Download Besicovitch and other generalizations of Bohr`s almost periodic

Besicovitch and other generalizations of Bohr’s almost periodic functions
Kevin Nowland
We discuss several classes of almost periodic functions which generalize the uniformly continuous almost
periodic (a.p.) functions originally defined by Harald Bohr. We have two goals, which we accomplish in two
ways from two different sources. The first is to develop the proper setting for a Riesz-Fischer type theorem for
almost periodic functions, which we do by following [3], and which leads to the definition of the Besicovitch
almost periodic functions. Then we switch in section 5 to showing that Besicovitch functions are naturally
occurring, but by defining such functions on the set of nonnegative numbers. For this we follow [2]. This note
is just a summary of results with few proofs.
A good reference on the subtle differences between the plethora of definitions of a.p. functions is [1].
Warning: Table 2 at the end of section 6 of that paper is an excellent summary of what the authors show,
but there is a lone arrow that points up and to the left that should point down to the right.
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Bohr almost periodic functions
The classical periodic functions on R can be characterized by f (x) = f (x + t) for all x and some period t.
The first generalization of this condition was due to Bohr (1924), who instead required that a function f at
a point x be approximated by f at x + t to an arbitrary degree of accuracy for a large set of t.
Definition 1.1 (almost periods). A continuous function f : R → C is Bohr almost periodic if for any
ε > 0, the set of ε-periods {τ : |f (x + τ ) − f (x)| < ε} is relatively dense in R, i.e., there exists an l = l(ε)
such that every interval of the form [x, x + l] intersects the set of ε-periods. The set of such functions is
denoted APB (R).
We can of course give an equivalent definition based on ε-periods for functions defined on other spaces; in
particular, we will consider functions in APB (N) with N the set of nonnegative integers.
Example 1.2. Periodic functions are in APB (R).
√
Example 1.3. f (x) = sin x + sin 2x is in APB (R).
Proposition 1.4. If f ∈ APB (R), then f is bounded and uniformly continuous.
For almost periodic functions, we have a mean.
Definition 1.5. Let the mean of f : R → C to be the value of
Z
1 l
f (x)dx,
M {f (x)} := lim
l→∞ l 0
if this limit exists, which it does for all f ∈ APB (R).
Definition 1.6. APB (R) is an inner product space if we define
n
o
hf, gi := M f (x)g(x) .
Using this mean and a given f , we can define a function af (λ) : R → C by
af (λ) := M f (x)e−iλx .
Note that the (periodic) functions eiλx form an orthonormal basis with respect to this mean, such that af (λ)
is zero for all but at most countably many λ.
P
Definition 1.7. We define the Fourier series of an a.p. function to be the formal sum n a(λn )e−iλn x
where the λn are the nonzero frequencies of f . We write
X
f (x) ∼
a(λn )e−iλn x .
n
1
Note that we are being coy by writing “a.p.” instead of “Bohr a.p.,” as the mean will exist for more general
classes of almost periodic functions.
It turns out that the “structural” definition given by Bohr can be replaced with the following equivalent
statement:
Definition 1.8 (approximation). A function f is in APB (R) if there exists an at most countable sequence
of real numbers (λn ) and complex numbers (An ) such that f (x) is the uniform limit of sums of trigonometric
PN
polynomials n=1 An eiλn x .
Another name for for Bohr’s almost periodic functions is uniformly almost periodic functions, as they are
the closure of the trigonometric polynomials under the uniform norm, which we denote by k · k∞ . This gives
us a second way to define almost periodic functions as the closure of the trigonometric polynomials under
various norms and seminorms.
This uniform approximation by polynomials allows us to recover a notion of Fourier series, and some
hope of recovering a Riesz-Fischer type theorem for almost periodic functions. However, the Riesz-Ficher
theorem is a statement about L2 functions on a compact interval, and we are thus far dealing with uniformly
continuous functions, so it appears we are not in the correct setting yet.
Before moving on, we note that there is a third equivalent definition to the above two.
Definition 1.9 (normality). A continuous function f : R → C is uniformly normal if for every sequence
(hn ) of real numbers, the set of translates {f (x + hn )} contains a uniformly convergent subsequence. In other
words, the given set of translates is pre-compact.
This definition is due to Bochner, and is a normality type definition. In a complete metric space, precompactness is equivalent to total boundedness.
Definition 1.10. A set in a metric space is called totally bounded if for every ε > 0 there exists a finite
number of ε-balls covering the set.
We do not a priori have a complete metric space, as the functions are only required to be continuous, but if
we assume that they are bounded (as they end up being) then we do.
Theorem 1.11. The definitions 1.1, 1.8, and 1.9 are equivalent. The function space APB (R) is an algebra.
2
Stepanov almost periodic functions
We have the above three equivalent definitions of the Bohr a.p. functions. These functions are all uniformly
continuous, which is a very strong statement. To relax this, and leave the continuous functions entirely, we
can generalize the three definitions. Stepanov was the first to provide such a generalization. We will switch
to the approximation by trigonemtric polynoimals being out main source of a definition, as we are keeping
an eye for a Riesz-Fischer theoroem.
Definition 2.1 (approximation). Let 1 ≤ p < ∞ and l > 0. We define the space APSlp (R) of Stepanov
(Stepanoff ) almost periodic functions to be the closure of the trigonmetric polynomials under the norm
kf k
Slp
:= sup
x∈R
1
l
Z
x+l
p
x
|f (t)| dt
!1/p
Note. These functions necessarily are in Lploc (R), which are only defined up to sets of Lebesgue measure zero,
such that limits are only unique on classes of functions.
Proposition 2.2. The Stepanov norms induce equivalent topologies, in the sense that for any l, l′ there exist
C, C ′ > 0 depending on l and l′ such that
Ckf kS p′ ≤ kf kSlp ≤ C ′ kf kS p′ .
l
l
We typically take l = 1, in which case we write S p in place of
2
S1p .
As a consequence of Hölder’s inequality,
Proposition 2.3. Let 1 ≤ p < p′ < ∞ and f ∈ APS p′ (R). Then
kf kS p ≤ kf kS p′ ,
which implies APS p′ (R) ⊆ APS p (R).
Even more simply,
Proposition 2.4. Let 1 ≤ p < ∞ and f ∈ APB (R). Then
kf kS p ≤ kf k∞ ,
which implies APB (R) ⊆ APS p (R).
The other two definitions are the following:
Definition 2.5 (almost periods). Let 1 ≤ p < ∞. A function f ∈ Lploc (R) is in APS p (R) if for every ε > 0
the set of all Stepanov ε-almost periods τ satisfying
kf (t + τ ) − f (t)kS p = sup
x∈R
Z
x+1
p
x
|f (t + τ ) − f (t)| dt
1/p
is relatively dense in R.
Definition 2.6 (normality). Let 1 ≤ p < ∞. A function f ∈ Lploc (R) is in APS p (R) if the set {f (x + τ )}t∈R
is pre-compact.
We have been sloppy in saying that these are all definitions of the same space APS p (R), but for the
Stepanov a.p. functions, we continue our good luck and have the following theorem.
Theorem 2.7. The definitions 2.1, 2.5, and 2.6 are equivalent. The function space APS p (R) is an algebra
for all 1 ≤ p < ∞.
Example 2.8. Let E be set of closed intervals of the form [2k, 2k + 1] where k is any integer and let O be
the complement of these intervals. Defined
(
1 x∈E
f (x) =
0 e ∈ O.
Clearly f is not Bohr almost periodic, as f is not continuous. However, it is in APS p for any p, as for any
ε > 0, we can take the relatively dense set of integers as the periods.
We also have the theorem due to Bochner.
Theorem 2.9 (Bochner’s Theorem). If f ∈ APS 1 (R) is uniformly continuous, then f ∈ APB (R).
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Weyl almost periodic functions
Though the ε-almost periodic, approximation by trigonometric polynomials, and normality / pre-compactness
definitions have to this point coincided, we will no longer have that luxury. However, we are not yet at the
point where we have the Riesz-Fischer theorem. We keep using trigonometric polynomials as our main
definition.
The definition of the Weyl a.p. functions is based on the following observation.
Lemma 3.1. The limit liml→∞ kf kSlp exists, in the sense that if the norm is infinite for any l > 0 then the
limit is infinite.
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Proof. We consider only the p = 1 case. This proof boils down to the fact that all values of l give rise to the
same topology. Note that if kf kSl is infinite for some l > 0, then it is infinite for all l > 0. Thus it suffices to
consider the case where all such are finite. Let l 6= l0 be greater than zero and let n be the positive integer
such that
(n − 1)l0 < l ≤ nl0 .
Note that
1
nl0
Z
x+nl0
x
1
|f (x)|dx =
n
1
l0
Z
x+l0
x
1
|f (x)|dx + · · · +
l0
Z
x+nl0
x+(n−1)l0
|f (x)|dx
!
≤ kf kSl0 .
Therefore kf kSnl0 ≤ kf kSl0 . We calculate
kf kSl ≤
nl0
l + l0
l + l0
kf kSnl0 ≤
kf kSnl0 ≤
kf kSl0 .
l
l
l
(1)
Therefore
lim sup kf kSl ≤ lim inf kf kSl0 = lim inf kf kSl ,
l0 →∞
l→∞
l→∞
as desired.
With this in hand, we make a definition.
Definition 3.2 (approximation). Let 1 ≤ p < ∞. We define the space of Weyl almost periodic functions, denoted APW p (R), to be the set of functions on R which can be approximated arbitrarily well by
triogonometric polynomials under the Weyl seminorm
kf kW p := lim kf kSlp .
l→∞
Example 3.3. There exist functions f on R which are strictly positive but satisfy kf kW p = 0. Thus the
uniqueness of elements in the closure is up to functions which may differ on sets of even infinite measure.
The second definition is in terms of Stepanov ε-almost periods.
Definition 3.4 (almost periods). Let 1 ≤ p < ∞. A function f ∈ Lploc (R) is in APW p (R) if for every
ε > 0 there exists an l = l(ε) such that that there is a relatively dense set {τ } of Stepanov ε-almost periods
satisfying
kf (t + τ ) − f (t)kSlp < ε.
It is clear that APS p (R) ⊂ APW p (R).
Remark. In the above definition, we are not using the Weyl seminorm but rather the Stepanov seminorm,
and the l is allowed to vary with ε, which is not the case for the Slp functions. This turns out to be crucial, as
if we insist on defining the ε-almost periods in terms of the Weyl seminorm, we find a strictly larger space.
Theorem 3.5. The definitions 3.2 and 3.4 are equivalent.
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Besicovitch periodic functions
Finally, we arrive at the spaces which give us a Riesz-Fischer theorem. We still do not have all three
definitions, however, but we do have two equivalent definitions, as with the Weyl a.p. functions.
Definition 4.1. Let 1 ≤ p < ∞. We define the space APB p (R) of Besicovitch almost periodic functions
as the functions on R which can be approximated arbitrarily well by trigonometric polynomials under the
seminorm
!1/p
Z
1 l
p
|f (x)| dx
.
kf kB p = lim sup
2l −l
l→∞
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Note. As with the Weyl seminorm, there are functions which are nonzero on all of R and nonetheless have
zero Besicovitch seminorm.
It is clear from the definition that kf kB p ≤ kf kW p , such that APW p (R) ⊆ APB p (R).
Proposition 4.2. Let p ≤ 1 < ∞. Then
APB (R) ⊂ APS p (R) ⊂ APW p (R) ⊂ APB p (R),
and all the inclusions are strict.
We wish to find another characterization based on ε-periods, but this is not easy to do. It turns out that
relatively dense sets are too broad to describee the elements of APB p (R).
Definition 4.3. A subset E of R is called satisfactorily uniform if there exists a a positive number l
such that the ratio of the maximum number of terms of E in an interval [x, x + l] to the minimum number
of terms of E in such an interval is less than 2.
Example 4.4. Every satisfactorily uniform set is relatively dense.
∞
Example 4.5. The set {1, 2, . . . , } ∪ n1 n=1 is relatively dense but not satisfactorily uniform.
Definition 4.6. Let 1 ≤ p < ∞ and f ∈ Lp (R). Then f ∈ APB p (R) if for any ε > 0, there corresponds a
satisfactorily uniform set · · · < τ−1 < t0 < t1 < · · · such that
kf (x + τ ) − f (x)kB p < ε
and for every c > 0,
lim sup
l→∞
1
2l
Z
l
−l
# !1/p
Z
n
X
1
1 x+c
p
lim sup
|f (t + τi ) − f (t)| dt dx
< ε.
c x
n→∞ 2n + 1
i=−n
"
Then we have the equivalence of definitions that we desire.
Theorem 4.7. The definitions 4.1 and 4.6 are equivalent.
Finally, we are in a position where we have a Riesz-Fischer theorem.
P
P
Theorem 4.8. To any (generalized) Fourier series n An eiλn x such that n |An |2 converges there corresponds a function f ∈ APB 2 (R) with this as its Fourier series.
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Almost periodicity on the nonnegative integers
We now shift focus from almost periodic functions on R to almost periodic functions on N = {0, 1, . . .}. Note
that we include zero when we use the symbol “N.” One way to define APG (N) for G one of B, Slp , W p , and
B p , is to replace the integrals with sums over the integers. However, a different approach is taken in [2].
We begin by definining APB (N), the Bohr almost periodic functions, in the same way as originally, by
using the relative density of the ε-periods with respect to the l(∞) norm. As before, elements in APB (N)
are contained in λ(∞).
Definition 5.1. Let f : N → C be such that f ∈ l(∞). Then we define W (f ) to be the set of functions
{f (x + a) : a ∈ N}, i.e., the set of translates of f .
Proposition 5.2. Let f ∈ APB (N). Then W (f ) is relatively compact (totally bounded) in l(∞).
Proof. To show that W (f ) is totally compact in the metric space l(∞), it suffices to find a finite number of
ε-balls which cover the set. Let ε > 0 be fixed, and let l = l(ε) be such that every interval of length at least
λ in N contains an ε-almost period of f . Let J = {j, j + 1, . . . , j + l}. We claim that W (f ) is contained in
the set of l + 1 ε-balls, each centered at f (x + k) for k ∈ J. Consider f (x + n) for some n ∈ N. If k ∈ J, we
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are done. If k < j, the consider the set {k − n : k ∈ J}. This is an interval of length l + 1 in N and hence
must contain an ε-almost period. Write
|f (x + n) − f (x + k)| = |f (x + n) − f (x + n + (k − n))|.
Then for some k − n, this will be less than ε for all x. Thus f (x + n) is contained in some ε-ball centered at
f (x + k) for k ∈ J, as required. If n > j + l, then we do the same trick but replace k − n with n − k.
With this proposition in hand, we can now define new classes of almost periodic functions on N.
Definition 5.3. We say that f : N → C with f ∈ l(∞) is Weyl almost periodic if the set of translates
W (f ) is relatively compact in l(∞). This set is denoted by APW (N).
From proposition 5.2, we have that
APB (N) ⊆ APW (N).
Example 5.4. The converse is false, such that the inclusion above is strict, as can be seen by looking at
the function
(
0 x = 0,
f (x) =
.
1 x > 0.
The set of translates of f consists of f and the constant function 1, such that W (f ) is relatively compact.
However, f is not Bohr almost periodic, as can be seen for any ε < 1.
Note. APW (Z) = APB (Z).
Fréchet gave the following characterization of almost periodic funtions which gives rise to many examples.
We state the result without proof.
Theorem 5.5 (Fréchet, 1941). A function f ∈ l(∞) is in APW (N) if and only if f admits a (unique)
decomposition f = p + w wehre p ∈ APB (N) and limn→∞ w(n) = 0.
This definition of APW p (N) is is sometimes called W p -normal.
Definition 5.6. The Eberlein or weakly almost periodic functions on N are bounded functions such
that W (f ) is weakly relatively compact. We denote this set by APw (N).
Eberlein characterized the weakly almost periodic functions in a theorem similar that that of Fréchet.
Theorem 5.7 (Eberlein, 1956). f ∈ l(∞) is weakly almost periodic on N if and only if f admits a (unique)
decomposition f = p + w where p ∈ APB (N) and w satisfies
n−1
1X
|ω(x + j)| = 0,
n→∞ n
j=0
lim
with the limit is uniform in x.
Example 5.8. Let w : N → C be defined as a sequence of ones and and zeros such that the number of
zeros between consecutive ones is increasing. Then w(n) does not tend to zero as n tends to infinity, but the
average about does tend to zero uniformly in x.
Corollary 5.9. APW (N) ⊂ APw (N), and the inclusion is proper.
The characterization theorems of Fréchet and Eberlein imply that APW (N) and APw (N) are algebras.
Finally, we return to the Besicovitch almost periodic functions. Instead of using topological properties of
translates to define these functions, we return to a definition based on the closure under a seminorm.
Definition 5.10. Let X be the set of sequences of the form (z k ) where z ∈ C, |z| = 1, and k = 0, 1, . . .. A
sequence is a trigonometric polynomial if it is a linear combination of a finite number of sequences in X.
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Definition 5.11. Let 1 ≤ p < ∞. We say that f is Besicovitch almost periodic on N if f is in the
closure of the trigonometric polynomials under the seminorm k · kp defined by
kf kpp = lim sup
n→∞
n−1
1X
|f (k)|p .
n
k=0
In this case we write f ∈ APB p (N). We say that f is bounded Besicovitch if f ∈ APB p (N) ∩ l(∞).
It turns out that APB 1 (N) ∩ l(∞) = APB p (N) ∩ l(∞) for 1 ≤ p < ∞. The idea behind the proof will be
explained below, but no proof will be given.
For any function f , we can define its mean M {f } to be
n−1
1X
f (j).
n→∞ n
j=0
M {f } := lim
The mean does not necessarily exist or is finite. Using this mean, we can define a function a : R → C by
af (λ) = M f e−iλx .
For any f ∈ APB p (N) ∩ l(∞), this function is well-defined for all λ ∈ R. If we take f to be the purely periodic
eiλx , then
(
1 λ = λ′ ,
′
af (λ ) =
.
0 λ 6= λ′
It follows that for any f ∈ APB p (N) ∩ l(∞), af (λ) is nonzero for an at most countable set of λ. (This relies
on standard inner-product space results, which we gloss over.)
Definition 5.12. Let f ∈ APB p (N)∩l(∞) and let (λn ) be the sequence of real numbers such that af (λ) 6= 0.
Then we write
X
f (x) ∼
af (λn )eiλn x ,
n
and call the right hand side of the above the (generalized) Fourier series for f .
Definition 5.13. Let f and (λn ) be as in the prevoius definition. Then a set of (at moust) countable real
numbers Bf is called a base of f if Bf forms a basis for (λn ) over Q, i.e., Bf consists of numbers such
that all finite subsets of Bf are linearly independent over Q and every λn can be written as a finite linear
combination of elements in Bf with coefficients in Q.
Let Kn (t) be the Fejér kernel defined as
Kn (t) =
X |λ|<n
|λ|
1−
n
e−iλt .
Then a Bochner-Fejér kernel
K(k) = Kn1 (b1 k) · · · knm (bm k)
is a finite product of m Fejér kernels with the bi linearly independent over Q. We take k ∈ N. Then, using
the base Bf for a function f in APB p (N) ∩ L1 (∞), it is possible to define a sequence of Bochner-Fejér kernels
Knf with the following properties.
(i) σnf (x) := M f (x + j)Knf (j) is a trigonometric polynomial;
(ii) kσnf kp ≤ kf kp for all n;
(iii) kσnf k∞ ≤ kf k∞ for all n;
(iv) kσnf − f kp → 0 as n → ∞.
Using these properties and Hölder’s inequality, we have that
APB 1 (N) ∩ l(∞) = APB p (N) ∩ l(∞)
for 1 ≤ p < ∞.
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6
Relation to Ergodic Theory
We now show that the bounded Besicovitch functions are naturally occurring in the context of ergodic theory
and prove a related theorem about pointwise convergence of a weighted sequence of operators acting on a
function.
Before stating the theorem which indicates that Bounded besicovitch functions occur naturally, we need
a definition.
Definition 6.1. Let T be a bimeasurable, measure-preserving, ergodic, bijection on a probability space
(Ω, A, µ). Let UT be the induced transformation on functions, i.e., UT f (x) = f (T x). Then T has discrete
spectrum if L2 (Ω, µ) has an orthonormal basis of eigenfunctions of T .
The ergodic functions T can be characterized as follows.
Theorem 6.2. Let T be as in the definition above. Then the following are equivalent:
(i) T has discrete spectrum.
(ii) If g ∈ L∞ (Ω), then for almost every ω ∈ Ω, the sequence (g(T n ω)) for n ∈ N is bounded Besicovitch.
Proof. (i)⇒(ii). Let {fi } be an orthonormal set of eigenfunctions for UT . Let g ∈ L∞ (Ω). This implies
g ∈ L1 (Ω), since (Ω, A, µ) is a probability space. Then there exists a sequence hn in the linear span of {fi }
such that
hn → g
in L1 . Linear span means finite linear combinations of the fi . By the individual (Birkhoff’s?) ergodic theorem,
for each n ∈ N, there exists a set Ωn of measure 1 such that ω ∈ Ωn implies
Z
n−1
1X
|g(T j ω) − hn (T j ω)| =
|g − hn |dµ.
k→∞ k
Ω
lim
k=0
Let Ωg be the intersection of the Ωn , and note that this set has full measure. If we consider the sequences
v = (g(T j ω)) and h(n) = (hn (T j ω)). Then, as n goes to infinity,
Z
k−1
1X
|g − hn |dµ → 0.
|g(T j ω) − hn (T j ω)| =
k→∞ k
Ω
j=0
kv − h(n)k1 := lim
It is claimed in [2] that h(n) is “clearly” a trigonometric polynomial, such that g is in the closure of these
polynomials by the above. g is bounded by assumption. Thus g ∈ APB 1 (N) ∩ l(∞), as claimed.
We leave the proof of the reverse direction to the interested reader, who may consult [2].
References
[1] J. Andres, A.M. Bersani, and R.F. Grande, Hierarchy of almost-periodic functions spaces, Rend. Mat.
Appl. (2006).
[2] A. Bellow and V. Losert, The weighted pointwise ergodic theorem and the individual ergodic theorem along
subsequences, T. Am. Math. Soc. (1985).
[3] A.S. Besicovitch, Almost periodic functions, Cambridge University Press, 1954.
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