# Some spaces of almost periodic functions.

1 The construction of the space A[P.sub.[phi]](R, C)

We will consider an arbitrary function [phi] [member of] K, and we will use the approach already used in our papers quoted above, namely, to start with a class of formal Fourier/trigonometric series of the form

[mathematical expression not reproducible], (1.1)

denoting by A[P.sub.[phi] ](R, C) the set/class of such series for which

[mathematical expression not reproducible]. (1.2)

As usual, in this paper, we shall assume

[mathematical expression not reproducible]. (1.3)

Now we can start providing some structure to the set/class A[P.sub.[phi]](R, C), asking one more condition for [phi]. Namely, the condition of subadditivity

[mathematical expression not reproducible] (1.4)

will be accepted. Functions [phi] satisfying (1.4) can be the linear ones, like [phi](u) = [lambda]u, [lambda], u [member of] [R.sub.+], but also nonlinear ones, like [phi](u) = [square root of (term)]u. The case [phi](u) = [u.sup.r], 1 [less than or equal to] r [less than or equal to] 2, is discussed in our papers quoted above, leading to the spaces A[P.sub.r](R, C), r [member of] [1, 2], which constitute a scale of function/spaces starting with A[P.sub.1] (R, C) = the space of Poincare of almost periodic functions (actually a Banach algebra, see Corduneanu [4], for instance), and ending with the Besicovitch space [[beta].sub.2](R, C).

On behalf of (1.4), one can easily find that A[P.sub.[phi]](R, C) is an additive group, with the usual operation of addition. Let us notice that condition (1.2) is the same for both series (1.1) and its opposite (i.e., with the coefficients -[a.sub.k], k [greater than or equal to] 1, instead of [a.sub.k].

Because of the nonlinearity of the function [phi], in the definition of the space A[P.sub.[phi]] (R, C), it is not (generally) possible to define a norm, by means of the function [phi]. The condition [parallel]c[sigma][parallel] = |c [sigma][parallel], with [sigma] denoting a series like (1.1), will be violated. We know, from the case of the spaces A[P.sub.r](R, C), 1 [less than or equal to] r [less than or equal to] 2, that a norm can be defined (the Minkowski norm) for the space [[??].sup.r], 1 [less than or equal to] r [less than or equal to] 2.

But, it turns out, that it is possible to define a distance function on A[P.sub.[phi]], thus obtaining a topological structure.

Consider two arbitrary series of the form (1.1), say (1.1) and

[mathematical expression not reproducible]. (1.5)

Now, define the distance function d(.,.) by the formula

[mathematical expression not reproducible], (1.6)

with d : A[P.sub.[phi]] x A[P.sub.[phi]] [right arrow] [R.sub.+].

From condition (1.2), one finds |[a.sub.k] - [b.sub.k]| [less than or equal to] |[a.sub.k]| + |[b.sub.k]|, k [greater than or equal to] 1, which leads to the inequality

[phi](|[a.sub.k] - [b.sub.k]|) [less than or equal to] [phi](|[a.sub.k]|) + [phi](|[b.sub.k]|), k [greater than or equal to] 1,

relying also on (1.2). This inequality shows that the right hand side in (1.6) is always defined. Let us prove now that d(.,.), defined by (1.6), is indeed a distance function on AP|[phi](R, C), i.e., the space of series A[P.sub.[phi]](R, C), subject to the conditions imposed above on the function [phi].

First, from (1.6) we see that d(.,.) [greater than or equal to] 0 for any couple of series (1.1), the possibility of having d(.,.) = 0 appearing only in the case [a.sub.k] = [b.sub.k], k [greater than or equal to] 1, which implies the identity of the series involved.

Second, taking into account the obvious inequalities

|[a.sub.k] - [c.sub.k]| [less than or equal to] |[a.sub.k] - [b.sub.k]| + |[b.sub.k] - [c.sub.k]|, k [greater than or equal to] 1,

one obtains the triangle inequality for the function d(.,.).

Third, the symmetry of the function d(.,.) is assured by the obvious relations |[a.sub.k] - [b.sub.k]| = |[b.sub.k] - [a.sub.k]|, k [greater than or equal to] 1.

Therefore, on behalf of the above considerations, we can conclude that the set of series of the form (1.1), under the main assumptions (1.2), (1.4), can be organized as a metric space, with the distance function given by (1.6).

Definition 1.1. One denotes also by A[P.sub.[phi]](R, C) the metric space whose elements are the trigonometric series of the form (1.1), under the main assumptions (1.2), (1.4), and with the distance function defined by (1.6).

Several elementary properties of the space A[P.sub.[phi]](R, C), about its algebraic and topological structures have been already mentioned above: algebraically it is an additive and commutative group of formal series and topologically (convergence) it is a metric space.

The next property of the space A[P.sub.[phi]](R, C), very important in making it a real tool in nonlinear analysis, is the following: the space A[P.sub.[phi]](R, C) is complete.

For instance, it will be possible to use, in A[P.sub.[phi]](R, C), the Banach or Schauder fixed point theorems.

Proof of completeness. Let us consider a sequence of series of the form (1.1), namely

[mathematical expression not reproducible], (1.7)

with k indicating the rank of the term in the series, while j standing for the number of the series.

Let us now assume that this sequence satisfies the Cauchy (necessary and sufficient) condition of convergence

[mathematical expression not reproducible]. (1.8)

From inequality (1.8) we derive the sequence of inequalities, for each fixed k [greater than or equal to] 1,

[mathematical expression not reproducible], (1.9)

if we keep in mind that [phi] [member of] K = Kamke functions. Moreover, since [[phi].sup.1] is also a Kamke function, (1.9) is leading to

[mathematical expression not reproducible], (1.10)

and k [member of] N. Since [member of] > 0 is arbitrary, small and [[phi].sup.1] is continuous with [[phi].sup.1](0) = 0, one obtains from (1.10) the Cauchy condition for each index k [member of] N. Therefore, for each k [member of] N, there exists a unique [??] [member of] C, such that

[mathematical expression not reproducible]. (1.11)

From (1.8) and (1.11), we obtain

[mathematical expression not reproducible]. (1.12)

therefore, the sequence (of series) in (1.7) is convergent to the series

[mathematical expression not reproducible], (1.13)

which proves the completeness of the space A[P.sub.[phi]](R, C).

Theorem 1.2. The space A[P.sub.[phi]](R, C), as described above, consists of generalized trigonometric series of the form (1.1) under conditions (1.2), (1.4), being organized algebraically as an additive commutative group and topologically as a complete metric space. The continuity of the addition operation is a simple consequence of the definition of the distance d, by formula (1.6).

2 Some elementary properties of the space A[P.[phi]](R, C)

Since the space A[P.sub.[phi]](R, C) is not a linear space, it appears natural to emphasize some of its properties that are not present in case of linear topological spaces, like Hilbert or Banach spaces, usually encountered in similar cases.

First property we want to mention, to compensate somehow the fact that, for a series a [member of] A[P.sub.[phi]](R, C), it does not follow always c[sigma] [member of] A[P.sub.[phi]](R, C), c [member of] C.

One can state, nevertheless, that c[sigma] [member of] A[P.sub.[phi]](R, C), provided |c| [less than or equal to] 1, c [member of] C. Indeed, from the definition of A[P.sub.[phi]](R, C) we know that (1.2) must be satisfied. But |c| [less than or equal to] 1 implies [phi](|c| |[a.sub.k]|) [less than or equal to] [phi](|[a.sub.k]|), k [greater than or equal to] 1, and taking into account the fact that [phi] is increasing, one sees, on behalf of (1.1), that the series [mathematical expression not reproducible] converges. Therefore, ca [member of] A[P.sub.[phi]](R, C), for |c| [less than or equal to] 1, i.e., (1.2) is valid for the series ca.

Second, some arithmetic properties hold for the space A[P.sub.[phi]](R, C). Since we deal with an additive group, [sigma] [member of] A[P.sub.[phi]](R, C) obviously implies -[sigma] [member of] A[P.sub.[phi]](R, C); one sees that [sigma][+ or -][[sigma].sup.1] [member of] A[P.sub.[phi]](R, C) if [sigma], [[sigma].sup.1] [member of] A[P.sub.[phi]](R, C). Also, any finite sum [[sigma].sup.1] + [[sigma].sup.2] + ...+[[sigma].sup.n] [member of] A[P.sub.[phi]](R, C), provided each [[sigma].sup.k] [member of] A[P.sub.[phi]](R, C). k [less than or equal to] [bar.[sigma]]. Since [bar.[sigma]] = the conjugate of a [member of] A[P.sub.[phi]](R, C), also belongs to A[P.sub.[phi]](R, C), one easily derives similar properties to those mentioned above.

On the other hand, based on the first property above, one can state that any linear combination [mathematical expression not reproducible], belongs to A[P.sub.[phi]](R, C). This makes, obviously, sense when we want to deal with convexity properties in the space A[P.sub.[phi]] (R, C).

Third, another property is related to the translates of the argument in a series in A[P.sub.[phi]](R, C). Namely, if we deal with the series, obtained from (1.1), when t [right arrow] t + h, h [member of] R, one obtains the series

[mathematical expression not reproducible], (2.1)

which can be rewritten in the form

[mathematical expression not reproducible], (2.2)

where [a'.sub.k] = [a.sub.k] exp(i[[lambda].sub.k]h), k [greater than or equal to] 1. Since |[a.sub.k]| = |[a'.sub.k]|, k [greater than or equal to] 1, there results, on behalf of (1.2), that the series (1.1) and (1.5) are simultaneously in A[P.sub.[phi]](R, C) or not.

Let us now estimate, on behalf of (1.6), the distance between the series (1.1) and (2.2). One obtains

[mathematical expression not reproducible]. (2.3)

We notice that [mathematical expression not reproducible], which implies

[mathematical expression not reproducible],

because

[mathematical expression not reproducible].

From the above sequence of inequalities and (2.3), we obtain

[mathematical expression not reproducible].

We choose now N sufficiently large, such that

[mathematical expression not reproducible].

Of course, N = N([epsilon]) > 0. For an arbitrary [member of] > 0, we can obtain easily, from the last inequality above,

[mathematical expression not reproducible],

with [delta] = [delta]([epsilon], N([epsilon])) = [[delta].sub.1]([epsilon]). Taking into account the inequalities above, starting with (2.3), one obtains

[mathematical expression not reproducible],

provided |h| < [[delta].sub.1]([epsilon]). The last inequality tells us that the map h [right arrow] [f.sub.h] (t) = f (t + h) from an interval [0, [h.sub.0]] into A[P.sub.[phi]](R, C), for fixed f [member of] A[P.sub.[phi]](R, C), is uniformly continuous. In this way, we have retrieved another property from the classical theory of almost periodic functions.

3 Three basic properties of A[P.sub.[phi]](R, C)

The following properties, known for the classical almost periodic functions, can be extended to the space A[P.sub.[phi]] (R, C). namely, we have in mind the approximation property, the Bohr property which served as definition (existence of the almost periods) and the Bochner property (the compactness of the family of translates).

Before we can state and prove these basic/fundamental properties for the space A[P.sub.[phi]] (R, C), we need to establish a Lemma which has the role to clarify the connection between the space A[P.sub.[phi]](R, C), regarded as a series space and the corresponding function space A[P.sub.[phi]](R, C), which we want to construct. Of course, an isomorphism, both algebraic and topologic must take place between the two realizations of the same abstract space.

Lemma 3.1. Let us consider the space A[P.sub.[phi]](R, C) which consists of the series of the form (1.1), under condition (1.2) for its coefficients, where [phi] is a function as described above. Besides the properties listed for [phi], we will assume (throughout this paper) that [phi] : [R.sub.+] [right arrow] [R.sub.+]. Then the series

[mathematical expression not reproducible], (3.1)

is uniformly and absolutely convergent on R. We shall take its sum as the correspondent in A[P.sub.1](R, C).

Proof. Taking in account condition (1.2), one sees that the series (3.1) is absolutely and uniformly convergent on R. Indeed, one has [phi](|[a.sub.k]|) [greater than or equal to] 0, k [greater than or equal to] 1, which allows the application of the Weierstrass criterion for uniform convergence. Actually, the sum of the series (3.1) is an almost periodic function in the sense of Bohr, belonging to the space AP1(R, C).

We conclude, on behalf of the above considerations, that [phi](|[a.sub.k]|) are the Fourier coefficients of (3.1), hence leading for k [greater than or equal to] 1 to the formulas

[mathematical expression not reproducible],

from which we obtain the | [a.sub.k]|'s, k [greater than or equal to] 1.

The correspondence series [right arrow] function is thus established and we notice the fact that it is not one-to-one, but we achieve that if we introduce in A[P.sub.[phi]](R, C) a relation of equivalence, the classes of equivalence containing each of the series with the same sum.

We shall use the lemma to advance in getting other properties of the space A[P.sub.[phi]] (R, C), as mentioned above in this section: approximation by trigonometric polynomials, the Bohr definition and existence of almost periodes, the last being Bochner property, stating the compactness of the translates of a given f [member of] A[P.sub.[phi]](R, C). Of course, we have to use the distance d (formula (1.6)) for A[P.sub.[phi]](R, C).

In regard to the approximation property, we notice, on behalf of Lemma 3.1, that the series

[mathematical expression not reproducible], (3.2)

is absolutely and uniformly convergent on R. One can write, for an arbitrary [epsilon] > 0,

[mathematical expression not reproducible],

and since

[mathematical expression not reproducible], (3.3)

on behalf of (1.2), there results

[mathematical expression not reproducible],(3.4)

which proves the approximation property for an arbitrary f [member of] A[P.sub.[phi]](R, C) = the function space. One may rewrite (3.4) as

[mathematical expression not reproducible],

which emphasizes the distance functions in A[P.sub.[phi]](R, C).

The next property, to be established, is the Bohr property for classical almost periodic functions, in case of the space A[P.sub.[phi]](R, C). Since the elements of A[P.sub.[phi]](R, C) are functions belonging to the space AP1 (R, C), we may expect that further discussion is superfluous. We shall look at to the validity of this property, using again the distance function d, defined by (1.6). This distance is defined by means of the function [phi] and we shall sketch here a short proof based on the knowledge of the theory of classical almost periodic functions. Namely, in order to clarify the problem of the existence of almost periodicity (Bohr), we shall notice that the function f(t), the sum of the series (3.2), has almost periods. Indeed, the terms of the series form a sequence of almost periodic functions (Bohr), and it is convergent to zero (because p(|[a.sub.k]|) [right arrow] 0 as k [right arrow] [infinity]. Therefore, the sequence of almost periodic functions {[phi](|[a.sub.k]|) exp(i[[lambda].sub.k]t); k [greater than or equal to] 1} is a compact set in A[P.sub.1](R, C) and in AP(R). This implies the existence of common e-almost periods for all terms in the sequence of [epsilon]-almost periods for the sum f(t), for each [epsilon] > 0.

The property of compactness of the family of translates of a function, say {f (t + h), h [member of] R}, for the function f (t) [member of] A[P.sub.[phi]](R, C), which has been discovered by Boehner, can be also proved for the A[P.sub.[phi]](R, C)-almost periodic functions. It is a characteristic property of Bohr almost periodic functions.

To prove this property, actually the relative compactness of {f(t + h); h [member of] R} for any f [member of] A[P.sub.[phi]](R, C), we send the reader to the book Corduneanu [4], in which the proof is carried out for the space AP(R, C) of Bohr almost periodic functions. The proof has three steps: first, when f (t) = Aexp(i[[lambda].sub.k]t), A [member of] C, A [member of] R; second step, when f (t) is a trigonometric polynomial, [mathematical expression not reproducible], [A.sub.k] [member of] C, [[lambda].sub.k] [member of] R, n [greater than or equal to] 2; third, one uses the approximation property by polynomials, of the form [mathematical expression not reproducible], for the final step which leads to any f [member of] A[P.sub.[phi]](R, C). It is important to use the distance function d : A[P.sub.[phi]] x A[P.sub.[phi]] [right arrow] [R.sub.+], instead of the absolute value |.| on C.

The above considerations end the proof of validity of the three basic properties of the space A[P.sub.[phi]](R, C): approximation of its elements by trigonometric polynomials; the existence of [epsilon]-almost periods for each [epsilon] > 0; the (relative) compactness of the set {f (t + h); h [member of] R} in A[P.sub.[phi]](R, C), for each f [member of] A[P.sub.[phi]](R, C). Each of these properties is characteristic in classical spaces, can it therefore be taken as definition for the space A[P.sub.[phi]] (R, C)? We invite the reader to check if they play the same role.

This paper presents an example of a space of almost periodic functions, from R [right arrow] C, the results being able for generalizations to other spaces, for instance, replacing C by [C.sup.n], n [greater than or equal to] 1, or any complex Banach space. The space A[P.sub.[phi]](R, C) constitutes a nonlinear space, endowed only with addition as a commutative group and a topology of a metric (and complete) space.

All the elements of A[P.sub.[phi]](R, C) belong to the classical space A[P.sub.1](R, C), the space of Bohr almost periodic functions, with absolutely (and uniformly) convergent Fourier series (we have called it the Poincare space of almost periodic functions, due to the fact that he produced the first example of an almost periodic function, in the sense of Bohr, showing how to find the coefficients of its Fourier series, by the introduction of the mean value on an infinite interval).

Let us notice the fact that A[P.sub.[phi]](R, C) is possessing different properties, depending on the choice of the topology accepted. For instance, in defining the space A[P.sub.r] (R, C), 1 [less than or equal to] r [less than or equal to] 2 (see, e. g., Corduneanu [1]), the topology was one of linear spaces, a situation made possible by the properties of the Minkovski spaces and the theory of [l.sup.p]-spaces, with complex elements, 1 [less than or equal to] p [less than or equal to] 2.

In the case of the space A[P.sub.[phi]](R, C), it does not appear possible, in general, to achieve the same goal, as in the case of A[P.sub.r]-spaces, for 1 [less than or equal to] r [less than or equal to] 2. Any successful trial would be welcome, enriching the theory of nonclassical spaces of almost periodic functions.

Several problems are worth being studied, in regards to the spaces A[P.sub.[phi]](R, C), accepting functions with diverse properties (still keeping those that provide the minimum for the construction of a metric space/topology).

Of great interest are those properties of the classical spaces of almost periodic functions, which may have a correspondent in the generalized spaces of such functions, including the A[P.sub.[phi]](R, C) spaces.

Since the topology of A[P.sub.[phi]](R, C) is different from the topology of A[P.sub.1](R, C), the last being considered as a series space, the sup-norm is dominated, for many p's, by the distance defined by means of p, from formula (1.6), even in the case [phi](u) = u. This suggests that the topology of generalized spaces is stronger than the sup-topology. For instance, from [mathematical expression not reproducible], in case of convergence of the series and with [equivalent] replaced by =, one gets sup [mathematical expression not reproducible]. This inequality between norms (the case of linear spaces being the beneficiary of such situations), shows that the sup-norm is generating a topology, generally weaker than the one generated by [mathematical expression not reproducible].

Other properties of the spaces A[P.sub.[phi]](R, C) will be examined in a forthcoming paper, as well as some applications.

First, showing (if possible) that the three basic properties are also characteristic for the space A[P.sub.[phi]](R, C). If not, what else should be added to the hypotheses. Second, the connection with almost periodic dynamical systems (on metric spaces).

References

[1] C. Corduneanu, A scale of almost periodic function spaces, Differential and Integral Equations, 24 (1-2) (2011), 1-27.

[2] C. Corduneanu, Elements d'une construction axiomatique de la theorie des fonctions presque periodiques, Libertas Mathematica, 32 (2012), 5-18.

[3] C. Corduneanu, Formal trigonometric series, almost periodicity and oscillatory functions, Nonlinear Dynamics and Systems Theory, 13 (2013), 367-388.

[4] C. Corduneanu, Almost Periodic Oscillations and Waves, Springer, New York, 2009.

[5] V.F. Osipov, Bohr-Fresnel Almost Periodic Functions, University of Sankt Petersburg Press (Russian), 1992.

[6] M.A. Shubin, Differential and pseudodifferential operators in spaces of almost periodic functions, Matematicheskii Sbornik (Russian), 137 (2) (1974), 560-587.

[7] M.A. Shubin, Almost periodic functions and partial differential operators, Uspekhi Matematicheskih Nauk (Russian), 33 (2) (1978), 3-47.

[8] Chuanyi Zhang, Strong limit power functions, The Journal of Fourier Analysis and Applications, 12 (2006), 291-307.

[9] Chuanyi Zhang, Generalized Kronecker's theorem and strong limit power functions. In V. Barbu and O. Carja (eds.), American Institute of Physics Proceedings, 1329 (2015), 281-299.

[10] Chuanyi Zhang, Almost Periodic Type Function and Ergodicity, Kluwer Academic, Dordrecht, 2003.

C. Corduneanu

The University of Texas at Arlington and The Romanian Academy

E-mail: cordun@exchange.uta.edu