Sampling recovery and Gelfand n-widths

The considerations above cover linear algorithms in the classical sense. Last but not least we consider an extension of this concept, so-called approximation using standard information, cf. [84, 85]. This means we consider algorithms that are defined as a composition of a linear information map and a possibly non-linear reconstruction operator. To avoid further technicalities we restrict to Banach spaces F that are either Sobolev spaces S_p^rW (T^d) or H¨older-Nikolskij spaces S_p,∞^r B(T^d) in this subsection. The non-linear sampling widths were defined in (1.4.2). The following relation clearly holds true

%_n(F , L_q(T^d) ≤ %^lin_n (F , L_q(T^d)) .

Therefore (possibly non-sharp) upper bounds for sampling widths are always provided by linear sampling widths. To consider questions on optimality of these bounds we consider Gelfand n-widths

cn(F , Lq(T^d)) := inf

B: F →Cⁿ

linear sup

kf |F k≤1 f ∈ker B

kf |Lq(T^d)k. (8.3.1)

Here B denotes a general linear mapping B : F → Cⁿ. This means c_n measures the minimal (over all information mappings) worst case distance of elements in the unit ball of F which can not be distinguished by the information mapping B. This immediately gives

c_n(F , L_q(T^d)) . %n(F , L_q(T^d)).

Note that (1.4.3) is actually the definition of the nth “Gelfand numbers”, which we call

“Gelfand n-width” here. For a thorough discussion on the relation between Gelfand numbers and suitable worst-case errors we refer to the recent paper [21, Rem. 2.3].

Since Gelfand widths for embeddings id : S_p^rW (T^d) → L_q(T^d) are not studied directly we use a duality relation to Kolmogorov n-widths, cf. (D.1).

Lemma 8.17. The following duality relation holds true d_n(T : X → Y ) = c_n(T⁰ : Y⁰ → X⁰),

where T⁰ denotes the adjoint operator of T and X⁰, Y⁰ the topological dual spaces of X and Y .

Proof. We refer to [91, Theorem 6.2].

Corollary 8.18. Let 1 < p, q < ∞ and r >







1

2 : 1 < p < q ≤ 2, 1 − ¹_q : p < 2 < q, (¹_p − ¹_q)₊ : otherwise, with

r₁ = . . . = r_µ< r_µ+1 ≤ . . . ≤ r_d < ∞. (8.3.2) Then

c_n(S_p^rW (T^d), L_q(T^d))  (n⁻¹log^µ−1n)^{r1−(min{1p,12}−1q)+} for all n ∈ N.

Proof. The proof follows by the duality relation stated in Lemma 8.17 and a lifting ar-gument. The topological dual spaces of S_p^rW (T^d) and L_q(T^d) are the spaces S_p^−r0 W (T^d) and L_q⁰(T^d) with 1 = ¹_p +_p¹0 = ¹_q + _q¹0. Lemma 8.17 yields

c_n(S_p^rW (T^d), L_q(T^d)) = d_n(L_q⁰(T^d), S_p^−r0 W (T^d)).

Finally we show the identity

d_n(L_q⁰(T^d), S_p^−r0 W (T^d))  d_n(S_q^r0W (T^d), L_p⁰(T^d)).

For that reason we consider the lifting operator Ir in D⁰(T^d) given by

I_r: f = X

k∈Z^d

f (k)eb ^ikx7→ X

k∈Z^d

f (k)b Y^d

i=1

(1 + |k_i|²)^−ri2 e^ikx.

It is easy to check that this is an isometry that maps f ∈ S_p^αW to I_rf ∈ S_p^α+rW , α ∈ R with (Ir)⁻¹ = I_−r. Therefore we may use the commutative diagram,

L_q⁰(T^d) S_p^−r0 W (T^d)

S_q^r0W (T^d) L_p⁰(T^d)

?

Ir

-id1

-id2

6I−r

which allows to describe the operators id1, id2 by

id₁ = I−r◦ id₂◦ I_r and id₂ = I_r◦ id₁◦ I−r.

Kolmogorov widths are s-numbers and fulfill a multiplicativity property that yields dn(id1) = dn(I−r◦ id2◦ Ir) ≤ kI−rkdn(id2)kIrk  dn(id2)

and

d_n(id₂) = d_n(L_r◦ id₁◦ L−r) ≤ kL_rkd_n(id₁)kI−rk  d_n(id₁).

Inserting the result from Theorem D.26 finishes the proof.

Surprisingly, a new result in [76] allows us to prove the following results for Gelfand n-widths of H¨older spaces S_p,∞^r B(T^d).

Theorem 8.19. Let 1 < p < q < ∞ and r with

(1/p − 1/q)₊ < r₁ = . . . = r_µ < r_µ+1 ≤ . . . ≤ r_d < ∞ then

c_n(S_p,∞^r B(T^d), L_q(T^d)) 







log^µ−1n n

r−¹₂+¹_q

(log n)^µ−1q : ¹_p + ¹_q < 1, p ≤ 2, r1 > 1 − ¹_q,

log^µ−1n n

r−¹_p+¹_q

(log n)^µ−1q : 2 ≤ p < q.

Proof. The upper bounds follow from the results for linear widths in [42]. The lower bounds are new. Malykhin and Ryutin proved in [76] the following bound on Kol-mogorov n-widths for finite dimensional normed spaces `<sup>M</sup><sub>p</sub> (`^N_q )

d_b^{N M}

2 c(`<sup>M</sup><sub>∞</sub>(`^N₁ ), `<sup>M</sup><sub>1</sub> (`^N₂ ))  M. (8.3.3) In the first case the technique for the lower bounds on linear widths presented in [42]

works well also for Gelfand n-widths. The discretization stated there yields c_n(S_p,∞^r B(T^d), L_q(T^d)) & 2^{u(−r+12−1q)}c_n(`<sup>u</sup><sub>∞</sub><sup>µ−1</sup>(`²₂^u), `^u_q^µ−12u).

The duality relation in Lemma 8.17 gives

cn(S_p,∞^r B(T^d), Lq(T^d)) & 2^{u(−r+12−1q)}dn(`<sup>u</sup><sub>q</sub><sup>0µ−12u</sup>, `^u₁^µ−1(`²₂^u)).

Applying H¨older’s inequality in finite dimensional spaces `<sup>M</sup><sub>p</sub> (`^N_q ) yields the following estimate

c_n(S_p,∞^r B(T^d), L_q(T^d)) & 2^{u(−r+12−1q)}u⁻

µ−1

q0 d_n(`<sup>u</sup><sub>∞</sub><sup>µ−1</sup>(`²₁^u), `<sup>u</sup><sub>1</sub><sup>µ−1</sup>(`²₂^u)).

Choosing n  u^µ−12^u then the relation in (8.3.3) implies

c_n(S_p,∞^r B(T^d), L_q(T^d)) & 2^{u(−r+12−1q)}u^µ−1q  log^µ−1n n

^r−1₂⁺¹_q

(log n)^µ−1q . The second case is obtained by the embedding

S^r−(

1 2−¹

p)

2,∞ B(T^d) ,→ S_p,∞^r B(T^d) together with the result from the first case.

Corollary 8.20. Let 2 ≤ p < q < ∞ and r > ¹_p fulfilling (8.1.1). Then (i)

%_n(S_p^rW (T^d), L_q(T^d))  c_n(S_p^rW (T^d), L_q(T^d))  %^lin_n (S_p^rW (T^d), L_q(T^d))

 λ_n(S_p^rW (T^d), L_q(T^d))  (n⁻¹log^µ−1n)^r1−1p+1q, (ii)

%_n(S_p,∞^r B(T^d), L_q(T^d))  c_n(S_p,∞^r B(T^d), L_q(T^d))  %^lin_n (S_p,∞^r B(T^d), L_q(T^d))

 λ_n(S_p,∞^r B(T^d), L_q(T^d))

 (n⁻¹log^µ−1n)^r1−1p+1q(log^µ−1q n), holds for all n ∈ N.

Proof. The proof follows by Theorems 8.3, 8.6, 8.19 and Corollary 8.18.

Remark 8.21. In the parameter range 2 < p < q < ∞ permitting non-linear re-construction operators does not yield better results. Optimal rates can be achieved by completely linear sampling algorithms.

We obtain the following counterpart of Theorem 8.10 for non-linear sampling.

Corollary 8.22. Let 1 < p < 2 < q < ∞ and r > max{¹_p, 1 − ¹_q} fulfilling (8.1.1).

Additionally let F denote either S_p^rW (T^d) or S_p,∞^r B(T^d). Then cn(F , Lq(T^d)) = o(%n(F , Lq(T^d))), or more precisely

cn(F , Lq(T^d))  n^{−(r1−1p+1q)}. %ⁿ(F , Lq(T^d)) holds for all n ∈ N.

Proof. The proof can be obtained by following the construction of the lower bound for the univariate situation in [83], where we recognize that the stronger inequality

%_n(F , L_q(T^d)) ≥ inf

(ξk)ⁿ_k=1⊂T^d sup

kf |F k≤1 f (ξ_k)=0, k=1,...,n

kf |L_q(T^d)k

holds. The estimates for cn(S_p^rW (T^d), Lq(T^d)) were obtained in Corollary 8.18. For S_p,∞^r B(T^d) we refer to Theorem 8.19. Gelfand numbers for more general Besov spaces were studied in [80].

Remark 8.23. As a consequence of the lower bound in Corollary 8.22 for %_n(F , L_q(T^d)), we obtain that in the parameter range 1 < p < 2 < q < ∞ even linear approximation behaves significantly better than sampling recovery with a possibly non-linear recon-structing operator.

Chapter 9

Outlook and open problems

We discuss some research aspects and questions that were left open at the end of our studies and require further research.

9.1 Sampling: same integrability in target and source space

Using a trigonometric sparse grid sampling operator Temlyakov [117] proved for r > ¹_p, 1 < p < ∞ that

%^lin_n (S_p,∞^r B(T^d), L_p(T^d)) . (n⁻¹log^d−1n)^r(log^d−1n) (9.1.1) holds. Later, Sickel [102, 103] contributed to the 2-dimensional case and Sickel, Ullrich [104] for general d > 1 with 1 ≤ θ ≤ ∞ the (best) today known upper bounds

%^lin_n (S_p,θ^r B(T^d), L_p(T^d)) . (n⁻¹log^d−1n)^r(log^d−1n)^1−1θ, r > 1 p,

%^lin_n (S_p^rW (T^d), L_p(T^d)) . (n⁻¹log^d−1n)^r(log^d−1n)¹², r > maxn1 p,1

2 o

. (9.1.2) The upper bounds in (9.1.1) and (9.1.2) have in common that the sharp estimates for linear widths λn (defined in (8.2.1))

λ_n(S_p,∞^r B(T^d), L_p(T^d))  (n⁻¹log^d−1n)^r(log^d−1n)¹², p ≥ 2, cf. [116]

λn(S_p^rW (T^d), Lp(T^d))  (n⁻¹log^d−1n)^r, 1 < p < ∞, cf. Theorem D.24 do not coincide with the estimates for %^lin_n , which are typically used to obtain lower bounds for %^lin_n . A logarithmic gap appears. In fact, it is unknown whether linear approximation based on information generated by general linear functionals behaves better as linear approximation by sampling values. As a consequence of Chapter 5 (Theorem 5.14) we know that linear operators which sample functions on sparse grids

behave worse compared to approximation with general linear information. For general point sets we have no indication concerning this phenomenon. Considering the limiting case r = 2 Bungartz, Griebel [8] proved for the Faber-Schauder sparse grid operator I_M defined in (5.1.1) the convergence rate

kf − I_Mf |L₂([0, 1]^dk . M^d−12^−2Mkf |S_p²W ([0, 1]^d)k

 (n⁻¹log^d−1n)²log^d−1nkf |S_p²W ([0, 1]^d)k,

see also Theorem 5.7. The method for the lower bound in Theorem 5.14 allows to prove for the sparse grid width

g_n^SG(S₂²W ([0, 1]^d), L₂[(0, 1)]^d) & (n⁻¹log^d−1n)²log^d−12 n.

In fact there is a gap of log^d−12 n for the knowledge of the exact asymptotic approxi-mation rate of IM. It would be interesting to know whether the limited regularity of the hat functions causes a little worse approximation rate in the limiting case.