Besicovitch and other generalizations of Bohr’s almost periodic functions

Kevin Nowland

We discuss several classes of almost periodic functions which generalize the uniformly continuous almost periodic (a.p.) functions originally defined by Harald Bohr. We have two goals, which we accomplish in two ways from two different sources. The first is to develop the proper setting for a Riesz-Fischer type theorem for almost periodic functions, which we do by following [3], and which leads to the definition of the Besicovitch almost periodic functions. Then we switch in section 5 to showing that Besicovitch functions are naturally occurring, but by defining such functions on the set of nonnegative numbers. For this we follow [2]. This note is just a summary of results with few proofs. A good reference on the subtle differences between the plethora of definitions of a.p. functions is [1]. Warning: Table 2 at the end of section 6 of that paper is an excellent summary of what the authors show, but there is a lone arrow that points up and to the left that should point down to the right.

1 Bohr almost periodic functions

The classical periodic functions on R can be characterized by f(x) = f(x + t) for all x and some period t. The first generalization of this condition was due to Bohr (1924), who instead required that a function f at a point x be approximated by f at x + t to an arbitrary degree of accuracy for a large set of t. Definition 1.1 (almost periods). A continuous function f : R C is Bohr almost periodic if for any ε> 0, the set of ε-periods τ : f(x + τ) f(x) <ε is relatively→ dense in R, i.e., there exists an l = l(ε) such that every interval of{ the| form [x,x− + l]| intersects} the set of ε-periods. The set of such functions is denoted APB(R). We can of course give an equivalent definition based on ε-periods for functions defined on other spaces; in particular, we will consider functions in APB(N) with N the set of nonnegative integers.

Example 1.2. Periodic functions are in APB(R).

Example 1.3. f(x) = sin x + sin √2x is in APB(R). Proposition 1.4. If f AP (R), then f is bounded and uniformly continuous. ∈ B For almost periodic functions, we have a mean. Definition 1.5. Let the mean of f : R C to be the value of → 1 l M f(x) := lim f(x)dx, { } l→∞ l Z0 if this limit exists, which it does for all f AP (R). ∈ B Definition 1.6. APB(R) is an inner product space if we define

f,g := M f(x)g(x) . h i n o Using this mean and a given f, we can define a function a (λ): R C by f → −iλx af (λ) := M f(x)e .

iλx Note that the (periodic) functions e form an orthonormal basis with respect to this mean, such that af (λ) is zero for all but at most countably many λ.

−iλnx Definition 1.7. We define the Fourier series of an a.p. function to be the formal sum n a(λn)e where the λn are the nonzero frequencies of f. We write P f(x) a(λ )e−iλnx. ∼ n n X

1 Note that we are being coy by writing “a.p.” instead of “Bohr a.p.,” as the mean will exist for more general classes of almost periodic functions. It turns out that the “structural” definition given by Bohr can be replaced with the following equivalent statement:

Definition 1.8 (approximation). A function f is in APB(R) if there exists an at most countable sequence of real numbers (λn) and complex numbers (An) such that f(x) is the uniform limit of sums of trigonometric N iλnx polynomials n=1 Ane . Another nameP for for Bohr’s almost periodic functions is uniformly almost periodic functions, as they are the closure of the trigonometric polynomials under the uniform norm, which we denote by ∞. This gives us a second way to define almost periodic functions as the closure of the trigonometric polynomialsk · k under various norms and seminorms. This uniform approximation by polynomials allows us to recover a notion of Fourier series, and some hope of recovering a Riesz-Fischer type theorem for almost periodic functions. However, the Riesz-Ficher theorem is a statement about L2 functions on a compact interval, and we are thus far dealing with uniformly continuous functions, so it appears we are not in the correct setting yet. Before moving on, we note that there is a third equivalent definition to the above two. Definition 1.9 (normality). A continuous function f : R C is uniformly normal if for every sequence → (hn) of real numbers, the set of translates f(x + hn) contains a uniformly convergent subsequence. In other words, the given set of translates is pre-compact{ . } This definition is due to Bochner, and is a normality type definition. In a complete metric space, pre- compactness is equivalent to total boundedness. Definition 1.10. A set in a metric space is called totally bounded if for every ε> 0 there exists a finite number of ε-balls covering the set. We do not a priori have a complete metric space, as the functions are only required to be continuous, but if we assume that they are bounded (as they end up being) then we do.

Theorem 1.11. The definitions 1.1, 1.8, and 1.9 are equivalent. The function space APB(R) is an algebra.

2 Stepanov almost periodic functions

We have the above three equivalent definitions of the Bohr a.p. functions. These functions are all uniformly continuous, which is a very strong statement. To relax this, and leave the continuous functions entirely, we can generalize the three definitions. Stepanov was the first to provide such a generalization. We will switch to the approximation by trigonemtric polynoimals being out main source of a definition, as we are keeping an eye for a Riesz-Fischer theoroem.

Definition 2.1 (approximation). Let 1 p < and l > 0. We define the space APSp (R) of Stepanov ≤ ∞ l (Stepanoff) almost periodic functions to be the closure of the trigonmetric polynomials under the norm

1/p x+l 1 p f Sp := sup f(t) dt k k l R l | | x∈ Zx ! p Note. These functions necessarily are in Lloc(R), which are only defined up to sets of Lebesgue measure zero, such that limits are only unique on classes of functions.

Proposition 2.2. The Stepanov norms induce equivalent topologies, in the sense that for any l,l′ there exist C,C′ > 0 depending on l and l′ such that

p p ′ p C f S ′ f S C f S ′ . k k l ≤ k k l ≤ k k l p p We typically take l = 1, in which case we write S in place of S1 . As a consequence of H¨older’s inequality,

2 ′ Proposition 2.3. Let 1 p

f p f p′ , k kS ≤ k kS which implies AP p′ (R) AP p (R). S ⊆ S Even more simply, Proposition 2.4. Let 1 p< and f AP (R). Then ≤ ∞ ∈ B

f p f , k kS ≤ k k∞

which implies AP (R) AP p (R). B ⊆ S The other two definitions are the following:

p Definition 2.5 (almost periods). Let 1 p< . A function f Lloc(R) is in APSp (R) if for every ε> 0 the set of all Stepanov ε-almost periods τ≤satisfying∞ ∈

x+1 1/p p f(t + τ) f(t) Sp = sup f(t + τ) f(t) dt k − k R | − | x∈ Zx  is relatively dense in R.

p Definition 2.6 (normality). Let 1 p< . A function f Lloc(R) is in APSp (R) if the set f(x + τ) t∈R is pre-compact. ≤ ∞ ∈ { }

We have been sloppy in saying that these are all definitions of the same space APSp (R), but for the Stepanov a.p. functions, we continue our good luck and have the following theorem.

Theorem 2.7. The definitions 2.1, 2.5, and 2.6 are equivalent. The function space APSp (R) is an algebra for all 1 p< . ≤ ∞

Example 2.8. Let E be set of closed intervals of the form [2k, 2k + 1] where k is any integer and let O be the complement of these intervals. Defined

1 x E f(x)= ∈ 0 e O. ( ∈

Clearly f is not Bohr almost periodic, as f is not continuous. However, it is in APSp for any p, as for any ε> 0, we can take the relatively dense set of integers as the periods.

We also have the theorem due to Bochner.

Theorem 2.9 (Bochner’s Theorem). If f AP 1 (R) is uniformly continuous, then f AP (R). ∈ S ∈ B 3 Weyl almost periodic functions

Though the ε-almost periodic, approximation by trigonometric polynomials, and normality / pre-compactness definitions have to this point coincided, we will no longer have that luxury. However, we are not yet at the point where we have the Riesz-Fischer theorem. We keep using trigonometric polynomials as our main definition. The definition of the Weyl a.p. functions is based on the following observation.

Lemma 3.1. The limit lim f p exists, in the sense that if the norm is infinite for any l> 0 then the l→∞ Sl limit is infinite. k k

3 Proof. We consider only the p = 1 case. This proof boils down to the fact that all values of l give rise to the same topology. Note that if f is infinite for some l> 0, then it is infinite for all l> 0. Thus it suffices to k kSl consider the case where all such are finite. Let l = l0 be greater than zero and let n be the positive integer such that 6 (n 1)l

1 x+nl0 1 1 x+l0 1 x+nl0 f(x) dx = f(x) dx + + f(x) dx f . nl | | n l | | ··· l | | ≤ k kSl0 0 Zx 0 Zx 0 Zx+(n−1)l0 ! Therefore f f . We calculate k kSnl0 ≤ k kSl0 nl l + l l + l f 0 f 0 f 0 f . (1) k kSl ≤ l k kSnl0 ≤ l k kSnl0 ≤ l k kSl0 Therefore

lim sup f Sl lim inf f Sl0 = liminf f Sl , l→∞ k k ≤ l0→∞ k k l→∞ k k as desired. With this in hand, we make a definition. Definition 3.2 (approximation). Let 1 p < . We define the space of Weyl almost periodic func- ≤ ∞ tions, denoted APW p (R), to be the set of functions on R which can be approximated arbitrarily well by triogonometric polynomials under the Weyl seminorm

f W p := lim f Sp . k k l→∞ k k l

Example 3.3. There exist functions f on R which are strictly positive but satisfy f W p = 0. Thus the uniqueness of elements in the closure is up to functions which may differ on sets of evenk k infinite measure. The second definition is in terms of Stepanov ε-almost periods.

p Definition 3.4 (almost periods). Let 1 p < . A function f Lloc(R) is in APW p (R) if for every ε> 0 there exists an l = l(ε) such that that≤ there is∞ a relatively dense∈ set τ of Stepanov ε-almost periods satisfying { } f(t + τ) f(t) Sp < ε. k − k l It is clear that AP p (R) AP p (R). S ⊂ W Remark. In the above definition, we are not using the Weyl seminorm but rather the Stepanov seminorm, p and the l is allowed to vary with ε, which is not the case for the Sl functions. This turns out to be crucial, as if we insist on defining the ε-almost periods in terms of the Weyl seminorm, we find a strictly larger space.

Theorem 3.5. The definitions 3.2 and 3.4 are equivalent.

4 Besicovitch periodic functions

Finally, we arrive at the spaces which give us a Riesz-Fischer theorem. We still do not have all three definitions, however, but we do have two equivalent definitions, as with the Weyl a.p. functions.

Definition 4.1. Let 1 p< . We define the space APBp (R) of Besicovitch almost periodic functions as the functions on R which≤ can∞ be approximated arbitrarily well by trigonometric polynomials under the seminorm 1/p l 1 p f p = limsup f(x) dx . k kB 2l | | l→∞ Z−l !

4 Note. As with the Weyl seminorm, there are functions which are nonzero on all of R and nonetheless have zero Besicovitch seminorm.

It is clear from the definition that f p f p , such that AP p (R) AP p (R). k kB ≤ k kW W ⊆ B Proposition 4.2. Let p 1 < . Then ≤ ∞

AP (R) AP p (R) AP p (R) AP p (R), B ⊂ S ⊂ W ⊂ B and all the inclusions are strict.

We wish to find another characterization based on ε-periods, but this is not easy to do. It turns out that relatively dense sets are too broad to describee the elements of APBp (R). Definition 4.3. A subset E of R is called satisfactorily uniform if there exists a a positive number l such that the ratio of the maximum number of terms of E in an interval [x,x + l] to the minimum number of terms of E in such an interval is less than 2.

Example 4.4. Every satisfactorily uniform set is relatively dense. ∞ Example 4.5. The set 1, 2,..., 1 is relatively dense but not satisfactorily uniform. { }∪ n n=1 p Definition 4.6. Let 1 p < andf L (R). Then f APBp (R) if for any ε > 0, there corresponds a satisfactorily uniform set≤ <τ∞

f(x + τ) f(x) p <ε k − kB and for every c> 0,

1/p l n x+c 1 1 1 p lim sup lim sup f(t + τi) f(t) dt dx < ε. 2l n→∞ 2n + 1 c | − | l→∞ −l " i=−n x # ! Z X Z Then we have the equivalence of definitions that we desire.

Theorem 4.7. The definitions 4.1 and 4.6 are equivalent. Finally, we are in a position where we have a Riesz-Fischer theorem.

Theorem 4.8. To any (generalized) Fourier series A eiλnx such that A 2 converges there corre- n n n | n| sponds a function f APB2 (R) with this as its Fourier series. ∈ P P 5 Almost periodicity on the nonnegative integers

We now shift focus from almost periodic functions on R to almost periodic functions on N = 0, 1,... . Note { p }p that we include zero when we use the symbol “N.” One way to define APG(N) for G one of B, Sl , W , and Bp, is to replace the integrals with sums over the integers. However, a different approach is taken in [2]. We begin by definining APB(N), the Bohr almost periodic functions, in the same way as originally, by using the relative density of the ε-periods with respect to the l( ) norm. As before, elements in APB(N) are contained in λ( ). ∞ ∞ Definition 5.1. Let f : N C be such that f l( ). Then we define W (f) to be the set of functions f(x + a): a N , i.e., the set→ of translates of f. ∈ ∞ { ∈ } Proposition 5.2. Let f AP (N). Then W (f) is relatively compact (totally bounded) in l( ). ∈ B ∞ Proof. To show that W (f) is totally compact in the metric space l( ), it suffices to find a finite number of ε-balls which cover the set. Let ε> 0 be fixed, and let l = l(ε) be such∞ that every interval of length at least λ in N contains an ε-almost period of f. Let J = j,j + 1,...,j + l . We claim that W (f) is contained in the set of l + 1 ε-balls, each centered at f(x + k) for{ k J. Consider}f(x + n) for some n N. If k J, we ∈ ∈ ∈

5 are done. If k

f(x + n) f(x + k) = f(x + n) f(x + n +(k n)) . | − | | − − | Then for some k n, this will be less than ε for all x. Thus f(x + n) is contained in some ε-ball centered at f(x + k) for k −J, as required. If n>j + l, then we do the same trick but replace k n with n k. ∈ − − With this proposition in hand, we can now define new classes of almost periodic functions on N.

Definition 5.3. We say that f : N C with f l( ) is Weyl almost periodic if the set of translates W (f) is relatively compact in l( ).→ This set is denoted∈ ∞ by AP (N). ∞ W From proposition 5.2, we have that AP (N) AP (N). B ⊆ W Example 5.4. The converse is false, such that the inclusion above is strict, as can be seen by looking at the function 0 x = 0, f(x)= . (1 x> 0. The set of translates of f consists of f and the constant function 1, such that W (f) is relatively compact. However, f is not Bohr almost periodic, as can be seen for any ε< 1.

Note. APW (Z)=APB(Z). Fr´echet gave the following characterization of almost periodic funtions which gives rise to many examples. We state the result without proof.

Theorem 5.5 (Fr´echet, 1941). A function f l( ) is in APW (N) if and only if f admits a (unique) decomposition f = p + w wehre p AP (N) and∈lim∞ w(n) = 0. ∈ B n→∞ p This definition of APW p (N) is is sometimes called W -normal. Definition 5.6. The Eberlein or weakly almost periodic functions on N are bounded functions such that W (f) is weakly relatively compact. We denote this set by APw(N). Eberlein characterized the weakly almost periodic functions in a theorem similar that that of Fr´echet.

Theorem 5.7 (Eberlein, 1956). f l( ) is weakly almost periodic on N if and only if f admits a (unique) decomposition f = p + w where p ∈AP∞(N) and w satisfies ∈ B 1 n−1 lim ω(x + j) = 0, n→∞ n | | j=0 X with the limit is uniform in x.

Example 5.8. Let w : N C be defined as a sequence of ones and and zeros such that the number of zeros between consecutive ones→ is increasing. Then w(n) does not tend to zero as n tends to infinity, but the average about does tend to zero uniformly in x.

Corollary 5.9. AP (N) AP (N), and the inclusion is proper. W ⊂ w The characterization theorems of Fr´echet and Eberlein imply that APW (N) and APw(N) are algebras. Finally, we return to the Besicovitch almost periodic functions. Instead of using topological properties of translates to define these functions, we return to a definition based on the closure under a seminorm.

Definition 5.10. Let X be the set of sequences of the form (zk) where z C, z = 1, and k = 0, 1,.... A sequence is a trigonometric polynomial if it is a linear combination of a∈ finite| number| of sequences in X.

6 Definition 5.11. Let 1 p < . We say that f is Besicovitch almost periodic on N if f is in the closure of the trigonometric≤ polynomials∞ under the seminorm defined by k · kp n−1 p 1 p f p = limsup f(k) . k k n→∞ n | | kX=0 In this case we write f AP p (N). We say that f is bounded Besicovitch if f AP p (N) l( ). ∈ B ∈ B ∩ ∞ It turns out that APB1 (N) l( )=APBp (N) l( ) for 1 p< . The idea behind the proof will be explained below, but no proof∩ will∞ be given. ∩ ∞ ≤ ∞ For any function f, we can define its mean M f to be { } 1 n−1 M f := lim f(j). { } n→∞ n j=0 X The mean does not necessarily exist or is finite. Using this mean, we can define a function a : R C by → −iλx af (λ)= M fe .

For any f APBp (N) l( ), this function is well-defined for all λ R. If we take f to be the purely periodic eiλx, then ∈ ∩ ∞ ∈ ′ ′ 1 λ = λ , af (λ )= . 0 λ = λ′ ( 6 It follows that for any f APBp (N) l( ), af (λ) is nonzero for an at most countable set of λ. (This relies on standard inner-product∈ space results,∩ ∞ which we gloss over.)

Definition 5.12. Let f APBp (N) l( ) and let (λn) be the sequence of real numbers such that af (λ) = 0. Then we write ∈ ∩ ∞ 6 f(x) a (λ )eiλnx, ∼ f n n X and call the right hand side of the above the (generalized) Fourier series for f.

Definition 5.13. Let f and (λn) be as in the prevoius definition. Then a set of (at moust) countable real numbers Bf is called a base of f if Bf forms a basis for (λn) over Q, i.e., Bf consists of numbers such that all finite subsets of Bf are linearly independent over Q and every λn can be written as a finite linear combination of elements in Bf with coefficients in Q.

Let Kn(t) be the Fej´er kernel defined as λ K (t)= 1 | | e−iλt. n − n |λX|

AP 1 (N) l( )=AP p (N) l( ) B ∩ ∞ B ∩ ∞ for 1 p< . ≤ ∞

7 6 Relation to Ergodic Theory

We now show that the bounded Besicovitch functions are naturally occurring in the context of ergodic theory and prove a related theorem about pointwise convergence of a weighted sequence of operators acting on a function. Before stating the theorem which indicates that Bounded besicovitch functions occur naturally, we need a definition.

Definition 6.1. Let T be a bimeasurable, measure-preserving, ergodic, bijection on a probability space (Ω, ,µ). Let UT be the induced transformation on functions, i.e., UT f(x) = f(Tx). Then T has discrete spectrumA if L2(Ω,µ) has an orthonormal basis of eigenfunctions of T . The ergodic functions T can be characterized as follows.

Theorem 6.2. Let T be as in the definition above. Then the following are equivalent:

(i) T has discrete spectrum.

(ii) If g L∞(Ω), then for almost every ω Ω, the sequence (g(T nω)) for n N is bounded Besicovitch. ∈ ∈ ∈

∞ Proof. (i) (ii). Let fi be an orthonormal set of eigenfunctions for UT . Let g L (Ω). This implies 1 ⇒ { } ∈ g L (Ω), since (Ω, ,µ) is a probability space. Then there exists a sequence hn in the linear span of fi such∈ that A { } h g n → 1 in L . Linear span means finite linear combinations of the fi. By the individual (Birkhoff’s?) ergodic theorem, for each n N, there exists a set Ω of measure 1 such that ω Ω implies ∈ n ∈ n n−1 1 j j lim g(T ω) hn(T ω) = g hn dµ. k→∞ k | − | Ω | − | kX=0 Z

Let Ωg be the intersection of the Ωn, and note that this set has full measure. If we consider the sequences j j v =(g(T ω)) and h(n)=(hn(T ω)). Then, as n goes to infinity,

k−1 1 j j v h(n) 1 := lim g(T ω) hn(T ω) = g hn dµ 0. k − k k→∞ k | − | | − | → j=0 Ω X Z It is claimed in [2] that h(n) is “clearly” a trigonometric polynomial, such that g is in the closure of these polynomials by the above. g is bounded by assumption. Thus g APB1 (N) l( ), as claimed. We leave the proof of the reverse direction to the interested reader,∈ who∩ may∞ consult [2].

References

[1] J. Andres, A.M. Bersani, and R.F. Grande, Hierarchy of almost-periodic functions spaces, Rend. Mat. Appl. (2006).

[2] A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences, T. Am. Math. Soc. (1985).

[3] A.S. Besicovitch, Almost periodic functions, Cambridge University Press, 1954.

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