Almost periodic function

Almost periodic functions are considered in the mathematical sub-area of harmonic analysis . These are functions that are defined in groups and that are periodic apart from a small deviation . They were introduced by Harald Bohr in 1924/1925 and proved to be an important tool for investigating the representation theory of groups, in particular their finite-dimensional representations. The latter was carried out with a slightly modified definition by Hermann Weyl , another variant goes back to John von Neumann .

Almost periodic functions according to Bohr

Bohr generalized the concept of the periodic function defined on the set of real numbers . As a reminder, a function is called periodic with period , if for all , as is known from the functions sine and cosine . Such a number is called a period. Obviously, whole-number multiples of such periods are also periods. Hence there is such a period in every closed interval of length . ${\displaystyle \mathbb {R} }$ ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {C} }$ ${\displaystyle \tau >0}$ ${\displaystyle f(x+\tau )=f(x)}$ ${\displaystyle x\in \mathbb {R} }$ ${\displaystyle \tau }$ ${\displaystyle [a,a+\tau ]}$ ${\displaystyle \tau }$

A continuous function is called almost periodic (according to Bohr) if there is a number for each such that each interval of length contains a number such that ${\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {C} }$ ${\displaystyle \varepsilon >0}$ ${\displaystyle L_{\varepsilon }>0}$ ${\displaystyle [a,a+L_{\varepsilon }]}$ ${\displaystyle L_{\varepsilon }}$ ${\displaystyle \tau }$

${\displaystyle |f(x+\tau )-f(x)|<\varepsilon }$for all real numbers . ${\displaystyle x}$

As stated above, every continuous, periodic function is evidently almost periodic. In the following, such functions are called more precisely almost periodic according to Bohr in order to distinguish them from the following variants.

Almost periodic functions according to Weyl

The variant presented here goes back to Hermann Weyl. The definition has a somewhat more complicated structure, but can be formulated for any groups.

definition

A function defined on a group is called almost periodic if there are finitely many pairwise disjoint sets for each with ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle \varepsilon >0}$ ${\displaystyle A_{1},\ldots ,A_{n}\subset G}$

${\displaystyle G=A_{1}\cup \ldots \cup A_{n}}$ and

${\displaystyle |f(axb)-f(ayb)|<\varepsilon }$for everyone . ${\displaystyle a,b\in G,\,x,y\in A_{i},\,i\in \{1,\ldots ,n\}}$

This estimate is valid if only and come from the same part of the group. ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle A_{i}}$

With this definition, it is also not clear that periodic functions are almost periodic, and for discontinuous functions this is even wrong. The relationship to Bohr's definition, which explicitly refers to continuous functions, looks like this: On the group, the almost periodic functions according to Bohr agree with the continuous, almost periodic functions, in particular continuous, periodic functions are almost periodic. ${\displaystyle G=\mathbb {R} }$ ${\displaystyle \mathbb {R} }$

Multiple, complex conjugates , sums and products of almost periodic functions are again almost periodic, as are uniform limit values of sequences of almost periodic functions. The set of almost periodic functions thus forms a closed function algebra , even a C * -algebra . ${\displaystyle AP(G)}$

Mean values

In the representation theory of finite groups , one forms sums averaged for functions . For infinite groups such mean values ​​can still be obtained for almost periodic functions; ${\displaystyle G=\{x_{1},\ldots ,x_{n}\}}$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle \textstyle {\frac {1}{n}}\sum _{i=1}^{n}f(x_{i})}$

Mean value theorem : For every almost periodic function there is a uniquely determined number , the so-called mean value of , so that for each there are finitely many with ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle M(f)}$ ${\displaystyle f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle x_{1},\ldots ,x_{n}\in G}$

${\displaystyle \left|M(f)-{\frac {1}{n}}\sum _{i=1}^{n}f(ax_{i}b)\right|<\varepsilon }$for everyone . ${\displaystyle a,b\in G}$

The proof uses a clever selection of subdivisions of the group, as in the definition of near-periodicity above; the marriage rate is used in this more or less combinatorial approach .

The mean value is linear and monotonic and it is , where the constant function with value 1 is denoted by. The mean can therefore be used like an integral. If there are about two almost periodic functions, then is through ${\displaystyle M(1_{G})=1}$ ${\displaystyle 1_{G}}$ ${\displaystyle G}$ ${\displaystyle f,g\colon G\rightarrow \mathbb {C} }$

${\displaystyle \langle f,g\rangle :=M(f\cdot {\overline {g}})}$

defines a scalar product that makes a prehilbert dream. ${\displaystyle AP(G)}$

Law on almost periodic functions

The group operates on through the formula ${\displaystyle G}$ ${\displaystyle AP(G)}$

${\displaystyle (x\cdot f)(y):=f(yx),\,x,y\in G,f\in AP(G)}$,

that is , it becomes a module that is complete with regard to uniform convergence. A sub-module is called invariant if it is closed under the group operation, it is called closed if it is closed with regard to uniform convergence, and it is called irreducible if it contains no further invariant sub-modules apart from the zero module and itself. By using the Prehilbert space structure introduced above, one can show the so-called main theorem about almost periodic functions: ${\displaystyle AP(G)}$ ${\displaystyle G}$

Every closed, invariant sub-module of is a uniform closure of a vector space sum of finite-dimensional, invariant, irreducible sub-modules. ${\displaystyle AP(G)}$

This is how you master representation theory, if only it is sufficiently rich. In extreme cases, however , it can only consist of the constant functions, then the main theorem is trivial. If there is a compact group , one can show that every continuous function is almost periodic, which then leads to the well-known representation theory of compact groups , in particular the case of finite groups is included. ${\displaystyle AP(G)}$ ${\displaystyle AP(G)}$ ${\displaystyle G}$ ${\displaystyle G\rightarrow \mathbb {C} }$

Almost periodic functions according to von Neumann

J. von Neumann has found another approach using Haar's measure , which was not yet available for the developments described so far, which clarifies in particular the nature of the above mean value.

If a mapping is on a group and is , then the functions and are defined by the formulas . A bounded function is then almost periodic if and only if the sets and in the metric space of the bounded functions are totally bounded with the metric defined by the supremum norm . ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle x\in G}$ ${\displaystyle f_{x}}$ ${\displaystyle {}_{x}f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle f_{x}(y):=f(xy),{}_{x}f(y):=f(yx),y\in G}$ ${\displaystyle G\rightarrow \mathbb {C} }$ ${\displaystyle \{f_{x}\colon x\in G\}}$ ${\displaystyle \{{}_{x}f\colon x\in G\}}$ ${\displaystyle G\rightarrow \mathbb {C} }$

This condition is von Neumann's definition. With this approach von Neumann was able to show, among other things, that every compact group, which is a ( finite-dimensional ) topological manifold as a topological space , is a Lie group , which solved Hilbert's fifth problem for compact groups.

The mean value that made the theory described above possible in the first place is as follows. First we show that for every topological group there is a compact group and a continuous group homomorphism with the following universal property : For every continuous group homomorphism in a compact group there is exactly one continuous group homomorphism , so that . Such a compact group is uniquely defined except for isomorphism and is called the compact group to be associated or the Bohr compactification of . Furthermore, one can show that a bounded function is almost periodic if and only if there is a function with . With these terms the following applies to an almost periodic function : ${\displaystyle G}$ ${\displaystyle \Sigma }$ ${\displaystyle \alpha \colon G\rightarrow \Sigma }$ ${\displaystyle {\tilde {\alpha }}\colon G\rightarrow {\tilde {\Sigma }}}$ ${\displaystyle {\tilde {\Sigma }}}$ ${\displaystyle \beta \colon \Sigma \rightarrow {\tilde {\Sigma }}}$ ${\displaystyle {\tilde {\alpha }}=\beta \circ \alpha }$ ${\displaystyle \Sigma }$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$ ${\displaystyle {\tilde {f}}\colon \Sigma \rightarrow \mathbb {C} }$ ${\displaystyle f={\tilde {f}}\circ \alpha }$ ${\displaystyle f\colon G\rightarrow \mathbb {C} }$

The closed, convex hull of all functions contains exactly one constant function, and this has the value . If this is Haar's measure standardized to 1, then applies . ${\displaystyle {}_{x}f,x\in G}$ ${\displaystyle M(f)}$ ${\displaystyle \mu _{\Sigma }}$ ${\displaystyle \textstyle M(f)=\int _{\Sigma }{\tilde {f}}\,\mathrm {d} \mu _{\Sigma }}$

This results in the mean value here in a very natural way. The further theory indicated above can now be built on this mean value.

An even more abstract approach can be found in. The set of bounded almost periodic functions on a group forms a commutative C * -algebra with one element, this is, according to the Gelfand-Neumark theorem, isometrically isomorphic to an algebra of continuous functions on a compact space , which corresponds to the space of all homomorphisms of the commutative C * -Algebra can be identified according to (see Gelfand transformation ). Since the score for each is such a homomorphism , a map is obtained . From this one can show that it is continuous and that the group operation continues from on . The group to be associated (see above) is thus constructed. ${\displaystyle AP(G)}$ ${\displaystyle G}$ ${\displaystyle C(\Sigma )}$ ${\displaystyle \Sigma }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle f\mapsto f(x)}$ ${\displaystyle x\in G}$ ${\displaystyle \varphi _{x}}$ ${\displaystyle \alpha \colon G\rightarrow \Sigma ,x\mapsto \varphi _{x}}$ ${\displaystyle G}$ ${\displaystyle \Sigma }$ ${\displaystyle G}$

For a locally compact , Abelian group the associated compact group can be constructed as follows. Let be the dual group , be the same group, but provided with the discrete topology , so that the mapping is continuous. If you apply the Pontryagin duality to this, you get a continuous mapping . According to Pontryagin's duality theorem , the left side is isomorphic to and the right side is compact as a dual group of a discrete group. The associated, compact group emerges again in a very natural way. ${\displaystyle G}$ ${\displaystyle {\hat {G}}}$ ${\displaystyle {\hat {G}}_{d}}$ ${\displaystyle \mathrm {id} _{G}:{\hat {G}}_{d}\rightarrow {\hat {G}}}$ ${\displaystyle \alpha \colon {\hat {\hat {G}}}\rightarrow {\widehat {{\hat {G}}_{d}}}}$ ${\displaystyle G}$

Further terms of almost periodic functions

The defining condition in Bohr's definition of the near-periodic function can be expressed as

${\displaystyle \|f_{\tau }-f\|_{\infty }=\sup _{x\in \mathbb {R} }\left|f(x+\tau )-f(x)\right|<\varepsilon }$

be written, where be defined by. By replacing the norm with other distance terms, one arrives at other definitions. This has been implemented by some authors who, in particular, pursued a generalization to discontinuous functions. ${\displaystyle f_{\tau }}$ ${\displaystyle f_{\tau }(x):=f(x+\tau )}$ ${\displaystyle \|\cdot \|_{\infty }}$

WW Stepanow used the term distance

${\displaystyle \|f-g\|:=\sup _{x\in \mathbb {R} }\left({\frac {1}{L}}\int _{x}^{x+L}|f(t)-g(t)|^{p}\,\mathrm {d} t\right)^{\frac {1}{p}}}$,

where and . H. Weyl used this concept of distance for the borderline case . Finally, the approach of AS Besikowitsch should be mentioned, he put the concept of distance ${\displaystyle 1\leq p<\infty }$ ${\displaystyle L>0}$ ${\displaystyle L\to \infty }$

${\displaystyle \|f-g\|:=\limsup _{L\to \infty }\left({\frac {1}{2L}}\int _{-L}^{L}|f(t)-g(t)|^{p}\,\mathrm {d} t\right)^{\frac {1}{p}}}$

fundamentally.

Individual evidence

1. ↑ H. Bohr: On the theory of almost periodic functions , Acta math. Volume 45 (1924), pages 29-127, Volume 46, pages 101-214
2. ↑ W. Maak : Fast-periodic functions , basic teaching of mathematical sciences, Volume 61 (1967), Chapter IV, §24
3. ↑ W. Maak: Fast-periodic functions , basic teaching of mathematical sciences, Volume 61 (1967), Chapter II, §7
4. ↑ W. Maak: Fastperiodische Functions , Grundlehren der Mathematischen Wissenschaften, Volume 61 (1967), Chapter II, §9
5. ↑ J. v. Neumann: Almost periodic functions in a group I , Transactions Amer. Math. Soc., Volume 36 (1934), pages 445-492
6. ↑ W. Maak: Fastperiodic Functions , Basic Teachings of Mathematical Sciences, Volume 61 (1967), Chapter VI, § 35, Sentence 3
7. ↑ W. Mak: Almost periodic functions , basic teachings of Mathematical Sciences, Volume 61 (1967), Chapter VI, § 37: At Hilbert V. problem
8. ^ J. Dixmier : C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , Theorem 16.1.1
9. ^ J. Dixmier: C * -algebras and their representations , North-Holland Publishing Company (1977), ISBN 0-7204-0762-1 , Theorem 16.2.1
10. ^ L. Loomis : Abstract Harmonic Analysis , Van Nostrand 1953, Chapter VIII: Compact Groups and Almost Periodic Functions
11. ↑ L. Loomis : Abstract Harmonic Analysis , Van Nostrand 1953, Chapter VIII, Section 41E
12. ^ VV Stepanov: Sur quelques généralisations des fonctions presque périodiques , CR Acad. Sci. Paris, Volume 181 (1925), pages 90-92
13. ↑ H. Weyl: Integral equations and almost periodic functions , Math. Annalen, Volume 97 (1927), pages 338-356
14. ^ AS Besicovitch: Almost periodic functions , Cambridge Univ. Press (1932)