Sparse approximation in (vector-valued) sequence spaces

Let us now discuss specific situations. The following lemma is well-known and usually referred as Stechkin’s lemma. For our knowledge the first reference for the stated generality is [112, Lemma 2.1, p. 97], see also [33, Section 7.4] and the references given there.

Lemma 6.2. Let 0 < p < q ≤ ∞ and (a_n)_n∈N ⊂ C be a sequence of complex numbers with the property

|a₁| ≥ |a₂| ≥ |a₃| ≥ . . . Then

 X^∞

n=m+1

|a_n|^q¹_q

≤ (m + 1)^−(1p−1q)X^∞

n=1

|a_n|^p¹_p

(6.2.1) holds for all m ∈ N0. As usual, for q = ∞ the sum on the left hand side is replaced by a supremum.

Let us now turn to the vector-valued situation. Here we have X_µ, Y_µ quasi-Banach (p-Banach) spaces, T_µ ∈ L(X_µ, Y_µ) linear operators and I be an index set. Let D_µ denote a dictionary in Y_µ and

D := [

µ∈I

[

eµ∈Dµ

{(0, . . . , 0, e_µ, 0, . . . , 0)}.

Definition 6.3. Let I be an index set and (Xµ)µ∈I be a sequence of quasi-Banach (q-Banach) spaces. We define the following sequence space

`_p(X_µ, I) = n

x = (x_µ)_µ∈I : x_µ∈ X_µ, kxk_`p(Xµ,I) = X

µ∈I

kx_µ|X_µk^p¹_p

< ∞o

with the usual modifications in case p = ∞.

Theorem 6.4. Let 0 < p < q ≤ ∞ and T = (T_µ)_µ∈I : `<sub>p</sub>(X<sub>µ</sub>, I) → `_q(Y_µ, I). Then σ_m(T, D) ≤ sup

µ∈I

sup

0≤s≤m

s + 1 m + 1

¹_p⁻¹_q

σ_s(T_µ, D_µ).

Proof. We have T = (T_µ)_µ∈I with T_µ : X_µ → Y_µ. Let m be given. Define m_µ = b(m+1)kx_µ|X_µk^pc for some x = (x_µ)_µ∈I with kx|`_p(X_µ)k < 1. Using m_µ-approximation in the component T_µx_µ we obtain the relation

σ_m(T x, D)_`q(Yµ)≤ X

µ∈I

σ_mµ(T_µx_µ, D_µ)^q_Yµ¹_q , since

X

µ∈I

m_µ≤ (m + 1)X

µ∈I

kx_µ|X_µk^p < m + 1.

We proceed estimating as follows additionally λ > 1. We use a limiting argument together with (6.2.2). Obviously

^x_λ|X

< 1. For that reason we obtain σ_m(T x, D)_`q(Yµ) = λσ_m This holds for all λ > 1 arbitrary close to 1. We obtain

σ_m(T x, D)_`q(Yµ) ≤ sup

Finally, taking the supremum over kx|`_p(X_µ)k ≤ 1 on both sides proves the theorem.

This theorem has some consequences. We consider some special cases and start with Lemma 6.2. Choosing Xµ = Yµ = C we can prove a similar result having the same convergence rate as in (6.2.1) immediately by applying Theorem 6.4. This gives us a slightly different selection procedure for the m terms in the m-term approximation.

Corollary 6.5. Let 0 < p < q ≤ ∞ and

Proof. Choosing X_µ = Y_µ = C we can prove the upper bound in (6.2.1) immediately by applying Theorem 6.4. Let a ∈ `p with ka|`pk < 1. ThenP

i∈A(m,a)aiei is a m-term approximation of a, since

|A(m, a)| ≤

The arguments provided in (6.2.4) yield the case ka|`_pk = 1.

This gives (6.2.5).

Remark 6.6. The case ka|`<sub>p</sub>k = 1 is based on a limiting argument. The above algo-rithm may not work in case ka|`_pk = 1. Replacing the definition of m_i by m_i = |a_i|^pm in 6.5 and accepting a constant C ≥ 1 in the approximation rates then we obtain an explicit algorithm that generates a m-term approximation for the case ka|`_pk = 1. Sim-ilarly m_µ in Theorem 6.4 can be replaced by m_µ = kx_µ|X_µk^pm. This gives us a more transparent approximation strategy. The price to pay is constant C ≥ 1.

Next we consider T = (T, . . . , T ) with T = id, X_µ = `<sup>d</sup><sub>u</sub>, Y<sub>µ</sub> = `^d_r, so X_µ = X, Y_µ = Y independent of µ, u < r. The next corollary generalizes [54, Theorem 4].

Corollary 6.7. Let u < r, p < q and 0 < ¹_u−¹_r ≤ ¹_p−¹_q. Let further T = (T₀, . . . , T₀) =

Proof. The case m ≥ bd is trivial, since dim `<sup>b</sup><sub>p</sub>(`^d_u) = bd. We consider the remaining cases. We apply Theorem 6.4. This gives

σ_m(T, D) ≤ sup

i=1,...,b

sup

0≤s≤m

s + 1 m + 1

¹_p−¹_q

σ_s(id : `<sup>d</sup><sub>u</sub> → `^d_r, D).

By Lemma 6.4 we have

σ_s(id : `<sup>d</sup><sub>u</sub> → `^d_r, D) ≤ (s + 1)^−(1u−1r) if 1 ≤ s ≤ d and

σ_s(id : `<sup>d</sup><sub>u</sub> → `^d_r, D) = 0, otherwise. Hence we have for m ≤ d

σ_m(T, D) ≤ sup

0≤s≤m

s + 1 m + 1

_p¹⁻¹_q

(s + 1)⁻



1 u−¹_r



= (m + 1)^−(1p−1q)(m + 1)^{(1p−1q)−(u1−1r)}

= (m + 1)^−(1u−1r). In case d ≤ m ≤ db we have

σ_m(T, D) ≤ sup

0≤s≤d−1

s + 1 m + 1

¹_p−¹_q

(d + 1)^−(1u−1r)

= d + 1 m + 1

¹_p−¹_q

(d + 1)^−(1u−1r). This gives the corollary.

Now we consider a more complicated situation. In a sense this represents a vector valued framework suitable for function space embeddings. Let

X_µ = `<sup>b</sup><sub>p</sub><sup>µ</sup>(d<sup>α</sup><sub>µ</sub>`^d_u^µ),

Y_µ = `<sup>b</sup><sub>q</sub><sup>µ</sup>(d<sup>β</sup><sub>µ</sub>`^d_r^µ), (6.2.6) where d_µ, b_µ are natural numbers which are growing with µ and α, β ≥ 0. Let us now study

σ_m(id : `<sub>p</sub>(X<sub>µ</sub>) → `_q(Y_µ), D), where

kxk_`p(Xµ)= X

µ∈I

kx_µ|X_µk^p¹_p ,

kyk_`p(Yµ)= X

µ∈I

ky_µ|Y_µk^q¹_q .

We define D as the set of unit vectors in `_p(X_µ). We are interested in the following Inserting the result from Corollary 6.7 the first term can be estimated by

sup

Taking the supremum with respect to µ we obtain sup Finally estimating the second term in (6.2.9) gives

sup

Corollary 6.9. Let 0 ≤ α − β with _u¹ − ¹_r ≤ ¹_p − ¹_q. Then we have

σ_m(id : `<sub>p</sub>(X<sub>µ</sub>, I) → `_q(Y_µ, I), D) ≤ 1 m + 1

(α−β)+(_u¹−¹_r)

if m + 1 ≤ inf_µ∈Id_µ. I ⊂ N^d0 denotes the index set of the outer sequence spaces.

Proof. We fix µ ∈ I. Due to m + 1 < d_µ Corollary 6.7 gives sup

0≤s≤m

 s + 1 m + 1

¹_p⁻¹_q 1 s + 1

¹_u⁻¹_r

d^−(α−β)_µ ≤ (m + 1)^−(u1−1r)d^−(α−β)_µ ≤ 1 m + 1

(α−β)+(¹_u−¹

r)

.

Corollary 6.10. Let p, q, u, r as in (6.2.7), (6.2.8) and X_µ, Y_µ as in (6.2.6). If α−β ≥ 0 we have

σm(id : `p(Xµ) → `q(Yµ), D) .

1 m

(α−β)+(_u¹−¹

r)

for all m ∈ N.

Proof. The proof follows immediately by Corollary 6.8 and 6.9 using for u = min{q, 1}

the decomposition

σ_2m(id : `<sub>p</sub>(X<sub>µ</sub>) → `_q(Y_µ), D)^q ≤ σ_m(id : `<sub>p</sub>(X<sub>µ</sub>, I<sub>1</sub>) → `_q(Y_µ, I₁), D)^q +σ_m(id : `<sub>p</sub>(X<sub>µ</sub>, I<sub>2</sub>) → `_q(Y_µ, I₂), D)^q where

I₁ := {µ ∈ N^d0 : d_µ≤ m + 1} and I₂ := {µ ∈ N^d0 : d_µ > m + 1}.

Next we consider best m-term approximation for discrete function spaces s^r,Ω_p,θf and s^r,Ω_p,θb with Ω = [0, 1]^d. We need some further notation. We introduce for µ ∈ N0 the following sets and quantities

M (µ, d) := n

j ∈ N^d−1 :

d

X

i=1

max{j_i, 0} = µo

, (6.2.10)

S(µ, d) = |M (µ, d)|,

∇_µ := {(j, k) : j ∈ M (µ, d), k ∈ D_j}, N (µ, d) := |∇_µ|.

Definition 6.11. We define the projection of the sequence a := (a_j,k)_(j,k)∈∇ to indices of the hyperbolic cross layer M (µ, d) by

R_µa := X

j∈M (µ,d)

X

k∈D_j

a_j,ke_j,k,

where e_j,k are the unit vectors with index (j, k). Such a projection fulfills the following properties.

Lemma 6.12. Let x ∈ {b, f }, 0 < p < q < ∞ (q = ∞ : x = b, ), 0 < θ < ν ≤ ∞ and t ≤ r. Then the following inequalities hold

(i)

kR_µa|s^r,Ω_p,θxk ≤ S(µ, d)^θ1−1νkR_µa|s^r,Ω_p,νxk, (ii)

kR_µa|s^r,Ω_p,θxk ≤ kR_µa|s^r,Ω_q,θxk, (iii)

kR_µa|s^r,Ω_p,θxk ≤ ka|s^r,Ω_p,θxk.

(iv) Additionally the identity

id : s^r,Ω_p,θx → s^t,Ω_p,θx =

∞

X

µ=0

R_µ holds.

Proof. (i) and (ii) can be proven using H¨older’s inequality. The estimates in (iii) and (iv) are obvious.

The case of large smoothness

The following result is due to Hansen and Sickel [56, Proposition 5.4]. We provide an alternative proof using pseudo s-number properties. D denotes the set of unit vectors in sequence spaces s^r,Ω_p,θf .

Theorem 6.13. Let x, y ∈ {b, f }, 0 < p, q ≤ ∞ (p = ∞ : x = b, q = ∞ : y = b) and 0 < θ, ν ≤ ∞. We denote by γ₀ = min{p, θ} and δ₁ = max{ν, q}. Further let r, t ≥ 0 with r − t > maxn

0,_γ¹

0 − _δ¹

1

o then σ_m(s^r,Ω_p,θx, D)_st,Ω

q,νy  C(m⁻¹log^d−1m)^r−t(log^d−1m)^1ν−1θ. (6.2.11) Proof. For the lower bound we refer to [56]. We give a new proof for the upper bound.

We start with the case γ₀ < δ₁. We denote by D_µ the set of unit vectors in s^r,Ω_p,θf restricted to the hyperbolic layer with |j|1 = µ. Let a ∈ s^r_p,θx with ka|s^r_p,θxk ≤ 1 then Lemma 6.1, (ii) with u := min{q, ν, 1} provides the decomposition

σ_m(a, D)^u

s^t,Ωq,νy ≤

M

X

µ=0

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy+

L

X

µ=M +1

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy (6.2.12) +

∞

X

µ=L+1

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy,

(6.2.13)

where

m_µ 







2^µµ^d−1 : 0 ≤ µ ≤ M, b2^µ2^{(M −µ)κ}µ^d−1c : M + 1 ≤ µ ≤ L,

0 : otherwise,

with κ > 1,

m  2^MM^d−1 and

L =lM κ + (d − 1) log M − 1 κ − 1

m . First show that m &PL

µ=0m_µ. Obviously

∞

X

µ=0

m_µ .

M

X

µ=0

µ^d−12^µ+ 2^{M κ}

L

X

µ=M +1

2^(1−κ)µµ^d−1. M^d−12^M  m.

The first sum in (6.2.13) vanishes, since |∇_µ|  µ^d−12^µ. So this part can be approxi-mated exactly. We continue dealing with the second sum. Applying Lemma 6.12, (i) gives

σ_mµ(R_µa, D_µ)_s^t,Ω

q,νy ≤ S(µ, d)^ν1−δ11 σ_mµ(R_µa, D_µ)_s^t

δ1,δ1b

. S(µ, d)^ν1−δ11 σ_mµ(R_µ : s^r,Ω_γ

0,γ0b → s^t,Ω_δ

1,δ1b, D_µ)kR_µa|s^r,Ω_γ

0,γ0xk

 S(µ, d)^ν1−δ11 2^{−µ(r−t−(γ01−δ11))}σmµ(Rµ : `<sup>µ</sup><sub>γ0</sub><sup>d−12µ</sup> → `^µ_δ1^d−12µ, Dµ) kR_µa|s^r,Ω_γ0,γ0xk.

Corollary 6.5 yields σ_mµ(R_µa, D_µ)_s^t,Ω

q,νy . S(µ, d)^1ν−δ11 2^{−µ(r−t−(γ01 −δ11))}m⁻⁽

1 γ0−¹

δ1)

µ kR_µa|s^r,Ω_γ0,γ0xk.

Lemma 6.12 allows to estimate this by σmµ(Rµa, Dµ)_s^t,Ω

q,νy . S(µ, d)^{1ν−δ11+γ01−1θ}2^{−µ(r−t−(γ01−δ11))}m⁻⁽

1 γ0−¹

δ1)

µ kRµa|s^r,Ω_p,θxk.

Choosing κ > 1 close to 1 such that κ(_γ¹

0 −¹_{δ 1}) < r − t then summing up yields

L

X

µ=M +1

σ_mµ(R_µa, D_µ)_s^t,Ω

q,νy . 2^{−M κ(γ01 −δ11)u}

L

X

µ=M +1

µ^{(d−1)(1ν−1θ)u}2^{µ(κ(γ01−δ11)−(r−t))u} . 2^{−M (r−t)u}M^{(d−1)(1ν−1θ)u}.

We estimate the last sum in (6.2.13). The choice of L yields m_µ = 0 for µ > L. Lemma 6.1 together with 6.12 gives

σ_mµ(R_µa, D_µ)_s^t,Ω

q,νy . kRµa|s^t,Ω_q,νyk ≤ µ^{(d−1)(ν1−1q)+}kR_µa|s^t,Ω_q,qbk . µ^{(d−1)(1ν−1q)+}2^{−(r−t−1p+1q))µ}kR_µa|s^r,Ω_p,pbk

. µ^{(d−1)[(ν1−1q)++(1p−1θ)+]}2^{−(r−t−1p+1q))µ}kR_µa|s^r,Ω_p,θxk.

Summing up with L > M yields

∞

X

µ=L+1

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy .

∞

X

µ=L+1

µ^{(d−1)[(1ν−1q)++(1p−θ1)+]u}2^{−(r−1p+1q))µu} . 2^{−(r−(1p−1q))Lu}L^{(d−1)[(1ν−1q)++(1p−1θ)+]u}. Finally, if κ is choosen close enough to 1 then L is sufficent large such that

∞

X

µ=L+1

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy . 2^{−(r−(1p−1q))Lu}L^{(d−1)[(1ν−1q)++(p1−1θ)+]u} . 2^{−M ru}M^{(d−1)(ν1−1θ)u} holds. Altogether, we obtain

σ_m(a, D)_s^t,Ω

q,νy . 2^{−M (r−t)}M^{(d−1)(ν1−1θ)}  (m⁻¹log^d−1m)^r−t(log^d−1m)^1ν−1θ. Finally we consider the case δ₁ < γ₀. Here we use (linear) hyperbolic cross approxima-tion. Again we choose M such that

m  2^MM^d−1 and

m_µ 

(µ^d−12^µ : µ ≤ M, 0 : otherwise.

Obviously

∞

X

µ=0

mµ . 2^MM^d−1. Lemma 6.12 yields

kR_µa|s^t,Ω_q,νyk . µ^{(d−1)(1ν−δ11)}kR_µa|s^t,Ω_δ

1,δ1bk

= µ^{(d−1)(1ν−δ11)}2^µ(t−δ11)kR_µa|`^µ_δ^d−12µ

1 k

. µ^{(d−1)(1ν−δ11)}2^µ(t−δ11)(2^µµ^d−1)^δ11−γ01 kR_µa|`^µ_γ^d−12µ

0 k

= µ^{(d−1)(1ν−γ01)}2^−µ(r−t)kR_µa|s^r,Ω_γ0,γ0bk . µ^{(d−1)(1ν−1θ)}2^−µ(r−t)kR_µa|s^r,Ω_p,θxk

Inserting this into the following estimate gives

σ_m(a, D)^u

s^t,Ωq,νy ≤

M

X

µ=0

σ_mµ(R_µa, D_µ)^u

s^t,Ωq,νy

| {z }

=0

+

∞

X

µ=M +1

σ₀(R_µa, D_µ)^u

s^t,Ωq,νy

| {z }

=kRµa|s^t,Ωq,νyk^u

.

∞

X

µ=M +1

µ^{(d−1)(1ν−1θ)}2^−µ(r−t)kR_µa|s^r,Ω_p,θxk . M^{(d−1)(1ν−θ1)}2^{−M (r−t)}

 (m⁻¹log^d−1m)^r−t(log^d−1m)^ν1−1θ. This concludes the proof.

Remark 6.14. Compared to [56] the analysis here uses ideas known from the Maiorov discretization technique, cf. [75], which is very well known for estimates on several s-numbers of classical function space embeddings (F and B spaces).The choice of para-meters is borrowed from [127, Theorem 3.19] where entropy numbers have been studied.

The case of small smoothness

In this section we consider the so called case of small smoothness. The small smoothness range is given by ¹_p−¹_q ≤ r ≤ ¹_θ−¹_ν. Here we recover some interesting effects concerning the logarithm. The next result originally goes back to [56]. In fact, it was obtained in a non-constructive way using interpolation theory. We contribute a constructive approximation method.

Theorem 6.15. Let 0 < p < q ≤ ∞, 0 < θ < ν ≤ ∞ and ¹_p−¹_q ≤ r − t ≤ ¹_θ−¹_ν. Then σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νb, D)  m^−(r−t). (6.2.14) Proof. For the lower bound we refer to [56, Corollary 5.11]. We prove the upper bound with a constructive method in case of the compact embedding with 0 < ε <

r − t − (¹_p−¹_q). For the non-compact embedding r − t = ¹_p−¹_q we refer to the comments in Remark 6.19. We set

L =l (r − t) log m r − t −¹_p +¹_q − ε

m

. (6.2.15)

Defining u := min{q, ν, 1} then Lemma 6.1, (i) together with Lemma 6.12, (iv) yields

σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νb, D)^u ≤ σ_mX^L

µ=0

R_µ: s^r,Ω_p,θb → s^t,Ω_q,νb, Du

+

∞

X

µ=L+1

σ₀(R_µ : s^r,Ω_p,θb → s^t,Ω_q,νb, D_µ)^u.

(6.2.16)

We consider the first term in (6.2.16). By Corollary 6.10 we have

σ_mµX^L

µ=0

R_µ, D

s^t,Ωq,νb

 σ_m(id : `<sub>θ</sub>(X<sub>µ</sub>) → `_ν(X_µ)) . m^−(r−t) (6.2.17)

where

Xµ= `<sup>µ</sup><sub>θ</sub><sup>d−1</sup>(2<sup>µ(r−p1)</sup>`²_p^µ) and Yµ= `<sup>µ</sup><sub>ν</sub><sup>d−1</sup>(2<sup>µ(t−1q)</sup>`²_q^µ).

Finally we deal with the last sum in (6.2.16). Let a ∈ s^r,Ω_p,θb with ka|s^r,Ω_p,θbk ≤ 1, then applying Lemma 6.1, (i) together with Lemma 6.12 gives

σ₀(R_µa, D_µ)_s^t,Ω

q,νb = kR_µa|s^t,Ω_q,νbk ≤ µ^{(d−1)(1θ−1ν)}kR_µa|s^t,Ω_q,θbk . µ^{(d−1)(1θ−ν1)}2^{−(r−t−1p+1q)µ}kRµa|s^r,Ω_p,θbk . µ^{(d−1)(1θ−ν1)}2^{−(r−t−1p+1q)µ}.

Taking supremum over ka|s^r,Ω_p,θbk ≤ 1 and summing up yields

∞

X

µ=L+1

σ0(Rµ : s^r,Ω_p,θb → s^t,Ω_q,νb, D)^u ≤

∞

X

µ=L+1

µ^{u(d−1)(θ1−1ν)}2^{−(r−t−1p+1q)µu} (6.2.18) . L^{u(d−1)(θ1−1ν)}2^{−(r−t−(1p−1q))Lu}.

(6.2.19) The choice of L in (6.2.15) gives

∞

X

µ=L+1

σ₀(R_µ: s^r,Ω_p,θb → s^t,Ω_q,νb, D)^u . m^−(r−t). (6.2.20)

Inserting the estimates from (6.2.20), (6.2.17) into (6.2.16) yields σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νb, D) . m^−(r−t). That proves the claim.

Theorem 6.16. Let 0 < p < q ≤ ∞ and 0 < θ < ν ≤ ∞ with ¹_p − ¹_q ≤ r − t ≤ ¹_θ − _ν¹ and q ≥ ν or θ ≤ p < q < ν. Then

σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νf, D)  m^−(r−t). (6.2.21) Proof. For the lower bound we refer to [54], Proposition 5.1. Similarly to Theorem 6.15 we prove the upper bound with a constructive method in case of the compact embedding with r > ¹_p − ¹_q. For the non-compact embedding r = ¹_p − ¹_q we refer to the comments in Remark 6.19. The case q ≥ ν follows from Theorem 6.15 using the decomposition provided in Figure 6.1 with Lemma 6.1, (iii). We prove the case q < ν with p ≥ θ. Let

N = blog mc and L as in (6.2.15).

s^r,Ω_p,θb s^t,Ω_q,νf

s^t,Ω_q,νb id

id

id

Figure 6.1: Trivial embedding in case q ≥ ν.

Defining u := min{q, ν, 1} then Lemma 6.1, (ii) yields

σ_2m(id : s^r,Ω_p,θb → s^t,Ω_q,νf, D)^u ≤ σ_mX^N

µ=0

R_µ : s^r,Ω_p,θb → s^t,Ω_q,νf, Du

(6.2.22)

+σ_m X^L

µ=N +1

R_µ: s^r,Ω_p,θb → s^t,Ω_q,νf, Du

+

∞

X

µ=L+1

σ₀(R_µ: s^r,Ω_p,θb → s^t,Ω_q,νf, D)^u.

(6.2.23)

s^r,Ω_p,θb s^t,Ω_q,νf

s^t,Ω_ν,νb PN

µ=0R_µ PN

µ=0R_µ

id

s^r,Ω_p,θb s^t,Ω_q,νf

s^t,Ω_q,qb PL

µ=N +1R_µ PL

µ=N +1R_µ

id

Figure 6.2: Decomposition of P R_µ in the b − f case.

In the first sum we apply the decomposition provided in the left commutative diagram of Figure 6.2. Lemma 6.1, (iii) yields

σ_mX^N

µ=0

R_µ : s^r,Ω_p,θb → s^t,Ω_q,νf, D

. σm

X^N

µ=0

R_µ : s^r,Ω_p,θb → s^t,Ω_ν,νb, D

kid : s^t,Ω_ν,νb → s^t,Ω_q,νf k

| {z }

≤1

.

The choice of N gives sup

µ=0,...,N

d_µ = sup

µ=0,...,N

2^µ≤ m and inf

µ=N +1,...,Ld_µ= inf

µ=N +1,...,L2^µ≥ m.

Additionally, we have r − t < ¹_θ−_ν¹. This allows us to apply Corollary 6.8 which yields

σm

X^N

µ=0

Rµ : s^r,Ω_p,θb → s^t,Ω_ν,νb, D



 σm(id : `p(Xµ) → `ν(Yµ), D) . m^−(r−t), (6.2.24)

where

X_µ= `<sup>µ</sup><sub>θ</sub><sup>d−1</sup>(2<sup>µ(r−1p)</sup>`²_p^µ) and Y_µ= `<sup>µ</sup><sub>ν</sub><sup>d−1</sup>(2<sup>µ(t−ν1)</sup>`²_ν^µ).

We estimate the second sum in (6.2.23) by using the right commutative diagram in Figure 6.2. Notice, that ¹_p −¹_q < r − t < ¹_θ − ¹_q. This allows us to apply Corollary 6.10 which yields

σ_mX^N

µ=0

R_µ: s^r,Ω_p,θb → s^t,Ω_q,qb, D

 σ_m(id : `<sub>p</sub>(X<sub>µ</sub>) → `_q(Y_µ), D) . m^−(r−t), (6.2.25)

where

X_µ= `<sup>µ</sup><sub>θ</sub><sup>d−1</sup>(2<sup>µ(r−p1)</sup>`²_p^µ) and Y_µ= `<sup>µ</sup><sub>q</sub><sup>d−1</sup>(2<sup>µ(t−1q)</sup>`²_q^µ).

Finally we deal with the last sum in (6.2.23). Let a ∈ s^r,Ω_p,θb with ka|s^r,Ω_p,θbk ≤ 1.

Proceeding by applying Lemma 6.1, (i) and Lemma 6.12 gives σ0(Rµa, Dµ)^u_st,Ω

q,νf . kR^µa|s^t,Ω_q,qf k ≤ µ^{(d−1)(1θ−1q)}kRµa|s^t,Ω_q,θbk . µ^{(d−1)(1θ−1q)}2^{−(r−t−1p+1q)µ}kRµa|s^r,Ω_p,θbk . µ^{(d−1)(1θ−1q)}2^{−(r−t−1p+1q)µ}.

Taking supremum over a and summing up shows

∞

X

µ=L+1

σ₀(R_µ: s^r,Ω_p,θb → s^t,Ω_q,νf, D_µ)^u ≤

∞

X

µ=L+1

µ^{u(d−1)(1θ−1q)}2^{−(r−t−1p+1q)µu} . L^{u(d−1)(1θ−1q)}2^{−(r−t−(1p−1q))Lu}.

Finally, the choice of L yields

∞

X

µ=L+1

σ₀(R_µ: s^r,Ω_p,θb → s^t,Ω_q,νf, D_µ)^u . L^{u(d−1)(1θ−1q)}2^{−(r−t−(1p−1q)Lu} . m^−(r−t). (6.2.26)

Inserting the estimates from (6.2.25), (6.2.24), (6.2.26) into (6.2.23) yields σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νf, D) . m^−(r−t).

This proves the claim.

s^r,Ω_p,θf s^t,Ω_q,νb

s^r,Ω_p,θb id

id

id

Figure 6.3: Trivial embedding in case θ ≥ p.

Theorem 6.17. Let 0 < p < q ≤ ∞ and 0 < θ < ν ≤ ∞ with ¹_p − ¹_q < r − t ≤ ¹_θ −_ν¹ and θ ≥ p or θ < p < q ≤ ν. Then

σm(id : s^r,Ω_p,θf → s^t,Ω_q,νb, D)  m^−(r−t). (6.2.27)

Proof. For the lower bound we refer to [54], Proposition 5.1. Again, we prove the upper bound with a constructive method in case of the compact embedding with r > ¹_p − ¹_q. For the non-compact embedding r = ¹_p −¹_q we refer to the comments in Remark 6.19..

The case θ ≥ p follows from Theorem 6.15 using the decomposition provided in Figure 6.3 with Lemma 6.1, (iii). We prove the case q ≤ ν with p ≥ θ. Let

N = blog mc and L as in (6.2.15).

Defining u := min{q, ν, 1} Lemma 6.1, (ii) yields

σ_2m(id : s^r,Ω_p,θf → s^t,Ω_q,νb, D)^u ≤ σ_mX^N

µ=0

R_µ : s^r,Ω_p,θf → s^t,Ω_q,νb, Du

+σ_m X^L

µ=N +1

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νb, Du

+

∞

X

µ=L+1

σ0(Rµ: s^r,Ω_p,θf → s^t,Ω_q,νb, Dµ)^u.

(6.2.28)

We consider the first sum where we use the decomposition presented in the left com-mutative diagram of Figure 6.4.

s^r,Ω_p,θf s^t,Ω_q,νb

s^r,Ω_θ,θb id

PN µ=0R_µ

PN µ=0R_µ

s^r,Ω_p,θf s^t,Ω_q,νb

s^r,Ω_p,pb id

PL

µ=N +1R_µ

PL

µ=N +1Rµ

Figure 6.4: Decomposition of P R_µ in the f − b case.

Then Lemma 6.1, (iii) provides σ_mX^N

µ=0

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νb, D

. σm

X^N

µ=0

R_µ: s^r,Ω_θ,θb → s^t,Ω_q,νb, D

kid : s^r,Ω_θ,θf → s^r,Ω_θ,θbk

| {z }

≤1

.

The choice of N gives sup

µ=0,...,N

d_µ= sup

µ=0,...,N

2^µ≤ m and inf

µ=N +1,...,Ld_µ = inf

µ=N +1,...,L2^µ ≥ m.

Additionally we have r − t < ¹_θ−¹_ν. This allows us to apply Corollary 6.8 which yields

σ_mX^N

µ=0

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νb, D

. σm(id : `<sub>θ</sub>(X<sub>µ</sub>) → `_ν(Y_µ), D) . m^−(r−t), (6.2.29) where

X_µ = `<sup>µ</sup><sub>θ</sub><sup>d−1</sup>(2<sup>µ(r−θ1)</sup>`²_θ^µ) and Y_µ= `<sup>µ</sup><sub>ν</sub><sup>d−1</sup>(2<sup>µ(t−1q)</sup>`²_q^µ).

We estimate the second sum in (6.2.28) by using the decomposition provided in the right commutative diagram in Figure 6.4. Note that ¹_p − ¹_q < r − t. This allows us to apply Corollary 6.9 that yields

σ_mX^N

µ=0

R_µ: s^r,Ω_p,pb → s^t,Ω_q,νb, D

. σm(id : `<sub>p</sub>(X<sub>µ</sub>) → `_q(Y_µ), D) . m^−(r−t), (6.2.30) where

X_µ= `<sup>µ</sup><sub>p</sub><sup>d−1</sup>(2<sup>µ(r−p1)</sup>`²_p^µ) and Y_µ= `<sup>µ</sup><sub>ν</sub><sup>d−1</sup>(2<sup>µ(t−1q)</sup>`²_q^µ).

Finally we deal with the third sum in (6.2.23). Let a ∈ s^r,Ω_p,θf with ka|s^r,Ω_p,θf k ≤ 1.

Applying Lemma 6.1, (i) and Lemma 6.12 gives σ₀(R_µa, D)^u_st,Ω

q,νb . kRµa|s^t,Ω_q,νf k ≤ µ^{(d−1)(1p−1ν)}kR_µa|s^t,Ω_q,pbk . µ^{(d−1)(1p−1ν)}2^{−(r−t−1p+1q)µ}kR_µa|s^r,Ω_p,pbk . µ^{(d−1)(1p−1ν)}2^{−(r−t−1p+1q)µ}kR_µa|s^r,Ω_p,θf k

| {z }

≤1

.

Taking supremum over a and summing up shows

∞

X

µ=L+1

σ0(Rµ: s^r,Ω_p,θf → s^t,Ω_q,νb, D)^u ≤

∞

X

µ=L+1

µ^{u(d−1)(1p−ν1)}2^{−(r−t−1p+1q)µu} . L^{u(d−1)(1p−ν1)}2^{−(r−t−(1p−1q))Lu}.

Finally, the choice of L yields

∞

X

µ=L+1

σ₀(R_µ : s^r,Ω_p,θf → s^t,Ω_q,νb, D)^u . L^{u(d−1)(p1−1ν)}2^{−(r−t−(1p−1q)Lu}. m^−(r−t). (6.2.31)

Inserting the estimates from (6.2.30), (6.2.29), (6.2.31) into (6.2.28) yields σ_m(id : s^r,Ω_p,θb → s^t,Ω_q,νf, D) . m^−(r−t),

which concludes the proof.

Theorem 6.18. Let 0 < p < q ≤ ∞ and 0 < θ < ν ≤ ∞ with ¹_p − ¹_q < r − t ≤ ¹_θ −_ν¹ and p ≤ θ, ν ≤ q or θ ≤ p < q ≤ ν. Then

σ_m(id : s^r,Ω_p,θf → s^t,Ω_q,νf, D)  m^−(r−t). (6.2.32) Proof. For the lower bound we refer to [54], Proposition 5.1. We prove the upper bound in case r > ¹_p − ¹_q with a constructive way. For the case r = ¹_p − ¹_q we refer to Remark 6.19. The case θ ≥ p, q ≥ ν follows from Theorem 6.15 using the the commutative diagram provided in Figure 6.5 together with Lemma 6.1, (iii). We prove the case

s^r,Ω_p,θf s^t,Ω_q,νf

s^r,Ω_p,θb s^t,Ω_q,νb id

id id

id

Figure 6.5: Trivial embeddings in case p ≤ θ, ν ≤ q.

q ≤ ν with p ≥ θ. Let

N = blog mc and L as in (6.2.15).

Defining u := min{q, ν, 1} Lemma 6.1, (ii) yields

σ_2m(id : s^r,Ω_p,θf → s^t,Ω_q,νf, D)^u ≤ σ_mX^N

µ=0

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νf, Du

(6.2.33)

+σ_m X^L

µ=N +1

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νf, Du

+

∞

X

µ=L+1

σ₀(R_µ : s^r,Ω_p,θf → s^t,Ω_q,νf, D)^u.

(6.2.34) Concerning the first sum we consider the left commutative diagram in Figure 6.6.

s^r,Ω_p,θf s^t,Ω_q,νf

s^r,Ω_θ,θb s^t,Ω_ν,νb id

PN µ=0R_µ PN

µ=0R_µ

id

s^r,Ω_p,θf s^t,Ω_q,νf

s^r,Ω_p,pb s^t,Ω_q,qb id

PL

µ=N +1R_µ PL

µ=N +1R_µ

id

Figure 6.6: Decomposition of P R_µ in the f − f case.

Lemma 6.1, (iii) provides

σ_mX^N

µ=0

R_µ: s^r,Ω_p,θb → s^t,Ω_q,νf, D

. kid : s^t,Ων,νf → s^r,Ω_θ,θbk

| {z }

≤1

σ_mX^N

µ=0

R_µ : s^r,Ω_θ,θb → s^t,Ω_ν,νb, D

× kid : s^r,Ω_p,θf → s^r,Ω_θ,θbk

| {z }

≤1

.

The choice of N gives sup

µ=0,...,N

d_µ= sup

µ=0,...,N

2^µ≤ m and inf

µ=N +1,...,Ld_µ = inf

µ=N +1,...,L2^µ ≥ m.

Additionally we have r − t ≤ ¹_θ − ¹_ν. This allows us to apply Corollary 6.8 that yields

σ_mX^N

µ=0

R_µ: s^r,Ω_p,θf → s^t,Ω_q,νf, D

. σm(id : `<sub>θ</sub>(X<sub>µ</sub>) → `_ν(Y_µ), D) . m^−(r−t), (6.2.35)

where

X_µ= `<sup>µ</sup><sub>θ</sub><sup>d−1</sup>(2<sup>µ(r−1θ)</sup>`²_θ^µ) and Y_µ= `<sup>µ</sup><sub>ν</sub><sup>d−1</sup>(2<sup>µ(t−ν1)</sup>`²_ν^µ).

We estimate the second sum in (6.2.34) by using the right commutative diagram in Figure 6.6. We recognize ¹_p − ¹_q < r − t. This allows us to apply Corollary 6.9 which yields

σ_m X^L

µ=N +1

R_µ: s^r,Ω_p,pb → s^t,Ω_q,qb, D

. σm(id : `<sub>p</sub>(X<sub>µ</sub>) → `_q(Y_µ), D) . m^−(r−t), (6.2.36)

where

X_µ = `<sup>µ</sup><sub>p</sub><sup>d−1</sup>(2<sup>µ(r−1p)</sup>`²_p^µ) and Y_µ = `<sup>µ</sup><sub>q</sub><sup>d−1</sup>(2<sup>µ(t−1q)</sup>`²_q^µ).

Finally we deal with the last sum in (6.2.23). Let a ∈ s^r,Ω_p,θf with ka|s^r,Ω_p,θf k ≤ 1.

Applying Lemma 6.1, (i) and Lemma 6.12 gives σ₀(R_µa, D)^u

s^t,Ωq,νf . kR^µa|s^t,Ω_q,νf k ≤ µ^{(d−1)(1θ−ν1)}kR_µa|s^t,Ω_q,pf k . µ^{(d−1)(1θ−1ν)}2^{−(r−t−1p+1q)µ}kRµa|s^r,Ω_p,θf k.

Taking supremum over a and summing up shows

∞

X

µ=L+1

σ₀(R_µ: s^r,Ω_p,θf → s^t,Ω_q,νf, D)^u .

∞

X

µ=L+1

µ^{u(d−1)(1θ−ν1)}2^{−(r−t−1p+1q)µu} . L^{u(d−1)(1θ−ν1)}2^{−(r−t−(1p−1q))Lu}.

Finally, the choice of L yields

∞

X

µ=L+1

σ₀(R_µ : s^r,Ω_p,θf → s^t,Ω_q,νb, D)^u . L^{u(d−1)(p1−1ν)}2^{−(r−t−(1p−1q)Lu}. m^−(r−t). (6.2.37)

Inserting the estimates from (6.2.36), (6.2.35), (6.2.37) into (6.2.34) yields σ_m(id : s^r,Ω_p,θf → s^t,Ω_q,νf, D) . m^−(r−t),

which concludes the proof.

Remark 6.19. Setting L = ∞ the proofs of Theorems 6.15, 6.16, 6.17 and 6.18 work also in case r − t = ¹_p − ¹_q (non-compact embedding). The price to pay is the constructivity of the underlying algorithm. The algorithm needs full knowledge of the coefficients on infinitely many hyperbolic layers M (µ, d).

In document Sparse representation of multivariate functions based on discrete point evaluations (Page 87-104)