View source for Generalized almost-periodic functions

Classes of functions that are various generalizations of almost-periodic functions. Each of them generalizes some aspects of Bohr and Bochner almost-periodic functions (cf. [[Bohr almost-periodic functions|Bohr almost-periodic functions]]; [[Bochner almost-periodic functions|Bochner almost-periodic functions]]). The following mathematical concepts (structures) occur in the definitions of Bohr and Bochner almost-periodicity: 1) the space of continuous functions defined on the whole line, regarded as a metric space with metric (distance)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

2) a mapping of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437602.png" /> into the complex plane <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437603.png" /> (a function); 3) the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437604.png" /> as a group; and 4) the line as a topological space.

The existing generalizations of almost-periodic functions can conveniently be classified according to these structures.

1) If instead of continuity one requires the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437605.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437606.png" />, to be measurable with summable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437607.png" />-th power on each bounded interval, then one of the following three expressions can be taken for the distance:

the Stepanov distance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437608.png" /></td> </tr></table>

the Weyl distance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g0437609.png" /></td> </tr></table>

the Besicovitch distance:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376010.png" /></td> </tr></table>

Corresponding to these distances one has the generalized Stepanov, Weyl and Besicovitch almost-periodic functions (cf. [[Stepanov almost-periodic functions|Stepanov almost-periodic functions]]; [[Besicovitch almost-periodic functions|Besicovitch almost-periodic functions]]; [[Weyl almost-periodic functions|Weyl almost-periodic functions]]).

2) Suppose the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376011.png" /> is mapped not into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376012.png" />, but into a Banach space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376013.png" />. Such a mapping is called an abstract function. Suppose that the abstract functions are continuous and that the distance between them is defined by formula (*) with the modulus replaced by the norm. Then the definitions of Bohr and Bochner can be generalized and lead to the so-called abstract almost-periodic functions.

A further generalization can be obtained by replacing the Banach space by a topological vector space. In this case for every neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376014.png" /> of zero a real number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376015.png" /> is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376017.png" />-almost-period of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376018.png" /> whenever <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376019.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376020.png" />.

If the norm-topology is replaced by the weak topology, then one obtains the so-called weak almost-periodic functions: A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376022.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376023.png" />, is called weakly almost-periodic if for any functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376025.png" /> is a numerical almost-periodic function.

3) Suppose that instead of the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376026.png" /> one considers an arbitrary (not necessarily topological) group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376027.png" /> and a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376028.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376029.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376030.png" /> into a topological vector space (in particular, into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376031.png" />). As a definition of almost-periodic functions it is convenient to take Bochner's definition: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376032.png" /> is called an almost-periodic function on the group if the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376034.png" /> (or, equivalently, the family <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376035.png" />), is conditionally compact with respect to uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376036.png" /> (cf. [[Almost-periodic function on a group|Almost-periodic function on a group]]).

4) In the definition of almost-periodic functions on a group, the important thing is not the group operation itself, but the displacement operator on functions: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376037.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376038.png" />), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376039.png" />. Hence a further generalization of almost-periodic functions is obtained by generalizing the displacement operator. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376040.png" /> be an abstract space (not necessarily a group) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376042.png" />, be a function defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376043.png" />. Linear operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376045.png" />, are called [[Generalized displacement operators|generalized displacement operators]] if the following axioms are satisfied:

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376046.png" />) associativity: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376047.png" />;

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376048.png" />) the existence of a neutral element, that is, an element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376050.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376051.png" /> is the identity operator.

A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376053.png" />, is called almost-periodic relative to the family of generalized displacement operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376054.png" /> if the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376055.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376056.png" /> a parameter) is conditionally compact with respect to uniform convergence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376057.png" />. It must be noted that the theory of such functions is still poorly developed, even relative to specific families of generalized displacement operators (see [[#References|[1]]], [[#References|[5]]]).

5) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376058.png" /> be a finite or countable set of real numbers. Suppose that the line <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376059.png" /> is made into a topological vector space by defining a neighbourhood of the origin as a set of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376060.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376062.png" /> (the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376063.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376064.png" /> are chosen arbitrarily and determine the neighbourhood). It turns out that the Bohr almost-periodic functions coincide with the functions that are uniformly continuous in this topology (for the numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376065.png" /> one may take the Fourier indices of the function or an integral basis of them). Functions that are continuous in this topology provide another generalization of almost-periodic functions. These are the so-called Levitan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376067.png" />-almost-periodic functions. The definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376068.png" />-almost-periodic functions can be carried over in an obvious way to functions defined on an Abelian group (and, less obviously, to non-commutative groups).

The so-called asymptotic almost-periodic functions introduced by M. Fréchet (see [[#References|[9]]], [[#References|[10]]]) in connection with certain problems of ergodic theory do not fit particularly well into the above classification of generalized almost-periodic functions. A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376069.png" /> is called an asymptotic almost-periodic function if for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376070.png" /> and every arbitrary sequence of real numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376071.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376072.png" />, there exists a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376074.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376075.png" /> converges uniformly for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376076.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.M. Levitan,   "Almost-periodic functions" , Moscow  (1953)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.S. Besicovitch,   "Almost periodic functions" , Cambridge Univ. Press  (1932)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Amerio,   G. Prouse,   "Almost-periodic functions and functional equations" , v. Nostrand-Reinhold  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S. Bochner,   "Abstrakte fastperiodische Funktionen"  ''Acta Math.'' , '''61'''  (1933)  pp. 149–184</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.A. Marchenko,   "Some questions in the theory of one-dimensional linear second-order differential operators"  ''Trudy Moskov. Mat. Obshch.'' , '''2'''  (1953)  pp. 3–83  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  B.Ya. Levin,   "On the almost-periodic functions of Levitan"  ''Ukrain. Mat. Zh.'' , '''1'''  (1949)  pp. 49–101  (In Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.S. Besicovitch,   H. Bohr,   "Almost periodicity and general trigonometric series"  ''Acta Math.'' , '''57'''  (1931)  pp. 203–292</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.M. Levitan,   V.V. Zhikov,   "Almost-periodic functions and differential equations" , Cambridge Univ. Press  (1982)  (Translated from Russian)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  M. Fréchet,   "Les fonctions asymptotiquement presque-périodiques continues"  ''C.R. Acad. Sci. Paris'' , '''213'''  (1941)  pp. 520–522</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  M. Fréchet,   "Les transformations asymptotiquement presque périodiques discontinues et le lemme ergodique I"  ''Proc. Roy. Soc. Edinburgh Sect. A'' , '''63'''  (1950)  pp. 61–68</TD></TR></table>



====Comments====
More about this topic can be found in the articles [[Almost-periodic function|Almost-periodic function]] and [[Almost-periodic function on a group|Almost-periodic function on a group]].

In addition to the notion of weak almost-periodicity as defined above, there is another one which applies to complex-valued functions on a topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376077.png" /> (but can easily be generalized to functions with values in an arbitrary Banach space): A bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376078.png" /> is called weakly almost-periodic whenever the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376079.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376080.png" />, is conditionally compact with respect to the weak topology in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376081.png" /> of all bounded continuous functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376082.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376083.png" />. See [[#References|[a3]]], [[#References|[a1]]] and [[#References|[a2]]]. In [[#References|[a6]]] it is shown that these definitions are not equivalent for vector-valued functions.

For almost-periodicity with respect to (specific) families of generalized displacement operators, see [[#References|[a5]]]. (In the above definition the sub-index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376084.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376085.png" /> denotes that the generalized displacement operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376086.png" /> is applied to a function of the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376087.png" />. Thus, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376088.png" /> is obtained by applying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376089.png" /> to the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376090.png" />.) Of the same flavour is the notion of an almost-periodic function on a transformation group: If a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376091.png" /> acts continuously on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376092.png" />, then a bounded continuous function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376093.png" /> is said to be (weakly) almost-periodic on the transformation group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376094.png" /> whenever the family of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376096.png" />, is conditionally compact with respect to the uniform (respectively, weak) topology in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376097.png" />. See e.g. [[#References|[a4]]].

More about Levitan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043760/g04376099.png" />-almost-periodic functions can be found in [[#References|[8]]] and [[#References|[a7]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R.B. Burckel,   "Weakly almost-periodic functions on semigroups" , Gordon &amp; Breach  (1970)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  K.S. de Leeuw,   I. Glicksberg,   "Almost periodic functions on semigroups"  ''Acta Math.'' , '''105'''  (1961)  pp. 99–140</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  W.F. Eberlein,   "Abstract ergodic theorems and weak almost periodic functions"  ''Trans. Amer. Math. Soc.'' , '''67'''  (1949)  pp. 217–240</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M.B. Landstadt,   "On the Bohr compactification of a transformation group"  ''Math. Z.'' , '''127'''  (1972)  pp. 167–178</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  B.M. Levitan,   "The application of generalized displacement operators to linear differential equations of the second order"  ''Transl. Amer. Math. Soc. (1)'' , '''10'''  (1950)  pp. 408–451  ''Uspekhi Math. Nauk'' , '''4''' :  1(29)  (1949)  pp. 3–112</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  P. Milnes,   "On vector-valued weakly almost periodic functions"  ''J. London Math. Soc. (2)'' , '''22'''  (1980)  pp. 467–472</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  A. Reich,   "Präkompakte Gruppen und Fastperiodicität"  ''Math. Z.'' , '''116'''  (1970)  pp. 216–234</TD></TR></table>