Stepanov and Weyl Almost Periodic Functions on Locally Compact

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Stepanov almot periodic function [18]. For more background on the connection between norm almost periodicity, cut and project schemes and pure point diffraction, we refer the reader to [4]. Let us briefly mention another application of Stepanov almost periodic functions, namely stable sampling. It is concerned with the problem of reconstrucing a function f from a subset Λ. Now, a uniformly discrete set Λ is a set of stable sampling with respect to Lp(K) and the Stepanov space Sp(Rd) if every function f Lp(K) is the restriction to K of a function ∈ F Sp(Rd) whose frequencies belong to Λ. This was studied in [14] for the Euclidean space ∈ Rd; see [15] for more general locally compact Abelian groups. Weyl almost periodic functions also play an important role in aperiodic order. Next to the diffraction measure, certain dynamical systems provide another powerful tool to study aperiodicity. Consider a translation bounded measure µ. It was shown in [10] that µ is Weyl almost periodic if and only if the corresponding hull X(µ) has a discrete (or pure point) dynamical spectrum, continuous eigenfunctions and is uniquely ergodic. This result was generalised in [11] from dynamical systems of translation bounded measures X(µ), which are of interest in aperiodic order, to arbitrary metric dynamical systems. This is connected to the diffraction measure, because X has a discrete dynamical spectrum if and only if γµ is a pure point measure. By definition, a measure is Stepanov or Weyl almost periodic if the function f µ is Stepanovc ∗ or Weyl almost periodic, for all f Cc(G). Therefore, the results mentioned above motivate ∈ to study these kind of almost periodic functions on locally compact Abelian groups.

A function f : R R is called periodic if there is a number t R such that → ∈ f(x t)= f(x) for all x R , − ∈ which is equivalent to

Ttf f = 0, k − k∞ where Ttf(x) := f(x t). Next, instead of the supremum norm, one can consider the norm − 1 y+ℓ f S1 := sup f(x) dx , k k R ℓ | | y∈ Zy and deﬁne f L1 (R) to be Stepanov almost periodic if the set ∈ loc t R f Ttf 1 < ε { ∈ | k − kS } is relatively dense in R. The space of Stepanov almost periodic functions is well studied in the case G = R, see [5].

It forms a Banach space with respect to the norm 1 . One of the main theorems in this k · kS area states three equivalent properties which can be used to characterise Stepanov almost periodic functions.

Theorem 1.1. Let f be a locally integrable function on R. Then, the following statements are equivalent:

(1) For each ε> 0, the set t R f Ttf 1 < ε is relatively dense in R. { ∈ | k − kS } STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 3

(2) f is the limit of a sequence of trigonometric polynomials with respect to the Stepanov norm 1 . k · kS (3) The orbit Ttf t R is precompact in the 1 -topology. { | ∈ } k · kS Every locally integrable function satisfying one (and thus all) of the above three properties is called a Stepanov almost periodic function. We will prove in this paper that Theorem 1.1 remains true for second countable locally compact Abelian groups. Similarly, one can deﬁne that a locally integrable function f is Weyl almost periodic if it is the limit of a sequence of trigonometric polynomials with respect to the seminorm 1 y+ℓ f W 1 := lim sup f(x) dx . k k ℓ→∞ R ℓ | | y∈ Zy We will see below that Weyl almost periodic functions on locally compact Abelian groups can also be characterised via a relatively dense set of ε-almost periods. From now on, G denotes a locally comact Abelian group. The corresponding Haar measure is denoted by θG.

2. Stepanov almost periodic functions First, we will study Stepanov almost periodic functions on second countable locally compact Abelian groups and establish some of their basic properties. We start by introducing a norm on the set of locally integrable functions f : G C. → Deﬁnition 2.1. Let K be a precompact set with non-empty interior, and let 1 6 p < . p ∞ We denote by BSK(G) the space of all f L (G) such that ∈ loc 1 p 1 p f Sp (G) := sup f(x) dθG(x) < . k k K θ (K) | | ∞ y∈G G Zy+K If G is clear from the context, we write f Sp instead of f Sp (G), or simply f Sp . The k k K k k K k k mapping f f Sp deﬁnes a norm on BSK(G). 7→ k k Next, let K , K G be two compact sets with non-empty interior. We will show that the 1 2 ⊆ norms Sp and Sp are equivalent. k · k K1 k · k K2 Lemma 2.2. For every two compact sets K , K G with non-empty interior, there are 1 2 ⊆ constants c1, c2 > 0 such that

c1 f Sp 6 f Sp 6 c2 f Sp k k K2 k k K1 k k K2 for all f BSp (G), j 1, 2 . ∈ Kj ∈ { } Proof. (i) First note that we can cover K1 with ﬁnitely many translates of K2, i.e. there are n n N and z ,...,zn G such that K (zj + K ). Then, we have ∈ 1 ∈ 1 ⊆ j=1 2 1 1 p p S 1 p θG(K2) 1 p f(x) dθG(x) 6 f(x) dθG(x) θG(K1) y+K | | θG(K1) θG(K2) Sn (z +K ) | | Z 1 Z j=1 j 2 ! 1 θG(K2) p 6 n f Sp θ (K ) k k K2 G 1 4 TIMO SPINDELER

1 θG(K2) p for all y G, which proves the second inequality with c2 := n . ∈ θG(K1) (ii) The ﬁrst inequality can be shown similarly.

p p Also, it follows from the triangle inequality for L -spaces that SK(G) is a vector space. In p fact, the pair BS (G), Sp is a Banach space. K k · k p Proposition 2.3. The pair BS (G), Sp is a Banach space. K k · k p Proof. Let (fn)n∈N be a Cauchy sequence in BS K(G). In that case, we can pick a subsequence

(fnk )k∈N such that 1 fn fn Sp < for all k N . k k+1 − k k 2k ∈ ∞ Let g(x) := fn (x) fn (x) , for all x G. Then, k=1 | k+1 − k | ∈ 1 P N p 1 p p 6 g S sup lim inf fnk+1 (x) fnk (x) dθG(x) k k N→∞ θG(K) | − | y∈G y+K k ! Z X=1 N 1 = sup lim inf fn fn 1 k+1 k p N→∞ p | − | L (y+K) y∈G θG(K) Xk=1 N 1 6 sup lim inf fn fn 1 k+1 k p y∈G N→∞ θ (K) p − L (y+K) k=1 G X N 6 lim inf fnk fnk 6 1 . N→∞ +1 − Sp k=1 X Therefore, g BSp (G) and hence g < + almost everywhere. Because of this and ∈ K | | ∞ N fn fn = fn fn , N+1 − 1 k+1 − k k=1 X there is a set A of full measure such that the sequence (fnN (x))N∈N converges to the limit ∞ fn (x) + fn (x) fn (x) for all x A. This means that it converges almost 1 k=1 k+1 − k ∈ everywhere to the function P ∞ fn (x)+ fn (x) fn (x) , if x A, f(x) := 1 k=1 k+1 − k ∈ (0, P otherwise.

It remains to show that (fn )N N converges to f with respect to Sp because every N ∈ k · k Cauchy sequence which contains a converging subsequence converges itself. Let ε> 0. Since (fn)n N is a Cauchy sequence, there is a number n N such that ∈ 0 ∈ fn fm Sp < ε for all n,m > n . k − k 0 Finally, this implies (using Fatous Lemma)

1 p 1 p f fn Sp = sup lim fn (x) fn (x) dθG(x) k − k k θ (K) m→∞ | m − k | y∈G G Zy+K STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 5

1 p 1 p 6 sup lim inf fn (x) fn (x) dθG(x) m→∞ θ (K) | m − k | y∈G G Zy+K 6 lim inf fn fn Sp < ε m→∞ k m − k k for all k > n0. Deﬁnition 2.4. Let K and p be as above. A function f Lp (G) is called Stepanov p-almost ∈ loc periodic (with respect to K) if, for all ε> 0, the set of all t G which satisfy ∈ f Ttf Sp < ε k − k is relatively dense in G. The space of all Stepanov p-almost periodic functions on G is denoted by Sp(G).

Again, we may sometimes write Sp instead of Sp(G). Note that, by Lemma 2.2, if f is Stepanov p-almost periodic with respect to some compact set K with non-empty interior, it is Stepanov p-almost periodic with respect to all compact sets with non-empty interior. Let us collect some basic properties of Stepanov almost periodic functions.

Lemma 2.5. Let f Sp(G). ∈ p (a) The function f is S -bounded, i.e. f Sp < . k k ∞ (b) The function f is Sp-uniformly continous, i.e. for any ε> 0, there is a neighbourhood V of 0 such that

s V = f Tsf Sp < ε . ∈ ⇒ k − k ′ (c) If 1 6 p′ 6 p< , we have Sp(G) Sp (G). More precisely, we have ∞ ⊆

f p′ 6 f Sp . k kS k k (d) One also has Sp(G) L∞(G)= S1(G) L∞(G) . ∩ ∩ (e) Let f † and f be deﬁned via f †(x) := f( x) and f(x) := f( x). Let t G. Then, − − ∈ p † p f S (G) = f, f , f, f , Ttf S (G) . e ∈ ⇒ | | ∈e Proof. (a) Since f is Stepanov almost periodic, theree is a relatively dense set R such that 1 p 1 p sup f(x) Ttf(x) dθG(x) < 1 θ (K) | − | y∈G G Zy+K for all t R. Also, by the relative denseness of R, there is a compact set K′ such that, for ∈ every y G, there is a t R such that y t K′. We obtain ∈ ∈ − ∈ p p p f(x) dθG(x) 6 f(x) Ttf(x) dθG(x)+ Ttf(x) dθG(x) | | | − | | | Zy+K Zy+K Zy+K p 6 θG(K)+ f(x) dθG(x) ′ | | ZK +K uniformly in y. Hence, f Sp < + . k k ∞ 6 TIMO SPINDELER

(b) Fix ε> 0. Then, there is a relatively dense set R = R(ε) such that

1 p 1 p ε sup f(x) Ttf(x) dθG(x) < θ (K) | − | 3 y∈G G Zy+K for all t R. Similar to (a), there is a compact set K′ such that, for every y G, there is a ∈ ∈ t R such that y t K′. Then, by [16, Thm. 1.1.5], there is a a neighbourhood V of 0 ∈ − ∈ such that

f Tsf Sp 6 f Ttf Sp + Ttf Ts tf Sp + Ts tf Tsf Sp k − k k − k k − + k k + − k 1 p ε 1 p ε < + sup Ttf(x) Ts tf(x) dθG(x) + 3 θ (K) | − + | 3 y∈G G Zy+K 1 p 2 ε 1 p 6 + f(x) Tsf(x) dθG(x) 3 θ (K) ′ | − | G ZK +K 2 ε ε < + = ε , 3 3 whenever s V . ∈ ′ p (c) This follows from Jensens inequality with the convex function x x p . 7→ (d) The inclusion Sp(G) L∞(G) S1(G) L∞(G) follows immediately from (c), while ∩ ⊆ ∩ p−1 1 p ∞ 1 ∞ p p S (G) L (G) S (G) L (G) is a consequence of f Sp 6 f ∞ f 1 . ∩ ⊇ ∩ k k k k k kS (e) This follows from the deﬁnition.

Lemma 2.6. Let f Sp(G), and let ε > 0. Then, there is a bounded function b Sp(G) ∈ ∈ such that

f b Sp < ε . k − k Proof. Let L N. Deﬁne fL : G C by ∈ → f(x), if f(x) 6 L, fL(x)= f(x) | | (L |f(x)| , otherwise. Since

fL(x) fL(y) 6 f(x) f(y) | − | | − | p for any pair x,y G, one has fL S (G). ∈ ∈ Next, let K be a compact subset of G. Then, there is a relatively dense set R such that ε fL TtfL Sp 6 f Ttf Sp < k − k k − k 3 for all t R. Since R is relatively dense, there is a compact set K′ such that ∈ f fL Sp 6 f Ttf Sp + Ttf TtfL Sp + TtfL fL Sp k − k k − k k − k k − k 1 p ε 1 p ε 6 + f(x) fL(x) dθG(x) + (2.1) 3 θ (K) ′ | − | 3 G ZK +K STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 7

1 p 2 ε 1 p = + f(x) fL(x) dθG(x) , 3 θ (K) ′ S | − | G Z(K +K) ∩ L ! where SL := x G f(x) > L . It follows from the following standard inequality from { ∈ | | | } measure theory 1 µ ( x X f(x) >L ) 6 f 1 { ∈ | | | } L k kL (X,µ) that there is L N such that 0 ∈ 1 p 1 p ε f(x) fL0 (x) dθG(x) < . (2.2) θG(K) (K′+K) ∩ S | − | 3 Z L0 ! Finally, Eqs. (2.1) and (2.2) imply

f fL Sp < ε . k − 0 k

The claim follows for b := fL0 . If G = R, Stepanov almost periodic functions are sometimes deﬁned as the closure of the

set of trigonometric polynomials with respect to the norm 1 . Then, it is shown that this k · kS is equivalent to the deﬁnition given above, which is also equivalent to the statement that the set Ttf t G is precompact in the 1 -topology. The next two propositions will show { | ∈ } k · kS that this equivalence is still true for second countable locally compact Abelian groups.

Proposition 2.7. For f Lp (G) the following statements are equivalent: ∈ loc (1) The function f is Stepanov p-almost periodic. (2) The orbit Ttf t G is precompact in the Sp -topology. { | ∈ } k · k Proof. First, let us recall that a set in a complete metric space is precompact if and only if it is totally bounded (that is, for every ε > 0, it can be covered by a union of ﬁnitely many balls of radius ε). (1) = (2): Let ε > 0. Since f Sp(G), there is a relatively dense set R of ε -almost ⇒ ∈ 2 periods. Let K be a corresponding compact set, i.e. R + K = G, and let r > 0 be such that K Br(0). By our choice of the metric, the ball Br(0) is precompact, and hence can ⊆ be covered by ﬁnitely many balls of radius, say, δ > 0. Consequently, due to Lemma 2.5(b), there are elements z ,...,zm Bδ(0) such that 1 ∈ ε inf Tx f Tz f Sp < 16j6m k 0 − j k 2 for all x Br(0). For an arbitrary x G, there is a t R and an x Br(0) such that 0 ∈ ∈ ∈ 0 ∈ x = t + x0. In that case, we obtain ε Txf Tx f Sp = Tt x f Tx f Sp = Ttf f Sp < . k − 0 k k + 0 − 0 k k − k 2 Therefore, we have ε ε inf Txf Tz Sp 6 Txf Tx f Sp + inf Tx f Tz f Sp < + = ε , 16j6m k − j k k − 0 k 16j6m k 0 − j k 2 2 8 TIMO SPINDELER

and Ttf t G is covered by the union of balls of radius ε, centered at Txj f, 1 6 j 6 m. { | ∈ } p Therefore, it is precompact because Lloc(G) is a complete metric space. ε (2) = (1): Fix ε> 0. Let B1,..., Bm be balls of radius 2 such that Ttf t G is covered ⇒ m { | ∈ } by the union Bj. Without loss of generality, we can assume that Bj Ttf t G = ∅. j=1 ∩{ | ∈ } 6 Hence, we can pick Tt f Bj, 1 6 j 6 m. The balls of radius ε centered at Tt f cover S j ∈ j Ttf t G . { | ∈ } Now, we claim that every ball B with radius ρ := 2 max16j6m d(tj, 0) contains an ε-almost period of f. If B is such a ball, denote by x its centre. There is a j0 such that

Txf Tt f Sp < ε . k − j0 k Writing t = x tj , it is clear that t B and − 0 ∈ Ttf f Sp = Tx−t f f Sp = Txf Tt f Sp < ε . k − k k j0 − k k − j0 k Therefore, t is an ε-almost period of f. Sometimes, Stepanov almost periodic functions on R are deﬁned diﬀerently, namely as limits of trigonometric polynomials with respect to 1 . As we will now see, these two k · kS properties are also equivalent for second countable locally compact Abelains groups.

Proposition 2.8. For f Lp (G), the following statements are equivalent: ∈ loc (1) The function f is Stepanov p-almost periodic. (2) f is the Sp -limit of a sequence of trigonometric polynomials. k · k Proof. (2) = (1): Let f be the Sp -limit of a sequence of trigonometric polynomials ⇒ k · k (pn)n N, and let ε> 0. Then, there is a number n N such that ∈ ∈ ε f pn Sp < . k − k 3 Also, since every trigonometric polynomial is Bohr almost periodic and hence Stepanov almost periodic, there is a relatively dense set of t G such that ∈ ε pn Ttpn Sp < for all such t G . k − k 3 ∈ This implies ε ε ε f Ttf Sp 6 f pn Sp + pn Ttpn Sp + Ttpn Ttfn Sp < + + = ε . k − k k − k k − k k − k 3 3 3 (1) = (2): Let f Sp(G). Consider the function ⇒ ∈ 1 f (x) := f(y) dθ (y) . η θ (B (x)) G G η ZBη (x) (i) We will first show that, for fixed η, the function fη is continuous. 1 1 Fix ε> 0, and let q be such that p + q = 1. By Lemma 2.5 and Hölders inequality, there is δ > 0 such that 1 1 fη(x) fη(x + h) = f(y) dθG(y) f(y) dθG(y) | − | θG(Bη(x)) Bη(x) − θG(Bη(x + h)) Bη(x+h) Z Z

STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 9

1 6 f(y) T hf(y) dθG(y) θ (B (x)) | − − | G η ZBη(x) 1 p 1 1 p 6 θG(Bη(x)) q f(y) T hf(y) dθG(y) θ (B (x)) | − − | G η ZBη(x) ! 6 f T−hf Sp < ε, k − k Bη(x) whenever h Bδ(0). Hence, fη is continuous. ∈ (ii) Next, we will show that fη is a Bohr almost periodic function. Let t be a Stepanov ε-almost period of f. Then, we have 1 1 fη(x) Ttfη(x) = f(y) dθG(y) f(y) dθG(y) | − | θG(Bη(x)) Bη(x) − θG(Bη(x t)) Bη (x−t) Z − Z 1 6 f(y) Ttf(y) dθG(y) θ (B (x)) | − | G η ZBη(x) 6 f Ttf Sp < ε, k − k independent of x. Consequently, fη is Bohr almost periodic.

(iii) We will now prove that limη f fη Sp = 0. →0 k − k Fix ε> 0. By Lemma 2.5, there is η > 0 such that

h Bη(0) = f Thf Sp < ε . ∈ ⇒ k − k This implies via Minkowski’s integral inequality 1 1 1 p p f fη Sp = sup f(x) f(z) dθG(z) dθG(x) k − k θ (K) − θ (B (x)) y∈G G Zy+K G η ZBη(x) !

1 1 = sup f(x) dθ (z) θ (K) θ (B (0)) G y∈G G Zy+K G η ZBη(0)

1 1 p p f(z + x) dθG(z) dθG(x) − θ (B (0)) G η ZBη(0) !

p 1 1 1 p 6 sup 1 f(x) f(x + z) dθG(z) dθG(x) p θ (B (0)) | − | y∈G θG(K) Zy+K G η ZBη (0) ! ! 1 p 1 1 p 6 sup 1 f(x) f(x + z) dθG(x) dθG(z) p θ (B (0)) | − | y∈G θG(K) G η ZBη(0) Zy+K ! 1 6 f T zf Sp dθG(z) θ (B (0)) k − − k G η ZBη (0) 1 < ε θG(Bη(0)) = ε . θG(Bη(0))

(iv) Finally, we can prove the claim. 10 TIMO SPINDELER

Fix ε> 0. By step (iii), there is η > 0 such that ε f fη Sp < . k − k 2

Moreover, fη is Bohr almost periodic by step (ii). Hence, there is a trigonometric polynomial P such that ε fη P Sp 6 fη P < . k − k k − k∞ 2 Therefore, we obtain

f P Sp 6 f fη Sp + fη P Sp < ε , k − k k − k k − k which ﬁnishes the proof.

Now, Propositions 2.8 and 2.7 establish the equivalence of the three statements about Stepanov almost periodic functions mentioned above.

Theorem 2.9. For f Lp (G) the following statements are equivalent. ∈ loc (1) The function f is Stepanov p-almost periodic. (2) f is the Sp -limit of a sequence of trigonometric polynomials. k · k (3) The orbit Ttf t G is precompact in the Sp -topology. { | ∈ } k · k Corollary 2.10. The space of Bohr almost periodic functions is dense in Sp(G).

Proof. This immediately follows from Proposition 2.8 because every element of Sp(G) can be approximated by trigonometric polynomials, which are Bohr almost periodic.

p Remark 2.11. As a consequence of Proposition 2.8, we ﬁnd that (S (G), Sp ) is a normed p k·k vector space. In fact, if f, g S (G) and if (pn)n N, (qn)n N are sequences of trigonometric ∈ ∈ ∈ polynomials which approximate f and g, respectively, one has p f + g = lim pn + lim qn = lim (pn + qn) S (G) , n→∞ n→∞ n→∞ ∈ and, in the same way, cf Sp(G) for every c C. ∈ ∈ p We can say even more. The pair (S (G), Sp ) is not only a normed vector space but also k · k a Banach space.

p Proposition 2.12. S (G), Sp is a Banach space. k · k Proof. Every element f Sp(G) is an element of BSp(G) by Lemma 2.5(a). Since the space p ∈ p (BS (G), Sp ) is a Banach space, it suﬃces to show that S (G) is closed. k · k p In order to do so, let (fn)n∈N be a sequence in S (G) that converges to some function f in the Sp -topology. First, this means that, given ε> 0, there is an element n N such that k · k 0 ∈ ε f fn Sp < for all n > n . k − k 3 0 Second, for such an n, there is a relatively dense set of elements t G such that ∈ ε fn Ttfn Sp < for all such t . k − k 3 STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 11

Finally, one has ε ε ε f Ttf Sp 6 f fn Sp + fn Ttfn Sp + Ttfn Ttf Sp < + + = ε , k − k k − k k − k k − k 3 3 3 = kf−fnkSp for all such t, which ﬁnishes the proof. | {z } p The space S (G), Sp is closed under addition, but it is not closed under multiplication. k·k 1 To see this, consider the function φ(x)= 1[− 1 , 1 [(x). Then, the function √|x| · 2 2 f(x)= φ(x + k) Z Xk∈ is an element of S1(R), but f 2 is not locally integrable. Hence, f 2 / S1(R). However, if one ∈ of the two functions is bounded, their product is Stepanov almost periodic. Proposition 2.13. Let f, g Sp(G), and let f be bounded. Then, f g Sp(G). ∈ · ∈ Proof. Let ε> 0. First, there is a trigonometric polynomial Q such that ε g Q Sp < . k − k 2 f k k∞ Note that we can assume that f > 0 (otherwise f(x) = 0 almost everywhere and the k k∞ statement is trivially true). As Q is a trigonometric polynomial, it is bounded. Hence, there is a trigonometric polynomial P such that ε f P Sp < . k − k 2 Q k k∞ Therefore, we obtain

fg PQ Sp 6 fg fQ Sp + fQ PQ Sp 6 f g Q Sp + Q f P Sp < ε . k − k k − k k − k k k∞ k − k k k∞ k − k Since PQ is a trigonometric polynomial, the claim follows. If we want the product of two Stepanov almost periodic functions to be an element of S1(G), we can apply H¨olders theorem. Proposition 2.14. Let f Sp(G), g Sq(G) such that 1= 1 + 1 . Then, fg S1(G). ∈ ∈ p q ∈ Proof. This is an immediate consequence of H¨olders inequality.

3. Weyl almost periodic functions After establishing basic properties of Stepanov almost periodic functions in the previous section, we will now focus on Weyl almost periodic functions. The main difference is that we don’t consider a fixed compact set K but a specific sequence of sets - a so called van Hove sequence.

Definition 3.1. A sequence (An)n∈N of precompact open subsets of G is called a van Hove sequence if, for each compact set K G, we have ⊆ K ∂ An lim | | = 0 , n→∞ An | | 12 TIMO SPINDELER where the K-boundary ∂K A of an open set A is defined as ∂K A := A + K A ((G A) K) A . \ ∪ \ − ∩ Note that every σ-compact locally compact Abelian group G admits the construction of van Hove sequences [17]. Let us proceed with the definition of Weyl almost periodic functions. p Definition 3.2. Let (An)n N be a van Hove sequence. A function f L (G) is called Weyl ∈ ∈ loc p-almost periodic (with respect to (An)n∈N) if there is a sequence of trigonometric polynomials (pm)m∈N such that 1 p 1 p lim W p,A(f pm) := lim lim sup sup f(x) pm(x) dθG(x) = 0 . m→∞ M − m→∞ n θ (A ) | − | →∞ y∈G G n Zy+An p We denote the space of Weyl almost periodic functions by WA(G). p Remark 3.3. A function f L (G) is Weyl p-almost periodic (with respect to (An)n N) if ∈ loc ∈ and only if, for each ε> 0, there is a trigonometric polynomial P such that 1 p 1 p lim sup sup f(x) P (x) dθG(x) < ε . n θ (A ) | − | →∞ y∈G G n Zy+An We will see soon that we can replace ‘lim supn→∞ supy∈G’ simply by ‘limn→∞’ in the above definition. To see this, we will first establish that every f W p (G) is amenable. ∈ A p Proposition 3.4. Let (An)n N be a van Hove sequence. Then, every f W (G) is amenable ∈ ∈ A (with respect to (An)n∈N), i.e. the limit 1 lim f(x) dθG(x) n→∞ θ (A ) G n Zy+An exists uniformly in y G. ∈ Proof. Fix ε> 0. Since f W p (G), we can write ∈ A f(x)= P (x)+ r(x) , where P is a trigonometric polynomial and r satisfies 1 p 1 p ε lim sup sup r(x) dθG(x) < . n θ (A ) | | 4 →∞ y∈G G n Zy+An Every trigonometric polynomial is Bohr almost periodic and (hence) amenable. Therefore, there are numbers N N and M(P ) R such that, by Jensens inequality, ∈ ∈ 1 1 f(x) dθG(x) M(P ) 6 P (x) dθG(x) M(P ) θ (A ) − θ (A ) − G n Zy+An G n Zy+An

1 + r(x) dθG(x) θ (A ) | | G n Zy+An 1 6 P (x) dθG(x) M(P ) θ (A ) − G n Zy+An

STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 13

1 p 1 p + r(x) dθG(x) θ (A ) | | G n Zy+An ε ε ε < + = 4 4 2 for all n > N, independently of y. Consequently, we obtain 1 1 f(x) dθG(x) f(x) dθG(x) < ε ′ ′′ θG(An ) y+A ′ − θG(An ) y+A ′′ Z n Z n for all n′,n′′ > N, independently of y. This ﬁnishes the proof.

p Proposition 3.5. Let (An)n N be a van Hove sequence, and let f, g W (G). Then, ∈ ∈ A f g, f , cf, χf W p (G) ± | | ∈ A for all c C and χ G. ∈ ∈ Proof. This is not obvious only for f : b | | Let ε> 0. Since f W p (G), there is a trigonometric polynomial P such that ∈ A 1 1 p ε lim sup sup f(x) P (x) p dx < . n θ (A ) | − | 2 →∞ y∈G G n Zy+An Note that P is Bohr almost periodic, and hence there is a trigonometric polynomial Q such | | that ε P Q < . k| |− k∞ 2 Therefore, we obtain 1 p 1 p W p,A( f Q) 6 lim sup sup f(x) P (x) dx M | |− n θ (A ) || | − | || →∞ y∈G G n Zy+An 1 1 p + limsup sup P (x) Q(x) p dx n θ (A ) || |− | →∞ y∈G G n Zy+An 1 p 1 p 6 lim sup sup f(x) P (x) dx + P Q ∞ n θ (A ) | − | k| |− k →∞ y∈G G n Zy+An ε ε < + = ε . 2 2

Corollary 3.6. Let f W p (G), and let χ G be a character. Then, χf is amenable. ∈ A ∈ Proof. This is a consequence of the previous twob propositions. Corollary 3.7. Let f Lp (G). Then, the function f is Weyl p-almost periodic if and only ∈ loc if there is a sequence of trigonometric polynomials (pm)m∈N such that 1 p 1 p lim lim sup f(x) pm(x) dθG(x) = 0 . (3.1) m→∞ n→∞ θ (A ) | − | y∈G G n Zy+An ! 14 TIMO SPINDELER

Proof. It is easy to see that f P is Weyl p-almost periodic if P is a trigonometric polynomial | − | and f W p (G). Hence, f P is amenable by Proposition 3.4, and the claim follows. ∈ A | − | The other direction is obvious.

Remark 3.8. Let us define 1 p 1 p f W p := lim f(x) dθG(x) , k k A n→∞ θ (A ) | | G n Zy+An for an arbitrary y G. Then, Eq. (3.1) becomes ∈ lim f pm W p = 0 m→∞ k − k A p because f p = p (f) for all f W (G), see Proposition 3.4 and Corollary 3.7. Note WA W A k k M p ∈ p that, in contrast to (S (G), Sp ), the pair (W (G), p ) is not a complete space. k · k A k · kWA Moreover, W p is not a norm but only a seminorm (in general) because f W p = 0 for k · k k k all f Lp(Rd). This leads to another difference between Stepanov and Weyl almost periodic ∈ functions. While f g Sp = 0 implies that f and g coincide almost everywhere, f and g k − k can differ on a set of positive (or even infinite) measure when f g p = 0. k − kWA Proposition 3.9. We have Sp(G) W p (G). More precisely, there is a compact set C such ⊆ A that

f W p 6 f Sp k k A k k C for every f Sp(G). ∈ p Proof. Let f S (G), and let (An)n∈N be a van Hove sequence. Consider the measure p ∈ µ := f θG. By [20, Lem. 3.3], there is a compact set C such that | | 1 µ (y + A ) µ p n 6 C p lim sup f(x) dθG(x) = limsup | | k k = f Sp n θ (A ) | | n θ (y + A ) θ (C) k k C →∞ G n Zy+An →∞ G n G for all y G. Since every f W p (G) is amenable by Proposition 3.4, we can replace ‘lim sup’ ∈ ∈ A by ‘lim’ in the above equation, and we obtain

f W p 6 f Sp , k k A k k C which ﬁnishes the proof.

Corollary 3.10. Every f Sp(G) is amenable. ∈ Proof. This is an immediate consequence of Propositions 3.4 and 3.9.

The next goal is to establish the analogon of Proposition 2.8 for Weyl p-almost periodic functions. Thus, we will show that Weyl almost periodicity can also be characterised by relatively dense sets of almost periods. For this, we will need the following lemmas.

Lemma 3.11. Let f Lp (G) be such that, for all ε′ > 0, there is a relatively dense set R ∈ loc and a number N N such that ∈ ′ f Ttf Sp < ε k − k An STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 15 for all t R and all n > N. Then, for all ε> 0, there is a neighbourhood V of 0 such that ∈ f Tsf Sp < ε k − k AN for all s V . ∈ Proof. Fix ε> 0. By assumption, there is a relatively dense set R and a number N N such ε ∈ p > that f Ttf S < 3 for all n N and all t R. So, we can write G = R + K for some k − k An ∈ compact set K. Due to [16, Thm. 1.1.5], there is a neighbourhood V of 0 such that

1 p 1 p ε f(x) Tsf(x) dθG(x) < θ (A ) | − | 3 G N ZK+AN for all s V . Consequently, one has ∈ f Tsf Sp 6 f Ttf Sp + Ttf Tt+sf Sp + Tt+sf Tsf Sp k − k AN k − k AN k − k AN k − k AN 1 p ε 1 p ε < + sup Ttf(x) Ts tf(x) dθG(x) + 3 θ (A ) | − + | 3 y∈G G N Zy+AN 1 p ε 1 p ε < + f(x) Tsf(x) dθG(x) + 3 θ (A ) | − | 3 G N ZK+AN < ε for all s V , which completes the proof. ∈ Lemma 3.12. Let f Lp (G) be such that, for every ε > 0, there is a relatively dense set ∈ loc R and a number N N such that ∈ f Ttf Sp < ε k − k An for all n > N and all t R. Fix ε′ > 0 and N ′ N. Then, there is a function : G C ∈ ∈ K → with the following properties: > 0, • K is bounded, • K 1 lim infn (z) dθG(z) = 1, →∞ θG(An) An • K ′ (z) = 0 only if f T−zf p < ε . SA • K 6 kR − k N′ Proof. Fix ε′ > 0. By assumption, there is a number N N such that ∈ ′ A := z G f T−zf Sp < ε { ∈ | k − k AN } and ε′ B := z G f T−zf Sp < { ∈ | k − k AN 2 } are relatively dense. Thus, there is a compact set K with G = K + B. Moreover, by Lemma 2.5, there is an open neighbourhood V of 0 such that ε′ f T−tf Sp < for all t V . k − k AN 2 ∈ 16 TIMO SPINDELER

Note that B + V A. Since K is compact, it can be covered by ﬁnitely many translates of ℓ ⊆ V , i.e. K (V + tj), t ,...,tℓ G. Hence, one has ⊆ j=1 1 ∈ S ℓ ℓ ℓ G = B + K B + (V + tj)= ((V + B)+ tj) (A + tj) . ⊆ ⊆ j j j [=1 [=1 [=1 This implies that

θG(A An) 1 c := lim inf ∩ = lim inf 1A(x) dθG(x) > 0 . n→∞ θ (A ) n→∞ θ (A ) G n G n ZAn 1 Now, the function (x)= 1A(x) satisﬁes the properties. K c Proposition 3.13. Let f Lp (G). Then, the following statements are equivalent: ∈ loc (1) f is Weyl p-almost periodic. (2) For every ε> 0, there is a relatively dense set R and a number N N such that ∈

f Ttf Sp < ε k − k An for all n > N and all t R. ∈ p Proof. (1) = (2): Fix ε> 0. Since f WA(G), there is a trigonometric polynomial P such ⇒ ε ∈ that f P p < . In particular, there is a number N N such that k − kWA 3 ∈ 1 p 1 p ε f P Sp = sup f(x) P (x) dθG(x) < k − k An θ (A ) | − | 3 y∈G G n Zy+An for all n > N, independently of y. This and the fact that every trigonometric polynomial is Bohr almost periodic imply that there is a relatively dense set R such that

f Ttf Sp 6 f P Sp + P TtP Sp + TtP Ttf Sp k − k An k − k An k − k An k − k An ε ε < + P TtP + < ε 3 k − k∞ 3 for all n > N and all t R. ∈ (2) = (1): Fix ε> 0 and y G. By assumption, there is a number N N and a relatively ⇒ ∈ ∈ dense set R such that 1 p 1 p ε f Ttf Sp = sup f(x) Ttf(x) dθG(x) < k − k An θ (A ) | − | 3 y∈G G n Zy+An for all n > N and t R, independently of y. ∈ Next, we consider a function : G C with the following properties: K → > 0, • K is bounded, • K 1 lim infn→∞ (z) dθG(z) = 1, • θG(An) An K (z) = 0 only if f T−zf Sp < ε. • K 6 k R − k AN STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 17

Such a function exists by Lemma 3.12. Moreover, we deﬁne the function φ : G C by → 1 φ(x) := lim inf f(x + z) (z) dθG(z) . n→∞ θ (A ) K G n ZAn (i) First, we will show that φ is continuous. To see this, only note that, by Lemma 3.11, for every ε′ > 0, there is a neighbourhood V of 0 such that 1 φ(x) T−δφ(x) 6 lim inf f(x + z) f(x + δ + z) (z) dθG(z) | − | n→∞ θ (A ) | − | K G n ZAn 6 f T δf W p kKk∞ k − − k (3.2) 6 c ∞ f T−δf Sp kKk k − k AN < ε′ , whenever δ V , where we made use of Proposition 3.9 and Lemma 2.2 in the penultimate ∈ step. Hence, φ is continuous. (ii) Additionally, φ is Bohr almost periodic. This immediately follows from Eq. (3.2), since δ = t gives − φ(x) Ttφ(x) 6 f Ttf W p | − | kKk∞ k − k for all x G. ∈ (iii) We have f φ p < ε. k − kWA This is a consequence of 1 f(x) φ(x) dθG(x) θ (A ) | − | G N Zy+AN 1 1 6 lim inf f(x) f(x + z) (z) dθG(z) dθG(x) θ (A ) n→∞ θ (A ) | − | K G N Zy+AN G n ZAn 1 1 6 lim inf f(x) f(x + z) dθG(x) (z) dθG(z) n→∞ θ (A ) θ (A ) | − | K G n ZAn G N Zy+AN 1 6 lim inf f T−zf Sp (z) dθG(z) n→∞ θ (A ) k − k AN K G n ZAn 1 < ε lim inf (z) dθG(z) = ε , n→∞ θ (A ) K G n ZAn where we applied Fatous lemma and Fubinis theorem. (iv) Finally, we can prove the claim. This follows exactly as in the last step of the proof of Proposition 2.8. Remark 3.14. Let f Lp (G). In this case, the property ∈ loc (I) For every ε> 0, there is a relatively dense set R and a number N N such that ∈ f Ttf Sp < ε k − k An for all n > N and all t R. ∈ 18 TIMO SPINDELER is stronger than (II) For every ε> 0, there is a relatively dense set R such that

lim f Ttf Sp < ε n→∞ k − k An for all t R. ∈ The reason for this is that (II) is equivalent to (II’) For every ε > 0, there is a relatively dense set R such that, for all t R, there is a ∈ number N N such that ∈ f Ttf Sp < ε k − k An for all n > N. The difference is that N can be chosen independently of t in (I), but it will depend on t in (II’). For this reason, some people prefer to call elements f W p (R) equi-Weyl almost ∈ A periodic. Next, we can state the analoga of Lemma 2.6, Proposition 2.13 and Proposition 2.14. Lemma 3.15. Let f W p (G), and let ε> 0. Then, there is a bounded function b W p (G) ∈ A ∈ A such that f b p < ε . k − kWA Proof. This is proved like Lemma 2.6 using the characterisation from Proposition 3.13. Proposition 3.16. Let f W p (G), g W q (G) such that 1= 1 + 1 . Then, fg W 1 (G). ∈ A ∈ A p q ∈ A Proof. This is an immediate consequence of Hölders inequality. At the end of this section, let us briefly mention some additional properties of bounded Weyl almost periodic functions. Proposition 3.17. Let f, g W p (G), and let f be bounded. Then, fg W p (G). ∈ A ∈ A Proof. Simply imitate the proof of Proposition 2.13. Proposition 3.18. ([10, Prop. 4.11]) Let f : G C be a bounded and measurable function. → p Let and be van Hove sequences. Then, f belongs to WA(G) if and only if it belongs to Ap B p p W ap (G). If f belongs to W (G) and W ap (G), then f p = f W p holds. B A B k kWA k k B The previous proposition says that a bounded Weyl p-almost periodic function is independent of the choice of the van Hove sequence.

4. Convolutions with Stepanov and Weyl almost periodic functions In this section, we will have a very brief look at convolutions with Stepanov almost periodic functions. We will consider the following three kinds of convolutions: (1) If f and g are functions from G to C, we deﬁne

(f g)(x) := f(x y) g(y) dθG(y) , ∗ − ZG whenever the integral exists. STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 19

(2) If µ is a measure on G, we deﬁne

(f µ)(x) := f(x y) dµ(y) , ∗ − ZG whenever the integral exists. (3) Last, we deﬁne the Eberlein convolution of f and g (with respect to the van Hove sequence (An)n∈N) by 1 (f ⊛ g)(x) := lim f(x y) g(y) dθG(y) , n→∞ θ (A ) − G n ZAn whenever the limit exists. Proposition 4.1. Let f Sp(G), and let µ be a ﬁnite measure. Then, f µ Sp(G). ∈ ∗ ∈ Proof. The statement follows from Minkowskis inequality for integrals because

f µ Tt(f µ) Sp = (f Ttf) µ Sp k ∗ − ∗ k k − ∗ k 1 p 1 p = sup ((f Ttf) µ)(x) dθG(x) θ (K) | − ∗ | y∈G G Zy+K p 1 1 p = sup (f Ttf)(x z) dµ(z) dθG(x) θ (K) − − y∈G G Zy+K ZG 1 p 1 p 6 sup (f Ttf)(x z) dθG (x) dµ(z) θ (K) | − − | y∈G ZG G Zy+K

6 f Ttf Sp dµ(z) k − k ZG = µ(G) f Ttf Sp . k − k The next corollary is an immediate consequence. Corollary 4.2. Let f Sp(G), and let g L1(G). Then, f g Sp(G). ∈ ∈ ∗ ∈ In general, the convolution of two Weyl almost periodic functions does not exist. For this reason, we consider its averaged version - the Eberlein convolution. Lemma 4.3. Let f W p (G), and let g W q(G) for q > 1 with 1 + 1 = 1. Then, the limit ∈ A ∈ p q f ⊛ g exists. Proof. Fix ε> 0. Since f W p (G) and g W q(G), we can write ∈ A ∈ f(x)= P (x)+ r(x) and g(x)= Q(x)+ s(x) , where P,Q are trigonometric polynomials and r and s satisfy 1 p 1 p ε lim sup sup r(x) dθG(x) < n→∞ θG(An) | | 8 Q y∈G Zy+An k k∞ and 1 q 1 q ε lim sup sup s(x) dθG(x) < . n→∞ θG(An) | | 8 P y∈G Zy+An k k∞ 20 TIMO SPINDELER

Therefore, there is an N N such that (via the Jensen and H¨older inequality) ∈ 1 f(x z) g(z) dθG(z) (P ⊛ Q)(x) θ (A ) − − G n ZAn

1 6 P (x z) Q(z) dθG(z) (P ⊛ Q)(x) θ (A ) − − G n ZAn

1 + Q(y) r(x z)+ P (x z) s(z )+ r(x z) s(z) dθG(z) θ (A ) − − − G n ZAn ε 1 1 < + Q r(x z) dθG(z)+ P s(z) dθG(z) 8 k k∞ θ (A ) | − | k k∞ θ (A ) | | G n ZAn G n ZAn 1 + r(x z) s(z) dθG(z) θ (A ) | − | G n ZAn 1 p ε 1 p 6 + Q r(x z) dθG(z) 8 k k∞ θ (A ) | − | G n ZAn 1 q 1 q 1 + P s(z) dθG(z) + r(x z) s(z) dθG(z) k k∞ θ (A ) | | θ (A ) | − | G n ZAn G n ZAn 1 1 q p 3 ε 1 q 1 p < + s(z) dθG(z) r(x z) dθG(z) 8 θ (A ) | | θ (A ) | − | G n ZAn G n ZAn 3 ε ε2 ε < + < 8 16 Q P 2 k k∞ k k∞ for all n > N. Consequently, we obtain

1 1 f(x y) g(y) dθG(y) f(x y) g(y) dθG(y) < ε ′ ′′ θG(An ) y+A ′ − − θG(An ) y+A ′′ − Z n Z n for all n′,n′′ > N. This ﬁnishes the proof.

Proposition 4.4. Let f W p (G), and let g W q(G) for q > 1 with 1 + 1 = 1. Then, f ⊛ g ∈ A ∈ p q is a Bohr almost periodic function.

Proof. By deﬁnition, there are sequences of trigonometric polynomials (Pn)n∈N and (Qn)n∈N such that

lim f Pn W p = 0 and lim g Qn W q = 0 . n→∞ k − k n→∞ k − k This immediately implies that there is a constant c> 0 such that Qn W q 6 c for all n N. k k ∈ Applying the H¨older inequality, we get

f ⊛ g Pn ⊛ Qn 6 f ⊛ (g Qn) + (f Pn) ⊛ Qn k − k∞ k − k∞ k − k∞ 6 f W p g Qn W q + f Pn W p Qn W q . k k k − k k − k k k Thus, the sequence (Pn ⊛ Qn)n∈N converges uniformly to f ⊛ g. The claim follows because Pn ⊛ Qn is Bohr almost periodic for all n N, see [2, Thm. 4.6.3], and uniform limits of Bohr ∈ almost periodic functions are Bohr almost periodic [2, Prop. 4.3.4]. STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 21

5. Fourier–Bohr series p Definition 5.1. Let f W (G), let y G, and let (An)n N be a van Hove sequence. We ∈ A ∈ ∈ define the Fourier–Bohr coefficients of f by 1 cχ(f) := M(χf) := lim χ(x) f(x) dx n→∞ An | | Zy+An if the limit exists. Moreover, we call the formal sum

cχ(f) χ χ G X∈ b the Fourier–Bohr series of f. Proposition 5.2. Let f W p (G). ∈ A (a) The Fourier–Bohr coefficients cχ(f) exist uniformly in y for all χ G. If in addi- ∈ tion f is uniformly continuous and bounded, cχ(f) is also independent of (An)n∈N. (b) The set χ G cχ(f) = 0 is at most countable. b { ∈ | 6 } Proof. (a) This immediatelyb follows from Corollary 3.6. (b) By definition, f can be approximated by a sequence (pn)n∈N of trigonometric polynomials. For fixed n N, the value cχ(pn) differs from 0 only for a finite number of χ, since M(χ) = 0 ∈ for all non-trivial characters χ. From this, we can infer the claim. Corollary 5.3. The Fourier–Bohr series exists for all f W p (G). ∈ A Now, since cχ(f) exists for all χ G, the next result follows from standard techniques. ∈ 2 Proposition 5.4. Let f W (G), and let χ ,...,χn G be distinct characters. Then, one ∈ b 1 ∈ has n n 2 2 b 2 M f cχ (f) χj = M f cχ (f) . − j | | − | j | j j X=1 X=1 n 2 2 Consequently, cχ (f) 6 M f . Hence, (we obtain one more time that) the set j=1 | j | | | χ G cχ(f) = 0 is at most countable, and one has Bessels inequality { ∈ | P6 } ∞ 2 2 b cχ (f) 6 M f < . | j | | | ∞ j=1 X Proof. See [2, p. 226]. To prove the main result of this section, we need some preparation.

2 Lemma 5.5. Let f W (G), and assume that An = An for all n. Then, we have ∈ − 2 cχ(f ⊛ f)= cχ(f) . | | 2 Proof. Since An = An, it is easy to see that f W (G). Then, the proof of this lemma − e ∈ is given in [2, p. 227]. The author assumes that f is weakly almost periodic, but the proof reveals that the above assumption suﬃces. e 22 TIMO SPINDELER

2 Theorem 5.6. (Parsevals equality) Let f W (G), and assume that An = An for all n. ∈ − Then, cχ(f) = 0 for at most countable many χ G, and one has 6 ∈ 2 2 M f = cχ(f) . | | |b | χ∈G Xb Proof. Proposition 4.4 tells us that f ⊛ f is a Bohr almost periodic function. Moreover, its 2 Fourier–Bohr coeﬃcients are given by cχ(f) , see Lemma 5.5. By Bessel’s inequality (see | e2 | Proposition 5.4), the series χ∈G cχ(f) converges absoultely. Hence, the Fourier–Bohr b | | series of f ⊛ f converges uniformly,P and f ⊛ f is its limit [2, Lem. 4.6.11]. Therefore, we obtain e 2 e 2 M f = (f ⊛ f)(0) = cχ(f ⊛ f) χ(0) = cχ(f) . | | | | χ G χ G X∈ b X∈ b e e We complete the section with a uniqueness result.

Theorem 5.7. Let f, g Sp(G) such that ∈ cχ(f)= cχ(g) for all χ G. ∈ Then, f and g coincide θG-almost everywhere. b A proof of this theorem will be given in the appendix.

6. Relations to other notions of almost periodic functions So far, we have only considered Stepanov and Weyl almost periodic functions. However, there are other notions of almost periodic functions. In the following, we want to see how they are connected. The space of Bohr almost periodic functions is denoted by SAP(G).

p Proposition 6.1. One has S (G) Cu(G)= SAP(G). ∩ Proof. : Since f is uniformly continuous, for every ε> 0, there is a neighbourhood V of 0 ⊆ such that ε f(x′) f(x′′) < , (6.1) | − | 4 whenever x′ x′′ V . Next, let t G be such that − ∈ ∈ 1 θG(V ) p ε f Ttf Sp < . (6.2) k − k θ (K) 2 G Now, we claim that

f Ttf < ε . k − k∞ Assume on the contrary that this statement is not true, i.e. there is an element x G such 0 ∈ that f(x ) Ttf(x ) > ε. Then, Eq. (6.1) implies | 0 − 0 | ε f(x) Ttf(x) > , | − | 2 STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 23 as long as x x0 V . Consequently, one has − ∈ 1 1 p p 1 p 1 p f(x) Ttf(x) dθG(x) > f(x) Ttf(x) dθG(x) θ (K) | − | θ (K) | − | G Zx0+K G Zx0+V 1 ε 1 > θ (x + V ) p 1 2 G 0 θG(K) p 1 θ (V ) p ε = G , θ (K) 2 G which contradicts Eq. (6.2). : On the one hand, every weakly almost periodic function (in particular, every Bohr almost ⊇ periodic function) is uniformly continuous, see [8, Thm. 13.1]. On the other hand, we have 1 p 1 p f Ttf Sp = sup f(x) Ttf(x) dθG(x) 6 f Ttf , k − k θ (K) | − | k − k∞ y∈G G Zy+K which implies SAP(G) Sp(G). ⊆ Remark 6.2. It is not possible to replace the uniform continuity by mere continuity. An example of a function which is continuous and Stepanov almost periodic but not Bohr almost periodic is given by 1 f : R R, x sin , → 7→ 2 + cos(αx) + cos(βx) where α, β R such that α, β and αβ−1 are irrational, see [12]. ∈ Deﬁnition 6.3. A function f C (G) is called weakly almost periodic if the closure of ∈ u Ttf t G is compact in the weak topology. The space of all weakly almost periodic { | ∈ } functions is denoted by WAP(G).

Deﬁnition 6.4. Let f Cu(G) and let = (An)n N be a van Hove sequence. We deﬁne the ∈ A ∈ upper absolute mean of f along by A 1 M A(f) := lim sup f(x) dx. n→∞ An | | | | ZAn Whenever the van Hove sequence is clear from the context, we will write M(f) instead of A M A(f).

Deﬁnition 6.5. Fix a van Hove sequence = (An)n N. A function f C (G) is called A ∈ ∈ u mean almost periodic (with respect to ) if, for all ε> 0, the set A t G M (Ttf f) < ε { ∈ | A − } is relatively dense in G. The space of all mean almost periodic functions is denoted by MAP(G). Proposition 6.6. We have the following chain of inclusions: SAP(G) S1(G) W 1 (G) MAP(G) . ⊆ ⊆ A ⊆ 24 TIMO SPINDELER

Proof. The ﬁrst inclusion follows from Proposition 6.1, the second inclusion follows from Proposition 3.9, and the third inclusion follows from Remark 3.14.

Definition 6.7. A function f WAP(G) is called null weakly almost periodic if M(f) = 0. ∈ The set of all null weakly almost periodic functions is denoted by WAP0(G). Proposition 6.8. We also have: SAP(G) WAP(G) W 1 (G) MAP(G) . ⊆ ⊆ A ⊆ Proof. The first inclusion is trivial. For the second inclusion, let f WAP(G). In that case, we can write ∈ f = fs + f0 with f SAP(G) and f WAP (G), see [2, Thm. 4.7.11]. Since f SAP(G), for every s ∈ 0 ∈ 0 s ∈ n N, there is a trigonometric polynomial pn such that ∈ 1 f pn < . k s − k∞ n Therefore, one obtains 1 f pn 1 = M(f pn) 6 M(f pn)+ M(f ) 6 f pn < , k − kW − s − 0 k s − k∞ n which implies f W 1 (G). ∈ A The last inclusion was proved in the previous proposition. One should note that there is no subset relation between WAP(G) and S1(G). The example from Remark 6.2 is Stepanov almost periodic but not weakly almost periodic (because it is not uniformly continuous). On the other hand, every function vanishing at infinity is weakly almost periodic but not Stepanov almost periodic (unless it is identical zero). To be more precise, one has WAP(G) S1(G) = SAP(G) , ∩ WAP (G) S1(G)= 0 . 0 ∩ { } Remark 6.9. The limit 1 lim f(x) dθG(x) n→∞ θ (A ) G n ZAn is in general not independent of (An)n N if f MAP (G). For this, consider the function ∈ ∈ 0, x< 0, h(x)= x, 0 6 x 6 1,  1, x> 1.

It is straightforward to check that h MAP( G). On the other side, h is not independent of ∈  (An)n∈N because 1 n 1 0 f(x) dx =1 and f(x) dx = 0 . n n Z0 Z−n STEPANOV AND WEYL ALMOST PERIODICITY IN LOCALLY COMPACT ABELIAN GROUPS 25

Remark 6.10. If f is a weakly almost periodic function, it is Weyl almost periodic by Proposition 6.8. Moreover, there is a unique decomposition

f = fs + f0 , where fs is a Bohr almost periodic function, and f0 is a null weakly almost periodic function. We trivially have f W p = 0, which implies k 0k f W p = f W p . k k k sk So, the seminorm W p can only see the strongly almost periodic component of a weakly k · k almost periodic function.

Appendix A. In this last section, we will have a look at Stepanov almost periodic measures. Proposition A.1. Let µ ∞(G), and let 1 p< . Then, µ (G) if and only if p ∈ M ≤ ∞ ∈ SAP µ f S (G) for all f Cc(G). ∗ ∈ ∈ Proof. = : If µ (G), we have µ f SAP(G) for all f Cc(G). Since every Bohr ⇒ ∈ SAP ∗ ∈ ∈ almost periodic function is stepanov almost periodic, µ f Sp(G) for all f Cc(G). ∗ ∈ ∈ p =: By assumption, µ f S (G) for all f Cc(G). Furthermore, we have µ f Cu(G), ⇐ ∗ ∈ ∈ ∗ ∈ for all f Cc(G), because µ ∞(G). Now, Proposition 6.1 implies the claim. ∈ ∈ M Deﬁnition A.2. A measure µ ∞(G) is called Stepanov p-almost periodic if ∈ M p µ f S (G) for all f Cc(G) . ∗ ∈ ∈ Remark A.3. Proposition A.1 says that µ is Stepanov almost periodic if and only if it is strongly almost periodic. This happens because µ f Cu(G) for every f Cc(G) (since µ is ∗ ∈ ∈ translation bounded) and uniformly continuous Stepanov almost periodic functions are Bohr almost periodic, see Proposition A.1.

Remark A.4. We know that a function f Cu(G) is Bohr almost periodic (weakly almost ∈ periodic/ null weakly almost periodic) if and only if the measure fθG is strongly almost periodic (weakly almost periodic/ null weakly almost periodic), see [2, Prop. 4.10.5]. Now, p if f Cu(G), then f S (G) if and only if fθG is an Stepanov p-almost periodic measure ∈ ∈ because p f Cu(G) S (G) f SAP(G) fθG (G) , ∈ ∩ ⇐⇒ ∈ ⇐⇒ ∈ SAP see Proposition 6.1 and Remark A.3.

p Proposition A.5. If f S (G), then fθG (G). ∈ ∈ SAP p Proof. By Corollary 4.2, µ φ S (G) for all φ Cc(G). Consequently, µ is an Stepanov p- ∗ ∈ ∈ almost periodic measure. Also, it is easy to verify that fθG is translation bounded. Therefore, it is strongly almost periodic, see Remark A.3.

Finally, let us give a proof of Theorem 5.7. 26 TIMO SPINDELER

p Proof. First note that, for any h S (G), we have cχ(h)= cχ(hθG) for all χ G. Together ∈ ∈ with the assumption, this gives b cχ(fθG)= cχ(gθG) for all χ G . (A.1) ∈ By Proposition A.5, the measures fθ and gθ are strongly almost periodic, which implies G G b that fθG gθG is strongly almost periodic, too. On the other hand, it follows from Eq. (A.1) − and [9, Thm. 8.1] that fθG gθG is null weakly almost periodic. But the only measure which − is strongly and null weakly almost periodic is the null measure, see [9, Prop. 5.7]. Thus, we have fθG = gθG, which is equivalent to f = g almost everywhere.

Acknowledgments. The author wishes to thank Nicolae Strungaru for interesting discus- sions. The work was supported by DFG via a Forschungsstipendium with grant 415818660, and the author is grateful for the support.

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Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, AB, T6G 2G1, Canada E-mail address: [email protected]