Revista de Matema´tica: Teor´ıa y Aplicaciones 2004 11(2) : 25–33
cimpa – ucr – ccss issn: 1409-2433
some aspects in n-dimensional
almost periodic functions iii
Vernor Arguedas∗ Edwin Castro†
Received/Recibido: 2 Apr 2004
Abstract
The properties of almost periodical functions and some new results have been
published in [CA1], [CA2] and [CA3] In this paper we show some new definitions in
order to analize some singularities. For this functions we find some uniqueness sets in
R and Rn. The paper finishes analizing the relation of this functions and the function
sinc.
Keywords: Almost periodic functions, structure theorem, Radon transform.
Resumen
Las propiedades de las funciones cuasiperio´dicas y algunos resultados nuevos se han
presentado en [CA1], [CA2] y [CA3]. En este art´ıculo variamos un poco la definicio´n
para incluir cierto tipo de singularidades y encontramos para estas funciones algunos
conjuntos numerables de unicidad en R y en Rn. El art´ıculo termina analizando la
relacio´n entre estas funciones y la funcio´n sinc.
Palabras clave: Funciones cuasiperio´dicas, teorema de estructura, transformada de
Radon.
Mathematics Subject Classification: 42A75,43A60,35A22,46F12.
∗CIMPA, Escuela de Matema´tica, Universidad de Costa Rica, 2060 San Jose´, Costa Rica. E-Mail:
vargueda@amnet.co.cr
†CIMPA, Escuela de Matema´tica, Universidad de Costa Rica, 2060 San Jose´, Costa Rica. E-Mail:
Hyperion32001@yahoo.com
25
26 V. Arguedas – E. Castro Rev.Mate.Teor.Aplic. (2004) 11(2)
1 Some notations and reminders
Elementary properties of some sets of almost periodic functions have been published in
[Ca], [CO], [A-P], [BO], [COR] This paper is a natural continuation of [CA1], [CA2] and
[CA3]. We keep the basic notations and results.
Let us summarize some important results:
f : RN → R is an almost periodic function if ∀ε > 0 there is a N−dimensional vector L
whose entries are positive and satisfies that ∀y in RN there is an T in the N−dimensional
box [y, y + L] (component wise) such that |f [x+ T ]− f [x]| < ε for all x in RN .
Let x ∈ RN , x[[i]] denotes the i-th component of x. We write x > 0 if x[[i]] > 0,
i = 1, . . . , N .
If x, y are in RN we write:
|x− y| :=
 |x[[1]] − y[[1]]|...
|x[[N ]]− y[[N ]]|
 .
In the case of the usual functions sin, cos, exp, sinc, we write: sin : RN → R as
sin
 x1...
xN
 := sin(x1) ∗ . . . ∗ sin(xN )
and the same definition holds for the other functions. In general we extend in the multi-
plicative way any finite family of functions.
A set E ⊂ RN is called relatively dense (r.d) if there is an L ∈ RN , L > 0 such that
for all a ∈ RN , [a, a+ L] ∩E 6= ∅.
There are many examples of r.d sets, for instance:
• Z and pZ, wsich that p ∈ R and p /∈ Z, are r.d in R.
• ZN , p1Z× . . .× pNZ, pi /∈ Z, i = 1, . . . , N are r.d in RN .
• If A is an r.d set in RN and B is an r.d set in RM then A×B is an r.d set in RN+M .
• If A is an r.d set in RN and pii : RN → R is the i-th projection then pii[A] is an r.d
set in R.
• If f : RN → RN is an isometry then f [A] is an r.d set for any A r.d set in RN .
• Let G in RN a discrete non trivial additive subgroup then G is r.d. also a+G is r.d.
for all a in RN .
Cb(RN ,R) denotes the set of all bounded functions from RN → R endowed with the
norm ‖ · ‖∞
f [x− +m] denotes the function x→ f [x+m], m fixed.
some aspects in n-dimensional almost periodic functions iii 27
We use the following definition:
Let f : RN → R be an almost periodic function; f is said to have Bochner compact range
(BCR) if for any N−dimensional sequence (xn)n∈N there is a subsequence (xnk)k∈N and
x0 ∈ RN such that f [x + xnk ]→ f [x + x0] uniformly when k →∞.
We proved in those papers results like:
• Let f : RN → R be a continuous function, f is almost periodic iff A = {f [x ± y],
y ∈ RN} is relatively compact in C(RN , ‖ · ‖∞).
• f is almost periodic iff for any sequence (yn)n∈N there is a subsequence (ynk)k∈N and
a function g : RN → R such that f [x + ynk ]→ g in C(RN , ‖ · ‖∞).
• Let f : RN → R be a uniformly continuous bounded function, (yn)n∈N ⊂ RN be a
sequence such that f [x +yn]→ g[x ] uniformly, and let (xn)n∈N ⊂ RN be a sequence
such that xn → x0. Then f [x + yn + xn]→ g[x + x0].
• Let f : RN → R be a continuous bounded function, and let E ⊂ RN , E r.d and⋃
y∈E{f [x +y]} relatively compact in Cb(RN , ‖·‖∞). Then f is uniformly continuous
.
• (Haraux condition) Let f : RN → R be a continuous bounded function, E ⊂ RN ,
E r.d and ∪y∈E{f [x + y]} relatively compact in Cb(RN , ‖ · ‖∞), then f is almost
periodic.
• Let f : RN → R be an almost periodic function that it attains its maximum and
minimum. Then for any sequence (xn)n∈N there is a subsequence (xnk)k∈N and
x0 ∈ RN such that f ′[x + xnk ]→ f [x + x0] uniformly.
• Let f : R → R be an almost periodic function, f is periodic if and only if f has
Bochner compact range.
2 Periodic and almost periodic functions and its relations
to some sets
It is well known that any non trivial additive subgroup G of RN such that for all x > 0,
there exists g ∈ G with 0 < g < x (lexicographic) is dense in RN . From that result it
follows immediately that {n +m ∗ r} is dense in R with n,m integers and r irrational.
Without difficulties it is easy to prove the same result in RN with n,m in ZN and r
in RN , r[[i]] irrational for i = 1, . . . , N , m ∗ r denotes the componentwise multiplication.
Interesting though is that from the above results it follows that:
• {sin(n), n ∈ Z} and {cos(n), n ∈ Z} are dense in [−1, 1].
• {|sin(n)| , n ∈ Z} and {|cos(n)| , n ∈ Z} are dense in [0, 1].
• {sin(n), n ∈ G} and {cos(n), n ∈ G} are dense in [−1, 1], where G is any non trivial
additive subgroup of R such that for all x > 0, there is g ∈ G with 0 < g < x.
28 V. Arguedas – E. Castro Rev.Mate.Teor.Aplic. (2004) 11(2)
The above statements can be formulated in RN , for example: {sin(n), n ∈ ZN} is dense
in [−1, 1].
Definition 1 Let G be any discreet non trivial additive group of RN . L ⊂ RN is called a
lattice —determined by G— if L = G or there exists a ∈ RN with L = a+G.
It is easy to prove that any n-dimensional lattice is r.d.
In R a lattice G has the form: G = a+ pZ, for a, p in R.
Let f, g : R → R be two periodic, non trivial, continuous functions, then f/g is a
continuous function except for a lattice L, L = {x ∈ R/g(x) = 0}.
If f, g have measurable periods T1, T2, then f/g is periodic−measurable means
T1/T2 ∈ Q−.
If f, g have no measurable periods then f/g is almost almost periodic (a.a.p). Here,
non measurable means T1/T2 /∈ Q−.
Let Ap := {g : R→ R, g continuous of period p}.
Theorem 1 If p in R is an irrational number then Z is a uniqueness set for Ap.
Proof: B = {n+m ∗ p/n,m ∈ Z} is dense in R. Then f(x = n+m ∗ p) = f(n) for all
n,m ∈ Z.
Theorem 2 Let f ∈ Ap, with a uniqueness set E, then f(x + z) ∈ Ap for all z ∈ R with
the same uniqueness set E.
As a matter of fact sometimes if f ∈ Ap, f an odd function, there is z ∈ R with
f(x + z) an even function.
Some examples are:
• sin(x ) and z = pi/2;
• ∑pk=0 ak sin((2k + 1)x) and z = pi/2, ak ∈ R, k = 0, . . . , p.
• For the odd function: sin(x ) + sin(2x ) + sin(3x ) + sin(4x ) there is not such a z.
Some graphics illustrate this situation in Figures 1, 2 and 3.
Theorem 3 If we take in consideration in Ap only the even functions we obtain that N0
is a uniqueness set for this class of functions.
As examples we have:
• {sin(n), n ∈ N0} is dense in [−1, 1].
• {cos(n), n ∈ N0} is dense in [−1, 1].
• {|sin(n)| , n ∈ N0} is dense in [0, 1].
• {|cos(n)| , n ∈ N0} is dense in [0, 1].
some aspects in n-dimensional almost periodic functions iii 29
Figure 1: sin(x) + sin(3 ∗ x). Figure 2. sin(x) + sin(3 ∗ x) + sin(5 ∗ x).
Figure 3: sin(x) + sin(2 ∗ x) + sin(3 ∗ x) + sin(4 ∗ x).
In the case p ∈ Q we get:
Theorem 4 If p in R is a rational number then Zr, r irrational, is a uniqueness set for
Ap.
Z and Zr are lattices. We may summarizes the result as: let f be a continuous function
of period p then there is a lattice L which is a uniqueness set for Ap.
This statement can be extended to the set of functions: Bp := {f/g|f, g ∈ Ap}. There
are discontinuous functions on this set.
We introduce now the sets:
APp := {f : R→ R|f almost periodic }
and the set of a.a. functions BBp,
BBp := {f/g|f, g ∈ APp}.
Actually, those sets are vector spaces over R
30 V. Arguedas – E. Castro Rev.Mate.Teor.Aplic. (2004) 11(2)
For instance we get: {tan(n), n ∈ N0} is dense in R.
In the n-dimensional case there are several definitions of the concept of periodic func-
tion, but we work with the R-periodic concept: f : RN → R is an R-periodic function if
there are N linearly independent vectors ek, k = 1, . . . , N such that: f(x + ek) = f(x),
∀x ∈ RN . The vectors ek k = 1, . . . , N are called periods of f .
We get that if f is R-periodic and all the ek in the definition are irrational then∑N
k=1 Zek is an uniqueness set for the set of functions: Aei,...,eN := {f : RN → R is a contin-
uous R-periodic function, with periods ek, k = 1, . . . , N} and for
Bei,...,eN := {f/g|, f, g ∈ Aei,...,eN}; of course there are discontinuous functions on this
set.
We have an inmediate generalization of Theorem 2.
Theorem 5 Let f ∈ Aei,...,eN with a uniqueness set E, then f(x + z) ∈ Aei,...,eN for all
z ∈ RN with the same uniqueness set E.
Theorem 6 Let f ∈ Aei,...,eN then there exists a lattice L such that L is a uniqueness set
of Aei,...,eN .
3 The relation between sinc and Ap, Bp, APp, and BBp
Theorem 7 Let L be a numerable uniqueness lattice of a function f in Ap or APp,
L = Zh. Then
∑
k∈L f(kh)sinc(
pi
h (x − k)) is convergent toward f . When f ∈ Ap this
convergence is uniform. When f ∈ APp this convergence is uniform when restricted to
compact sets. Over RN it holds the same result.
Proof: A detailed proof will appear elsewhere.
In an schematic way we proceed as follows: We associate to f a function fc ∈ Cc(R) and
apply the Fourier band limited theory and Wiener-Paley like theorem.
A point wise proof in one variable is: Let f : R→ R be a continuous periodic function
of period pi, let us consider the case f even.
Let an(x ) := f(n)sinc(pi(x − n)) + f(−n)sinc(pi(x + n)), n ∈ N, then
an(x ) = (−1)n2f(n)pi sin(pix) xx2−n2 from this follows the convergence over compact sets of∑∞
n=0 an(x ) toward a function g. It follows immediately that g(n) = f(n) for all n ∈ Z
then f = g.
In the odd case we have: an(x ) := f(n)sinc(pi(x− n)) + f(−n)sinc(pi(x+ n)), n ∈ N,
then: an(x ) = (−1)n2f(n)pi sin(pix) nx2−n2 from this follows the point wise convergence.
In the general case of a continuous periodic function f of period pi we get that:
f(x ) = f(x)+f(−x)2 +
f(x)−f(−x)
2 ,
f(x)+f(−x)
2 is an even periodic function and
f(x)−f(−x)
2
is an odd periodic function, by using the preceding method we get the result. The choice
of the period pi is irrelevant, the same with respect to the choice of the lattice Z.
At this moment we do not know what happens to
∑
k∈L=Z∗p f(kp)sinc(
pi
p (x−k)) when
f belongs to Bp or BBp.
However, it is that a function f in BBp has not necessarily the property that for any
sequence (xn) ∈ R there is a subsequence (xnk) such that f(x + xnk)→ g.
some aspects in n-dimensional almost periodic functions iii 31
An easy counterexample is: f(x ) := sin(
√
2x)
sin(x) .
We define: x1 = b2pic, x2 = b2 ∗ 2pic + 0.d1,. . . ,xn = bn ∗ 2pic + 0.d1 . . . dn−1, where
0.d1 . . . dn−1 denotes the n− 1 decimal expansion of the number n ∗ 2pi.
4 Some graphical examples
Let us see the graphics in the interval [−2pi, 2pi].
Figure 4:
5∑
k=−5
sin(k) ∗ sin(pi ∗ (x− k))
pi ∗ (x− k) . Figure 5. sin(x).
Figure 6:
5∑
k=−5
sin(k) ∗ sin(pi ∗ (x− k))
pi ∗ (x− k) . Figure 7.
10∑
k=−10
sin(k) ∗ sin(pi ∗ (x− k))
pi ∗ (x− k) .
See the case of the tangent in (−pi/2, pi/2) in Figure 10.
32 V. Arguedas – E. Castro Rev.Mate.Teor.Aplic. (2004) 11(2)
Figure 8:
10∑
k=−10
sin(k) ∗ sin(pi ∗ (x− k))
(pi ∗ (x− k) . Figure 9.
5∑
k=−5
tan(k) ∗ sin(pi ∗ (x− k))
(pi ∗ (x− k) .
Figure 10: tan(x). Figure 11.
100∑
k=−100
tan(k) ∗ sin(pi ∗ (x− k))
pi ∗ (x− k) .
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