , pp. 360-369.
2. ^ Anton Deitmar: Analysis . 2nd Edition. Springer Spectrum, Tübingen, S. 147 .
3. ↑ Seidel: Note on a property of the series, which represent discontinuous functions . In: Treatises of the Mathem.-Physikalische Class of the Royal Bavarian Academy of Sciences , Volume 5, 1847, pp. 381–394. Reissued by Heinrich Liebmann in 1900 in the Ostwalds Klassiker series with an essay by Dirichlet (1837).
4. ^ Stokes: On the critical values ​​of sums of periodic series . 1847. In: Stokes, Mathematical and Physical Papers , Volume 1, Cambridge UP, 1880, p. 237, archive.org
5. ↑ This is doubted in the book by Klaus Viertel, just like the conclusion by Alfred Pringsheim in the Encyclopedia of Mathematical Sciences (1899) Cauchy would have sharply defined the term uniform convergence in 1853 and found it independent of Seidel and Stokes.
6. ↑ Klaus Viertel: History of Uniform Convergence . Springer, 2014
7. ↑ ^{a b c} H. Heuser: Textbook of Analysis. BG Teubner, Stuttgart 1984, ISBN 3-519-22221-3 , Part 1, XIII., 103., 106.
8. ^ V. Zorich: Analysis II. Springer, 2007, ISBN 978-3-540-46231-6 .
9. ↑ ^{a b c} F. Hausdorff: Fundamentals of set theory. 1914, Chelsea Publishing Co., New York 1949, chap. IX, § 4.