Norming Sets and Polynomial Reproduction

R^ku

W_p^j(D) =

u−Q^ku

W_p^j(D)≤C_γ,k,dδ_D^k−j|u|_Wp^k_(D) j= 0,1, . . . , k, (3.3.3) where C_γ,k,d = C_k,d(1 + 2γ)^d. Here, γ denotes again the chunkiness parameter from Definition 3.3.8.

3.4 Norming Sets and Polynomial Reproduction

This section deals with general Banach spaces. In the following letV be a finite dimensional linear space endowed with the normk·k_V. We denote byV^∗the dual space ofV, i.e., the space of all bounded linear functionals on V, and byZ ⊂ V^∗ a finite subset ofV^∗ with

|Z|=N.

Definition 3.4.1 Z is called anorming setforV if the map T : V →T(V)⊂R^N

v 7→T(v) = (z(v))_z∈Z is injective.T will be called thesampling operator.

To explain why Z is called a norming set we have to make a detour. First we have to introduce a normk·k^RN on theR^N and a corresponding dual normk·k₍^RN₎^∗ on R^N_∗ ∼= R^N. Note thatZ being a norming set means thatT is injective, which implies that T is invertible on this image, i.e., there is an inverse mapT⁻¹ :T(V)→V. The quantity

T⁻¹

:= sup

x∈T(V) x6=0

T⁻¹x V

kxk^RN = sup

v∈Vv6=0

kvkV

kT vk^RN (3.4.1)

is callednorming constant. Now we can explain the namenorming set, which is due to the fact thatZ allows us to introduce an equivalent norm onV bykT(·)kRN. The equivalence constants are given by

1

kTkkT vk^RN ≤ kvkV ≤ T⁻¹

kT vk^RN for allv ∈V .

The norming constant plays an essential rôle as can be seen in the next theorem.

3.4. NORMING SETS AND POLYNOMIAL REPRODUCTION 35 Theorem 3.4.2 SupposeV is a finite dimensional normed linear space, and the set of func-tionalsZ ={z1, . . . , zN}is a norming set forV with sampling operatorT. Then there is for everyψ∈V^∗a vectoru=u(ψ)∈R^N, such that

ψ(v) = XN j=1

u_jz_j(v) , and kuk(RN)^∗ ≤ kψkV^∗

T⁻¹ for everyv∈V.

Proof:[65, Theorem 3.4]. 2

Now we introduce the central idea of norming sets, namely that we can control the norms ofT and its inverse by enlarging the size ofZ. This will be calledoversampling. We shall exemplify this idea in a special case, see [63]. LetV be the restriction of π_m R^d

toΩ whereπ_m R^d

again denotes the space of all algebraic polynomials onR^dwith degree at mostm, i.e.,V =πm(Ω). LetZ ={δx1, . . . , δx_N}consist of point evaluation functionals.

Then we have the following result, where we choose thek·kℓ_∞-norm on theR^N with dual normk·kℓ1. In the following we shall denote byα∈N^d

0a multi-index.

Theorem 3.4.3 SupposeΩis compact and satisfies an interior cone condition with radius r > 0 and angle θ ∈ (0, π/2). Let m ∈ N be fixed, and suppose h > 0. If X = {x₁, . . . , x_N} ⊂Ωsatisfies

1. h≤ _4(1+sin^rsin_θ)m^θ 2

2. for everyB(x, h)⊂Ωthere is a centerx_j ∈X∩B(x, h),

thenZ = {D^α◦δx1, . . . , D^α◦δx_N} is for every multi-index|α| ≤ m a norming set for V =π_m(Ω). The norming constant is bounded by2

2m² rsinθ

_|α|

.

Proof:See [65, Proposition 11.7]. 2

If we in particular choose ψ = D^α ◦δx for some x ∈ Ω, this implies that for every p∈π_m(Ω)and any|α| ≤mthere are real numbersa^(α)_j (x)such that

D^αp(x) = XN j=1

a^(α)_j (x)p(x_j) , where

XN j=1

a^(α)_j (x) ≤2

2m² rsinθ

|α|

.

We point out that the constant arises from a Bernstein inequality for multivariate polyno-mials. Furthermore, all of this implies that recovery of functions by polynomials from

function values at scattered locations can be made with bounded Lebesgue constants if moderate oversampling is allowed. Therefore, the functionsa^(α)_j (·)are called apolynomial reproduction formula of degreem. Finally this yields a finite version of a Markov-Bernstein inequality.

Theorem 3.4.4 Suppose that the domainΩ⊂R^dis compact and satisfies an interior cone condition with radiusr >0and angleθ. Ifp∈πm(Ω)andα∈N^d

0with|α| ≤m, we have kD^αpk_L∞(Ω) ≤

2m² rsinθ

_|α|

kpk_L∞(Ω) .

Proof:See [65, Proposition 11.6]. 2

This is only a brief detour through the theoretical basics we need throughout the thesis. For more details we refer to the literature.

Chapter 4

Infinitely Smooth Functions

We derive in this chapter sampling inequalities for infinitely smooth functions where the sampling orders turn out to depend exponentially on the fill distanceh.

We are handling infinitely smooth functions by normed linear function spacesH(Ω)on a domainΩ ⊂ R^d that can for a fixed1 ≤ p < ∞ be continuously embedded into every classical Sobolev spaceW_p^k(Ω). More precisely, for a fixedp ∈[1,∞)and allk∈Nwe assume that there are embedding operatorsI_k^(p)and constantsE(k)such that

I_k^(p):H(Ω)→W_p^k(Ω) with

I_k^(p)

{^{H(Ω)→Wpk(Ω)}} ≤E(k) for allk∈N₀. (4.0.1) There are various examples of spaces with this property, e.g., Sobolev spaces of infinite order as they occur for instance in the study of partial differential equations of infinite order [1], or reproducing kernel Hilbert spaces of Gaussians and inverse multiquadrics (see Section 4.4).

In the case of infinitely smooth functions, the shape of the domainΩcrucially influences our sampling inequalities. For general Lipschitz domains Ω, which satisfy interior cone conditions, we use a polynomial reproduction [65], which accepts slight oversampling to bound the Lebesgue-constants. This results in a good behavior of the term with the discrete norm. A typical result in this case is that for sufficiently small fill distance h there are generic constantsc >0such that withq˜∈ {q,∞}the inequality

kD^αukLq(Ω)≤ce^{clog(ch)/√h}kuk_H(Ω)+ch^−|α|ku|Xkℓq˜(X)

holds for allu ∈ H(Ω). The best approximation orders for the first term can be obtained on compact cubes because we then may use a polynomial reproduction from [32]. Un-fortunately this approach is limited to cubes and cannot cope with derivatives on the left hand-side of our sampling inequalities. Nevertheless we obtain as a typical result, which applies for instance to the functions from the native space of Gaussian kernels, that there are generic constantsc >0, such that the inequality

kuk_Lq(Ω) ≤e^clog(ch)/hkuk_H(Ω)+c^1/hku|Xk_ℓq˜(X) 37

holds for allu∈ H(Ω)withq˜∈ {q,∞}if the fill distancehis sufficiently small.

It is an open research problem to improve the polynomial reconstruction results in [26] and [65]. There is some further discussion on this point in the outlook (see Chapter 9). Our main examples deal with reconstruction problems in Hilbert spaces. Therefore, in the second part we will focus on the native Hilbert spaces of Gaussian and inverse multiquadric kernels. In this case, we supposeuto be an error functionu=f−Rf, wheref denotes the function we would like to reconstruct, andRf is the reconstruction. To obtain optimal order error bounds one needs againstabilityandconsistencyof the reconstruction, namely

kRfk_H≤Ckfk_H , and k(Rf−f)|Xkℓp(X) ≤g(f, h),

wheregdetermines the expected approximation order. This can be used to show that the the-ory presented here reproduces the well-known exponential error estimates for the standard interpolation problem in the native Hilbert space of the inverse multiquadrics and Gaussian kernels.

4.1 Estimates on General Lipschitz Domains

Following [38], we first obtain estimates on local domains D ⊂ R^d and use a covering argument to get global results. We assume a domainDthat is is star-shaped with respect to a ballB_r(x_c), and that is contained in a ballB_R(x_c). In this case we know from [38]

thatD satisfies an interior cone condition as well. We denote the associated chunkiness parameter with

γ = δ_D ρ_max,

whereρ_max= sup{ρ:Dis star-shaped with respect to a ball of radiusρ}, andδ_Ddenotes the diameter ofD.

Let n

a^(α)_j :j = 1, . . . , No

be a polynomial reproduction of degree k with respect to a discrete setX ={x₁, . . . , x_N} ⊂ D, i.e.,

D^αq(x) = XN j=1

a^(α)_j (x)q(x_j) holds for everyα ∈N^d

0 with|α| ≤k, allx ∈ Dand allq ∈π_k^d(D)whereπ_k^ddenotes the space of alld-variate polynomials of degree not exceedingk. Then we have

|D^αu(x)| ≤ |D^αu(x)−D^αp(x)|+|D^αp(x)|

≤ kD^αu−D^αpk_L∞(D)+ XN j=1

a^(α)_j (x)

|p(xj)|

≤ kD^αu−D^αpk_L∞(D)+ XN j=1

a^(α)_j (x)

kp|Xk_ℓ∞(X)

≤ kD^αu−D^αpkL_∞(D)

+ XN j=1

a^(α)_j (x)

ku−pk_L∞(D)+ku|Xk_ℓ∞(X)

(4.1.1)

4.1. ESTIMATES ON GENERAL LIPSCHITZ DOMAINS 39 for arbitraryu∈W_p^k(D)and any polynomialp∈π_k^d(D). As a polynomial approximation we use again averaged Taylor polynomials. We recall the definition (3.3.2)

Q^ku(x) := X

|α|<k

1 α!

Z

B

D^αu(y) (x−y)^αφ(y)dy,

where B is a ball with radius ≥ 1/2ρ_max, relative to which D is star-shaped, and φ ∈ C^∞ is a bump function supported on B¯ satisfying both R

Bφ(y)dy = 1 andmaxφ ≤ C_d diam(B)^−d. For the remainder R^k := u −Q^ku there is the following bound from [11], where the explicit constants can be found in [38]. This bound differs slightly from the bound in Lemma 3.3.11. We state it here, because we need to calculate the explicit dependence of the constants onk.

Lemma 4.1.1 Foru∈W_p^k(D)with1< p <∞andk >|α|+d/por in the casep= 1 andk≥ |α|+dwe get

D^αu−D^αQ^ku

L_∞(D)≤C_d,θ d^k−|α|

(k− |α|)!δ^k_D^{−|α|−d/p}|u|_Wp^k_(D) , where the constantC_d,θdepends only on the space dimensiondand the angleθ.

Proof:We use the identity [11]D^βQ^ku=Q^k−|β|D^βu, for all|β| ≤k. This leads to

D^αu−D^αQ^ku

L_∞(D) =

D^αu−Q^k−|α|D^αu L_∞(D)

≤ C_d(1 +γ)^d d^k−|α|

(k− |α|)!δ_D^{k−|α|−d/p}|D^αu|_W^k−|α|

p (D)

≤ C_d,θ d^k−|α|

(k− |α|)!δ_D^{k−|α|−d/p}|u|W_p^k(D) .

Here we used the fact [38] that the chunkiness parameterγ can be bounded by 1 ≤ γ ≤ csc ^θ₂

. 2

We shall use the following local polynomial reproduction from [65], which we introduced in the remarks below Theorem 3.4.3.

Theorem 4.1.2 LetΩ⊂R^dsatisfy an interior cone condition with angleθ∈(0, π/2)and radiusr,ℓ∈N₀andα ∈N^d

0with|α| ≤ ℓ. Then there are constantsc₀, c^(α)₁ , c₂ >0, such that for allX = {x₁, . . . , x_N} ⊂ Ωwithh_X,Ω ≤ h₀ := c₀/ℓ² and allx ∈ Ωthere exist numbers˜a^(α)₁ (x), . . . ,a˜^(α)_N (x)with

1. P_N

j=1p(x_j) ˜a^(α)_j (x) =D^(α)p(x)forx∈Ωandp∈π^d_ℓ(Ω) 2. P_N

j=1

˜a^(α)_j (x)

≤c^(α)₁ h^−|α|_X,Ω for allx∈Ω, 3. ˜a^(α)_j (x) = 0ifkx−x_jk₂ > c^(α)₂ h_X,Ωandx∈Ω.

The condition θ ∈ (0, π/2) implies sinθ ∈ (0,1), i.e.,

1 2(1+sinθ)

_|α|

≤ _2(1+sin¹ _θ) for allα ∈ N^d

0. Therefore, we can choose all the constants independent ofα, i.e., there exist constantscθdepending only onθsuch that [65]

c^(α)₁ ≤c_θ2^−|α|≤c_θ, c₂ :=c_θℓ². (4.1.2) Inserting the bounds of Lemma 4.1.1 and Theorem 4.1.2 into (4.1.1) leads to the following local estimate.

Theorem 4.1.3 SupposeDsatisfies an interior cone condition with angleθand radiusr, letα ∈N^d

0such thatk >|α|+d/pfor1< p <∞, ork≥difp= 1. Then kD^αuk_L∞(D) ≤ C_d,θd^k

(k− |α|)!δ^k_D^−d/p

δ^−|_D^α|+h^−|α|

|u|_Wp^k_(D) +C_d,θh^−|α|ku|Xk_ℓ∞(X)

holds for allu∈W_p^k(D).

Corollary 4.1.4 Under the assumptions from Theorem 4.1.3, we get for1≤q ≤ ∞ kD^αuk_Lq(D) ≤ vol(D)^1/qkD^αuk_L∞(D)≤δ_D^d/qkD^αuk_L∞(D)

≤c_d,θ d^k

(k− |α|)!δ^k−d

“1 p−¹_q” D

δ_D^−|α|+h^−|α|

|u|_Wp^k_(D)+ + C_d,θδ^d/q_D h^−|α|ku|Xk_ℓ∞(X) .

Now we consider a ‘global‘ domain Ω ⊂ R^d that is bounded, has a Lipschitz boundary and satisfies an interior cone condition with maximum radiusR and angle φ ∈ (0, π/2).

To cover Ωwith smaller star-shaped domains {Dt}we use the construction described in Theorem 3.3.10.

Theorem 4.1.5 Letα∈N^d

0 andk∈Nbe fixed with|α|< k,k > d/pfor1 < p <∞or k≥dforp= 1and setC_min:= min_c0

2, Q_θ,R with the constantc₀ from Theorem 4.1.2.

Suppose a discrete setX ⊂ Ωwith fill distanceh ≤C_min/k². Then for allu ∈ W_p^k(Ω) the inequality

kD^αukLq(Ω) ≤ c^kh^−|α|

(k− |α|)! hk^2k−d“

1 p−¹_q”

+|u|W_p^k(Ω)

+ch^−|α| hk²d/q

ku|Xkℓq(X) (4.1.3) holds for 1 ≤ q ≤ ∞ with generic positive constants c, which may depend only on d, R, φ, p, qandα.

4.1. ESTIMATES ON GENERAL LIPSCHITZ DOMAINS 41 Proof: For u ∈ W_p^k(Ω)we may use the decomposition from Theorem 3.3.10, Corollary 4.1.4 and the estimateδ_D^−|α|≤C_φ,αh^−|α|, which gives We can restate this Theorem measuring the discrete term in theℓ_∞-norm.

Corollary 4.1.6 Under the assumptions from Theorem 4.1.5 we get with an analogous cal-culation

We shall now relatehandkto derive main result of this section, i.e., exponential estimates.

The actual orders depend on the asymptotic behavior of the constantE(k)from (4.0.1) for k→ ∞.

are constantsc, h₀ >0depending ond, p, q, R, φ, α, C_E such that for all data setsX⊂Ω with fill distanceh≤h0, the inequality

kD^αukLq(Ω)≤e^−h1/(s+1)c kuk_H(Ω)+ch^−|α|ku|Xkℓq(X)

holds for allu∈ H(Ω)and all1≤q≤ ∞. Proof:We use Stirling’s formula to estimate

1

(k− |α|)! ≤ k^|α|

k! ≤ k^|α|e^k k^k .

Ifkuk_Wp^k_(Ω) ≤ C_E^kk^(1−ǫ)kkuk_H(Ω) holds for all k ∈ N, we can bound the first term of (4.1.3) for arbitraryk∈Nby

˜

chk^2−ǫk

h⁻¹k_|α|

hk²−d“

1 p−¹_q”

+kuk_H(Ω) .

We setB = min{c_min,1/˜c}and choosek ∈ Nsuch that _2k^B2 ≤h ≤ _k^B2 holds. Then the first term can be bounded by

ck^−ǫkh^−3|α|/2kuk_H(Ω)≤e^{clog(ch)/√h}kuk_H(Ω) , where the constantsc >0may depend ond, p, q, R, φ, α, C_E andǫ.

With this choice ofkthe second term of (4.1.3) can be bounded by ch^−|α| hk²d/q

ku|Xk_ℓ∞(X)≤ch^−|α|ku|Xk_ℓ∞(X) . IfE(k)≤C_E^kk^sk, we can bound the first term of (4.1.3) for arbitraryk∈Nby

˜

chk^1+sk

h⁻¹k_|α|

hk²−d“

1 p−¹_q”

+kuk_H(Ω) . We setB = min

c_min,_e˜¹_c and choose k ∈ Nsuch that _2k^B_1+s ≤ h ≤ _k1+s^B holds. We point out that the conditionh≤ ^cmin_k2 is satisfied sinces≥1. Thenhk² ≤B, and therefore the first term can be bounded by

ce^−kh^−3|α|/2kuk_H(Ω)≤e^−h1/(s+1)c kuk_H(Ω) ,

where the constantc > 0 may depend on d, p, q, R, φ, α, C_E. The second term is again bounded by

ch^−|α| hk²d/q

ku|Xk_ℓ∞(X)≤ch^−|α|ku|Xk_ℓ∞(X) .

2 Again we get the following result for theℓ_∞-norm.

Corollary 4.1.8 If we use Corollary 4.1.6 instead of Theorem 4.1.5, we get in the case E(k)≤C_E^kk^(1−ǫ)kfor allk∈Nwith constantsǫ, c >0, that for allu∈ H(Ω)

kD^αukLq(Ω)≤e^{clog(ch)/√h}kuk_H(Ω)+ch^−|α|ku|Xkℓ_∞(X) . IfE(k)≤C_E^kk^sk for allk∈Nwith a constants≥1, we find for allu∈ H(Ω)

kD^αuk_Lq(Ω) ≤e^−h1/(1+s)1 kuk_H(Ω)+ch^−|α|ku|Xk_ℓ∞(X) .

Im Dokument Sampling Inequalities and Applications (Seite 38-47)