On the completeness of oscillation spaces
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irk-123456789-1780142021-02-18T01:27:34Z On the completeness of oscillation spaces Ben Slimane, M. 2005 Article On the completeness of oscillation spaces / M. Ben Slimane // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 435-443. — Бібліогр.: 12 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/178014 517.9 en Нелінійні коливання Інститут математики НАН України |
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Ben Slimane, M. On the completeness of oscillation spaces Нелінійні коливання |
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Ben Slimane, M. |
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On the completeness of oscillation spaces |
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On the completeness of oscillation spaces |
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On the completeness of oscillation spaces |
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On the completeness of oscillation spaces |
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On the completeness of oscillation spaces |
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on the completeness of oscillation spaces |
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2005 |
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On the completeness of oscillation spaces / M. Ben Slimane // Нелінійні коливання. — 2005. — Т. 8, № 4. — С. 435-443. — Бібліогр.: 12 назв. — англ. |
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Нелінійні коливання |
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AT benslimanem onthecompletenessofoscillationspaces |
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2025-07-15T16:21:55Z |
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UDC 517 . 9
ON THE COMPLETENESS OF OSCILLATION SPACES*
ПРО ПОВНОТУ ПРОСТОРIВ КОЛИВАНЬ
M. BEN SLIMANE
Campus Univ.
2092 El Manar, Tunis, Tunisia
e-mail: mourad.benslimane@fst.rnu.tn
The oscillation spaces Os,s′
p (Rd), introduced by Jaffard, are a variation on the definition of Besov spaces
for either s ≥ 0 or s ≤ −d/p. On the contrary the spaces Os,s′
p (Rd) for −d/p < s < 0 cannot be sharply
imbedded between Besov spaces with almost the same exponents, and thus are new spaces of really di-
fferent nature. Their norms take into account correlations between the positions of large wavelet coeffici-
ents through the scales. Several numerical studies have uncovered such correlations in several settings inclu-
ding turbulence, image processing, traffic, finance,. . . These spaces allow to capture oscillatory behaviors
which are left undetected by Sobolev or Besov spaces. Unlike Sobolev spaces (resp. Besov spaces Bs,q
p (Rd))
which are expressed by simple conditions on wavelet coefficients as `p norms (resp. mixed `p − `q norms),
oscillation spaces are written as `p averages of local Cs′
norms. In this paper, we prove the completeness
of oscillation spaces in spite of such a mixture of two norms of different kinds.
Простори коливань Os,s′
p (Rd), введенi Джаффаром, є варiацiями означення просторiв Бєсова
для s ≥ 0 або s ≤ −d/p. Але, якщо −d/p < s < 0, простори Os,s′
p (Rd) не можуть бути строго
включенi мiж просторами Бєсова з майже такими ж показниками, i тому є новими простора-
ми, що мають дiйсно iншу природу. Значення норми в цих просторах залежить вiд кореляцiї
положення коефiцiєнтiв при великих вейвлетах у послiдовностi просторiв. Декiлька чисель-
них дослiджень вiдкрили таку кореляцiю в кiлькох випадках, що охоплюють турбулентнiсть,
обробку зображень, рух машин, фiнанси i т. д. Цi простори дозволяють помiтити коливання,
якi залишаються непомiтними у просторах Соболєва та Бєсова. На вiдмiну вiд просторiв Со-
болєва (вiдповiдно Бєсова, Bs,q
p (Rd)), якi визначаються простими умовами на коефiцiєнти при
вейвлетах у термiнах норм `p (вiдповiдно `p−`q), простори коливань визначаються `p-середнiми
локальних норм Cs′
. У статтi доведено повноту просторiв коливань, незважаючи на таке по-
єднання норм рiзних типiв.
1. Introduction. We will use a family of 2d−1 smooth wavelets Ψ(i), such that the Ψ(i) and their
partial derivatives have fast decay. The 2dj/2Ψ(i)(2jx − k) (i = 1, . . . , 2d − 1, j ∈ Z, k ∈ Zd)
form an orthonormal basis of L2(Rd). We will use a L∞ normalization for wavelets, so that we
write
f(x) =
∑
i,j,k
C
(i)
j,kΨ
(i)(2jx− k), (1)
where
C
(i)
j,k = C
(i)
j,k(f) = 2dj
∫
f(t)Ψ(i)(2jt− k)dt.
∗ The author is supported by the research project CMCU 99/F1506.
c© M. Ben Slimane, 2005
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4 435
436 M. BEN SLIMANE
We will use the following simpler notations; λ and λ′ will denote respectively the cubes λj,k =
= k2−j + [0, 2−j [d and λj′,k′ = k′2−j′ + [0, 2−j′ [d, Cλ will denote the coefficient C
(i)
j,k, and Ψλ
will denote the wavelet Ψ(i)(2jx− k) (note that we forget the index i of the wavelet which is of
no consequence).
Recall that f belongs to the Besov space Bs,q
p (Rd) with p > 0 and q > 0 if
2sj2−dj/p
(∑
k
|Cλ|p
)1/p
:= εj with εj ∈ lq (2)
(which follows directly from [1, p. 50, 197] and [2, p. 45]). Note that Bs,2
2 is the Sobolev space
Hs and that Bs,∞
∞ is the Hölder space Cs. Recall that f ∈ Cs(Rd) for s > 0 if there exist a
polynomial P of degree smaller than s and a constant C such that
∀x ∈ Rd ∀x0 ∈ Rd : |f(x)− P (x− x0)| ≤ C|x− x0|s. (3)
The spaces Os,s′
p (Rd) are function spaces that have been introduced by Jaffard in [3] in
order to quantify the degree of correlations between positions of large wavelet coefficients
through the scales. Several numerical studies have uncovered such correlations in several set-
tings including turbulence [4], image processing, traffic [5], finance [6], . . . . Oscillation spaces
allow to capture oscillatory behaviors which are left undetected by Sobolev or Besov spaces.
Definition 1. Let p > 0, and s, s′ ∈ R; a function f belongs to the oscillation space Os,s′
p (Rd)
if its wavelet coefficients satisfy
sup
j
2sj
(∑
k
sup
λ′⊂λ
|Cλ′2s′j′ |p
)1/p
< ∞ (4)
(modified if p = +∞).
The left-hand side defines the Os,s′
p (Rd) quasinorm. Note that this definition is independent
on the wavelet basis which is chosen (see [3]).
In [7], Jaffard proved that, for either s ≥ 0 or s ≤ −d/p, the Os,s′
p (Rd) are a variation
on the definition of Besov spaces (Os,s′
p = B
s+s′+d/p,∞
p if s > 0 and Cs′ if s ≤ −d/p, and
B
s′+d/p,p
p ↪→ O0,s′
p ↪→ B
s′+d/p,∞
p ). On the contrary the spacesOs,s′
p for−d/p < s < 0 cannot be
sharply imbedded between Besov spaces with almost the same exponents (in fact B
s′+d/p,p
∞ ↪→
↪→ Os,s′
p ↪→ Cs′ and Cs+s′+d/p ↪→ Os,s′
p and the imbeddings are optimal), and thus are new
spaces of really different nature.
In [8], Jaffard proved several related results concerning the genericity (in the sense of Baire’s
categories) of multifractal functions. One result asserts that, if s >
d
p
, generically, functions
of the Besov space Bs,q
p (Rd) are multifractal. The completeness of Besov spaces was a key
topological property in his proof. (Note that the validity of the multifractal formalism never
holds in complete generality, but it has also been checked under an additional self-similarity
assumption in [9 – 11].)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE COMPLETENESS OF OSCILLATION SPACES 437
In the next section, we will first briefly recall the proof of the completeness of Besov spaces.
We will then see that oscillation spaces can be written as `p averages of local Cs′ norms. We
will prove the completeness of these spaces in spite of such a mixture of two norms of different
kinds. So, the main statement of the this paper is the following theorem.
Theorem 1. The oscillating spaces Os,s′
p for p > 0, and s, s′ ∈ R; are quasi-Banach spaces
(Banach spaces if p ≥ 1).
2. The completeness. 2.1. Completeness of Besov spaces. Recall that if A is a (real or
complex) linear vector space, ‖ · ‖ is said to be a quasinorm if ‖ · ‖ satisfies the usual condi-
tions of a norm with the exception of the triangle inequality, which will be replaced by
∃L ≥ 1 ∀(a1, a2) ∈ A2 : ‖a1 + a2‖ ≤ L(‖a1‖+ ‖a2‖). (5)
(If L = 1, then A is a normed space). A quasinormed space is said to be a quasi-Banach space
if it is complete (i.e., any Cauchy sequence in A with respect to ‖ · ‖ converges).
It is well known that Lp = Lp(Rd) (the set of all Borel measurable complex valued functions
on Rd such that
∫
|f(x)|p dx < ∞) is a quasi-Banach space (a Banach space if p ≥ 1).
The vector space `p of all sequences b = (bk)k∈N of complex numbers such that
‖b‖`p := ‖(bk)k‖`p =
( ∞∑
k=0
|bk|p
)1/p
< ∞
(modified if p = ∞) is a quasi-Banach space (a Banach space if p ≥ 1). Using the normalization
we choose, the wavelet characterization of Besov spaces Bs,q
p = Bs,q
p (Rd) for (s ∈ R, p > 0,
q > 0) can be written
‖f‖Bs,q
p
=
∥∥∥∥(2(s− d
p
)j ‖(Cj,k)k‖`p
)
j
∥∥∥∥
`q
< ∞. (6)
The completeness of Besov spaces Bs,q
p can be deduced from the continuous isometry that
relates it to quasi-Banach spaces `q(`p) (a mixed `p − `q norm) (see [2, p. 14, 48]).
2.2. Completeness of oscillation spaces. Let us begin by some remarks.
Remark 1. If T (f) :=
∑
λ
ω(λ)Ψλ, with ω(λ) = sup
λ′⊂λ
|Cλ′(f)|, then we can easily check that
f ∈ Os,0
p ⇔ T (f) ∈ Bs+d/p,∞
p .
Nevertheless, the mapping T is not linear.
Remark 2. In [3], Jaffard proved that ‖(−∆)α/2f‖Os,0
p
and ‖f‖Os,α
p
are equivalent norms. It
follows thatOs,s′
p is complete if and only ifOs,0
p is complete. Therefore, we can restrict the proof
of the completeness to the space Os,0
p .
Remark 3. The left-hand side of (4) is an `∞ norm (on the scale j) of the sequence 2sj
multiplied by a `p norm of the quantities
sup
λ′⊂λ
|Cλ′ | 2s′j′ (7)
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
438 M. BEN SLIMANE
and these quantities clearly look like a local Hölder Cs′ norm. Indeed a function belongs to Cs′
if its wavelet coefficients satisfy sup
λ′
|Cλ′ |2s′j′ < ∞, but in (7) the supremum is restricted to the
subcubes of λ. Hence Os,s′
p can be written as a `p average of local Cs′ norms. Such a mixture
of two norms of different kinds does not allow us to find an isometry similar to the above one
for Besov spaces, in order to check the completeness of Os,s′
p . We will instead use the following
result which can be deduced from [12, p. 58].
Proposition 1. Let A be a quasinormed vector space. Denote L a constant which appears in
the generalized triangle inequality (5). The space A is complete if and only if any sequence (an)n
of elements of A satisfies the following property:
“if there exists a constant D > L such that for any n ‖an‖ ≤ D−n then the series
∑
n
an converges
in A”.
Proof. Assume that A is a complete vector space. Let L be a constant which appears in the
generalized triangle inequality (5). Let (an)n be a sequence of A. Assume that there exists a
constant D > L such that ‖an‖ ≤ D−n ∀n. Let SN =
N∑
n=1
an. For M > N
‖SM − SN‖ =
∥∥∥ M∑
n=N+1
an
∥∥∥.
If A is normed then
‖SM − SN‖ ≤
M∑
n=N+1
‖an‖ ≤
M∑
n=N+1
D−n ≤ D−(N+1)
D − 1
.
Since D > L ≥ 1, (SN ) is a Cauchy sequence of A. So, (SN ) converges. If A is quasinormed
then
‖SM − SN‖ =
∥∥∥ M∑
n=N+1
an
∥∥∥ ≤
≤ L‖aN+1‖+ L
∥∥∥ M∑
n=N+2
an
∥∥∥ ≤
≤ L‖aN+1‖+ L2‖aN+2‖+ . . . + LM−N‖aM‖ ≤
≤ LD−(N+1) + L2D−(N+2) + . . . + LM−ND−M ≤
≤ D−N
1− L
D
(because D > L).
So, (SN ) converges.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE COMPLETENESS OF OSCILLATION SPACES 439
Now, for the converse part of Proposition 1, assume that A is a quasinormed vector space,
and that any sequence (an)n of elements of A satisfies the following property: “if there exists
a constant D > L such that ‖an‖ ≤ D−n ∀n, then the series
∑
n
an converges in A” . We will
prove that A is complete; let (bn) be a Cauchy sequence of A. It suffices to show that there exists
a subsequence (bnk
)k which converges in A; we extract a subsequence (bnk
)k such that
∀k ≥ 1 : ‖bnk+1
− bnk
‖ ≤ D−k. (8)
Denote ak = bnk+1
− bnk
. Relation (8) implies that ∀k ≥ 1 : ‖ak‖ ≤ D−k. Hence, by the
assumption, the series
∑
k
ak converges in A. Denote by S its limit. Denote SK =
K∑
k=1
ak. We
have SK = bnK+1 − bn1 . So, (bnk
)k converges to S + bn1 .
The proof of Proposition 1 is now finished.
We will now pursue the proof of completeness of the oscillation spaces using Proposition 1.
As mentioned in Remark 2, we only have to do this for Os,0
p (Rd). Recall that f ∈ Os,0
p (Rd) if
there exists a constant C > 0 such that(∑
k
(ω(λ))p
)1/p
≤ C2−sj ∀j, (9)
where ω(λ) = sup
λ′⊂λ
|Cλ′(f)|. We will use Proposition 1; we take A = Os,0
p (Rd), D = 2 if p ≥ 1,
and D = L + 1 if 0 < p < 1 (with L a constant which appears in the generalized triangle
inequality (5)). Let (fn)n be a sequence in Os,0
p such that
‖fn‖Os,0
p
≤ D−n ∀n. (10)
We will prove that the series
∑
n
fn converges in Os,0
p . We will divide the proof into three steps.
First step. In this step we will prove that for any cube λ the series
∑
n
Cλ(fn) converges;
relation (10) is equivalent to (∑
k
(ωn(λ))p
)1/p
≤ D−n2−js ∀j (11)
with
ωn(λ) = sup
λ′⊂λ
|Cλ′(fn)|. (12)
It follows that
|Cλ(fn)| ≤ D−n2−js ∀j ∀k. (13)
Since D > 1 we deduce that the series
∑
n
Cλ(fn) converges.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
440 M. BEN SLIMANE
Second step. For any cube λ we write
∞∑
n=0
Cλ(fn) := C̃λ. (14)
We denote f̃ =
∑
λ
C̃λΨλ. We will prove that f̃ ∈ Os,0
p (and in the third step, we will prove that
the series
∑
n
fn converges to f̃ in Os,0
p ). For that we will first show that
(
f̃N :=
N∑
n=0
fn
)
N
is a
bounded sequence in Os,0
p .
If p ≥ 1 then
‖f̃N‖ ≤
N∑
n=0
‖fn‖ ≤
N∑
n=0
2−n ≤ 2 ∀N, (15)
if 0 < p < 1 then
‖f̃N‖ ≤ L‖f0‖+ L2‖f1‖+ . . . + LN+1‖fN‖ ≤
≤ L(L + 1)−0 + L2(L + 1)−1 + . . . + LN+1(L + 1)−N ∀N.
Hence
‖f̃N‖ ≤ L(L + 1) ∀N. (16)
Consequently, (f̃N )N is a bounded sequence inOs,0
p . This property, together with (14), will allow
us to show that f̃ =
∑
λ
C̃λΨλ ∈ Os,0
p . Let (C̃N (λ))λ denote the wavelet coefficients of f̃N , i.e.,
C̃N (λ) =
N∑
n=0
Cλ(fn).
Relation (14) can be written
lim
N→+∞
C̃N (λ) = C̃λ ∀λ. (17)
Let w̃N (λ) denote sup
λ′⊂λ
|C̃N (λ′)|, and w̃(λ) denote sup
λ′⊂λ
|C̃λ′ |. We have
w̃(λ) = sup
λ′⊂λ
∣∣∣ ∞∑
n=0
Cλ′(fn)
∣∣∣ ≤ ∞∑
n=0
sup
λ′⊂λ
|Cλ′(fn)| =
∞∑
n=0
wn(λ).
Relation (11) implies that wn(λ) ≤ D−n2−js. Whence w̃(λ) is finite.
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
ON THE COMPLETENESS OF OSCILLATION SPACES 441
For ε > 0 there exists λ′ε ⊂ λ such that
w̃(λ) ≤ |C̃λ′
ε
|+ ε.
It follows from (17) that for N large enough,
w̃(λ) ≤ |C̃N (λ′ε)|+ 2ε ≤ w̃N (λ) + 2ε.
Therefore
w̃(λ) ≤ lim inf
N→+∞
w̃N (λ) ∀λ. (18)
The previous relation implies that∑
k
(w̃(λ))p ≤ lim inf
N→+∞
∑
k
(w̃N (λ))p ∀j.
From (9) and the fact that (f̃N )N is bounded in Os,0
p , we deduce that(∑
k
(w̃(λ)p
)1/p
≤ C2−sj ∀j.
Whence f̃ ∈ Os,0
p .
Third step. We will prove that the series
∑
n
fn converges to f̃ in Os,0
p :
if p ≥ 1 then, for M > N,
‖f̃M − f̃N‖ =
∥∥∥ M∑
n=N+1
fn
∥∥∥ ≤ M∑
n=N+1
‖fn‖ ≤
M∑
n=N+1
2−n
(where ‖ · ‖ = ‖ · ‖Os,0
p
), hence ∥∥∥f̃M − f̃N
∥∥∥ ≤ 2−N ; (19)
if 0 < p < 1 then, for M > N,
‖f̃M − f̃N‖ =
∥∥∥ M∑
n=N+1
fn
∥∥∥ ≤
≤ L‖fN+1‖+ L
∥∥∥ M∑
n=N+2
fn
∥∥∥ ≤
≤ L‖fN+1‖+ L2‖fN+2‖+ . . . + LM−N‖fM‖ ≤
≤ L(L + 1)−(N+1) + L2(L + 1)−(N+2) + . . . + LM−N (L + 1)−M .
ISSN 1562-3076. Нелiнiйнi коливання, 2005, т . 8, N◦ 4
442 M. BEN SLIMANE
Hence
‖f̃M − f̃N‖ ≤ L(L + 1)−N . (20)
Using similar arguments to those of the previous step, properties (19), (20) and (17) will imply
that (f̃N )N converges to f̃ in Os,0
p ; let N ≥ 1. For ε > 0 there exists λ′ε,N ⊂ λ such that
sup
λ′⊂λ
|C̃λ′ − C̃N (λ′)| ≤ ε + |C̃λ′
ε,N
− C̃N (λ′ε,N )|.
It follows from (17) that for M large enough
sup
λ′⊂λ
|C̃λ′ − C̃N (λ′)| ≤ 2ε + |C̃M (λ′ε,N )− C̃N (λ′ε,N )| ≤ 2ε + sup
λ′⊂λ
|C̃M (λ′)− C̃N (λ′)|.
Therefore
sup
λ′⊂λ
|C̃λ′ − C̃N (λ′)| ≤ lim inf
M→+∞
[
sup
λ′⊂λ
|C̃M (λ′)− C̃N (λ′)|
]
∀λ. (21)
The previous relation implies that
∑
k
( sup
λ′⊂λ
|C̃λ′ − C̃N (λ′)|)p ≤ lim inf
M→+∞
[∑
k
( sup
λ′⊂λ
|C̃M (λ′)− C̃N (λ′)|)p
]
∀j.
Hence sup
j
[
2sj
(∑
k
( sup
λ′⊂λ
|C̃λ′ − C̃N (λ′)|)p
)1/p
]
is smaller than
lim inf
M→+∞
sup
j
2sj
(∑
k
( sup
λ′⊂λ
|C̃M (λ′)− C̃N (λ′)|)p
)1/p
.
From properties (19) and (20), we deduce that ‖f̃N − f̃‖ ≤ 2−N if p ≥ 1, and L(L + 1)−N
if 0 < p < 1. Whence (f̃N )N converges to f̃ in Os,0
p .
The proof of Theorem 1 is now achieved.
Acknowledgments. The author is thankful to Stéphane Jaffard for suggesting the problem
studied in this paper. The author is grateful to Yves Meyer for his valuable help. The author
thanks the Department of Mathematics of Paris XII University, where a part of this work was
done, for its kind hospitality.
1. Meyer Y. Ondelettes et opérateurs. — Paris: Hermann, 1990.
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