Translation properties of time scales and almost periodic functions

Mathematical and Computer Modelling 57 (2013) 1165–1174

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Translation properties of time scales and almost periodic functions Yongjia Guan a,∗ , Ke Wang a,b a

Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai 264209, PR China

b

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, PR China

article

info

Article history: Received 23 June 2011 Received in revised form 8 October 2012 Accepted 14 October 2012 Keywords: Time scales Almost periodic function Translation invariants Primitive translation invariants

abstract In this paper, we study the structure of time scales under translations. We define several kinds of time scales such as the two-way translation invariant time scale and investigate their properties. Then we develop the almost periodic function theory on time scales, with examples given along the way. © 2012 Elsevier Ltd. All rights reserved.

1. Introduction The theory of time scales was first developed by Stefan Hilger [1] in order to unify continuous and discrete analysis. It has shown great potential for application recently, such as in the fields of economics [2,3], physics [4] and population dynamics [5,6]. Many important results have been obtained regarding time scales [7]. However, to the author’s best knowledge, almost periodic function theories have not been extended to time scales. The goal of this paper is to study almost periodic functions on time scales. The concept of almost periodic function was first studied by Harald Bohr [8] and later generalized by Abram Samoilovitch Besicovitch [9], Vyacheslav Stepanov and Hermann Weyl. It is useful in studying natural phenomenons that appear to retrace their paths, but not exactly, such as the planetary system. By studying the theory of almost periodic functions on time scales, we can broaden the fields of application for almost periodic functions. Almost periodic functions on R and almost periodic sequences on Z have been studied extensively, by developing almost periodic function theory on time scales, one can unify the two concepts and avoid proving results twice. This paper begins the investigation by analyzing the structure of a time scale under translation. In Section 2, we first define translation invariants for the time scale T and the set of translation invariants Υ (T), then investigate the properties of Υ (T). Next, according to the structure of time scales, we define two-way translation invariant time scales, positive translation invariant time scales and negative translation invariant time scales, and investigate their properties respectively. Since τ ∈ Υ (T) implies nτ ∈ Υ (T) for n ∈ N+ , we introduce the concept of primitive translation invariants. Section 3 introduces the concept of odd translation time scales, and proves that an odd translation time scale T either has no primitive translation invariants (in this case we have T = R), or has exactly one positive primitive translation invariant. Finally we get the important result that T is an odd translation time scale if and only if it is a two-way translation invariant time scale. Section 4 defines almost periodic functions and uniformly almost periodic functions on two-way translation invariant time scales and positive translation invariant time scales respectively, with examples given by theorems.

∗

Corresponding author. Tel.: +86 0631 5687086; fax: +86 0631 5687572. E-mail addresses: [email protected] (Y. Guan), [email protected] (K. Wang).

0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.10.018

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2. Translation invariance of time scales In order to define almost periodic functions on time scales, we need to investigate how time scales change under translations. We begin with the definition below. Definition 1. For a given time scale T, if a real number τ satisfies t + τ ∈ T,

for all t ∈ T,

(1)

then τ is called a translation invariant for T. We denote the set of all translation invariants of T as Υ (T). The following properties are obvious. (i) For a given time scale T, if τ ∈ Υ (T) then nτ ∈ Υ (T) for n ∈ N+ . (ii) For any time scale T, 0 ∈ Υ (T). We give several examples of Υ (T) below. Example 1. Denote [0, +∞) as R+ , (−∞, 0] as R− , then we can get

Υ (R) = R,

Υ (Z) = Z,

Υ (R+ ) = R+ ,

Υ (R− ) = R− .

Υ (T) may not be a subset of T. See the example given below. Example 2. Let

T = Z − 3Z = {. . . , −5, −4, −2, −1, 1, 2, 4, 5, . . .}. Apparently, Υ (T) = 3Z, which is not a subset of T. For some time scales, Υ (T)={0}. Example 3. Let

T = {1, 4, 9, . . . , n2 , . . .}. Here we have Υ (T) = {0}. Lemma 1. Υ (T) is a closed set. Proof. Suppose τ is an element of the closure of Υ (T). Then there exists a real number sequence {τi } whose limit is τ , with τi ∈ Υ (T). For a given t ∈ T, we have t + τi ∈ T. It is easy to see that t + τi → t + τ as i → ∞. According to the definition of time scales, T is a closed set, so we get t + τ ∈ T. Because t is arbitrary in T, we obtain τ ∈ Υ (T). The proof is complete.  Since 0 is always an element of Υ (T), and it does not change the original time scale during the translation, we denote the set of nonzero translation invariants Υ (T) − {0} as Υ (T). Definition 2. If Υ (T) ̸= ∅, and for all τ ∈ Υ (T), τ > 0 is satisfied, then T is called positive translation invariant time scale. If Υ (T) ̸= ∅, and for all τ ∈ Υ (T), τ < 0 is satisfied, then T is called negative translation invariant time scale. If there exist α, β ∈ Υ (T), with α > 0, β < 0, then T is called two-way translation invariant time scale. We have the following results. Lemma 2. (i) If T is a positive translation invariant time scale, then sup T = +∞. (ii) If T is a negative translation invariant time scale, then inf T = −∞. (iii) If T is a two-way translation invariant time scale, then sup T = +∞, inf T = −∞. Proof. Suppose T is a positive translation invariant time scale. Then there exists a τ1 ∈ Υ (T) with τ1 > 0 and a t1 ∈ T. So we get t1 + nτ ∈ T, n ∈ N+ . This implies sup T = +∞. This completes the proof of (i). The proofs of (ii) and (iii) are quite similar.  Definition 3. Suppose α ̸= 0, α ∈ Υ (T). We define



N+ (α) = n ∈ N+ :

α n

 ∈ Υ (T) .

It is easy to see that 1 ∈ N (α). +

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For a given n ∈ N+ (α), define

γ

α  n

=

   α 2α 3α : m ∈ N+ = , , , . . . ⊂ Υ (T).

 mα n

n

n

n

Define the set Υ (α) as

Υ (α) =



γ

α 

n∈N+ (α)

n

⊂ Υ (T).

Obviously, we have nα ∈ Υ (α) for n ∈ N+ . There is also the following lemma regarding Υ (α). Lemma 3. If β ∈ Υ (α), we have the following results. β

(i) α, β ∈ Υ (T), α ∈ Q. (ii) α ∈ Υ (β). Proof. Since β ∈ Υ (α), there exist n1 ∈ N+ (α) ⊂ N+ , β ∈ γ ( nα ). Thus we can find a k ∈ N+ satisfying β = knα . Since 1

α ̸= 0, α ∈ Υ (α), Υ (α) ⊂ Υ (T), we have α, β ∈ Υ (T), βα ∈ Q. β Since β = knα , we get k = nα ∈ Υ (T). This implies k ∈ N+ (β). Together with α = 1 1 thus α ∈ Υ (β). 

1

n1 β k

, n1 ∈ N+ , we obtain α ∈ γ ( βk ),

Lemma 4. If sup N+ (α) = ∞, α > 0, then R+ = [0, +∞) ⊂ Υ (T). If sup N+ (α) = ∞, α < 0, then R− = (−∞, 0] ⊂ Υ (T). Proof. Assume sup N+ (α) = ∞, α > 0. It is easy to see that Υ (α) is dense in R+ . Since Υ (α) ⊂ Υ (T), Υ (T) is dense in R+ . Since Υ (T) is a closed set, we have R+ ⊂ Υ (T). This completes the proof of our first statement. The proof for α < 0 is quite similar.  Lemma 5. If there exists α > 0, α ∈ Υ (T) satisfying sup N+ (α) < +∞, then for any β > 0, β ∈ Υ (T) we have sup N+ (β) < +∞. Proof. If sup N+ (β) = +∞, using Lemma 4 we get R+ ⊂ Υ (T), this implies sup N+ (α) = +∞. This contradicts with sup N+ (α) < +∞. The proof is complete.  In order to further investigate the structure of time scales under translations, we introduce the concept of primitive translation invariant below. α Definition 4. Suppose α ∈ Υ (T). If there exists n ∈ N+ satisfying αn ∈ Υ (T), but for any m ∈ N+ , m > n, m is not an α element of Υ (T), then β = n is called a primitive translation invariant of time scale T.

It is obvious that a primitive translation invariant does not equal 0. We give several examples of primitive translation invariants below. Example 4. The time scale R does not have primitive translation invariants. The primitive translation invariants for time scale Z are 1 and −1. The primitive translation invariant for time scale N is 1. The primitive translation invariants for time scale T = {m + nπ : m, n ∈ N} are 1 and π . For time scales that do not have primitive translation invariants, we have the following theorem. Theorem 1. Suppose time scale T does not have primitive translation invariants, Υ (T) ̸= ∅. (i) If T is a positive translation invariant time scale, then

T = [a, +∞),

a = inf T.

(ii) If T is a negative translation invariant time scale, then

T = (−∞, b],

b = sup T.

(iii) If T is a two-way translation invariant time scale, then T = R.

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Proof. (i) If T is a positive translation invariant time scale, then there is a α > 0, α ∈ Υ (T). Since T does not have primitive translation invariants, we have sup N+ (α) = ∞. Using Lemma 4 we get R+ ⊂ Υ (T). Since T is a positive translation invariant time scale, there is τ ≥ 0 for all τ ∈ Υ (T). So we get R+ = Υ (T). Thus

T = [inf T, +∞). If inf T = −∞, then T = R, −1 ∈ Υ (T). This contradicts the fact that T is a positive translation invariant time scale. So inf T = a is a finite number. This concludes the proof for statement (i). (ii) The proof is similar to (i). (iii) Similar to the proof of (i), we get R+ ⊂ Υ (T) and R− ⊂ Υ (T). So R ⊂ Υ (T). This obviously implies T = R.  Theorem 2. If there exists a primitive translation invariant sequence {αn } satisfying αn > 0 and αn → 0 as n → ∞, then R+ ⊂ Υ (T). Proof. Obviously, for k ∈ N, kαi ∈ Υ (T). This implies Υ (T) is dense in R+ . Since Υ (T) is a closed set, we get R+ ⊂ Υ (T).



Corollary 1. If R+ is not a subset of Υ (T), then there does not exist a primitive translation invariant sequence {αn } satisfying αn > 0 and αn → 0 as n → ∞. 3. Odd translation time scales This section is dedicated to the study of odd translation time scales. Definition 5. If Υ (T) ̸= ∅, and α ∈ Υ (T) ⇒ −α ∈ Υ (T), then T is called an odd translation time scale. It is obvious that if T is an odd translation time scale, then α ∈ Υ (T) implies nα ∈ Υ (T), n ∈ Z. Lemma 6. If T is an odd translation time scale, then sup T = +∞, inf T = −∞. Proof. If T is an odd translation time scale, then T is also a two-way translation invariant time scale. Applying Lemma 2 we have sup T = +∞, inf T = −∞.  Lemma 7. If T is an odd translation time scale, then Υ (T) is an abelian group. Proof. Take the regular addition operation as the operation of Υ (T), 0 as the identity element, −α as the inverse element of α , then it is clear that Υ (T) satisfies the definition of an abelian group.  Lemma 8. If

a b

is an irrational number, then for any given ϵ > 0, there exist n, m ∈ Z satisfying 0 < ma + nb < ϵ .

Proof. Without loss of generality, assume a > 0, b < 0. We have |b| = k1 a + b1 , with k1 ∈ N, b1 ∈ (0, a). We denote min{b1 , a − b1 } as a1 , then a1 < 2a . We have a = k2 a1 + b2 , with k2 ∈ N+ , b2 ∈ (0, a1 ). We denote min{b2 , a1 − b2 } as a2 , then a2 <

.. .

a1 . 2

We have ai = ki+2 ai+1 + bi+2 , with ki+2 ∈ N+ , bi+2 ∈ (0, ai+1 ). We denote min{bi+2 , ai+1 − bi+2 } as ai+2 , then a ai+2 < i2+1 (i ∈ N). So we get a sequence {ai } whose limit is 0. Also, ai > 0. Obviously, there are mi , ni ∈ Z satisfying mi a + ni b = ai . Since ai < 2ai , we get our desired result.  Using Lemma 8, we can prove the following important property of odd translation time scales. Theorem 3. If T is an odd translation time scale and T ̸= R, then for any α, β ∈ Υ (T) we have βα ∈ Q. Proof. This proof is by contradiction. If there exists α, β ∈ Υ (T) with βα being an irrational number, by the proof of Lemma 8 we can see that ai ∈ Υ (T). This leads to nai ∈ Υ (T), n ∈ Z. So Υ (T) is dense in R. But Υ (T) is closed, so we have Υ (T) = R. This leads to T = R. But we have already assumed T ̸= R, this is a contradiction.  Definition 6. For mk ∈ N+ , k = 1, 2, . . . , n, denote their least common multiple as [m1 , m2 , . . . , mn ], and their greatest common divisor as (m1 , m2 , . . . , mn ). q Suppose there are positive rational numbers rk = pk , k = 1, 2, . . . , n (here we assume pk , qk ∈ N+ and (pk , qk ) = 1 for k k = 1, 2, . . . , n). r If a rational number t satisfies tk ∈ N+ , k = 1, 2, . . . , n, we say that t is a common divisor of rk , k = 1, 2, . . . , n. Define the greatest common divisor of rk , k = 1, 2, . . . , n as

(r1 , r2 , . . . , rn ) =

(q1 , q2 , . . . , qn ) . [p 1 , p 2 , . . . , p n ]

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(r1 , r2 , . . . , rn ) is the largest common divisor of rk , k = 1, 2, . . . , n. See the following lemma. k Lemma 9. (i) (r ,r ,..., ∈ N+ for k = 1, 2, . . . , n. rn ) 1 2 (ii) If s is a rational number with s > (r1 , r2 , . . . , rn ), then for k = 1, 2, . . . , n,

r

rk s

are not all positive integers.

Proof. (i) It is easy to see that rk

(r1 , r2 , . . . , rn )

qk

(q1 , q2 , . . . , qn )

[p 1 , p 2 , . . . , p n ]

·

pk

∈ N+ .

are all positive integers for k = 1, 2, . . . , n. Since s is a positive rational number, we can write s as s =

rk s

(ii) Suppose

=

with ps , qs ∈ N

and (ps , qs ) = 1. Since

+

(ps , qs ) = 1 and (p1 , q1 ) = 1, we obtain From here we can easily get This implies

(q1 , q2 , . . . , qn ) qs

·

q1 qs

r1 s

+

∈ N , we have ∈ N+ and pp1s ∈ N+ .

q1 ps qs p1

+

∈ N . Thus

q1 ps qs

+

∈ N and

q1 ps p1

+

qs , ps

∈ N . Together with

qn ) s ∈ N+ and ppks ∈ N+ for k = 1, 2, . . . , n. So we obtain (q1 ,q2q,..., ∈ N+ and [p1 ,p2p,..., ∈ N+ . pn ] s

qk qs

ps

[p1 , p2 , . . . , pn ]

=

(q1 , q2 , . . . , qn ) 1 (r1 , r2 , . . . , rn ) · = ∈ N+ . [p 1 , p 2 , . . . , p n ] s s

However, this contradicts s > (r1 , r2 , . . . , rn ). Thus we have proved (ii).



Definition 7. If positive rational numbers r1 , r2 , . . . , rn satisfy

(r1 , r2 , . . . , rn ) < min{r1 , r2 , . . . , rn }, then we say that r1 , r2 , . . . , rn are irreducible. Example 5. (i) We have ( 21 , 31 ) =

(ii) We have ( , ) = 1 4

1 2

1 4

< min{ 21 , 31 }. So 21 , 31 are irreducible. = min{ , }. So 21 , 41 are not irreducible. 1 2

1 6 1 4

To prove the main result of this section, we need the following lemma. Lemma 10. Suppose T is an odd translation time scale, α1 , α2 , . . . , αn are positive primitive translation invariants of T, with αi ̸= αj when i ̸= j. If for each rij =

αi , αj

1 ≤ i , j ≤ n,

we have rij ∈ Q, then we have the following. (i) r11 , r21 , . . . , rn1 are irreducible. (ii) 0 < (r11 , r21 , . . . , rn1 )α1 < αk for k = 1, 2, . . . , n. (iii) (r11 , r21 , . . . , rn1 )α1 ∈ Υ (T). Proof. (i) If r11 , r21 , . . . , rn1 are not irreducible, we can assume ri1 = min{r11 , r21 , . . . , rn1 } = (r11 , r21 , . . . , rn1 ). Then we have 1 ri1

=

r11 ri1

∈ N+ and r11 = 1, which imply

α1 ∈ N+ . αi α

If i ̸= 1, then α1 ∈ N+ − {1}. This contradicts with the fact that αi and α1 are primitive translation invariants. i r If i = 1, then r21 = r21 ∈ N+ − {1}, which contradicts with the fact that α2 and α1 are primitive translation invariants. 11 (ii) Since r11 , r21 , . . . , rn1 are irreducible, we have

(r11 , r21 , . . . , rn1 ) < min{r11 , r21 , . . . , rn1 }. So we get

(r11 , r21 , . . . , rn1 ) < rk1 for k = 1, 2, . . . , n. α

Since rk1 = αk and α1 > 0, we get (r11 , r21 , . . . , rn1 )α1 < αk . 1 The result (r11 , r21 , . . . , rn1 )α1 > 0 is obvious.

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(iii) Assume rk1 = pk for k = 1, 2, . . . , n, with pk , qk ∈ N+ and (pk , qk ) = 1. For rational numbers k k = 1, 2, . . . , n, it is easy to see that their greatest common divisor is q

qk pk

· [p1 , p2 , . . . , pn ],

(q1 , q2 , . . . , qn ) · [p1 , p2 , . . . , pn ] = (q1 , q2 , . . . , qn ). [p 1 , p 2 , . . . , p n ] · [p1 , p2 , . . . , pn ] ∈ N+ for k = 1, 2, . . . , n and (q1 , q2 , . . . , qn ) ∈ N+ . Thus there exist mi ∈ Z, i = 1, 2, . . . , n satisfying

Here we also have

qk pk

(q1 , q2 , . . . , qn ) = m1

q1 p1

· [ p1 , p2 , . . . , pn ] + m2

q2 p2

· [p1 , p2 , . . . , pn ] + · · · + mn

qn pn

· [p1 , p2 , . . . , pn ].

Divided by [p1 , p2 , . . . , pn ] we get

(r11 , r21 , . . . , rn1 ) = m1 r11 + m2 r21 + · · · + mn rn1 . This means

(r11 , r21 , . . . , rn1 )α1 = m1 α1 + m2 α2 + · · · + mn αn . Since αi ∈ Υ (T) for i = 1, 2, . . . , n and T is an odd translation time scale, we have (r11 , r21 , . . . , rn1 )α1 ∈ Υ (T).



We present the main result of this section below. Theorem 4. Suppose T is an odd translation time scale and T ̸= R. Then T has exactly one positive primitive translation invariant. Proof. Since T is an odd translation time scale, there is an α0 ∈ Υ (T) with α0 ̸= 0. If T has no positive primitive translation α invariants, we have n0 ∈ Υ (T) for all n ∈ N+ . This implies Υ (T) is dense in R. Since Υ (T) is a closed set we have Υ (T) = R; this contradicts T ̸= R. So there exists at least one positive primitive translation invariant for T. β Suppose T has two positive primitive translation invariants α, β with α < β . By Theorem 3 we get α ∈ Q. Making use β

β

of Lemma 10 we have (1, α )α ∈ Υ (T) and 0 < (1, α )α < α . By Lemma 9 there is 1



β

1, α

 ∈ N+

which leads to

α 

β

1, α



α

∈ N+ .

So we have



1,

β α



α ∈ Υ (T),

0<



1,

β α



α < α and 

1,

α β α



α

∈ N+ .

But this contradicts with the fact that α is a primitive translation invariant.



Corollary 2. An odd translation time scale T either has no primitive translation invariants (in this case we have T = R), or has exactly one positive primitive translation invariant, denoted as γT > 0 (in this case we have Υ (T) = {kγT , k ∈ Z}). For A ⊂ R, a ∈ R, define A + a = {b + a : b ∈ A}. We have the following corollary regarding the structure of odd translation time scales. Corollary 3. Suppose T is an odd translation time scale with T ̸= R. Let M = T

T=

 [0, γT ]; then we have

 (M + kγT ). k∈Z

Theorem 5. T is an odd translation time scale if and only if it is a two-way translation invariant time scale. Proof. If T is an odd translation time scale, then there exists a α ̸= 0, α ∈ Υ (T) with −α ̸= 0, − α ∈ Υ (T). So T is a two-way translation invariant time scale. Suppose T is a two-way translation invariant time scale. Then there exist α > 0, α ∈ Υ (T) and β < 0, β ∈ Υ (T).

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First we assume that βα is an irrational number. Then according to Lemma 8, for > 0, k ∈ N+ there exist k min{α,−β} nk , mk ∈ Z satisfying 0 < mk α + nk β < . Since we have α > 0, β < 0, for a given k0 , nk0 , mk0 ∈ N+ or k + −nk0 , −mk0 ∈ N . So we have a sequence {ki } satisfying min{α,−β}

0 < |mki |α + |nki |β <

min{α, −β} ki

nki , mki ∈ N+ ,

,

or have a sequence {kj } satisfying 0 > |mkj |α + |nkj |β > −

min{α, −β} kj

,

−nkj , −mkj ∈ N+ .

Since |mk |α + |nk |β ∈ Υ (T), either R+ or R− is dense in the closed set Υ (T). So either R+ ⊂ Υ (T) or R− ⊂ Υ (T). Together with sup T = +∞, inf T = −∞ we have T = R. So T is an odd translation time scale. q If T ̸= R, then for any α > 0, β < 0, α, β ∈ Υ (T), there is βα ∈ Q. In this case we can assume βα = − p , q, p ∈

N+ , (q, p) = 1. For a given t ∈ T, we have t − α = t + qβ + (p − 1)α ∈ T, and t − β = t + pα + (q − 1)β ∈ T.

So −α ∈ Υ (T) and −β ∈ Υ (T). Since α and β are arbitrary, T is an odd translation time scale.



Theorem 6. Assume B is a bounded closed subset of R with B ̸= ∅. Let inf B = a and sup B = b, a ≤ b. Suppose γ > b − a ≥ 0. Then the following set

 (B + k γ ) k∈Z

γ

is an odd translation time scale with γ ∈ N+ . T Proof. For T = k∈Z (B + kγ ), it is obvious that γ ∈ Υ (T) and −γ ∈ Υ (T). So T is a two-way translation invariant time scale, which means that T is an odd translation time scale. Obviously, we have T ̸= R, and by Corollary 2 we have Υ (T) = {kγT , k ∈ Z}. Together with γ ∈ Υ (T), γ > 0 we get γγ ∈ N+ . 



T

4. Periodic functions and almost periodic functions on time scales Periodic functions defined on R have been studied extensively and extended to time scales [10,11]. Definition 8. Let T be a two-way translation invariant time scale. If f : T → R satisfies f (t + T ) = f (t )

for all t ∈ T

then we say f is a periodic function with period T on T. Note that we must have T ∈ Υ (T) in the above definition. Theorem 7. Let f : R → R be a periodic function with a period ω > 0. T is a two-way translation invariant time scale with a positive primitive translation invariant γT > 0. Define fT : T → R as fT (t ) = f (t ), If

ω γT

t ∈ T.

∈ Q, then fT is a periodic function with period

Proof. Let γω = T

q p

ωγT (ω,γT )

on T.

with q, p ∈ N+ and (q, p) = 1. Then for t ∈ T, we have

fT (t + qγT ) = f (t + qγT ) = f (t + pω) = f (t ) = fT (t ). So fT is a periodic function, and qγT = pω is a period of fT . It is easy to see that

ω = q(ω, γT ), and

γT = p(ω, γT ). So

ωγT (ω,γT )

= pω is a period of fT .



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Definition 9. Let T be a positive translation invariant time scale. If f : T → R satisfies f (t + T ) = f (t ) for all t ∈ T, then we say f is a periodic function with period T on T. Example 6. For T = f (t ) = sin

πt 3

,



k∈N

[3k + 1, 3k + 2], define f : T → R as

t ∈ T,

then f is a periodic function with period 6 on T. The following definition of almost periodic function on R and almost periodic sequence on Z comes from [12]. Definition 10. Let f : R → R be a continuous function. If for every ϵ > 0 there exists a number lϵ > 0 with the property that any interval of length lϵ contains a τ such that

|f (x + τ ) − f (x)| < ϵ for all x ∈ R, then f is called an almost periodic function. Definition 11. A sequence {x(n)} : Z → R is called an almost periodic sequence, if for every ϵ > 0 there exists a number lϵ > 0 with the property that any interval of length lϵ contains a τ such that

|x(n + τ ) − x(n)| < ϵ for all n ∈ Z. We need two important lemmas. For their proof, see [12]. Lemma 11. If f is an almost periodic function on R, f (n) = an , n ∈ Z, then {an } is an almost periodic sequence. Lemma 12. Trigonometric polynomials are almost periodic functions on R. Definition 12. Let T be a two-way translation invariant time scale. For f : T → R, if for every ϵ > 0 there exists a number lϵ > 0 with the property that any interval of length lϵ contains a τ such that

|f (x + τ ) − f (x)| < ϵ for all x ∈ T, then f is called an almost periodic function on T. Theorem 8. Let T be a two-way translation invariant time scale and γ > 0, γ ∈ Υ (T). Suppose a0 ∈ T. Denote a0 + kγ = ak for k ∈ Z. Let f (x) be an almost periodic function on R, define g : T → R as g (x) =



f (x), when x = ak , k ∈ Z, g (ak ) + (x − ak )(g (ak+1 ) − g (ak )),

when ak < x < ak+1 ,

then g (x) is an almost periodic function on T. Proof. Since f (x) is an almost periodic function on R, for a given ϵ > 0 there exists a number lϵ > 0, any interval of length lϵ contains a τ such that

|f (x + τ ) − f (x)| < ϵ for all x ∈ R. Let h(x) = f ( γ 0 ). We will prove that h(x) is also an almost periodic function on R. For the above ϵ > 0 there exists a number γ lϵ > 0, any interval of length γ lϵ contains a γ τ , with x −a

|f (x + τ ) − f (x)| < ϵ for all x ∈ R. Thus

      x − a0 x − a0   < ϵ, |h(x + γ τ ) − h(x)| = f +τ −f  γ γ

x ∈ R.

So h(x) is also an almost periodic function on R. According to Lemma 11, h(k) is an almost periodic sequence. Since h(k) = f (ak ) = g (ak ), we know g (ak ) is an almost periodic sequence. Thus, for any ϵ0 > 0, there exists a number lϵ0 > 0 with the property that any interval of length lϵ0 contains a k0 ∈ Z such that

|g (ak+k0 ) − g (ak )| < ϵ0 for all k ∈ Z.

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This means

|g (ak + k0 γ ) − g (ak )| < ϵ0 for all k ∈ Z. From the definition of g (x) we get g (x) = g (ak ) + (x − ak )(g (ak+1 ) − g (ak )),

when ak ≤ x < ak+1 .

So we have g (x + k0 γ ) = g (ak + k0 γ ) + (x + k0 γ − ak+k0 )(g (ak+k0 +1 ) − g (ak+k0 )),

when ak ≤ x < ak+1 .

Thus g (x + k0 γ ) = g (ak + k0 γ ) + (x − ak )(g (ak+1 + k0 γ ) − g (ak + k0 γ )),

when ak ≤ x < ak+1 .

For x ∈ T, ak ≤ x < ak+1 we have

|g (x + k0 γ ) − g (x)| ≤ |g (ak + k0 γ ) − g (ak )| + (x − ak )[|g (ak+1 + k0 γ ) − g (ak+1 )| + |g (ak + k0 γ ) − g (ak )|]. For the right hand side of the above inequality we have

|g (ak + k0 γ ) − g (ak )| + (x − ak )[|g (ak+1 + k0 γ ) − g (ak+1 )| + |g (ak + k0 γ ) − g (ak )|] < (1 + 2γ )ϵ0 . So we have for any ϵ0 > 0, there exists a number γ lϵ0 > 0 with the property that any interval of length γ lϵ0 contains a k0 γ such that

|g (x + k0 γ ) − g (x)| < (1 + 2γ )ϵ0 for all x ∈ T. So g (x) is an almost periodic function on T.



From the above theorem we know that we can define almost periodic functions on two-way translation invariant time scales. Furthermore, we can extend the concept of almost periodic functions on positive translation invariant time scales. Definition 13. Let T be a positive translation invariant time scale. For f : T → R, if for every ϵ > 0 there exists a number lϵ > 0 with the property that any interval I of length lϵ , with I ⊂ R+ , contains a τ such that

|f (x + τ ) − f (x)| < ϵ for all x ∈ T, then f is called an almost periodic function on T. Theorem 9. Let T be a two-waytranslation invariant time scale and f (x) be an almost periodic function on T. Suppose there is a constant A ∈ R. Then TA = T [A, +∞) is a positive translation invariant time scale. Denote f restricted on TA as fA (x), then fA (x) is an almost periodic function on TA . Proof. First, it is easy to see that TA is a nonempty closed subset of R, so it is a time scale. T is a two-way translation invariant time scale, so there exists a τ satisfying τ > 0, τ ∈ Υ (T). Obviously, τ ∈ Υ (TA ). And for any p < 0, (inf TA ) + p is not an element of TA , so p is not an element of Υ (TA ). Thus TA is a positive translation invariant time scale. For a given ϵ > 0, there exists a number lϵ > 0 with the property that any interval I of length lϵ contains a τ such that

|f (x + τ ) − f (x)| < ϵ for all x ∈ T. So, any interval I of length lϵ , I ⊂ R+ , contains a τ such that

|fA (x + τ ) − fA (x)| < ϵ for all x ∈ TA . The proof is complete.



Definition 14. (i) Let T be a two-way translation invariant time scale, D ⊂ Rn . For f (t , x) : T × D → R, if for every ϵ > 0 there exists a number lϵ > 0 (lϵ does not depend on x) with the property that any interval I of length lϵ contains a τ such that

|f (t + τ , x) − f (t , x)| < ϵ for all (t , x) ∈ T × D, then f is called a uniformly almost periodic function on T × D. (ii) Let T be a positive translation invariant time scale, D ⊂ Rn . For f (t , x) : T × D → R, if for every ϵ > 0 there exists a number lϵ > 0 (lϵ does not depend on x) with the property that any interval I of length lϵ , with I ⊂ R+ , contains a τ such that

|f (t + τ , x) − f (t , x)| < ϵ for all (t , x) ∈ T × D, then f is called a uniformly almost periodic function on T × D.

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Theorem 10. Let f (t ) be an almost periodic function on T, g (x) is bounded on D, then h(t , x) = f (t )g (x) is a uniformly almost periodic function on T × D. Proof. Since g (x) is bounded on D, we can assume |g (x)| < B for x ∈ D. For a given ϵ > 0, there exists a number lϵ > 0 with the property that any interval I of length lϵ (if T is a positive translation invariant time scale, let I ⊂ R+ ) contains a τ such that

|f (t + τ ) − f (t )| < ϵ for all x ∈ T. Thus we have

|h(t + τ , x) − h(t , x)| = |f (t + τ )g (x) − f (t )g (x)| ≤ |g (x)| |f (t + τ ) − f (t )| < Bϵ for all (t , x) ∈ T × D. So h(t , x) is a uniformly almost periodic function on T × D.



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