Homogeneous Approximation Property in H(E)

In this section we prove that the Homogeneous Approximation Property (HAP) holds for the reproducing kernel in H(E), we then use this result to prove the Comparison Theorem which will hold for frames formed by the reproducing kernels of the space. The Homogeneous Approximation Property and the Comparison Theorem were introduced by Ramanathan and Steger [42] in the context of Gabor frames. A version of the HAP for frames of translates of band-limited functions was proved by Gr¨ochenig and Razafinjatovo in [21].

Let Γ = {γn}_n∈Z ⊂ R be a sequence such that the corresponding normalized reproducing kernels k_γn = _kK(γ^K(γn,.)

n,.)k form a Riesz basis in H(E), and M = {µ_n}_n∈Z ⊆ R be such that the corresponding normalized reproducing kernels k_µn = _kK(µ^K(µn,.)

n,.)k form a frame in H(E). Let r > 0, and y ∈ R, and define the index sets

Jr(y) = {n ∈ Z : |γn− y| ≤ r}, Ir(y) = {n ∈ Z : |µn− y| ≤ r}

Define the subspaces

V_r(y) := span{k_γn : n ∈ J_r(y)}

and

W_r(y) := span{˜k_µn : n ∈ I_r(y)}

where {˜kµn} is the canonical dual frame of {k_µn}.

Denote the corresponding orthogonal projections by

Py,r : H(E) −→ Vr(y), Qy,r : H(E) −→ Wr(y)

For all f ∈ H(E):

kf − Q_y,rf k = inf

cn

f − X

n∈Ir(y)

cn˜kµn

The following simple lemma will be needed in the proof of the Homogeneous Approximation Property in H(E).

Lemma 4.2.1. Let H(E) be a de Branges space with reproducing kernel K(w, z). If {µn}_n∈Z⊂ R is a sequence such that {kµn(z)}_n∈Z is a frame for H(E), then for each x ∈ R the sequence {k_µn(x)} ∈ l².

Proof. Since {k_µn(z)}_n∈Z is a frame for H(E), then there exists A, B > such that

Akf k² ≤X

n∈Z

|hf (t), k_µn(t)i_E|² ≤ Bkf k², (4.2.1)

for all f (z) ∈ H(E). In particular, let x ∈ R, and set f (z) = K(x, z), then f ∈ H(E). Moreover,

f (x) = K(x, x) = kK(x, .)k²_E < ∞.

Hence, applying (4.2.1) for the function f , and using the reproducing kernel property (3.2.4) we get

AkK(x, .)k²_E ≤X

n∈Z

|k_µn(x)|² ≤ BkK(x, .)k²_E. That is, the sequence {k_µn(x)}_n∈Z∈ l².

Theorem 4.2.2. (Homogeneous Approximation Property in H(E)).

Let H(E) be a de Branges space such that the phase function of E(z) satisfying 0 < δ ≤ ϕ⁰(x) for all x ∈ R. Let {µn}_n∈Z ⊂ R be a separated sequence such that {kµn(z)}_n∈Z is a frame in H(E). Then given  > 0 there exists R = R() > 0 such that for all y ∈ R and all r > 0

sup

|x−y|≤r

k_x(.) − Q_y,r+Rk_x(.)

< , (4.2.2)

where kx(z) = _kK(x,.)k^K(x,z) , x ∈ R.

Proof. Let {˜k_µn(z)}_n∈Z be the canonical dual frame of {k_µn(z)}_n∈Zin H(E). Let y ∈ R, r > 0, and x ∈ R such that |x − y| ≤ r. First we will show that (4.2.2) holds when the function kx(z) is replace by the function ^K(x,z)

E(x) . Since the function ^K(x,z)

E(x) ∈ H(E) (as a function of z) for all x ∈ R, then it can be expanded as

K(x, z) E(x) =X

n∈Z

hK(x, t)

E(x) , K(µ_n, t)

kK(µ_n, .)ki ˜k_µn(z) =X

n∈Z

K(µ_n, x) E(x)kK(µn, .)k

˜k_µn(z)

Let R > 0 (to be determined). Since Qy,r+R is an orthogonal projection, then we have

n,.)k} is a frame then by inequality (2.1.10) there exists a constant C > 0 such that

where the sum in the right hand side of the last inequality is finite by Lemma 4.2.1.

Let L_x(R) := {n ∈ Z : |x − µn| ≤ R}. By the assumption |x − y| ≤ r, if n /∈ I_r+R(y), then

|µ_n− y| > r + R, so we have |x − µ_n| > R, i.e., n /∈ L_x(R). This implies that Lx(R) ⊆ I_r+R(y) whenever |x − y| ≤ r. Hence inequality (4.2.3) becomes

On the other hand, by the definition of the phase function ϕ of E(z) in (3.4.1), we have E(x)/E(x) = e^−2iϕ(x), which implies that



Using the last inequality together with inequalities (4.2.4) and (4.2.5) we get

Note that since the sequence {µ_n}_n∈Z is separated, then for each x ∈ R and for each finite R > 0, the index set Lx(R) is finite. Hence, we may assume that Lx(R) = {n1, n2, . . . , nL} then we should have µn< x. Let m be a nonnegative integer such that µn+mbe the first element to the left of x, of the sequence {µ_n}_n∈Z, such that |x − µ_n+m| > R. We also have

= Inequality (4.2.6) implies that

So, since the latter sum is finite then we can choose R = R() > 0 sufficiently large so that the latter sum is less than πδ/2C. That is, the homogeneous approximation property holds for the function K(x, z)/E(x), for all x ∈ R.

The Homogeneous Approximation Property has several implications on the geometry of the sequence M, in particular, it shows a relationship between the Beurling densities of a frame and the Beurling densities of any orthonormal basis or Riesz basis in H(E). This further yields necessary conditions for the existence of sampling and interpolation sequences as we will see later.

The following theorem is consistent with the fact that frames provide redundant non-orthogonal expansions in Hilbert space, accordingly, they should be “denser” than orthonormal bases (Riesz basis).

Theorem 4.2.3. (Comparison Theorem).

Let H(E) be a de Branges space, and the corresponding phase function of E satisfies 0 < δ ≤ ϕ⁰(x) for all x ∈ R. Suppose that M = {µn}, Γ = {γ_n} ⊆ R are two separated sequences, such that {k_µn(z)}_n∈Z is a frame in H(E), and {k_γn(z)}_n∈Z is a Riesz basis for a closed subspace of H(E). Then for every  > 0, there exists R = R() > 0, such that for all r > 0 and y ∈ R, we have

(1 − ) ] Γ ∩ [y − r, y + r)
≤ ] M ∩ [y − r − R, y + r + R)
.

Therefore,

D⁻(Γ) ≤ D⁻(M), and D⁺(Γ) ≤ D⁺(M) Proof. Let k_γn(z) = _kK(γ^K(γn,z)

n,.)k, n ∈ Z, and ˜k_γn denote the biorthogonal basis of k_γn, that is, hk_γn, ˜kγmi = δ_n,m. Since any Riesz basis is uniformly bounded and the biorthogonal sequence of a Riesz basis is also a Riesz basis, then there exists a constant C_o> 0 such that k˜k_γnk ≤ C_o, for all n ∈ Z.

Given  > 0, choose R = R() > 0 such that the homogeneous approximation property holds for the function k_x(z), x ∈ R, with /Co, i.e., for all r > 0 and y ∈ R

sup

|x−y|≤r

k_x(.) − Q_y,r+Rk_x(.)

< /C_o.

Given r > 0 and y ∈ R we define the operators Ty,r : V_r(y) → V_r(y) by

T_y,r = P_y,rQ_y,r+R. (4.2.7)

By definition, the sequence {kγn}_n∈Jr(y)is a basis for Vr(y) (since {kγn} is Riesz sequence and hence linearly independent). Moreover, the dual basis of {k_γn}_n∈Jr(y)in V_r(y) is the projection of {˜k_γn}_n∈Jr(y) under P_y,r, that is {P_y,rk˜_γn}_n∈Jr(y). Indeed, for n, m ∈ J_r(y) we have

hk_γn, P_y,r˜k_γmi = hP_y,r^∗ k_γn, ˜k_γmi = hP_y,rk_γn, ˜k_γmi = hk_γn, ˜k_γmi = δ_nm.

Note that since dom(T_y,r) = V_r(y) then

T_y,r= P_y,rQ_y,r+R= P_y,rQ_y,r+RP_y,r, (4.2.8)

hence, T_y,r is self-adjoint. Moreover, since V_r(y) is finite dimensional then the trace of T_y,r is finite. By the biorthogonality of the sequences {kγn}_n∈Jr(y) and {Py,rk˜γn}_n∈Jr(y) and Lemma 2.1.6, the trace of Ty,r can be written as

tr(Ty,r) = X

n∈Jr(y)

hT_y,rkγn, Py,r˜kγni

= X

n∈Jr(y)

hP_y,rT_y,rk_γn, ˜k_γni

= X

n∈Jr(y)

hT_y,rk_γn, ˜k_γni

Since P_y,r is an orthogonal projection, it is self adjoint, i.e., P_y,r^∗ = P_y,r, so we have

hT_y,rk_γn, ˜k_γni = hP_y,rQ_y,r+Rk_γn, ˜k_γni

= hQ_y,r+Rk_γn, P_y,r˜k_γni

= hQ_y,r+Rk_γn − k_γn+ k_γn, P_y,r˜k_γni

= hk_γn, Py,rk˜γni + hQ_y,r+Rkγn− k_γn, Py,r˜kγni

= hP_y,rkγn, ˜kγni + hQ_y,r+Rkγn− k_γn, Py,r˜kγni

= hk_γn, ˜kγni + hQ_y,r+Rkγn− k_γn, Py,rk˜γni

= 1 + hQ_y,r+Rkγn− k_γn, Py,rk˜γni

where we used the fact that P_y,rk_γn = k_γn, for all n ∈ J_r(y). So we have

hT_y,rk_γn, ˜k_γni − 1 = hQ_y,r+Rk_γn− k_γn, P_y,rk˜_γni (4.2.9)

Applying the Cauchy Schwarz inequality to the right hand side of the previous equation, and using the fact that kP_y,rk = 1 to get kP_y,r˜k_γnk ≤ kP_y,rkk˜k_γnk = k˜k_γnk ≤ C_o, we obtain

|hQ_y,r+Rk_γn− k_γn, P_y,rk˜_γni| ≤ kk_γn(.) − Q_y,r+Rk_γn(.)k kP_y,rk˜_γnk

≤ sup

|x−y|≤r

k_x(.) − Q_y,r+Rk_x(.)

kP_y,rkk˜k_γnk

< (/C_o).C_o

=  whenever |γn− y| ≤ r. Therefore, by (4.2.9) we get

|hT_y,rkγn, ˜kγni − 1| = |hQ_y,r+Rkγn− k_γn, Py,rk˜γni| ≤  (4.2.10)

Now, note that  

X

n∈Jr(y)

1 − X

n∈Jr(y)

hT_y,rkγn, ˜kγni  = 

 X

n∈Jr(y)

1 − hTy,rkγn, ˜kγni
 

≤ X

n∈Jr(y)

1 − hT_y,rk_γn, ˜k_γni  Hence, by the definition of the trace of T_y,r and (4.2.10) we have

X

n∈Jr(y)

1 − tr(Ty,r) ≤ 

X

n∈Jr(y)

1 − X

n∈Jr(y)

hT_y,rkγn, ˜kγni

≤ X

n∈Jr(y)



Therefore, we can estimate a lower bound to the trace of Ty,r by tr(T_y,r) ≥ X

n∈Jr(y)

(1 − ) = (1 − )](Γ ∩ [y − r, y + r])

On the other hand, since the operator norm of Ty,r satisfies kTy,rk = kP_y,rQy,r+Rk ≤ kP_y,rkkQ_y,r+Rk = 1, all the eigenvalues of T_y,r have modulus less than or equal to 1, this in turn provides us with an upper bound for the trace of Ty,r. Indeed,

tr(T_y,r) =X

(non-zero eigenvalues of T_y,r) ≤ rank(T_y,r)

Also, since rank(Ty,r) = dim(range(Ty,r)) = dim(range(Py,rQ_y,r+R)) ≤ dim(W_r+R), then tr(Ty,r) ≤ dim(W_r+R) ≤ ]{µn: |µn− y| ≤ r + R}

= ](M ∩ [y − r − R, y + r + R])

Therefore, combining these two estimates of the trace of Ty,r we get

(1 − )](Γ ∩ [y − r, y + r]) ≤ ](M ∩ [y − r − R, y + r + R]) for all r > 0 and all y ∈ R. Moreover,

(1 − )](Γ ∩ [y − r, y + r])

2r ≤ (2r + 2R)

2r

](M ∩ [y − r − R, y + r + R]) (2r + 2R)

taking the infimum over all y ∈ R for both sides (1 − ) inf

y∈R

](Γ ∩ [y − r, y + r])

2r ≤ (2r + 2R)

2r inf

y∈R

](M ∩ [y − r − R, y + r + R]) (2r + 2R)

and by taking liminf as r → ∞ yields the estimates

(1 − )D⁻(Γ) ≤ D⁻(M) Since  is arbitrary, we conclude that

D⁻(Γ) ≤ D⁻(M) A similar calculations shows that

D⁺(Γ) ≤ D⁺(M)

It should be noted that if the phase function ϕ satisfying 0 < δ ≤ ϕ⁰ ≤ M , then by Proposition4.1.5, the result of the Comparison Theorem can be stated, equivalently, in terms of the density of values of the phase function at the sequences Γ and M in the theorem. More precisely, we have the following:

Corollary 4.2.4. Let H(E) be a de Branges space, and the corresponding phase function of E satisfies 0 < δ ≤ ϕ⁰(x) ≤ M for all x ∈ R. Suppose that M = {µn}, Γ = {γ_n} ⊆ R are two separated sequences, such that {k_µn(z)}_n∈Z is a frame in H(E), and {k_λn(z)}_n∈Z is a Riesz basis for a closed subspace of H(E). Then

D⁻(ϕ(Γ)) ≤ M

δ D⁻(ϕ(M)), and

D⁺(ϕ(Γ)) ≤ M

δ D⁺(ϕ(M)).

In document Sampling and interpolation in Hilbert spaces of entire functions (Page 84-93)