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Almost-Periodic Single-Valued and Multivalued Functions

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Almost-Periodic Single-Valued and Multivalued Functions APPENDIX 1 599 APPENDIX 1 ALMOST-PERIODIC SINGLE-VALUED AND MULTIVALUED FUNCTIONS In this supplement, the hierarchy of almost-periodic function spaces is mainly clarified. The various types of definitions of almost-periodic functions are examined and compared. Apart from the standard definitions, we also introduce new classes and comment some, less traditional, definitions, to make a picture as much as pos­ sible complete. All is related to possible applications to nonlinear almost-periodic oscillations. The material for this exposition is based on the papers [AndBeGr], [AndBeL]. In the theory of almost-periodic (a.p.) functions, there are used many various definitions, mostly related to the names of H. Bohr, S. Bochner, V. V. Stepanov, H. Weyland A. S. Besicovitch ([AP-M], [BH-M], [BJM-M], [Bes-M], [BeBo], [Bo­ M], [BF], [BPS-M], [Cor-M], [Fav-M], [GKL-M], [Lev-M], [LZ-M], [Muc], [NG-M], [Pn-M]). On the other hand, it is sometimes difficult to recognize whether these defi­ nitions are equivalent or if one follows from another. It is well-known that, for example, the definitions of uniformly a.p. (u.a.p. or Bohr-type a.p.) functions, done in terms of a relative density ofthe set of almost-periods (the Bohr-type criterion), a compactness of the set of translates (the Bochner-type criterion, sometimes called normality), the closure of the set of trigonometric polynomials in the sup-norm metric, are equivalent (see, for example, [Bes-M], [Cor-M]). The same is true for the Stepanov class of a.p. functions ([Bes-M], [BF], [GKL­ M], [Pn-M]), but if we would like to make some analogy for, e.g., the Besicovitch class of a.p. functions, the equivalence is no longer true. For the Weyl class, the situation seems to be even more complicated, because in the standard (Bohr-type) definition, the Stepanov-type metric is used, curiously, instead of the Weyl one. Moreover, the space of the Weyl a.p. functions is well-known [BF], unlike the other classes, to be incomplete in the Weyl metric. In [Fol], E. Foolner already pointed out these considerations, without arriving at a clarification of the hierarchies. 600 ALMOST-PERIODIC SINGLE-VALUED AND MULTIVALUED FUNCTIONS Besides these definitions, there exists a lot of further characterizations, done, e.g., by J.-P. Bertrandias [Ber3], R. Doss [Dos1], [Dos2], [Dos3], A. S. Kovanko [KovlO], [Kov7], [Kov1], [Kov12], [Kov13], [Kov6], [Kov2], B. M. Levitan [Lev-MJ, A. A. Pankov [Pn-MJ and the references therein, A. C. Zaanen [Zaa-MJ, which are, sometimes, difficult to compare with more standard ones. Moreover, because of possible applications, the relations collected in Table 1 in Part 5 are crucial. This appendix is divided into the following parts: (A1.1) Uniformly almost-periodicity definitions and horizontal hierarchies, (A1.2) Stepanov almost-periodicity definitions and horizontal hierarchies, (A1.3) Weyl and equi-Weyl almost-periodicity definitions and horizontal hierar- chies, (A1.4) Besicovitch almost-periodicity definitions and horizontal hierarchies, (A1.5) Vertical hierarchies, (A1.6) Some further generalizations. (Al.l) Uniformly almost-periodicity definitions and horizontal hier­ archies. The theory of a.p. functions was created by H. Bohr in the twenties, but it was restricted to the class of uniformly continuous functions. Let us consider the space C(~, ~) of all continuous functions, defined on ~and with the values in~' endowed with the usual sup-norm. In this part, the definitions of almost-periodicity will be based on this norm. (Al.1.1) DEFINITION. A set X c ~is said to be relatively dense (r.d.) ifthere exists a number l > 0 (called the inclusion intervaQ, s.t. every interval [a, a+ l] contains at least one point of X. (A1.1.2) DEFINITION (see, for example, [AP-M, p. 3], [Bes-M, p. 2], [GKL­ M, p. 170])[Bohr-type definition]. A function f E C(~,~) is said to be uniformly almost-periodic (u.a.p.) if, for every s > 0, there corresponds a r.d. set {T}c s.t. (Al.l.3) llf(x + T)- f(x)llco =sup lf(x + T)- f(x)l < s, for all T E {T}c· xER Each number T E { T} c is called an s -uniformly almost-period (or a uniformly s-translation number) of f(x). (Al.l.4) PROPOSITION ([AP-M, p. 5], [Bes-M, p. 2], [Kat-M, p. 155]). Every u.a.p. function is uniformly continuous. (A1.1.5) PROPOSITION ([AP-M, p. 5), [Bes-M, p. 2], [Kat-M, p. 155]). Every u.a.p. function is uniformly bounded. APPENDIX 1 601 (A1.1.6) PROPOSITION ([AP-M, p. 6], [Bes-M, p. 3]). If a sequence of u.a.p. functions fn(x) converges uniformly in JR. to a function f(x), then f(x) is u.a.p., too. In other words, the set of u.a.p. functions is closed w.r.t. the uniform conver­ gence. Since it is a closed subset of a Banach space, it is Banach, too. Actually, it is easy to show that the space is a commutative Banach algebra, w.r.t. the usual product of functions (see, for example, [Sh-M, pp. 186-188]). (ALl. 7) DEFINITION ([Bes-M, p. 10], [Cor-M, p. 14], [LZ-M, p. 4]) [normality or Bochner-type definition]. A function f E C(IR., JR.) is called uniformly normal if, for every sequence {hi} of real numbers, there corresponds a subsequence {hnJ s.t. the sequence of functions {f(x + hn;)} is uniformly convergent. The numbers hi are called translation numbers and the functions f(x +hi) are called translates. In other words, f is uniformly normal if the set of translates is precompact in Cb := C n L00 (see [AP-M], [Pn-M]). Let us recall that a metric space X is compact (precompact) if every sequence {xn}nEN of elements belonging to X contains a convergent (fundamental) subse­ quence. Obviously, in a complete metric space X, the notions of precompactness, relative compactness (i.e. the closure is compact) and compactness coincide. The necessary (and, in a complete metric space, also sufficient) condition for the com­ pactness, or an equivalent condition for the precompactness, of a set X can be characterized by means of (see, for example, [KolF-M], [LS-M]): the total boundedness (Hausdorff Theorem): for every c > 0, there exists a finite number of points {xkh=l, ... ,n s.t. where (xk,c) denotes a spherical neighbourhood of Xk with radius c; the set {xkh=l, ... ,n is called an €-netfor X. (A1.1.8) REMARK. Every trigonometric polynomial n P(x) = :~::::akei>.k'\ (ak E JR., Ak E JR.) k=l is u.a.p. Then, according to Proposition (Al.l.6), every function f(x), obtained as the limit of a uniformly convergent sequence of trigonometric polynomials, is u.a.p. It is then natural to introduce the third definition: 602 ALMOST-PERIODIC SINGLE-VALUED AND MULTIVALUED FUNCTIONS (A1.1.9) DEFINITION ([BeBo, p. 224], [BF, p. 36], [Cor-M, p. 9]) [approxima­ tion]. We call Cap(IR, JR) the (Banach) space obtained as the closure of the space 'P(JR, JR) of all trigonometric polynomials w.r.t. the sup-norm. (Al.1.10) REMARK. Definition (Al.l.9) may be expressed in other words: A function f(t) belongs to Cap(IR, JR) if, for any c: > 0, there exists a trigonometric polynomial Te(t), s.t. sup lf(x)- Te(x)l < c:. zER It is easy to show that Cap, like C, is invariant under translations, that is Cap contains, together with f(x), the functions ft(x) := f(x + t), for all t E lR (see, for example, [LZ-M, p. 4]). The three main definitions, (Al.l.2), (Al.l.7) and (Al.l.9), are shown to be equivalent: (Al.l.ll) THEOREM ([AP-M, p. 8], [Bes-M, pp. 11-12], [Lev-M, pp. 23-27], [LZ-M, p. 4], [Pn-M, pp. 7-8]) [Bochner criterion]. A continuous function f(x) is u.a.p. iff it is uniformly normal. (Al.l.12) THEOREM ([BeBo, p. 226], [Pn-M, p. 9]). A continuous function f(x) is u.a.p. iff it belongs to Cap(JR,JR). (A1.1.13) REMARK. To show the equivalence among Definitions (Al.l.2), (Al.l.7) and (Al.l.9), in his book [Cor-M, pp. 15-23], C. Corduneanu follows another way that will be very useful in the following Sections: he shows that (Al.l.9) ==> (Al.l.7) ==> (Al.l.2) ==> (Al.l.9). In order to satisfy the S. Bochner criterion, L. A. Lusternik has proved an Ascoli-Arzela-type lemma, introducing the notion of equi-almost-periodicity. (A1.1.14) THEOREM ([Cor-M, p. 143], [LZ-M, p. 7], [LS-M, pp. 72-74]) [Lus­ ternik]. The necessary and sufficient condition for a family :F of u.a.p. functions to be precompact is that (1) :F is equicontinuous, i.e. for any c: > 0, there exists 8(c:) > 0 s.t. (2) :F is equi-almost-periodic, i.e. for any c: > 0 there exists l(c:) > 0 s.t., for any interval whose length is l(c:), there exists~ < l(c:) s.t. lf(x + ~)- f(x)l < c:, for all f E F, x E JR, (3) for any x E JR, the set of values f(x) of all the functions in :F is precompact. APPENDIX 1 603 (Al.l.15) REMARK. As already seen, for numerical almost-periodic functions, condition (3) in the Lusternik theorem coincides with the following: (3') for any x E R, the set of values f(x) of all the functions in :F is uniformly bounded.
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