(1)Stanisław Stoiński On the quasi N–almost periodic functions Abstract

Stanisław Stoiński

On the quasi N–almost periodic functions

Abstract. In this note we present the definition, examples and some properties of quasi N–almost periodic functions, i. e. certain almost periodic functions in the sense of Levitan.

2000 Mathematics Subject Classification: 42A75.

Key words and phrases: almost periodic function, sequence, superposition.

1. Preliminaries. Let f :ℝ → ℝ be a continuous function. If for N > 0, ε > 0 there is

DN(f − f±τ) = max{|f(x) − f±τ(x)| : −N ¬ x ¬ N} ¬ ε,

where fτ(x) ≡ f(x + τ), then the number τ ∈ ℝ is called an (N, ε)–almost period ((N, ε)–a.p.) of the function f. Let NE{ε; f} denote the set of (N, ε)–a.p. of f. The continuous function f :ℝ → ℝ is called N–almost periodic (N–a.p.) iff there exists a uniformly a.p. (B–a.p.; see:[1],[2]) function ϕ, adominant function of f, such that for any two positive numbers N, ε there exists a δ > 0 such that E{δ; ϕ} ⊂ NE{ε; f}, where E{δ; ϕ} = {τ ∈ ℝ : sup{|ϕ(x) − ϕ^τ(x)| : x ∈ ℝ} ¬ ε}. By eN we denote the set of N–a.p. functions. The above definitions were introduced by B. M. Levitan in 1937 (see:[2]). N–almost periodicity of functions– not necessarily continuous– was considered by B. M. Levitan, W. A. Marchenko, B. Ja. Levin, and recently by A. Michałowicz and S. Stoiński (see:[3],[4]).

2. Main results. A set E ⊂ ℝ is called relatively dense iff there exists a positive number l such that in every open interval (ω, ω + l), ω ∈ ℝ, there is at least one element of the set E. The continuous function f :ℝ → ℝ is called quasi N–almost periodic (N^∗–a.p.) iff there exists a uniformly a.p. function ϕ, a quasi dominant function of f, such that for any two positive numbers N, ε there exist a δ > 0 and a relatively dense set E_f^∗⊂ E{δ; ϕ} ∩ NE{ε; f} and 0 ∈ Ef^∗.

By fN^∗ we denote the set of N^∗–a.p. functions. If for f ∈ N^∗ we have E_f^∗ = E{δ; ϕ}, then f ∈ eN .

Theorem 2.1 If fn, n = 1, 2, . . . , are N –a.p. functions, then the function f of the form

f (x) = X∞ n=1

|fⁿ(x)|

2ⁿ(1 + |fⁿ(x)|) for x ∈ ℝ is an N^∗–a.p. function.

Proof Let ϕn, n = 1, 2, . . . , be the dominant functions of fn, n = 1, 2, . . . The func- tion ϕ of the form

ϕ(x) = X∞ n=1

|ϕⁿ(x)|

2ⁿ(1 + |ϕⁿ(x)|) for x ∈ ℝ

is continuous onℝ. Fix arbitrarily N > 0, ε > 0. For all x ∈ ℝ and τ ∈ ℝ we have

|ϕ(x + τ) − ϕ(x)| ¬

n₀

X

n=1

1

2ⁿ|ϕⁿ(x + τ) − ϕⁿ(x)| +ε 2, where we choose n0 = n0(ε) such that P∞

n=n0+1 1

2ⁿ < ε₂. For B–a.p. functions ϕ1, . . . , ϕn0 there exists a relatively dense set E0 = E0{ ε2; ϕ1, . . . , ϕn0} ⊂ ℝ of common ε₂
–almost periods for ϕ1, . . . , ϕn0. For τ ∈ E⁰ we obtain

|ϕ(x + τ) − ϕ(x)| ¬ ε for x ∈ ℝ, and so τ ∈ E{ε; ϕ}. Hence ϕ is a B–a.p. function.

Since fn, n = 1, 2, . . . , are N –a.p. functions, we see that there exist δn = δn(N, ε) > 0, n = 1, 2, . . . , such that E{δⁿ; ϕn} ⊂ NE{ ε2; fn}, n = 1, 2, . . . Write δ0= min{δⁿ: n = 1, . . . , n0}. The set E^∗= E^∗{δ⁰; ϕ1, . . . , ϕn₀} ⊂ ℝ is a relatively dense set of common δ0–almost periods for ϕ1, . . . , ϕn₀. Then E^∗⊂ E{δ; ϕ}, where δ = δ0+ ε₂. Moreover, for τ ∈ E^∗ we have τ ∈ E{δⁿ; ϕn}, n = 1, 2, . . . , n⁰. Hence for |x| ¬ N and τ ∈ E^∗ we obtain τ ∈ NE{ε2; fn}, n = 1, . . . , n⁰, and

|f(x + τ) − f(x)| ¬

n0

X

n=1

1

2ⁿ|fⁿ(x + τ) − fⁿ(x)| + X∞ n=n₀+1

1 2ⁿ < ε,

i.e. τ ∈ NE{ε; f} , and so f is an N^∗–a.p. function with the quasi dominant

function ϕ. _■

We say that the function f : ℝ → ℝ is N–bounded iff for every N > 0 there exist a relatively dense set E ⊂ ℝ and a positive constant M such that for every x∈S

{[τ − N, τ + N] : τ ∈ E} the inequality |f(x)| ¬ M holds.

Remark 2.2 Let l = l(N) > 0 be a number which characterizes the relative density of the set E. If l ¬ N, then |f(x)| ¬ M = M(N) for every x ∈ ℝ. If, moreover, sup{M(N) : N > 0} < +∞, then f is bounded on ℝ

Theorem 2.3 If f is an N^∗–a.p. function, then f is N–bounded.

Proof Fix arbitrarily N > 0, ε > 0. Since f is continuous on the interval [−N, N], there exists a constant M = M(N) > 0 such that

(1) |f(x)| ¬ M for x ∈ [−N, N].

By the assumption, it follows that there exist a δ = δ(N, ε) > 0 and a relatively dense set E_f^∗⊂ E{δ; ϕ}∩NE{ε; f}, where the B–a.p. function ϕ is a quasi dominant function of f. For τ ∈ Ef^∗ and |x| ¬ N we have |f(x + τ)| ¬ |f(x)| + ε, and so by (1) we obtain

|f(x + τ)| ¬ M + ε for x ∈ [−N, N].

Hence for x ∈S

[τ − N, τ + N], τ ∈ Ef^∗, we have the estimation |f(x)| ¬ M + ε, i.e.

f is N –bounded. _■

We say that the function f :ℝ → ℝ is N–continuous iff for an arbitrary ε > 0 and for every N > 0 there exist a relatively dense set E ⊂ ℝ and a δ = δ(N, ε) such that for every x⁰, x⁰⁰∈ [τ −N, τ +N], τ ∈ E, |x⁰−x⁰⁰| < δ, we have |f(x⁰)−f(x⁰⁰)| < ε.

Remark 2.4 If l ¬ N, where l = l(N, ε) > 0 is a number which characterizes the relative density of the set E, and inf{δ(N, ε) : N > 0} = δ0(ε) > 0, then f is uniformly continuous onℝ.

Theorem 2.5 If f is an N^∗–a.p. function, then f is N–continuous.

Proof Fix arbitrarily N > 0, ε > 0. The function f is uniformly continuous on the closed interval [−N, N], and so there exists a δ = δ(N, ε) > 0 such that for arbitrary x₁, x2∈ [−N, N] we have

(2) |f(x¹) − f(x²)| < ε

3 for |x¹− x²| < δ.

By the assumption, it follows that there exist a δ0= δ0(N, ε) > 0 and a relatively dense set E^∗_f ⊂ E{δ⁰; ϕ} ∩ NE{ε3; f}, where the B–a.p. function ϕ is a quasi dominant function of f. For every x⁰, x⁰⁰ ∈ [τ − N, τ + N], τ ∈ E^∗f, such that

|x⁰ − x⁰⁰| < δ, there exist x¹ = x⁰− τ, x² = x⁰⁰− τ ∈ [−N, N]. Hence for every x⁰, x⁰⁰∈ [τ − N, τ + N], τ ∈ Ef^∗, where |x⁰− x⁰⁰| < δ, by (2) we obtain

|f(x⁰) − f(x⁰⁰)| ¬ |f(x1+ τ) − f(x1)| + |f(x1) − f(x2)| + |f(x2) − f(x2+ τ)| < ε,

and so f is N–continuous. _■

It is known (see:[2]) that if a sequence of B–a.p. functions is uniformly convergent on the whole real axis, then the limit of this sequence is a B–a.p. function.

Let us consider a sequence (fk) of the N^∗–a.p. functions. Let ϕk, k = 1, 2, . . . , be the quasi dominant functions of fk, k = 1, 2, . . . The function ϕ of the form

(3) ϕ(x) =

X∞ k=1

|ϕ^k(x)|

2^k(1 + |ϕ^k(x)|) for x ∈ ℝ

is B–a.p. Fix arbitrarily N > 0, ε > 0. Since fk, k = 1, 2, . . . , are N^∗–a.p. functions, there exist a δk = δk(N, ε) > 0, k = 1, 2, . . . , and relatively dense sets E_f^∗_k, k = 1, 2, . . . , such that E_f^∗_k⊂ E{δ^k; ϕk} ∩ NE{ε; f^k}, k = 1, 2, . . . Set

(4) δ0= δ0(N, ε) = min(δk: k = 1, 2, . . . , k0), where we choose k0 such thatP∞

k=k₀+1 1

2^k < ε. The set E_k^N₀= E_k^N₀{δ⁰; ϕ1, . . . , ϕk0} is a relatively dense set of common δ0–almost periods for ϕ1, . . . , ϕk0. Then E^N_k0 ⊂ E{δ; ϕ}, where δ = δ⁰+ ε.

We say that the N^∗–a.p. functions fk, k = 1, 2, . . . , satisfies the condition (V) iff E_f^∗

k0 ⊂ Ek^N0∩ NE{ε; f^k0}, whereP∞ k=k0+1 1

2^k < ε.

We call the sequence (fk) of the N^∗–a.p. functions, which satisfies the con- dition (V), N–convergent to a continuous function f : ℝ → ℝ iff for an ar- bitrary ε > 0, N > 0 there exists k₀ such that P∞

k=k0+1 1

2^k < ε and for x ∈ S[τ − N, τ + N], τ ∈ Ef^∗_k0, we have

|f^k0(x) − f(x)| < ε.

Remark 2.6 If l ¬ N, where l = l(N, ε) > 0 is a number which characterizes the relative density of the set E_f^∗

k0 and sup{k⁰(N, ε) : N > 0} = k⁰(ε), then the N –convergence is the uniform convergence of the sequence (fk).

Theorem 2.7 If a sequence of N^∗–a.p. functions, which satisfies the condition (V), is N–convergent to the continuous function f :ℝ → ℝ, then f is N^∗–a.p.

Proof For an arbitrary ε > 0, N > 0 there exists k0, whereP∞ k=k₀+1 1

2^k < ε₃, for which we have for x ∈ [−N, N] and for τ ∈ Ef^∗_k0 ⊂ E^Nk0∩ NE ε

3; fk0

|f^k0(x) − f(x)| < ε

3 and |f^k0(x + τ) − f(x + τ)| < ε 3, and hence

(5) |f(x) − f(x + τ)| < 2

3ε +|f^k0− f^k0(x + τ)|.

It is easily seen that E_f^∗

k0 ⊂ E{δ; ϕ}, where ϕ is of the form (3) and δ = δ⁰+ ε₃ (see: (4)). Because τ ∈ NE ε

3; f_k0 , we have (6) |f^k0(x) − f^k0(x + τ)| < ε

3 for |x| ¬ N.

By the estimations (5) and (6) it follows that for |x| ¬ N and for τ ∈ Ef^∗_k0

|f(x) − f(x + τ)| < ε,

i.e.τ ∈ NE{ε; f}, and so Ef^∗_k0 ⊂ E{δ; ϕ} ∩ NE{ε; f}. Therefore f is an N^∗–a.p.

function with the quasi dominant function ϕ. _■

Let f, g ∈ fN^∗. This means that there exist the B–a.p. functions ϕ and ψ such that for an arbitrary ε > 0, N > 0 there exist δf > 0, δg > 0 and the relatively dense sets E_f^∗ and E_g^∗ such that E_f^∗⊂ E{δ^f; ϕ} ∩ NE ε

2; f and E_g^∗⊂ E{δ^g; ψ} ∩ N E ε

2; g . We denote by M(λ1, λ2, . . .) the least common modulus of the Fourier exponents of the B–a.p. functions ϕ and ψ. Let us consider the uniformly a.p.

function

(7) γ(x) =

X∞ n=1

1

n²e^iλnx for x ∈ ℝ.

Write δ = min(δf, δg). From Theorems 2.1.1 and 2.1.2 in [2] it follows that there exists η = η(δ) such that

E{η; γ} ⊂ E{δ; ϕ} ∩ E{δ; ψ}.

We say that the functions f, g ∈ fN^∗ satisfies the condition (W) iff the set E^∗= E_f^∗∩ E^∗g is relatively dense, E^∗⊂ E{η; γ} and 0 ∈ E^∗.

Theorem 2.8 The linear combination of the N^∗–a.p. functions f, g, which satisfies the condition (W), is an N^∗–a.p. function.

Proof Assume that the B–a.p. functions ϕ and ψ are the quasi dominant functions of f and g, respectively.

For a constant c 6= 0 and for an arbitrary ε > 0, N > 0 there exist a δ > 0 and a relatively dense set E^∗_cf ⊂ E{δ; ϕ} ∩ NE{ε; cf}, and so ϕ is the quasi dominant function of cf. Therefore cf ∈ fN^∗.

Since the functions f, g satisfy the condition (W), we have E^∗⊂ NEn ε

2; fo

∩ NEn ε 2; go

⊂ NE{ε; f + g}.

This means that the B–a.p. function γ of the form (7) is the quasi dominant of the

sum f + g, and so f + g ∈ fN^∗. _■

It is known (see:[4]) that if g is a B–a.p. function and the function f : Yg→ ℝ, where Yg is the set of values of g, is continuous, then the superposition f(g) is N –a.p.

Theorem 2.9 If the function f : Yg→ ℝ is continuous on the set Y^g of values of an N^∗–a.p. function g, then the superposition f(g) is N^∗–a.p.

Proof Fix arbitrarily N > 0, ε > 0. Let us write mN = min{g(x) : −N ¬ x ¬ N}, M^N = max{g(x) : −N ¬ x ¬ N}.

If the set Ygis bounded, i.e.

m = inf

x∈ℝg(x), M = sup

x∈ℝ

g(x),

then we put

IN =





















[mN− a^N, MN+ aN], where aN = ¹₂min(mN − m, M − M^N), if mN > m, MN < M ,

[mN, MN + aN], where 0 < aN =¹₂(M − M^N), if mN = m, MN < M ,

[mN− a^N, MN], where 0 < aN =¹₂(mN − m), if mN > m, MN = M.

Instead, if the set Yg is unbounded, then we write

IN =

















[mN− 1, M^N+ 1], if Yg= (−∞, +∞), [mN, MN + 1], if mN = m,

[mN− a^N, MN+ 1], where 0 < aN =¹₂(mN − m), if m < m^N, [mN− 1, M^N], if MN = M,

[mN− 1, M^N+ aN], where 0 < aN =¹₂(M − M^N), if MN < M . The function g is N^∗–a.p., i.e. there exist a uniformly a.p. function ϕ and a relatively dense set E_g^∗ ⊂ E{δ; ϕ} ∩ NE{∆; g}, where δ = δ(N, ε) > 0 and ∆ = ∆(N, ε) ∈ (0, min(aN, 1)) exists by the definition of the uniform continuity of f on the closed interval IN. For x ∈ [−N, N] and for τ ∈ E^∗g we obtain

|g(x + τ) − g(x)| ¬ ∆ and g(x), g(x + τ )∈ I^N, and so

|f(g(x + τ)) − f(g(x))| < ε.

Hence the superposition f(g) is an N^∗–a.p. function, because ϕ is the quasi domi-

nant function of f(g). _■

Let ϕ be the function

ϕ(x) = 2 + cos x + cos(√

2x) for x ∈ ℝ.

It is known (see:[2]) that ϕ(x) > 0 for every x ∈ ℝ and inf{ϕ(x) : x ∈ ℝ} = 0. In addition, _ϕ¹ is the N–a.p. function. For the function g of the form

g(x) = 1 ϕ(x)−1

4 for x ∈ ℝ

we have g(x) ­ 0 for every x ∈ ℝ and g(0) = 0. Moreover, g is N–a.p. Let us put for n = 1, 2, . . .

gn(x) = n

 1 ϕ(x)−1

4



, x∈ ℝ.

For every x ∈ ℝ we have gⁿ(x) ­ 0 and sup{gⁿ(x) : x ∈ ℝ} = +∞. The functions gn, n = 1, 2, . . . , are N –a.p. From Theorem 2.1 it follows that the function

f (x) = X∞ n=1

gn(x)

2ⁿ(1 + gn(x)) , x∈ ℝ,

is N^∗–a.p. Since g(x)/(1 + g(x)) ¬ f(x) < 1, we have sup{f(x) : x ∈ ℝ} = 1.

Applying Theorem 2.9 we conclude that the unbounded function h(x) = tg π

2f (x)

for x ∈ ℝ is N^∗–a.p.

Acknowledgement. I would like to express my gratitude to the Reviewer for his invaluable critical remarks and suggestions.

References

[1] H. Bohr, Zur Theorie der fastperiodischen Funktionen, I Teil: Eine Verallgemeinerung der The- orie der Fourierreihen, Acta math.45 (1925), 29–127; II Teil: Zusammenhang der fastperiodi schen Funktionen mit Funktionen von unendlichvielen Variablen; gleichm¨assige Approximation durch trigonometrische Summen, Acta math.46 (1925), 101–214.

[2] B. M. Levitan, Almost periodic functions, Moscow 1953 (in Russian) .

[3] A. Michałowicz, S. Stoiński, On the almost periodic functions in the sense of Levitan, Ann.

Soc. Math. Pol., Ser. I: Commentat. Math.47 (2)(2007), 149–159.

[4] S. Stoiński, A note on N-almost periodic functions and (NI)-almost periodic functions, Ann.

Soc. Math. Pol., Ser. I: Commentat. Math.44 (2) (2004), 199–204.

Stanisław Stoiński

Faculty of Mathematics and Computer Science, Adam Mickiewicz University Umultowska 87, 61-614 Poznań, Poland

E-mail: stoi@amu.edu.pl

(Received: 26.01.2010)