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; 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)

$$ \tag{* } \rho \{ f , g \} = \ \sup _ {x \in \mathbf R ^ {1} } \ | f ( x) - g ( x) | ; $$

2) a mapping of the line $ \mathbf R ^ {1} $ into the complex plane $ \mathbf C ^ {1} $( a function); 3) the line $ \mathbf R ^ {1} $ 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 $ f ( x) $, $ x \in \mathbf R ^ {1} $, to be measurable with summable $ p $- th power on each bounded interval, then one of the following three expressions can be taken for the distance:

the Stepanov distance:

$$ \rho _ {S _ {l} ^ {p} } \{ f , g \} = \ \sup _ {x \in \mathbf R ^ {1} } \ \left \{ \frac{1}{l} \int\limits _ { x } ^ { x+ } l | f ( x) - g ( x) | ^ {p} \ d x \right \} ^ {1/p} ; $$

the Weyl distance:

$$ \rho _ {W ^ {p} } \{ f , g \} = \ \lim\limits _ {l \rightarrow \infty } \ \rho _ {S _ {l} ^ {p} } \{ f , g \} ; $$

the Besicovitch distance:

$$ \rho _ {B ^ {p} } \{ f , g \} = \ \left \{ \overline{\lim\limits}\; _ {T \rightarrow \infty } \ \frac{1}{2T} \int\limits _ { - } T ^ { + } T | f ( x) - g ( x) | ^ {p} d x \right \} ^ {1/p} . $$

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

2) Suppose the line $ \mathbf R ^ {1} $ is mapped not into $ \mathbf C ^ {1} $, but into a Banach space $ B $. 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 $ U $ of zero a real number $ \tau = \tau _ {U} $ is called an $ U $- almost-period of $ f $ whenever $ f ( x + \tau ) - f ( x) \in U $ for all $ x \in \mathbf R $.

If the norm-topology is replaced by the weak topology, then one obtains the so-called weak almost-periodic functions: A function $ f ( x) $, $ x \in \mathbf R ^ {1} $, $ f \in B $, is called weakly almost-periodic if for any functional $ \phi \in B ^ {*} $, $ \phi ( f ( x) ) $ is a numerical almost-periodic function.

3) Suppose that instead of the line $ \mathbf R ^ {1} $ one considers an arbitrary (not necessarily topological) group $ G $ and a mapping $ f ( x) $, $ x \in G $, of $ G $ into a topological vector space (in particular, into $ \mathbf C ^ {1} $). As a definition of almost-periodic functions it is convenient to take Bochner's definition: $ f $ is called an almost-periodic function on the group if the family of functions $ f ( x h ) $, $ h \in G $( or, equivalently, the family $ f ( h x ) $), is conditionally compact with respect to uniform convergence on $ G $( cf. 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: $ T ^ {h} f ( x) = f ( x h ) $( or $ f ( h x ) $), $ x , h \in G $. Hence a further generalization of almost-periodic functions is obtained by generalizing the displacement operator. Let $ \Omega $ be an abstract space (not necessarily a group) and let $ f ( x) $, $ x \in \Omega $, be a function defined on $ \Omega $. Linear operators $ T ^ {h} $, $ h \in \Omega $, are called generalized displacement operators if the following axioms are satisfied:

$ \alpha $) associativity: $ T _ {h} ^ {g} T _ {x} ^ {h} f ( x) = T _ {x} ^ {h} T _ {x} ^ {g} f ( x) $;

$ \beta $) the existence of a neutral element, that is, an element $ h _ {0} \in \Omega $ such that $ T ^ {h _ {0} } = I $, where $ I $ is the identity operator.

A function $ f ( x) $, $ x \in \Omega $, is called almost-periodic relative to the family of generalized displacement operators $ T ^ {h} $ if the family of functions $ T ^ {h} f ( x) $( $ h $ a parameter) is conditionally compact with respect to uniform convergence on $ \Omega $. It must be noted that the theory of such functions is still poorly developed, even relative to specific families of generalized displacement operators (see [1], [5]).

5) Let $ \lambda _ {1} \dots \lambda _ {n} \dots $ be a finite or countable set of real numbers. Suppose that the line $ \mathbf R ^ {1} $ is made into a topological vector space by defining a neighbourhood of the origin as a set of real numbers $ x $ satisfying $ | e ^ {i \lambda _ {n} x } - 1 | < \epsilon $, $ n = 1 \dots N $( the numbers $ \epsilon $ and $ N $ 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 $ \{ \lambda _ {k} \} $ 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 $ N $- almost-periodic functions. The definition of $ N $- 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 [9], [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 $ f : \mathbf R ^ {1} \rightarrow \mathbf C ^ {1} $ is called an asymptotic almost-periodic function if for every $ \alpha \in \mathbf R ^ {1} $ and every arbitrary sequence of real numbers $ \{ h _ {n} \} $, with $ h _ {n} \rightarrow \infty $, there exists a subsequence $ \{ k _ {n} ^ {( \alpha ) } \} $ of $ \{ h _ {n} \} $ for which $ f ( x + k _ {n} ^ {( \alpha ) } ) $ converges uniformly for all $ x > \alpha $.

References

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

Comments

More about this topic can be found in the articles Almost-periodic function and 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 $ G $( but can easily be generalized to functions with values in an arbitrary Banach space): A bounded continuous function $ f: G \rightarrow \mathbf C $ is called weakly almost-periodic whenever the family of functions $ x \mapsto f ( xh) $, $ h \in G $, is conditionally compact with respect to the weak topology in the space $ C ( G, \mathbf C ) $ of all bounded continuous functions from $ G $ to $ \mathbf C $. See [a3], [a1] and [a2]. In [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 [a5]. (In the above definition the sub-index $ x $ in $ T _ {x} ^ {g} $ denotes that the generalized displacement operator $ T ^ {g} $ is applied to a function of the variable $ x $. Thus, $ T _ {h} ^ {g} T _ {x} ^ {h} f ( x) $ is obtained by applying $ T ^ {g} $ to the function $ h \mapsto ( T ^ {h} f ) ( x) $.) Of the same flavour is the notion of an almost-periodic function on a transformation group: If a group $ G $ acts continuously on a space $ X $, then a bounded continuous function $ f: X \rightarrow \mathbf C $ is said to be (weakly) almost-periodic on the transformation group $ ( G, X) $ whenever the family of functions $ x \mapsto f ( tx) $, $ t \in G $, is conditionally compact with respect to the uniform (respectively, weak) topology in the space $ C ( X, G) $. See e.g. [a4].

More about Levitan $ N $- almost-periodic functions can be found in [8] and [a7].

References

[a1]	R.B. Burckel, "Weakly almost-periodic functions on semigroups" , Gordon & Breach (1970)
[a2]	K.S. de Leeuw, I. Glicksberg, "Almost periodic functions on semigroups" Acta Math. , 105 (1961) pp. 99–140
[a3]	W.F. Eberlein, "Abstract ergodic theorems and weak almost periodic functions" Trans. Amer. Math. Soc. , 67 (1949) pp. 217–240
[a4]	M.B. Landstadt, "On the Bohr compactification of a transformation group" Math. Z. , 127 (1972) pp. 167–178
[a5]	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
[a6]	P. Milnes, "On vector-valued weakly almost periodic functions" J. London Math. Soc. (2) , 22 (1980) pp. 467–472
[a7]	A. Reich, "Präkompakte Gruppen und Fastperiodicität" Math. Z. , 116 (1970) pp. 216–234