Tractability of approximation in the weighted Korobov space in the worst-case setting -- a complete picture
aa r X i v : . [ m a t h . NA ] F e b Tractability of approximation in the weighted Korobovspace in the worst-case setting — a complete picture
Adrian Ebert and Friedrich Pillichshammer ∗ Abstract
In this paper, we study tractability of L -approximation of one-periodic functionsfrom weighted Korobov spaces in the worst-case setting. The considered weightsare of product form. For the algorithms we allow information from the class Λ all consisting of all continuous linear functionals and from the class Λ std , which onlyconsists of function evaluations.We provide necessary and sufficient conditions on the weights of the functionspace for quasi-polynomial tractability, uniform weak tractability, weak tractabilityand ( σ, τ )-weak tractability. Together with the already known results for strongpolynomial and polynomial tractability, our findings provide a complete picture ofthe weight conditions for all current standard notions of tractability. Keywords: L -approximation; tractability; Korobov space. MSC 2020:
We study tractability of L -approximation of multivariate one-periodic functions fromweighted Korobov spaces of finite smoothness α in the worst-case setting. This problemhas already been studied in a vast number of articles and a lot is known for the twoinformation classes Λ all and Λ std , in particular for the primary notions of strong polynomialand polynomial tractability, but also for weak tractability; see, e.g., [4, 5, 12, 13] andalso the books [7, 9]. However, there are also some newer tractability notions such asquasi-polynomial tractability (see [1]), ( σ, τ )-weak tractability (see [11]) or uniform weaktractability (see [10]) which have not yet been considered for the approximation problemfor weighted Korobov spaces. Indeed, in [9, Open Problem 103] Novak and Wo´zniakowskiasked for appropriate weight conditions that characterize quasi-polynomial tractability.It is the aim of the present paper to close this gap and to provide matching necessaryand sufficient conditions for quasi-polynomial, ( σ, τ )-weak and uniform weak tractability ∗ The authors are supported by the Austrian Science Fund (FWF), Projects F5506-N26 (Ebert) andF5509-N26 (Pillichshammer), which are parts of the Special Research Program “Quasi-Monte CarloMethods: Theory and Applications”. all and Λ std , and therefore to extend and complete the alreadyknown picture regarding tractability of L -approximation in weighted Korobov spaces. Inparticular, we show that for the information class Λ all the notions of quasi-polynomialtractability, uniform weak tractability and weak tractability are equivalent and any ofthese holds if and only if the weights become eventually less than one (see Theorem 1).For the class Λ std we show that polynomial tractability and quasi-polynomial tractabilityare equivalent and additionally provide matching sufficient and necessary conditions forthe considered notions of weak tractability (see Theorem 3).The remainder of this article is organized as follows. In Section 2, we recall theunderlying function space setting of weighted Korobov spaces with finite smoothness andprovide the basics about L -approximation for such spaces. Furthermore, we give thedefinitions of the considered tractability notions. The obtained results are presented inSection 3. Finally, the corresponding proofs can be found in Section 4. Function space setting
The Korobov space H s,α, γ with weight sequence γ = ( γ j ) j ≥ ∈ R N is a reproducing kernelHilbert space with kernel function K s,α, γ : [0 , s × [0 , s → R given by K s,α, γ ( x , y ) := X k ∈ Z s r s,α, γ ( k ) exp(2 π i k · ( x − y ))and corresponding inner product h f, g i s,α, γ := X k ∈ Z s r s,α, γ ( k ) b f ( k ) b g ( k ) and k f k s,α, γ = q h f, f i s,α, γ . Here, the Fourier coefficients are given by b f ( k ) = Z [0 , s f ( x ) exp( − π i k · x ) d x and the used decay function equals r s,α, γ ( k ) = Q sj =1 r α,γ j ( k j ) with α > r α,γ ( k ) := ( k = 0 , γ | k | α for k ∈ Z \ { } . The kernel K s,α, γ is well defined for α > x , y ∈ [0 , s , since | K s,α, γ ( x , y ) | ≤ X k ∈ Z s r s,α, γ ( k ) = s Y j =1 (1 + 2 ζ ( α ) γ j ) < ∞ , where ζ ( · ) is the Riemann zeta function (note that α > ζ ( α ) < ∞ ).Furthermore, we assume in this article that the weights satisfy 1 ≥ γ ≥ γ ≥ · · · ≥ pproximation in the weighted Korobov space We consider the operator APP s : H s,α, γ → L ([0 , s ) with APP s ( f ) = f for all f ∈ H s,α, γ .In order to approximate APP s with respect to the L -norm k · k L over [0 , s , we willemploy linear algorithms A n,s that use n information evaluations and are of the form A n,s ( f ) = n X i =1 T i ( f ) g i for f ∈ H s,α, γ (1)with functions g i ∈ L ([0 , s ) and bounded linear functionals T i ∈ H ∗ s,α, γ for i = 1 , . . . , n .We will assume that the considered functionals T i belong to some permissible class ofinformation Λ. In particular, we study the class Λ all consisting of the entire dual space H ∗ s,α, γ and the class Λ std , which consists only of point evaluation functionals. Rememberthat H s,α, γ is a reproducing kernel Hilbert space, which means that point evaluations arecontinuous linear functionals and therefore Λ std is a subclass of Λ all .The worst-case error of an algorithm A n,s as in (1) is then defined as e ( A n,s ) := sup f ∈H s,α, γ k f k s,α, γ ≤ k APP s ( f ) − A n,s ( f ) k L and the n -th minimal worst-case error with respect to the information class Λ is given by e ( n, APP s ; Λ) := inf A n,s ∈ Λ e ( A n,s ) . We are interested in how the approximation error of algorithms A n,s depends on thenumber of used information evaluations n and how it depends on the problem dimension s .To this end, we define the so-called information complexity as n ( ε, APP s ; Λ) := min { n ∈ N : e ( n, APP s ; Λ) ≤ ε } with ε ∈ (0 ,
1) and s ∈ N . We note that it is well known and easy to see that the initialerror equals one for the considered problem and therefore there is no need to distinguishbetween the normalized and the absolute error criterion. Notions of tractability
In order to characterize the dependency of the information complexity on the dimension s and the error threshold ε , we will study several notions of tractability which are given inthe following definition. Definition 1.
Consider the approximation problem
APP = (APP s ) s ≥ for the informa-tion class Λ . We say we have: (a) Polynomial tractability (PT) if there exist non-negative numbers τ, σ, C such that n ( ε, APP s ; Λ) ≤ C ε − τ s σ for all s ∈ N , ε ∈ (0 , . Strong polynomial tractability (SPT) if there exist non-negative numbers τ, C suchthat n ( ε, APP s ; Λ) ≤ C ε − τ for all s ∈ N , ε ∈ (0 , . (2) The infimum over all exponents τ ≥ such that (2) holds for some C ≥ is calledthe exponent of strong polynomial tractability and is denoted by τ ∗ (Λ) . (c) Weak tractability (WT) if lim s + ε − →∞ ln n ( ε, APP s ; Λ) s + ε − = 0 . (d) Quasi-polynomial tractability (QPT) if there exist non-negative numbers t, C suchthat n ( ε, APP s ; Λ) ≤ C exp( t (1 + ln s )(1 + ln ε − )) for all s ∈ N , ε ∈ (0 , . (3) The infimum over all exponents t ≥ such that (3) holds for some C ≥ is calledthe exponent of quasi-polynomial tractability and is denoted by t ∗ (Λ) . (e) ( σ, τ ) -weak tractability (( σ, τ )-WT) if there exist positive σ, τ such that lim s + ε − →∞ ln n ( ε, APP s ; Λ) s σ + ε − τ = 0 . (f) Uniform weak tractability (UWT) if ( σ, τ ) -weak tractability holds for all σ, τ ∈ (0 , . We obviously have the following hierarchy of tractability notions:SPT ⇒ PT ⇒ QPT ⇒ UWT ⇒ ( σ, τ )-WT for all ( σ, τ ) ∈ (0 , . Furthermore, WT coincides with ( σ, τ )-WT for ( σ, τ ) = (1 , Here we state our results about quasi-polynomial-, weak- and uniform weak tractability ofapproximation in the weighted Korobov space H s,α, γ for information from Λ all . In orderto provide a complete picture of all instances at a glance, we also include the alreadyknown results for (strong) polynomial tractability which were first proved by Wasilkowskiand Wo´zniakowski in [12]. Theorem 1.
Consider the approximation problem
APP = (APP s ) s ≥ for the informationclass Λ all and let α > . Then we have the following conditions: . (Cf. [12]) Strong polynomial tractability for the class Λ all holds if and only if s γ < ∞ ,where for γ = ( γ j ) j ≥ the sum exponent s γ is defined as s γ = inf ( κ > ∞ X j =1 γ κj < ∞ ) , with the convention that inf ∅ = ∞ . In this case the exponent of strong polynomialtractability is τ ∗ (Λ all ) = 2 max (cid:18) s γ , α (cid:19) .
2. (Cf. [12]) Strong polynomial tractability and polynomial tractability for the class Λ all are equivalent.3. Quasi-polynomial tractability, uniform weak tractability and weak tractability for theclass Λ all are equivalent and hold if and only if the weights ( γ j ) j ≥ become eventuallyless than 1, i.e., if and only if there exists some index j ∈ N such that γ = · · · = γ j = 1 and γ j +1 < . Setting γ ∗ := γ j +1 , we then have γ j ≤ γ ∗ < for all j > j .4. If we have quasi-polynomial tractability, then the exponent of quasi-polynomial tractabil-ity equals t ∗ (Λ all ) = 2 max (cid:18) α , γ − ∗ (cid:19) , where γ ∗ is the first weight in the weight sequence γ = ( γ j ) j ≥ that is strictly lessthan . If γ ∗ = 0 , then we set (ln γ − ∗ ) − := 0 . Remark 2.
We remark that in [7] a different formulation of the necessary and sufficientcondition for weak tractability is given. In particular, according to [7, Theorem 5.8] theapproximation problem APP = (APP s ) s ≥ for Λ all is weakly tractable if and only iflim s + ε − →∞ k ( ε, s, γ ) s + ε − = 0 , (4)where k ( ε, s, γ ) is defined as the element k ∈ { , . . . , s } such that k Y j =1 γ j > ε and k +1 Y j =1 γ j ≤ ε . If such a k does not exist, we set k ( ε, s, γ ) = s . In the following, we show that thiscondition is equivalent to our condition that the weights γ j become eventually less than 1.Assume that there exists an index j ∈ N such that γ = · · · = γ j = 1 and γ j +1 =: γ ∗ <
1. Then we see that for k > j we have k +1 Y j =1 γ j = k +1 Y j = j +1 γ j ≤ k +1 Y j = j +1 γ ∗ = γ k − j +1 ∗ . ε > s is large enough such that there exists a k ∗ ∈ { , . . . , s } with γ k ∗ − j +1 ∗ ≤ ε . This implies in particular that k ( ε, s, γ ) ≤ k ∗ .Elementary transformations show that the inequality γ k ∗ − j +1 ∗ ≤ ε is equivalent to k ∗ ≤ ε − ln γ − ∗ + j − . Therefore, we obtain thatlim s + ε − →∞ k ( ε, s, γ ) s + ε − ≤ lim s + ε − →∞ k ∗ s + ε − ≤ lim s + ε − →∞ ε − ln γ − ∗ + j − s + ε − = 0and thus the condition in (4) is satisfied.On the other hand, assume that (4) is satisfied but γ j = 1 for all j ∈ N . Then,according to the definition we obviously have that k ( ε, s, γ ) = s for all ε ∈ (0 , ε ∈ (0 ,
1) thatlim s →∞ k ( ε, s, γ ) s + ε − = lim s →∞ ss + ε − = 1and this contradicts (4). Hence the γ j have to become eventually less than 1.In the next theorem we present the respective conditions for tractability of approxi-mation in the weighted Korobov space for the information class Λ std . In order to providea detailed overview, we also include the already known results for (strong) polynomialtractability, see, e.g., [5]. Theorem 3.
Consider multivariate approximation
APP = (APP s ) s ≥ for the informationclass Λ std and α > . Then we have the following conditions:1. (Cf. [5]) Strong polynomial tractability for the class Λ std holds if and only if ∞ X j =1 γ j < ∞ (which is equivalent to s γ ≤ ). In this case the exponent of strong polynomialtractability satisfies τ ∗ (Λ std ) = 2 max (cid:18) s γ , α (cid:19) .
2. (Cf. [5]) Polynomial tractability for the class Λ std holds if and only if lim sup s →∞ s s X j =1 γ j < ∞ .
3. Polynomial and quasi-polynomial tractability for the class Λ std are equivalent. . Weak tractability for the class Λ std holds if and only if lim s →∞ s s X j =1 γ j = 0 . (5)
5. Weak ( σ, τ ) -tractability for the class Λ std holds if and only if lim s →∞ s σ s X j =1 γ j = 0 . (6)
6. Uniform weak tractability for the class Λ std holds if and only if lim s →∞ s σ s X j =1 γ j = 0 for all σ ∈ (0 , . (7)The proofs of the statements in Theorems 1 and 3 are given in the next section.The results in Theorems 1 and 3 provide a complete characterization for tractability ofapproximation in the weighted Korobov space H s,α, γ with respect to all commonly studiednotions of tractability and the two information classes Λ all and Λ std . We summarize theconditions in a concise table (Table 1) below.Λ all Λ std SPT s γ < ∞ P ∞ j =1 γ j < ∞ PT s γ < ∞ lim sup s →∞ s P sj =1 γ j < ∞ QPT ∃ j ∈ N : γ j < ∀ j > j lim sup s →∞ s P sj =1 γ j < ∞ UWT ∃ j ∈ N : γ j < ∀ j > j lim s →∞ s σ P sj =1 γ j = 0 ∀ σ ∈ (0 , σ, τ )-WT ∃ j ∈ N : γ j < ∀ j > j lim s →∞ s σ P sj =1 γ j = 0WT ∃ j ∈ N : γ j < ∀ j > j lim s →∞ s P sj =1 γ j = 0Table 1: Overview of the conditions for tractability of approximation in H s,α, γ . In this section we present the proofs of Theorem 1 and Theorem 3.7 he information class Λ all It is commonly known that the n -th minimal worst-case errors e ( n, APP s ; Λ) are directlyrelated to the eigenvalues of the self-adjoint operator W s := APP ∗ s APP s : H s,α, γ → H s,α, γ . In the following lemma, we derive the eigenpairs of the operator W s . For this purpose, wedefine, for x ∈ [0 , s , k ∈ Z s , the vectors e k ( x ) = e k ,α, γ ( x ) := p r s,α, γ ( k ) exp(2 π i k · x ). Lemma 4.
The eigenpairs of the operator W s are ( r s,α, γ ( k ) , e k ) with k ∈ Z s . This result is well known; see, e.g., [7, p. 215]. For the sake of completeness we addthe following short proof.
Proof of Lemma 4.
We find that for any f, g ∈ H s,α, γ we have h APP s ( f ) , APP s ( g ) i L = h f, APP ∗ s APP s ( g ) i s,α, γ = h f, W s ( g ) i s,α, γ and hence, due to the orthonormality of the Fourier basis functions, h e k , W s ( e h ) i s,α, γ = h e k , e h i L = q r s,α, γ ( k ) r s,α, γ ( h ) δ k , h , where the Kronecker delta δ k , h is 1 if k = h , and 0 otherwise. For k = h this gives h e k , W s ( e k ) i s,α, γ = r s,α, γ ( k ) which in turn implies that W s ( e h ) = X k ∈ Z s h W s ( e h ) , e k i s,α, γ e k = r s,α, γ ( h ) e h and proves the lemma.In order to exploit the relationship between the eigenvalues of W s and the informationcomplexity, we define the set A ( ε, s ) := { k ∈ Z s : r s,α, γ ( k ) > ε } . It is commonly known (see [7]) that then the following identity holds n ( ε, APP s ; Λ all ) = |A ( ε, s ) | . We will use this fact also in the proof of Theorem 1, which is presented below.
Proof of Theorem 1.
We prove the necessary and sufficient conditions for each of the listednotions of tractability. For the sake of completeness, we also include the proofs for items1 and 2 of Theorem 1. 8. In order to give a necessary and sufficient condition for strong polynomial tractabilityfor Λ all , we use a criterion from [7, Section 5.1]. From [7, Theorem 5.2] we find thatthe problem APP is strongly polynomially tractable for Λ all if and only if thereexists a τ > s ∈ N X k ∈ Z s ( r α, γ ( k )) τ ! /τ < ∞ (8)and then τ ∗ (Λ all ) = inf { τ : τ satisfies (8) } . Assume that s γ < ∞ . Then take τ such that τ > max( s γ , α ) and thus P ∞ j =1 γ τj isfinite. For the sum in (8) we then obtain X k ∈ Z s ( r α, γ ( k )) τ = s Y j =1 ∞ X k = −∞ ( r α,γ j ( k )) τ ! = s Y j =1 γ τj ∞ X k =1 k ατ ! = s Y j =1 (cid:0) ζ ( ατ ) γ τj (cid:1) (9) ≤ exp ζ ( ατ ) ∞ X j =1 γ τj ! < ∞ , where we also used that τ > /α and hence ζ ( ατ ) < ∞ . This implies that we havestrong polynomial tractability and that τ ∗ (Λ all ) ≤ s γ , α ) . (10)On the other hand, assume we have strong polynomial tractability. Then thereexists a finite τ such that (8) holds true. From (9) we see that we obviously requirethat τ > α . Then, again using (9), we obtain that X k ∈ Z s ( r α, γ ( k )) τ = s Y j =1 (1 + 2 ζ ( ατ ) γ τj ) ≥ ζ ( ατ ) s X j =1 γ τj . Again, since (8) holds true, we require that P ∞ j =1 γ τj < ∞ and hence s γ < τ < ∞ .Combining both results yields that τ > max( s γ , α ) and hence also τ ∗ (Λ all ) ≥ s γ , α ) . (11)Equations (10) and (11) then imply that τ ∗ (Λ all ) = 2 max( s γ , α ).9. We use ideas from [12]. In order to prove the equivalence of strong polynomialtractability and polynomial tractability it suffices to prove that polynomial tractabil-ity implies strong polynomial tractability. So let us assume that APP is polynomiallytractable, i.e., there exist numbers C, p > q ≥ n ( ε, APP s , Λ all ) ≤ C s q ε − p for all ε ∈ (0 ,
1] and s ∈ N . Without loss of generality we may assume that q is an integer. Take s ∈ N suchthat s ≥ q + 1 and choose vectors k ∈ Z s with s − q − q + 1 components equal to 1. The total number of such vectors is (cid:0) sq +1 (cid:1) . Nowchoose ε ∗ = γ ( q +1) / s . Assume that k ∈ Z s is of the form as mentioned above anddenote by u ⊆ { , . . . , s } the set of indices of k which are equal to 1. Then we have r s,α, γ ( k ) = Y j ∈ u γ j ≥ γ q +1 s > ε ∗ . Hence all the (cid:0) sq +1 (cid:1) vectors k of the form mentioned above belong to A ( ε ∗ , s ) andthis implies that |A ( ε ∗ , s ) | ≥ (cid:18) sq + 1 (cid:19) ≥ ( s − q ) q +1 ( q + 1)! ≥ s q +1 ( q + 1)!( q + 1) q +1 =: s q +1 c q . This now yields s q +1 c q ≤ |A ( ε ∗ , s ) | = n ( ε ∗ , APP s ; Λ all ) ≤ C s q ε − p ∗ = 2 p C s q γ − ( q +1) p/ s , which in turn implies that γ ( q +1) p/ s ≪ p,q s and hence γ s ≪ p,q s / (( q +1) p ) . This estimate holds for all s ≥ q + 1. Hence the sum exponent s γ of the sequence γ = ( γ j ) j ≥ is finite, s γ < ∞ , and this implies by the first statement that we havestrong polynomial tractability.3. We use the following criterion for QPT taken from [9, Sec. 23.1.1] (see also [3]),which states that QPT holds if and only if there exists a τ > C := sup s ∈ N s ∞ X j =1 λ τ (1+ln s ) s,j ! /τ < ∞ , (12)where λ s,j is the j -th eigenvalue of the operator W s in non-increasing order.Assume that we have γ ∗ ∈ (0 ,
1) and j ∈ N such that γ = · · · = γ j = 1 and γ j ≤ γ ∗ < j > j . (13)10or the weighted Korobov space H s,α, γ we have by Lemma 4 that ∞ X j =1 λ τ (1+ln s ) s,j = X k ∈ Z s ( r s,α, γ ( k )) τ (1+ln s ) = s Y j =1 ∞ X k =1 ( r α,γ j ( k )) τ (1+ln s ) ! = s Y j =1 (cid:16) ζ ( ατ (1 + ln s )) γ τ (1+ln s ) j (cid:17) . In order that ζ s := ζ ( ατ (1 + ln s )) < ∞ for all s ∈ N , we need to require from nowon that τ > /α . Furthermore, we have that1 s ∞ X j =1 λ τ (1+ln s ) s,j ! /τ = 1 s s Y j =1 (cid:16) ζ s γ τ (1+ln s ) j (cid:17)! /τ = exp τ s X j =1 ln (cid:16) ζ s γ τ (1+ln s ) j (cid:17) − s ! ≤ exp τ ζ s s X j =1 γ τ (1+ln s ) j − s ! , where we used that ln(1 + x ) ≤ x for all x ≥
0. Now we use the well-known factthat ζ ( x ) ≤ x − for all x > ζ s ≤ ατ −
1) + ατ ln s . Then we obtain1 s ∞ X j =1 λ τ (1+ln s ) s,j ! /τ ≤ exp τ (cid:18) ατ −
1) + ατ ln s (cid:19) s X j =1 γ τ (1+ln s ) j − s ! . Using assumption (13) we obtain for every s ∈ N that s X j =1 γ τ (1+ln s ) j ≤ j + γ τ (1+ln s ) ∗ max( s − j , j + γ τ ∗ max( s − j , s τ ln γ − ∗ ≤ j + 1 , as long as τ ≥ (ln γ − ∗ ) − . Thus, if τ > /α and τ ≥ (ln γ − ∗ ) − we have1 s ∞ X j =1 λ τ (1+ln s ) s,j ! /τ ≤ exp (cid:18) τ (cid:18) ατ −
1) + ατ ln s (cid:19) ( j + 1) − s (cid:19) = exp( O (1)) < ∞ , s ∈ N . By the characterization in (12), this implies quasi-polynomialtractability. Of course, quasi-polynomial tractability implies uniform weak tractabil-ity, which in turn implies weak tractability.It remains to show that weak tractability implies condition (13). Assume on thecontrary that (13) does not hold, i.e., γ j = 1 for all j ∈ N . Then we have forall k ∈ {− , , } s that r s,α, γ ( k ) = 1. This means that for all ε ∈ (0 ,
1) we have {− , , } s ⊆ A ( ε, s ) and hence n ( ε, APP s ; Λ all ) ≥ s . This means that the ap-proximation problem suffers from the curse of dimensionality and, in particular, wecannot have weak tractability. This concludes the proof of item 3.4. Again from [9, Theorem 23.2] we know that the exponent of quasi-polynomialtractability is t ∗ (Λ all ) = 2 inf { τ : τ for which (12) holds } . From the above part of the proof we already know that τ satisfies (12) as longas τ > /α and τ ≥ (ln γ − ∗ ) − , where we put (ln γ − ∗ ) − := 0 whenever γ ∗ = 0.Therefore, t ∗ (Λ all ) ≤ (cid:18) α , γ − ∗ (cid:19) . Assume now that we have quasi-polynomial tractability. Then (12) holds true forsome τ >
0. Considering the special instance s = 1 this means C ≥ ∞ X j =1 λ τ ,j ! /τ = (1 + 2 ζ ( ατ ) γ τ ) /τ and hence we must have τ > /α .Now, again according to (12), there exists a τ > /α such that for all s ∈ N we have C ≥ s s Y j =1 (cid:16) ζ ( ατ (1 + ln s )) γ τ (1+ln s ) j (cid:17)! /τ ≥ exp τ s X j =1 ln (cid:16) γ τ (1+ln s ) j (cid:17) − s ! . Taking the logarithm leads toln C ≥ τ s X j =1 ln (cid:16) γ τ (1+ln s ) j (cid:17) − s for all s ∈ N . According to item 3, quasi-polynomial tractability forces the weights γ j to be of the form (13). Therefore we also have QPT for the weight sequence(1 , , . . . , | {z } j times , γ ∗ , γ ∗ , . . . ) . s ≥ j thatln C ≥ τ ( s − j ) ln (cid:0) γ τ (1+ln s ) ∗ (cid:1) − s. Since γ ∗ ∈ (0 ,
1) and since ln(1 + x ) ≥ x ln 2 for all x ∈ [0 , s ≥ j we haveln C ≥ ( s − j ) ln 2 τ γ τ (1+ln s ) ∗ − s = γ τ ∗ ( s − j ) ln 2 τ s τ ln γ − ∗ − s. This implies that τ ≥ (ln γ − ∗ ) − . Therefore, we also have that t ∗ (Λ all ) ≥ (cid:18) α , γ − ∗ (cid:19) and the claimed result follows. The information class Λ std Below, we provide the remaining proof of Theorem 3.
Proof of Theorem 3.
The necessary and sufficient conditions for polynomial and strongpolynomial tractability (items 1 and 2) have already been proved in [5]. See also [7,p. 215ff.], where the exact exponent of strong polynomial tractability τ ∗ (Λ std ) is given.We will therefore only provide proofs for items 3 to 6.We start with a preliminary remark about the relation between integration and approx-imation. It is well known that multivariate approximation is not easier than multivariateintegration INT s ( f ) = R [0 , s f ( x ) d x for f ∈ H s,α, γ , see, e.g., [5]. In particular, necessaryconditions for some notion of tractability for the integration problem are also necessaryfor the approximation problem. We will use this basic observation later on. Now wepresent the proof of item 3.3. Obviously, it suffices to prove that quasi-polynomial tractability implies polynomialtractability. Assume therefore that quasi-polynomial tractability for the class Λ std holds for approximation. Then we also have quasi-polynomial tractability for theintegration problem. Now we apply [8, Theorem 16.16] which states that integrationis T -tractable if and only iflim sup s + ε − →∞ P sj =1 γ j + ln ε − T ( ε − , s ) < ∞ . (14)We do not require the definition of T -tractability here (see, e.g., [7, p. 291]). For ourpurpose it suffices to know that the special case T ( ε − , s ) = exp((1+ln s )(1+ln ε − ))corresponds to quasi-polynomial tractability. But for this instance condition (14) isequivalent to the criterion lim sup s →∞ s s X j =1 γ j < ∞ . (15)13rom item 2, we know that condition (15) implies polynomial tractability and thiscompletes the proof of item 3.For the remaining conditions in items 4 to 6, note that since α > W s ,denoted by trace( W s ), is finite for all s ∈ N . Indeed, we havetrace( W s ) = X k ∈ Z s r α, γ ( k ) = s Y j =1 (1 + 2 γ j ζ ( α )) < ∞ . (16)In this case, we can use relations between notions of tractability for Λ all and Λ std whichwere first proved in [13] (see also [9, Section 26.4.1]).4.-6. We prove the three statements in one combined argument. If any of the threeconditions (5), (6) or (7) holds, then this implies that the weights ( γ j ) j ≥ have tobecome eventually less than 1 since otherwise, for every σ ∈ (0 , s →∞ s σ s X j =1 γ j = lim s →∞ ss σ = lim s →∞ s − σ ≥ . Therefore, we have by Theorem 1 that uniform weak tractability (and even quasi-polynomial tractability) holds for the class Λ all . Furthermore, from (16) we obtainln(trace( W s )) s σ = 1 s σ ln s Y j =1 (1 + 2 γ j ζ ( α )) ! = 1 s σ s X j =1 ln(1 + 2 γ j ζ ( α )) ≤ ζ ( α ) s σ s X j =1 γ j , and thus if s σ P sj =1 γ j converges to 0 as s goes to infinity, with σ ∈ (0 , s →∞ ln(trace( W s )) s σ ≤ lim s →∞ ζ ( α ) s σ s X j =1 γ j = 0 . By the same argument as in the proof of [9, Theorem 26.11], we obtain that (5)implies weak tractability for the class Λ std . The proof for the other two notionsof weak tractability can be obtained analogously by appropriately modifying theargument used in the proof of [9, Theorem 26.11].It remains to prove the necessary conditions for the three notions of weak tractabil-ity. From our preliminary remark we know that necessary conditions on tractabilityfor integration are also necessary conditions for approximation. Hence it suffices tostudy integration INT s .Due to, e.g., [14], we know that weak tractability of integration for H s,α, γ holds ifand only if lim s →∞ s s X j =1 γ j = 014nd thus this is also a necessary condition for weak tractability of approximation.We are left to prove the necessity of the respective conditions for uniform weaktractability and ( σ, τ )-weak tractability for integration. These follow from a similarapproach as used in [14] for weak tractability. We just sketch the argument whichis more or less an application and combination of results from [2] and [6].In [2, Theorem 4.2] Hickernell and Wo´zniakowski showed that integration in a suit-ably constructed weighted Sobolev space H Sob s,r, b γ of smoothness r = ⌈ α/ ⌉ and withproduct weights b γ is no harder than in the weighted Korobov space H s,α, γ . Theproduct weights of the Korobov and Sobolev spaces are related by γ j = b γ j G r witha multiplicative non-negative factor G r . Hence, it suffices to study necessary con-ditions for tractability of integration in H Sob s,r, b γ . To this end we proceed as in [2,Section 5].The univariate reproducing kernel K , b γ of H Sob1 ,r, b γ (case s = 1) can be decomposed as K , b γ = R + b γ ( R + R ) , where each R j is a reproducing kernel of a Hilbert space H ( R j ) of univariate func-tions. In our specific case, we have R = 1 and H ( R ) = span(1) (cf. [2, p. 679]). Itis then shown in [2, Section 5] that all requirements of [6, Theorem 4] are satisfied.For the involved parameter α , we have α = k h , k H ( R ) = 1 (this is easily shown,since R = 1). Furthermore, we have that the parameter α in [6, Theorem 4] (notto be confused with the smoothness parameter α of the Korobov space) satisfies α ∈ [1 / , h , , (0) = 0 and h , , (1) = 0, as shown in [2, p. 681] (where h , , ( j ) is called η , , ( j ) for j ∈ { , } ). In order to avoid any misunderstanding, we denotethe α in [6, Theorem 4] by e α from now on. Then, we apply [6, Theorem 4] and ob-tain for the squared n -th minimal integration error in the considered Sobolev spacethat e ( n, INT s ) ≥ X u ⊆{ ,...,s } (1 − n e α | u | ) + α | u | Y j ∈ u b γ j Y j u (1 + b γ j α ) , where α , α are positive numbers (cf. [6, p. 425]) and ( x ) + := max( x, e ( n, INT s ) ≥ X u ⊆{ ,...,s } (1 − n e α | u | ) α | u | Y j ∈ u b γ j = s Y j =1 (1 + α b γ j ) − n s Y j =1 (1 + α e α b γ j ) , which in turn yields that n ( ε, INT s ) ≥ Q sj =1 (1 + α b γ j ) − ε Q sj =1 (1 + α e α b γ j ) . n ( ε, INT s ) ≥ ln s Y j =1 (1 + α b γ j ) ! + ln − ε Q sj =1 (1 + α b γ j ) ! − ln s Y j =1 (1 + α e α b γ j ) ! ≥ s X j =1 ln(1 + α b γ j ) − α e α s X j =1 b γ j + ln(1 − ε ) , where we used that ln(1 + x ) ≤ x for any x ∈ R .Recall that e α < c := (1 + e α ) /
2. Then c ∈ ( e α,
1) and sincelim x → ln(1 + x ) x = 1 , it follows that ln(1 + x ) ≥ cx for sufficiently small x > σ, τ )-weak tractability for integration in the consideredSobolev space. Then the weights b γ j necessarily tend to zero for j → ∞ (see [6,Theorem 4, Item 4]). In particular, there exists an index j >
0, such that for all j ≥ j we have ln(1 + α b γ j ) ≥ c α b γ j . Hence for s ≥ j , we haveln n ( ε, INT s ) ≥ α ( c − e α ) s X j = j b γ j + ln(1 − ε ) + O (1) . Note that c − e α >
0. Since we assume ( σ, τ )-weak tractability, we have that0 = lim s + ε − →∞ ln n ( ε, INT s ) s σ + ε − τ ≥ lim s + ε − →∞ α ( c − e α ) P sj = j b γ j + ln(1 − ε ) s σ + ε − τ . This, however, implies that lim s →∞ s σ s X j =1 b γ j = 0 , and thus, since γ j and b γ j only differ by a multiplicative factor, thatlim s →∞ s σ s X j =1 γ j = 0 . Now the claimed results follow. 16 eferences [1] M. Gnewuch, H. Wo´zniakowski.
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Adrian EbertJohann Radon Institute for Computational and Applied Mathematics (RICAM)Austrian Academy of SciencesAltenbergerstr. 69, 4040 Linz, AustriaE-mail: [email protected]
Friedrich PillichshammerInstitut f¨ur Finanzmathematik und Angewandte ZahlentheorieJohannes Kepler Universit¨at LinzAltenbergerstr. 69, 4040 Linz, AustriaE-mail: [email protected]@jku.at