Stanisław Stoiński
On unbounded almost periodic functions in Hausdorff metric
Abstract. In this article we give the definition and some properties of unbounded almost periodic functions in Hausdorff metric. Moreover, we examine almost peri- odicity of the inverse of a generalized trigonometric polynomial of constant sign.
2000 Mathematics Subject Classification: 42A75.
Key words and phrases: Hausdorff metric, almost periodic function, generalized tri- gonometric polynomial.
1. Introduction. We shall now define and study almost periodic functions (bounded or not) in the sense of Hausdorff metric. This metric, defined by F. Haus- dorff [3], was used in approximation theory by B. Sendov and others [5]-[7], whereas almost periodicity of bounded or bounded on every bounded interval functions with respect to this metric was first studied in Poznań in the early seventies of last century ([8],[9],[12]-[14]) and slightly later in Baku [2].
2. Main result. Let F∆ be the class of subsets of the plane Oxy whose pro- jections onto the x-axis coincide with the closed interval ∆ (bounded or not) and such that the intersection of each line x = x0, where x0∈ ∆, with every F ∈ F∆is a bounded closed interval or unbounded interval: (−∞, a], [a, +∞), (−∞, +∞). Let A, B ∈ F∆. TheHausdorff distance between A and B is the quantity
r∆(A, B) = max
sup
X∈A
Yinf∈B||X − Y ||0, sup
X∈B
Yinf∈A||X − Y ||0
, where
||X − Y ||0= ||X(x1, y1) − Y (x2, y2)||0= max (|x1− x2|, |y1− y2|) . If ∆ =R, we write r∆= r.
It is easily checked that for all A, B, C ∈ F∆ the following hold:
r∆(A, B) = 0 ⇔ A = B, r∆(A, B) = r∆(B, A),
r∆(A, B) ¬ r∆(A, C) + r∆(C, B).
The following lemmas are useful in estimating Hausdorff distance:
Lemma 2.1 Let A, B ∈ F∆ and δ > 0. Then r∆(A, B) ¬ δ if and only if the following conditions hold:
1. for every X ∈ A there is a Y ∈ B such that ||X − Y ||0¬ δ, 2. for every X ∈ B there is a Y ∈ A such that ||X − Y ||0¬ δ.
Lemma 2.2 Let A, B ∈ F∆. If there is an X0∈ A such that ||X0− Y ||0> δ for all Y ∈ B, then
r∆(A, B) > δ.
Proofs can be found in [5] and [6].
Let f : ∆ → R. The lower and upper Baire functions of f are defined by If(x) = lim
δ→0 inf
|x−x0|¬δf (x0) and Sf(x) = lim
δ→0 sup
|x−x0|¬δ
f (x0), resp. Acomplete graph of f is the set
f =e {(x, y) : x ∈ ∆ and If(x)5 y 5 Sf(x)}.
Remark 2.3
a5 b ⇔ (a ¬ b if a, b ∈ R, or a < b if a = −∞ or b = +∞).
Then ef ∈ F∆. TheHausdorff distance between functions f, g : ∆ → R is defined to be the Hausdorff distance between their complete graphs, i.e. r∆(f, g) = r∆( ef ,eg).
A non-empty set E ⊂ R is called relatively dense if there exists a number l > 0 such that every open interval in R of length l contains at least one element of E.
Let f : R → R. If for ε > 0 the Hausdorff distance between f and fτ, where fτ(x) ≡ f(x + τ), satisfies r(f, fτ) ¬ ε, then τ is called an (H, ε)-almost period of f . The set of all (H, ε)-almost periods of f is denoted EH{ε; f}.
A function f :R → R is called H-almost periodic if EH{ε; f} is relatively dense for all ε > 0. Denote by eH the space of all H-almost periodic functions.
In the following, in case when the function f is discontinuous at the point x0, we denote with f(x0) one of the members from interval [If(x0), Sf(x0)] (or:
(−∞, Sf(x0)], [If(x0), +∞), (−∞, +∞)).
Theorem 2.4 If g is H-almost periodic and f : Ω → R is uniformly continuous on Ω = {y ∈ R : (x, y) ∈ eg, x ∈ R}, then the composite function h = f ◦ g is H-almost periodic.
Proof For an arbitrary ε > 0 there exists a δ > 0 such that for every y0, y00 ∈ Ω, |y0−y00| ¬ δ, we have |f(y0)−f(y00)| ¬ ε. Let τ ∈ EH{δ0; g}, where δ0= min(ε, δ).
By Lemma 2.1, we obtain for every x ∈ R
gτ(x) = g(x + ζ1(x, τ)δ0) + ζ2(x, τ)δ0,
where |ζi(x, τ)| ¬ 1, i = 1, 2. For each X = X(x1, y1) ∈ ehτ there exists a Y = Y (x2, h(x2)) ∈ eh, where x2= x1+ ζ1(x1, τ )δ0, such that
||X − Y ||0= max(|x1− x2|, |f(gτ(x1)) − f(g(x1+ ζ1δ0))|) ¬ ε.
Analogously, for each X ∈ eh there exists a Y ∈ ehτ such that ||X − Y ||0 ¬ ε. By Lemma 2.1, we thus have r(hτ, h) ¬ ε, and so EH{δ0; g} ⊂ EH{ε; h}. Hence h is
H-almost periodic.
Corollary 2.5 If f is H-almost periodic, then so are |f|. If, moreover, inf{|f(x)| : x∈ R} > 0, then 1f is H-almost periodic.
Lemma 2.6 Let f and g be H-almost periodic. For each ε > 0 there exists a rela- tively dense set EH{2ε; f, g} such that if τ ∈ EH{ε2; f, g}, then τ is H,ε2
-almost period of f and g.
The proof can be found in [14].
Theorem 2.7 A linear combination of H-almost periodic functions is H-almost periodic.
Proof Assume f and g are H-almost periodic functions and c 6= 0 is an arbitrary constant.
For τ ∈ R and X0τ = X0τ(xτ0, y0τ) ∈ ](cf)τ there is X1τ = X1τ(xτ0, yτ1) ∈ efτ such that y0τ = cy1τ. Let us denote by Xi = Xiτ for τ = 0 and i = 0, 1. The following estimation
||X0− X0τ||0¬ max(1, |c|)||X1− X1τ||0 holds. Hence we have
(1) sup
X0τ∈g(cf)τ
inf
X0∈cfe||X0− X0τ||0¬ max(1, |c|) sup
X1τ∈feτ inf
X1∈fe||X1− X1τ||0. Analogously, we obtain
(2) sup
X0∈cfe inf
X0τ∈^(cf)τ
||X0− X0τ||0¬ max(1, |c|) sup
X1∈fe inf
X1τ∈feτ||X1− X1τ||0.
Thus, by (1) and (2) we have
r(cf, (cf )τ) ¬ max(1, |c|)r(f, fτ), and so for τ ∈ EHn
max(1,|c|)ε ; fo
we obtain r(cf, (cf)τ) ¬ ε. Hence cf is H-almost periodic.
For τ ∈ R and X0τ = X0τ(xτ0, yτ0) ∈ ^(f + g)τ there exists X1τ = X1τ(xτ0, yτ1) ∈ efτ
and X2τ = X2τ(xτ0, y2τ) ∈ egτ such that y0τ = y1τ + yτ2. Let us denote Xi = Xiτ for τ = 0 and i = 0, 1, 2. The following estimation
||X0− X0τ||0¬ ||X1− X1τ||0+ ||X2− X2τ||0
holds. Hence we obtain sup
X0τ∈ ^(f+g)τ
inf
X0∈]f+g||X0− X0τ||0¬ sup
X1τ∈feτ inf
X1∈fe||X1− X1τ||0+ sup
X2τ∈g(g)τ inf
X2∈eg||X2− X2τ||0. (3)
Analogously, we have sup
X0∈]f+g
inf
X0τ∈ ^(f+g)τ
||X0− X0τ||0¬
¬ sup
X1∈fe inf
X1τ∈feτ||X1− X1τ||0+ sup
X2∈eg inf
X2τ∈geτ||X2− X2τ||0. (4)
From (3) and (4) it follows that
r(f + g, (f + g)τ) ¬ r(f, fτ) + r(g, gτ).
Applying Lemma 2.6, for an arbitrary ε > 0 we obtain a relatively dense set EHε
2; f, g of common H,ε2
-almost periods of f and g. For τ ∈ EHε
2; f, g we have r(f + g, (f + g)τ) ¬ ε, i.e. f + g is H-almost periodic.
Example 2.8 Let
f (x) =
(tg x for x ∈ kπ −π2, kπ +π2 ,
0 for x = kπ ±π2, k∈ Z,
g(x) = (tg(√
2x) for x ∈
k√22π−√42π, k√22π +√42π ,
0 for x = k√22π±√42π, k∈ Z.
By Theorem 2.7 it follows that f + g is the unbounded H-almost periodic function.
We say that a sequence (Fn) in F∆ is H-convergent to an F ∈ F∆ if for each ε > 0 there is an N > 0 such that r∆(Fn, F )¬ ε for all n > N.
Theorem 2.9 If a sequence (fn) of H-almost periodic functions is H-convergent to a function f, then f is H-almost periodic.
Proof For each ε > 0 there is an n0 ∈ N such that r(fn0, f ) ¬ ε3. Let τ ∈ EHε
3; fn0
. Then
r(f, fτ) ¬ r(f, fn0) + r(fn0, fn0τ) + r(fn0τ, fτ) ¬ ε,
where fn0τ(x) ≡ fn0(x + τ). This means that the set EH{ε; f} is relatively dense,
i.e. f is H-almost periodic.
It is known [14] that if f is a real B-almost periodic function [1],[4],[14] and T is a generalized trigonometric polynomial of constant sign, then the composite f ◦ T−1, where T−1(x) ≡T(x)1 , is H-almost periodic. The following holds:
Theorem 2.10 If T is a generalized trigonometric polynomial such that (5) |T (x)| > 0 for x ∈ R and inf{T (x)sgnT (x) : x ∈ R} = 0, then f = T1 is µ-almost periodic [10].
Proof Since f is unbounded, hence f is not B-almost periodic.
For τ ∈ E{δ; T }, where δ > 0, and x ∈ R we have
|f(x + τ) − f(x)| ¬ δ
|T (x + τ)T (x)|.
Let CEα(u, u + 1) = {x ∈ [u, u + 1] : |T (x)| ¬ α} for u ∈ R. It is known [4] that for each ε > 0 there is an α0> 0 such that µ(CEα(u, u + 1)) < ε for all α ∈ (0, α0) uniformly with respect to u ∈ R, where µ is the Lebesgue measure. In the following, we assume that δ < α < α0. Then for x ∈ R such that |T (x)| > α we obtain
|f(x + τ) − f(x)| < δ (α − δ)α.
Given ε > 0, η > 0 and 0 < α < α0. We choose δ < α such that (α−δ)αδ < η.
Then for u ∈ R we have
{x ∈ [u, u + 1] : |f(x + τ) − f(x)| η}
= {x ∈ [u, u + 1] : |f(x + τ) − f(x)| η ∧ |T (x)| > α}
∪ {x ∈ [u, u + 1] : |f(x + τ) − f(x)| η ∧ |T (x)| ¬ α}
and
µ({x ∈ [u, u + 1] : |f(x + τ) − f(x)| η}) ¬ µ({x ∈ [u, u + 1] : |T (x)| ¬ α}) < ε Thus, D(η; f, fτ) = sup {µ({x ∈ [u, u + 1] : |f(x + τ) − f(x)| η}) : u ∈ R} ¬ ε, and so there is a relatively dense set E{ε, η; f} ⊃ E{δ; T }, i.e. f is µ-almost perio-
dic.
Remark 2.11 It is known [11] that the composite f ◦ g of a real B-almost periodic g and f : Yg → R continuous on the set Yg of values of g is N-almost periodic [4]. Hence, f = T1, where T is a generalized trigonometric polynomial satisfies the condition (5), is N-almost periodic.
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Stanisław Stoiński
Faculty of Mathematics and Computer Science, Adam Mickiewicz University Umultowska 87, 61-614 Poznań, Poland
E-mail: stoi@amu.edu.pl
(Received: 21.11.2012)
