Diametral dimension and property Omega Bar for spaces Snu

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Diametral dimension and property Ω for spaces Sν

Loïc Demeulenaere (FRIA-FNRS Grantee)

FNRS Group Functional Analysis: Han-sur-Lesse

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Introduction: spaces Sν _{and diametral dimension}

The concave case

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity}

The concave case

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Spaces S

ν

Context

Multifractal analysis: study of signals (Holderian regularity, spectrum of singularities, wavelet decomposition, ...).

Denition

An admissible prole is a map ν : R → {−∞} ∪ [0, 1], increasing, right-continuous and s.t. αmin:=inf{α ∈ R : ν(α) ≥ 0} is nite. Some notations

α_max:=inf{α ∈ R : ν(α) = 1} ∈ R ∪{∞},

Λ := ∪_{j ∈N}₀(j, k) : k ∈ {0, ..., 2j −1} ,

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Spaces S

ν

Context

Introduction of spaces Sν _{(S. Jaard, 2004)}_.

Denition

α_max:=inf{α ∈ R : ν(α) = 1} ∈ R ∪{∞},

Λ := ∪_{j ∈N}₀(j, k) : k ∈ {0, ..., 2j −1} ,

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Spaces S

ν

Context

Introduction of spaces Sν _{(S. Jaard, 2004)}_. Denition

An admissible prole is a map ν : R → {−∞} ∪ [0, 1], increasing, right-continuous and s.t. αmin:=inf{α ∈ R : ν(α) ≥ 0} is nite.

Λ := ∪_{j ∈N}₀(j, k) : k ∈ {0, ..., 2j −1} ,

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Spaces S

ν

Context

α_max:=inf{α ∈ R : ν(α) = 1} ∈ R ∪{∞},

Λ := ∪_{j ∈N}₀(j, k) : k ∈ {0, ..., 2j −1} ,

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Spaces S

ν

Context

α_max:=inf{α ∈ R : ν(α) = 1} ∈ R ∪{∞},

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Spaces S

ν

Context

α_max:=inf{α ∈ R : ν(α) = 1} ∈ R ∪{∞},

Λ := ∪_{j ∈N}₀(j, k) : k ∈ {0, ..., 2j −1} ,

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Denition and properties of spaces S

ν

Denition

The space Sν _{is the set of all ~c ∈ Ω s.t.}

∀α ∈ R, ∀ε > 0, ∀C > 0, ∃J ∈ N0 :

#{k : |cj ,k| ≥ C2−αj} ≤2(ν(α)+ε)j ∀j ≥ J.

Metric spaces: topological vector spaces (tvs), separable, complete, Schwartz, non-nuclear.

If p₀ :=min 1, inf α_min≤α<α_max∂ +_ν(α)

with ∂+_{ν(α) =}_{lim inf}

h→0+ ν(α+h)−ν(α)

h , then

I if p0>0, Sν is exactly locally p0-convex;

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Denition and properties of spaces S

ν

Denition

∀α ∈ R, ∀ε > 0, ∀C > 0, ∃J ∈ N0 :

#{k : |cj ,k| ≥ C2−αj} ≤2(ν(α)+ε)j ∀j ≥ J. Properties

h→0+ ν(α+h)−ν(α)

h , then

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Denition and properties of spaces S

Denition

∀α ∈ R, ∀ε > 0, ∀C > 0, ∃J ∈ N0 :

#{k : |cj ,k| ≥ C2−αj} ≤2(ν(α)+ε)j ∀j ≥ J. Properties

h→0+ ν(α+h)−ν(α)

h , then

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Open question:

Are the spaces Sν _{isomorphic (when same local convexity)?}

⇒ study of the diametral dimension... ⇒ study of the property Ω?

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Open question:

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Open question:

⇒ study of the diametral dimension... ⇒ study of the property Ω?

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Diametral dimension

Let E be a tvs and U be a basis of 0-neighbourhoods.

∆(E ) :=nξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ}

nδn(V , U) →0

o , with δn(V , U) :=inf {δ > 0 : ∃L ⊆ E, dim L ≤ n, s.t. V ⊆ δU + L}. Remark

∆is a topological invariant characterizing Schwartz and nuclear lcs.

Theorem(J.M. Aubry, F. Bastin, 2010) If Sν _{is locally p-convex, then}

∆ (Sν) =nξ ∈ CN0 _{: ∀s >}_{0, ξ}

n(n +1)−s →0

o .

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Diametral dimension

Denition

The diametral dimension of E is

∆(E ) :=nξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ}

nδn(V , U) →0

o , with δn(V , U) :=inf {δ > 0 : ∃L ⊆ E, dim L ≤ n, s.t. V ⊆ δU + L}.

Remark

∆ (Sν) =nξ ∈ CN0 _{: ∀s >}_{0, ξ}

n(n +1)−s →0

o .

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Diametral dimension

Denition

∆(E ) :=nξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ}

nδn(V , U) →0

∆ (Sν) =nξ ∈ CN0 _{: ∀s >}_{0, ξ}

n(n +1)−s →0

o .

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Diametral dimension

Denition

∆(E ) :=nξ ∈ CN0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ}

nδn(V , U) →0

∆ (Sν) =nξ ∈ CN0 _{: ∀s >}_{0, ξ}

n(n +1)−s →0

o .

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The concave case

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When ν is concave... it's nice! ,

Denition (Besov spaces)

For p > 0 and s ∈ R, b_p,∞s :=      ~ c ∈ Ω : k~ckbs p,∞ := sup j ∈N0   2 (s−1_p)j   2j₋₁ X k=0 |c_{j ,k}|p   1/p  < ∞      .

Proposition (J.M. Aubry, F. Bastin, S. Dispa, S. Jaard, 2006) If (pn)n∈N is a dense sequence of (0, ∞) and εm →0+, then

Sν = \ n∈N \ m∈N b η(pn) pn −εm pn,∞ ,

where η : p > 0 7→ infα≥α_min{αp − ν(α) +1}.

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When ν is concave... it's nice! ,

For p > 0 and s ∈ R, b_p,∞s :=      ~ c ∈ Ω : k~ckbs p,∞:= sup j ∈N0   2 (s−1_p)j   2j₋₁ X k=0 |c_{j ,k}|p   1/p  < ∞      . Sν = \ n∈N \ m∈N b η(pn) pn −εm pn,∞ ,

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When ν is concave... it's nice! ,

For p > 0 and s ∈ R, b_p,∞s :=      ~ c ∈ Ω : k~ckbs p,∞:= sup j ∈N0   2 (s−1_p)j   2j₋₁ X k=0 |c_{j ,k}|p   1/p  < ∞      .

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When ν is concave... it's nice! ,

For p > 0 and s ∈ R, b_p,∞s :=      ~ c ∈ Ω : k~ckbs p,∞:= sup j ∈N0   2 (s−1_p)j   2j₋₁ X k=0 |c_{j ,k}|p   1/p  < ∞      .

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When ν is concave... it's nice! ,

Topology dened by Pm(I )(~c) =sup i ∈I sup j ∈N0   2 _{η(pi )} pi −pi1−εm j   2j₋₁ X k=0 |cj ,k|pi   1/pi  

when I ⊆ N is nite and m ∈ N.

Numbers for weights j =0 (0, 0)

j =1 (1, 0) (1, 1)

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When ν is concave... it's nice! ,

Numbers for weights j =0 (0, 0)

j =1 (1, 0) (1, 1)

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When ν is concave... it's nice! ,

Topology dened by Pm(I )(~c) =sup i ∈I j ∈Nsup0   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi  

Numbers for weights

j =0 0

j =1 (1, 0) (1, 1)

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When ν is concave... it's nice! ,

Numbers for weights

j =0 0

j =1 1 (1, 1)

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When ν is concave... it's nice! ,

Numbers for weights

j =0 0

j =1 1 //2

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When ν is concave... it's nice! ,

Numbers for weights

j =0 0 j =1 1 //2 j =2 3 //4 //5 //6 2 −1 ≤n≤2 −2.

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When ν is concave... it's nice! ,

Numbers for weights

j =0 0

j =1 1 //2

j =2 3 //4 //5 //6

Scalej associated to the numbern: j (n)s.t.

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A rst property

Proposition If I ⊆ N is nite and if k0 ≥ m, δn B P(I ) k0 , B Pm(I ) ≤2(ε_k0−εm)j (n)_.

Reminder: δn(V , U) :=inf {δ > 0 : ∃L, dim L ≤ n, s.t. V ⊆ δU + L}.

Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)  2 _{η(pi )} pi −pi1−εm j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(εk0−εm)j (n)_. ⇒ ~c = ~c − πn(~c) + πn(~c) ∈2(εk0−εm)j (n)B_P(I ) m + πn(S ν₎_.

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A rst property

Reminder: δn(V , U) :=inf {δ > 0 : ∃L, dim L ≤ n, s.t. V ⊆ δU + L}. Proof: πn: Sν → Sν projection on the rst n −→ej ,k (from −→e0,0) and

~ c ∈ B P(I ) k0. Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)   2 _{η(pi )} pi −pi1−εm j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(εk0−εm)j (n)_. ⇒ ~c = ~c − πn(~c) + πn(~c) ∈2(εk0−εm)j (n)B_P(I ) m + πn(S ν₎_.

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A rst property

~ c ∈ B P(I ) k0. Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)   2 _{η(pi )} pi −pi1−εm j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(εk0−εm)j (n)_. ⇒ ~c = ~c − πn(~c) + πn(~c) ∈2(εk0−εm)j (n)B_P(I ) m + πn(S ν₎_.

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A rst property

~ c ∈ B P(I ) k0. Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)   2 (ε_k0−εm)j2 _{η(pi )} pi −pi1−εk0 j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(εk0−εm)j (n)_. ⇒ ~c = ~c − πn(~c) + πn(~c) ∈2(εk0−εm)j (n)B_P(I ) m + πn(S ν₎_.

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A rst property

~ c ∈ B P(I ) k0. Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)   2 (ε_k0−εm)j2 _{η(pi )} pi −pi1−εk0 j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(ε_k0−εm)j (n)_P(I ) k0(~c) ≤2 (ε_k0−εm)j (n)_.

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A rst property

~ c ∈ B P(I ) k0. Pm(I )(~c − πn(~c)) ≤sup i ∈I sup j ≥j (n)   2 (ε_k0−εm)j2 _{η(pi )} pi −pi1−εk0 j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2(ε_k0−εm)j (n)_P(I ) k0(~c) ≤2 (ε_k0−εm)j (n)_. ⇒ ~c = ~c − πn(~c) + πn(~c) ∈2(εk0−εm)j (n)B_P(I ) m + πn(S ν₎_.

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A rst property

Proposition If I ⊆ N is nite and if k0 ≥ m, δn B P(I ) k0 , B Pm(I ) ≤2(εk0−εm)j (n)_. Corollary n ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→_{0 if n → ∞}o_{⊆ ∆(S}ν_). Reminder: ∆(E ) :=n_{ξ ∈ C}N0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ} nδn(V , U) →0o. δn B_P(I ) k0 , B Pm(I ) ≤2(εk0−εm)j (n)_≤2εm−ε_k0_{(n +}2)ε_k0−εm | {z } becausen ≤ 2j (n)+1−₂ ≤2εm−ε_k0_{(n +}1)ε_k0−εm_.

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A rst property

Proposition If I ⊆ N is nite and if k0 ≥ m, δn B P(I ) k0 , B Pm(I ) ≤2(εk0−εm)j (n)_. Corollary n ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→_{0 if n → ∞}o_{⊆ ∆(S}ν_). Reminder: ∆(E ) :=n_{ξ ∈ C}N0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ} nδn(V , U) →0o. Proof: If k0 > m, δn B P(I ) k0 , B Pm(I ) ≤2(εk0−εm)j (n)_≤2εm−ε_k0_{(n +}2)ε_k0−εm | {z } becausen ≤ 2j (n)+1−₂ ≤2εm−ε_k0_{(n +}1)ε_k0−εm_.

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And for the other inclusion?

We need another assumption...

lim

p→0+

η(p)

p = αmax.

Reminders: η(p) = infα≥α_min{αp − ν(α) +1}, αmax=inf{α ∈ R : ν(α) = 1}.

Assumption: α_max< ∞

Examples

When Sν _{is locally p-convex, α}

max < ∞. 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0

Pseudoconvexity, with αmax < ∞ (blue) and αmax= ∞(orange).

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And for the other inclusion?

Lemma (L.D., 2017) lim

p→0+

η(p)

p = αmax.

Examples

max < ∞. 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0

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And for the other inclusion?

p→0+

η(p)

p = αmax.

When S is locally p-convex, αmax .

0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0

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And for the other inclusion?

p→0+

η(p)

p = αmax.

Examples

max< ∞. 0.5 1.0 1.5 2.0 0.2 0.4 0.6 0.8 1.0

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Construction of the I

ε

's

We x ε ∈ Q+_. 0 < p < pi0 ⇒ _p − i0 pi0 ≤ ε. NB: p > 0 7→ η(p) p =infα≥αmin n α +1−ν(α)_p o is decreasing and (pn)n∈N≡ Q+. 2. ∃` ∈ N0 s.t. `ε < _p1 i₀ ≤ (` +1)ε. For k = 0, ..., `, we dene ik ∈ N by 1 pik = 1 pi₀ − kε. 3. Iε:= {i0, ..., i`}.

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Construction of the I

ε

's

We x ε ∈ Q+_.

1. Because αmax < ∞, ∃i0 ∈ N s.t.

0 < p < pi0 ⇒ η(p) p − η(pi0) pi₀ ≤ ε. NB: p > 0 7→ η(p) p =infα≥αmin n α +1−ν(α)_p o is decreasing and (pn)n∈N≡ Q+. 2. ∃` ∈ N0 s.t. `ε < _p1 i₀ ≤ (` +1)ε. For k = 0, ..., `, we dene ik ∈ N by 1 pik = 1 pi₀ − kε. 3. Iε:= {i0, ..., i`}.

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Construction of the I

ε

's

We x ε ∈ Q+_.

0 < p < pi0 ⇒ η(p) p − η(pi0) pi₀ ≤ ε. NB: p > 0 7→ η(p) p =infα≥αmin n α +1−ν(α)_p o is decreasing and (pn)n∈N≡ Q+. 2. ∃` ∈ N0 s.t. `ε <_p1 i₀ ≤ (` +1)ε. For k = 0, ..., `, we dene ik ∈ N by 1 pik = 1 pi0 − kε.

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Construction of the I

ε

's

We x ε ∈ Q+_.

0 < p < pi0 ⇒ η(p) p − η(pi0) pi₀ ≤ ε. NB: p > 0 7→ η(p) p =infα≥αmin n α +1−ν(α)_p o is decreasing and (pn)n∈N≡ Q+. 2. ∃` ∈ N0 s.t. `ε <_p1 i₀ ≤ (` +1)ε. For k = 0, ..., `, we dene ik ∈ N by 1 pik = 1 pi0 − kε. 3. Iε:= {i0, ..., i`}.

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The main property of the I

's

Proposition (L.D., 2017) If m, n ∈ N and ~c ∈ Sν_, k~ck bpn,∞η(pn)/pn−εm ≤sup i ∈Iε sup j ∈N0   2 η(pi ) pi −pi1+ε−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi  . Proof: Admitted.

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Consequences

Proposition

Let m, k0∈ N be given, k0 ≥ m. If Iε⊆ J ⊆ N, J nite, then,

δn B_P(J) k0 , B_P(Iε) m ≥2(εk0−εm−ε)j(n)_.

Proof: Follows from:

Proposition (A. Pietsch, 1972)

Let (E, k.k) be a normed space, with closed unit ball U, and B be bounded. If ∃ P : E → E proj. with kPk ≤ 1, dim P(E) = n + 1, then ∃δ >0 : δU ∩ P(E) ⊆ B ⇒ δ_n(B, U) ≥ δ. Here: U = B Pm(Iε),B = BP (J) k0, P = πn+1 projection on the rst n + 1 −→ ej ,k andδ =2(εk0−εm−ε)j(n).

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Consequences

Proposition

Let (E, k.k) be a normed space, with closed unit ball U, and B be bounded. If ∃ P : E → E proj. with kPk ≤ 1, dim P(E) = n + 1, then

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Consequences

Proposition

Let (E, k.k) be a normed space, with closed unit ball U, and B be bounded. If ∃ P : E → E proj. with kPk ≤ 1, dim P(E) = n + 1, then

∃δ >0 : δU ∩ P(E) ⊆ B ⇒ δ_n(B, U) ≥ δ. Here: U = B

Pm(Iε),B = BP_k0(J),P = πn+1 projection on the rst n + 1 −→

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} Thesis 2(ε_k0−εm−ε)j(n)_B Pm(Iε) ∩ π_n+₁(Sν) ⊆ B P(J) k0 ⇔ P_k(J) 0 (~c) ≤2 (εm+ε−ε_k0)j (n)_P(Iε) m (~c) ∀~c ∈ πn+1(Sν). P_k(J) 0 (~c) ≤sup i ∈Iε sup j ∈N0  2 η(pi ) pi −pi1+ε−εk0 j  2−₁ X k=0 |c_{j ,k}|pi    ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} Thesis 2(ε_k0−εm−ε)j(n)_B Pm(Iε) ∩ π_n+₁(Sν) ⊆ B P(J) k0 ⇔ P_k(J) 0 (~c) ≤2 (εm+ε−ε_k0)j (n)_P(Iε) m (~c) ∀~c ∈ πn+1(Sν).

But, by last Proposition, if ~c ∈ πn+1(Sν)

P_k(J) 0 (~c) ≤sup i ∈Iε sup j ∈N0   2 η(pi ) pi −pi1+ε−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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Thesis 2(ε_k0−εm−ε)j(n)_B Pm(Iε) ∩ π_n+₁(Sν) ⊆ B P(J) k0 ⇔ P_k(J) 0 (~c) ≤2 (εm+ε−ε_k0)j (n)_P(Iε) m (~c) ∀~c ∈ πn+1(Sν).

P_k(J) 0 (~c) ≤sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1+ε−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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P_k(J) 0 (~c) ≤sup i ∈Iε sup j ≤j (n)   2( εm+ε−ε_k0)j2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1+εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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Thesis 2(ε_k0−εm−ε)j(n)_B Pm(Iε) ∩ π_n+₁(Sν) ⊆ B P(J) k0 ⇔ P_k(J) 0 (~c) ≤2 (εm+ε−ε_k0)j (n)_P(Iε) m (~c) ∀~c ∈ πn+1(Sν).

P_k(J) 0 (~c) ≤sup i ∈Iε sup j ≤j (n)   2( εm+ε−ε_k0)j2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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P_k(J) 0 (~c) ≤sup i ∈Iε sup j ≤j (n)   2( εm+ε−ε_k0)j2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2(εm+ε−ε_k0)j (n)sup i ∈Iε sup j ≤j (n)   2 η(pi ) pi −pi1−εm j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   =2(εm+ε−ε_k0)j (n)_P(Iε) m (~c).

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Diametral dimension and spaces S

ν Theorem(L.D., 2017)

If ν is concave and αmax< ∞,

∆ (Sν) = n ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→₀o_. Reminder: ∆(E ) :=n_{ξ ∈ C}N0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ} nδn(V , U) →0o.

By def., ∃k0≥ m and Iε⊆ J ⊆ N, J nite, s.t.

ξnδn B P(J) k0 , B Pm(Iε) →0. But, by last Proposition,

δn B P(J) k0 , B Pm(Iε) ≥2(εk0−εm−ε)j(n)_.

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Diametral dimension and spaces S

∆ (Sν) = n ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→₀o_. Reminder: ∆(E ) :=n_{ξ ∈ C}N0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ} nδn(V , U) →0o. Proof: ⊇ OK! ⊆ ξ ∈ ∆(Sν₎_{, s > 0. We take m ∈ N s.t. ε} m≤ s/2 and ε := εm.

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Diametral dimension and spaces S

∆ (Sν) = n ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→₀o_. Reminder: ∆(E ) :=n_{ξ ∈ C}N0 _{: ∀U ∈ U , ∃V ∈ U , V ⊆ U,} _{s.t. ξ} nδn(V , U) →0o. Proof: ⊇ OK! ⊆ ξ ∈ ∆(Sν₎_{, s > 0.} ξnδn B P(J) k0 , B Pm(Iε) →0. But, by last Proposition,

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Diametral dimension and spaces S

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Diametral dimension and spaces S

ξnδn B P(J) k0 , B Pm(Iε) →0. δn B_P(J) k0 , B Pm(Iε) ≥2 .

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Diametral dimension and spaces S

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Diametral dimension and spaces S

Theorem(L.D., 2017)

δn B P(J) k0 , B Pm(Iε) ≥2(εk0−2εm)j (n)_.

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Diametral dimension and spaces S

δn B P(J) k0 , B Pm(Iε) ≥2−2εmj (n)_.

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Diametral dimension and spaces S

Theorem(L.D., 2017)

δn B P(J) k0 , B Pm(Iε) ≥2−sj (n).

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Diametral dimension and spaces S

δn B P(J) k0 , B Pm(Iε) ≥2−sj (n)≥ (n +1)−s (2j (n)−1 ≤ n).

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Property Ω

Denition

A Fréchet space E, with a fundamental system of semi-norms (k.kn)n∈N, has the property Ω if

∀m ∃k ∀j ∃C >0 : kx0k∗_k2

≤ C kx0k∗_mkx0k∗_j ∀x0 ∈ E0 where k.k∗

m is the dual norm of k.km.

the property Ω i

∀m ∃k ∀j ∃C >0 : Uk ⊆ rUj +

C

r Um ∀r >0. ; Property (Ωid)

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Property Ω

Denition

A Fréchet space E, with a fundamental system of semi-norms (k.kn)n∈N, has the property Ω if

∀m ∃k ∀j ∃C >0 : kx0k∗_k2

≤ C kx0k∗_mkx0k∗_j ∀x0 ∈ E0 where k.k∗

m is the dual norm of k.km. Characterization

A Fréchet space E, with a basis of 0-neighbourhoods (Un)n∈N, has

the property Ω i

∀m ∃k ∀j ∃C >0 : Uk ⊆ rUj +

C

r Um ∀r >0. ; Property (Ωid)

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Property Ω and spaces S

If ν is concave and αmax< ∞, then Sν has the property (Ω_id).

ThesisIf j0 ∈ N and Ij₀ ⊆ N, Ij₀ nite, then

B P(Ik0) k0 ⊆ rB P(Ij0) j0 + 1 rBPm(Im) ∀r >0. 1. If r ≤ 1, B P(Ik0) k0 ⊆ B Pm(Im) ⊆ 1_rB Pm(Im) ⊆ rB P(Ij0) j0 +1_rB Pm(Im).

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Property Ω and spaces S

Proof: We x m ∈ N and Im ⊆ N nite.

∃k₀ s.t. ε_k₀ < εm/2: we put Ik0 := Iε∪ Im, with ε := εm/2 − εk0. ThesisIf j0 ∈ N and Ij₀ ⊆ N, Ij₀ nite, then

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Property Ω and spaces S

∃k₀ s.t. ε_k₀ < εm/2: we put Ik0 := Iε∪ Im, with ε := εm/2 − εk0. B P(Ik0) k0 ⊆ rB P(Ij0) j0 + 1 rBPm(Im) ∀r >0. 1. If r ≤ 1, B P(Ik0) k0 ⊆ B Pm(Im) ⊆ 1_rB Pm(Im) ⊆ rB P(Ij0) j0 +1_rB Pm(Im).

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Property Ω and spaces S

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Property Ω and spaces S

Theorem(L.D., 2017)

B P(Ik0) k0 ⊆ rB P(Ij0) j0 + 1 rBPm(Im) ∀r >0. 1. If r ≤ 1, B P(Ik0) k0 ⊆ B Pm(Im) ⊆ 1 rBPm(Im)⊆ rB_P(Ij0) j0 +1_rB Pm(Im).

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} 2. If r ≥ 1.∃J ∈ N0 s.t. 2εm2 J ≤ r ≤2εm2 (J+1). For ~c ∈ B P(Ik0) k0 , ~ c₁ := J X j =0 2j₋₁ X k=0 cj ,k−→ej ,k and c~2:= ∞ X j =J+1 2j₋₁ X k=0 cj ,k−→ej ,k.

So, since Iε⊆ Ik0 and ε = εm/2 − εk0,

P_j(Ij0) 0 ( ~c1) ≤sup_{i ∈I} ε sup j ≤J   2 η(pi ) pi −pi1+ε−εj0 j   2j₋₁ X k=0 |cj ,k|pi   1/pi   ≤2εm2 JP_k(Ik0) 0 (~c) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} 2. If r ≥ 1.∃J ∈ N0 s.t. 2εm2 J ≤ r ≤2εm2 (J+1). For ~c ∈ B P(Ik0) k0 , ~ c₁ := J X j =0 2j₋₁ X k=0 cj ,k−→ej ,k and c~2:= ∞ X j =J+1 2j₋₁ X k=0 cj ,k−→ej ,k. P_j(Ij0) 0 ( ~c1) ≤sup_{i ∈I} ε sup j ≤J  2 η(pi ) pi −pi1+ε−εj0 j  2−1 X k=0 |cj ,k|pi   ≤2εm2 JP_k(Ik0) 0 (~c) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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P_j(Ij0) 0 ( ~c1) ≤sup_{i ∈I} ε sup j ≤J   2 η(pi ) pi −pi1+ε−εj0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2εm2 J_P(Ik0) k0 (~c) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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For ~c ∈ B P(Ik0) k0 , ~ c₁ := J X j =0 2j₋₁ X k=0 cj ,k−→ej ,k and c~2:= ∞ X j =J+1 2j₋₁ X k=0 cj ,k−→ej ,k.

P_j(Ij0) 0 ( ~c1) ≤_{i ∈}sup_I k0 sup j ≤J   2 η(pi ) pi −pi1+ε−εj0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2εm2 J_P(Ik0) k0 (~c) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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P_j(Ij0) 0 ( ~c1) ≤_{i ∈}sup_I k0 sup j ≤J   2 (ε+ε_k0−ε_j0)j₂ η(pi ) pi −pi1−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2εm2 J_P(Ik0) k0 (~c) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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P_j(Ij0) 0 ( ~c1) ≤_{i ∈}sup_I k0 sup j ≤J   2 (ε+ε_k0−ε_j0)j₂ η(pi ) pi −pi1−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2εm2 J_P(Ik0) k0 (~c) (ε + εk0− εj₀ ≤ εm/2) ≤ r , so ~c1∈ rB P(Ij0) j0 .

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} Because Im⊆ Ik₀ and εk₀ < εm/2, P(Im) m ( ~c2) ≤ sup i ∈I_k0j ≥J+sup1   2 (ε_k0−εm)j2 η(pi ) pi −pi1−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2−εm2 (J+1)P_k(Ik0) 0 (~c) ≤ 1 r (2 εm 2 J ≤ r ≤2εm2 (J+1)), so ~c2∈ 1rBPm(Im). Finally,~c = ~c₁+ ~c₂ ∈ rB P(Ij0) j0 +1_rB Pm(Im).

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} Because Im⊆ Ik₀ and εk₀ < εm/2, P(Im) m ( ~c2) ≤ sup i ∈I_k0j ≥J+sup1   2 (ε_k0−εm)j2 η(pi ) pi −pi1−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2−εm2 (J+1)P_k(Ik0) 0 (~c) ≤ 1 r (2 εm 2 J ≤ r ≤2εm2 (J+1)), so ~c2∈ 1rBPm(Im).

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Introduction: spaces Sν_{and diametral dimension} _{The concave case} _{Local p-convexity} Because Im⊆ Ik₀ and εk₀ < εm/2, P(Im) m ( ~c2) ≤ sup i ∈I_k0j ≥J+sup1   2 (ε_k0−εm)j2 η(pi ) pi −pi1−εk0 j   2j₋₁ X k=0 |c_{j ,k}|pi   1/pi   ≤2−εm2 (J+1)P_k(Ik0) 0 (~c) ≤ 1 r (2 εm 2 J ≤ r ≤2εm2 (J+1)), so ~c2∈ 1rBPm(Im). Finally,~c = ~c₁+ ~c₂∈ rB P(Ij0) j0 +1_rB Pm(Im).

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The concave case

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When p

0

>

0...

Reminder: Sν_{locally p}

0-convex i p0=min 1, infα_min≤α<α_max∂+ν(α) >0. Denition For ~c ∈ Sν_{, α, s ∈ R,} k~ckα,s :=inf n k~c0_k bs p0,∞+ k ~c 00_k bα ∞,∞: ~c = ~c 0_{+ ~}_c00o where k~c0_k bs p0,∞ =_{j ∈N}sup 0   2 (s−1 p0)j   2j₋₁ X k=0 |cj ,k0 | p₀   1/p0  , k ~c00_k bα ∞,∞=sup j ∈N0 sup 0≤k≤2j−₁ 2αj_|c00 j ,k| .

Proposition (J.M. Aubry, F. Bastin, 2010)

The topology of Sν _{is dened by the pseudonorms} |||~c|||A,ε:=sup

α∈A

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When p

0

0...

Reminder: Sν_{locally p}

0-convex i p0=min 1, infα_min≤α<α_max∂+ν(α) >0. Denition For ~c ∈ Sν_{, α, s ∈ R,} k~ckα,s :=inf n k~c0_k bs p0,∞+ k ~c 00_k bα ∞,∞: ~c = ~c 0_{+ ~}_c00o where k~c0_k bs p0,∞ =_{j ∈N}sup 0   2 (s−1 p0)j   2j₋₁ X k=0 |cj ,k0 | p₀   1/p0  , k ~c00_k bα ∞,∞=sup j ∈N0 sup 0≤k≤2j−₁ 2αj_|c00 j ,k| .

Proposition (J.M. Aubry, F. Bastin, 2010)

The topology of Sν _{is dened by the pseudonorms} |||~c|||A,ε:=sup

α∈A

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Diametral dimension and property Ω

Key ideas:

Denition of a set Aε₀ for any ε0>0 (≡ Iε) Gives a proof for

∆ (Sν) =ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→_0 . If p0>0, Sν has (Ω_id) (L.D., 2017).

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Diametral dimension and property Ω

Key ideas:

Denition of a set Aε₀ for any ε0>0 (≡ Iε)

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Diametral dimension and property Ω

Key ideas:

Denition of a set Aε₀ for any ε0>0 (≡ Iε) Gives a proof for

∆ (Sν) =ξ ∈ CN0 _{: ∀s >}_{0, ξ}_n_{(n +}₁₎−s _→_0 . If p0>0, Sν has (Ω_id) (L.D., 2017).

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References I

J.-M. Aubry and F. Bastin.

Advanced topology on the multiscale sequence space Sν_.

J. Math. Anal. Appl., 350:439454, 2009.

J.-M. Aubry and F. Bastin.

Diametral dimension of some pseudoconvex multiscale spaces. Studia Math., 197(1):2742, 2010.

J.-M. Aubry, F. Bastin, S. Dispa, and S. Jaard.

Topological properties of the sequence spaces Sν_.

J. Math. Anal. Appl., 321:364387, 2006.

F. Bastin and L. Demeulenaere.

On the equality between two diametral dimensions. Functiones et Approximatio, Commentarii Mathematici, 56(1):95107, 2017.

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References II

L. Demeulenaere.

Dimension diamétrale, espaces de suites, propriétés (DN) et (Ω). Master's thesis, University of Liège, 2014.

L. Demeulenaere.

Spaces Sν_{, diametral dimension and property Ω.}

J. Math. Anal. Appl., 449(2):13401350, 2017.

L. Demeulenaere, L. Frerick, and J. Wengenroth.

Diametral dimensions of Fréchet spaces. Studia Math., 234(3):271280, 2016.

C. Esser.

Les espaces de suites Sν _{: propriétés topologiques, localement}

convexes et de prévalence.

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References III

S. Jaard.

Beyond Besov spaces, Part I : Distribution of wavelet coecients.

J. Fourier Anal. Appl., 10(3):221246, 2004.

H. Jarchow.

Locally Convex Spaces.

Mathematische Leitfäden. B.G. Teubner, Stuttgart, 1981.

A. Pietsch.

Nuclear Locally Convex Spaces.

Ergebnisse der Mathematik und ihrer Grenzgebiete. Band 66. Springer-Verlag, Berlin, 1972.

Diametral dimension and property Omega Bar for spaces Snu

Spaces S

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Diametral dimension and spaces S

Diametral dimension and spaces S

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Diametral dimension and property Ω

Diametral dimension and property Ω

References I

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Denition and properties of spaces S

Denition and properties of spaces S

Denition and properties of spaces S

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