Digital inversive vectors can achieve strong polynomial tractability for the weighted star discrepancy and for multivariate integration
Josef Dick, Domingo Gomez-Perez, Friedrich Pillichshammer, Arne Winterhof
aa r X i v : . [ m a t h . NA ] D ec Digital inversive vectors can achieve strongpolynomial tractability for the weighted stardiscrepancy and for multivariate integration
Josef Dick, Domingo Gomez-Perez,Friedrich Pillichshammer, Arne Winterhof ∗ In memory of Joseph Frederick Traub (1932–2015)andOscar Moreno de Ayala (1946–2015)
Abstract
We study high-dimensional numerical integration in the worst-case setting.The subject of tractability is concerned with the dependence of the worst-caseintegration error on the dimension. Roughly speaking, an integration problem istractable if the worst-case error does not explode exponentially with the dimen-sion. Many classical problems are known to be intractable. However, sometimestractability can be shown. Often such proofs are based on randomly selected inte-gration nodes. Of course, in applications true random numbers are not availableand hence one mimics them with pseudorandom number generators. This mo-tivates us to propose the use of pseudorandom vectors as underlying integrationnodes in order to achieve tractability. In particular, we consider digital inversevectors and present two examples of problems, the weighted star discrepancy andintegration of H¨older continuous, absolute convergent Fourier- and cosine series,where the proposed method is successful.
Keywords:
Weighted star discrepancy, pseudo-random numbers, tractability, quasi-Monte Carlo.
MSC 2000:
We study numerical integration of multivariate functions f defined on the s -dimensionalunit cube by means of quasi-Monte Carlo (QMC) rules, that is, I s ( f ) := Z [0 , s f ( x ) d x ≈ N N − X n =0 f ( x n ) =: Q N ( f ) , ∗ The research of J. Dick was supported under the Australian Research Councils Discovery Projectsfunding scheme (project number DP150101770). The research of D. Gomez-Perez is supported by theMinisterio de Economia y Competitividad research project MTM2014-55421-P. The last two authorsare supported by the Austrian Science Fund (FWF): Projects F5509-N26 (Pillichshammer) and F5511-N26 (Winterhof), respectively, which are part of the Special Research Program “Quasi-Monte CarloMethods: Theory and Applications”. x , . . . , x N − are well chosen integration nodes from the unitcube and where Q N is the QMC-rule based on these nodes. General references for QMCintegration are [5, 6, 15, 18].Usually one studies integrands from a given Banach space ( F , k · k F ) of functions.As quality criterion we consider the worst-case error e ( Q N , F ) = sup f ∈Fk f kF ≤ | I s ( f ) − Q N ( f ) | . For many function classes the problem of QMC integration is well-studied and onecan achieve optimal asymptotic convergence rates for the worst-case error which are of-ten of the form O ( N − ε ) for ε >
0, with some implicit dependence on the dimension s .Although this is the best possible convergence rate in N , the dependence on the dimen-sion can be crucial if s is large. This question is the subject of tractability. Tractabilitymeans that we control the dependence of the worst-case error on the dimension. Tractability.
For the numerical integration problem in F the N th minimal worst-caseerror is defined as e ( N, s ) = inf A N,s sup f ∈Fk f kF≤ | I s ( f ) − A N,s ( f ) | , where the infimum is extended over all integration rules (not necessarily QMC rules)which are based on N function evaluations f ( x n ), n = 0 , , . . . , N −
1. For ε ∈ (0 , information complexity N ( ε, s ) is then defined as the minimal number of functionevaluations which are required in order to achieve a worst-case error of at most ε . Inother words , N ( ε, s ) = min { N ∈ N : e ( N, s ) ≤ ε } . We say that we have: • The curse of dimensionality if there exist positive numbers C , τ and ε such that N ( ε, s ) ≥ C (1 + τ ) s for all ε ≤ ε and infinitely many s ∈ N . • Polynomial tractability if there exist non-negative numbers
C, τ , τ such that N ( ε, s ) ≤ Cs τ ε − τ for all s ∈ N , ε ∈ (0 , . • Strong polynomial tractability if there exist non-negative numbers C and τ suchthat N ( ε, s ) ≤ Cε − τ for all s ∈ N , ε ∈ (0 , . The exponent τ ∗ of strong polynomial tractability is defined as the infimum of τ for which strong polynomial tractability holds. Here we speak about the absolute error criterion. Sometimes one uses the so-called initial error e (0 , s ) = k I s k as a reference value and asks for the minimal N in order to reduce this minimal error bya factor of ε . In this case one speaks about the normalized error criterion. weak tractability , which means thatlim ε − + s →∞ log N ( ε, s ) ε − + s = 0 . Problems that are not weakly tractable (that is, the information complexity dependsexponentially on ε − or s ) are said to be intractable . The current state of the artin tractability theory is very well summarized in the three volumes of Novak andWo´zniakowski [20, 21, 22].It is known that many multivariate problems defined over standard spaces of func-tions suffer from the curse of dimensionality, as for example, integration of Lipschitzfunctions, monotone functions, convex functions (see [11]), or smooth functions (see[9, 10]). The reason for this disadvantageous behavior may be found in the fact that forstandard spaces all variables and groups of variables are equally important. As a wayout, Sloan and Wo´zniakowski [24] suggested to consider weighted spaces, in which theimportance of successive variables and groups of variables is monitored by correspond-ing weights, to vanquish the curse of dimensionality and to obtain polynomial or evenstrong polynomial tractability, depending on the decay of the weights.It is possible to construct QMC rules for some cases which achieve the one or othernotion of tractability, for example (polynomial) lattice rules for integration in weightedKorobov spaces or Sobolev spaces. In such cases the point sets heavily depend on thechosen weights and can generally not be used for other weights. A further disadvantageis, that there is in general no explicit construction for point sets which can achievetractability error bounds and thus one relies on computer search algorithms (for example,the fast component-by-component constructions, see [5, 15] and the references therein).On the other hand, for those instances of problems which are tractable this prop-erty is often proved with randomly selected point sets. A particular example is the“inverse of star discrepancy problem” for which Heinrich, Novak, Wasilkowski, andWo´zniakowski [12] showed with the help of random point sets that the star discrep-ancy is polynomially tractable (see also [1]). In [2] it is even shown that with veryhigh probability (say 99%), a randomly selected point set satisfies the aforementionedbounds. However, if we generate a random point set on a computer using a pseudoran-dom number generator, this result does not apply since the pseudo-random numbers aredeterministically constructed. Thus a fundamental question is whether known pseudo-random generators can be used to generate point sets which satisfy discrepancy boundswhich imply polynomial tractability.Thus in this paper we consider point sets generated by a certain pseudorandomnumber generator as candidates for point sets which achieve tractability for certainproblems. First attempts in this direction have been made in [4, 8] where so-called p -sets are used (see also the survey [7]).In the present paper we consider another choice of pseudorandom numbers obtainedfrom explicit inversive pseudorandom number generators. We show that such pointsets can be used to achieve tractability for two problems, namely the weighted stardiscrepancy problem (Section 3) and integration of functions from a subclass of theWiener algebra which has some additional smoothness properties (Section 4).In the subsequent section we introduce the proposed pseudorandom vectors andprove an estimate on an exponential sum from which we can derive discrepancy- andworst-case error bounds. 3 Explicit inversive vectors
Let F q be the finite field of order q = p k with a prime p and an integer k ≥
1. Furtherlet { β , . . . , β k } be an ordered basis of F q over F p .From a finite vector set in F sq { z n = ( z n, , z n, , . . . , z n,s ) ∈ F sq : n = 0 , , . . . , N − } , we can derive a point set in the s -dimensional unit interval. More precisely, if z n = c (1) n β + c (2) n β + · · · + c ( k ) n β k (1)with all c ( j ) n = ( c ( j ) n, , c ( j ) n, , . . . , c ( j ) n,s ) ∈ F sp , then we define an s -dimensional digital point set P s = ( x n = k X j =1 c ( j ) n p − j ∈ [0 , s : n = 0 , , . . . , N − ) . (2)The following point set was essentially introduced in [19]. Definition 1 (Set of explicit inversive points of size q ) . Put z = (cid:26) z − if z ∈ F ∗ q , z = 0 . For 1 ≤ s ≤ q we choose a subset S ⊆ F q of cardinality s . We consider the vector set S = { z , . . . , z q − } = { ( u + v ) v ∈ S : u ∈ F q } ⊂ F sq (3)of size q and derive P s = { x , . . . , x q − } ∈ [0 , s from S by (1) and (2). Note that here N = |P s | = q .Our second point set was essentially introduced in [23] and is defined as follows. Definition 2 (Set of explicit inversive points of period T ) . Let 0 = θ ∈ F q be an elementof multiplicative order T (hence T | ( q − S ⊆ F q be of cardinality 1 ≤ s ≤ q .Then we define S = { z , . . . , z T − } = { ( θ n + v ) v ∈ S : n = 0 , . . . , T − } ⊂ F sq (4)of size T and derive P s = { x , . . . , x T − } ∈ [0 , s from S by (1) and (2). We remarkthat in this case N = |P s | = T and T divides q − • For s ∈ N put [ s ] := { , , . . . , s } . • For a vector x = ( x , x , . . . , x s ) and for a nonempty u ⊆ [ s ], let x u be theprojection of x onto the components whose index belongs to u , that is, for u = { u , u , . . . , u w } with u < u < . . . < u w we have x u = ( x u , x u , . . . , x u w ). • Let ψ denote the canonical additive character of F q . • For vectors x , y ∈ F sq let x · y ∈ F q denote their standard inner product.4ow, we are ready to state the first character sum bound. Lemma 1.
Let S = { z , . . . , z q − } be given by (3) and let ∅ 6 = u ⊆ [ s ] . Then we have max w ∈ F | u | q \{ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q − X n =0 ψ ( w · z n, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 | u | − q / + | u | + 1 . Proof.
Note that the sums to be estimated are of the form S q := q − X n =0 ψ ( w · z n, u ) = X u ∈ F q ψ | u | X i =1 w i u + v i for some ( w , . . . , w | u | ) ∈ F | u | q \ { } and ( v , . . . , v | u | ) ∈ F | u | q with pairwise distinct coordi-nates v i = v j if i = j .We proceed as in the proof of [19, Theorem 1]. We have | S q | ≤ | u | + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X u ∈ F q ,g ( u ) =0 ψ (cid:18) f ( u ) g ( u ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where f ( x ) = | u | X i =1 w i | u | Y j =1 j = i ( x + v j )and g ( x ) = | u | Y j =1 ( x + v j ) . Since at least one of the w i is non-zero and the v i are distinct, we have f ( − v i ) = w i Q j = i ( v j − v i ) = 0 and f is not the zero polynomial. Since deg( f ) < deg( g ), by [19,Lemma 2] the rational function f ( x ) g ( x ) is not of the form A p − A with some rational functionover the algebraic closure of F q . Hence, we can apply a bound of Moreno and Moreno[17, Theorem 2] (see also [19, Lemma 1]) and the result follows.For the second point set, which was essentially studied in [3, 26], we also give ananalogous character sum bound. Lemma 2.
Let { z , . . . , z T − } be given by (4) of size T and let ∅ 6 = u ⊆ [ s ] . Then wehave max w ∈ F | u | q \{ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) T − X n =0 ψ ( w · z n, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | u | q / + | u | . Proof.
The proof is analogous to the proof of [26, Theorem 1].5
The weighted star discrepancy
For an N -element point set P s in [0 , s the local discrepancy ∆ P s is defined as∆ P s ( α ) = |P s ∩ [ , α ) | N − Volume([ , α ))for α = ( α , . . . , α s ) ∈ [0 , s . The star discrepancy is then the L ∞ -norm of the localdiscrepancy, D ∗ N ( P s ) = k ∆ P s k L ∞ . We consider the weighted star discrepancy. The study of weighted discrepancy hasbeen initiated by Sloan and Wo´zniakowski [24] in 1998 in order to overcome the curse ofdimensionality. Their basic idea was to introduce a set of weights γ = { γ u : ∅ 6 = u ⊆ [ s ] } which consists of non-negative real numbers γ u . A simple choice of weights are so-calledproduct weights ( γ j ) j ≥ , where γ u = Q j ∈ u γ j . In this case, the weight γ j is associatedwith the variable x j . Definition 3 (Weighted star discrepancy) . For given weights γ and for a point set P s in [0 , s the weighted star discrepancy is defined as D ∗ N, γ ( P s ) = max ∅6 = u ⊆ [ s ] γ u sup α ∈ [0 , s | ∆ P s (( α u , )) | , where for α = ( α , . . . , α s ) ∈ [0 , s and for u ⊆ [ s ] we put ( α u , ) = ( y , . . . , y s ) with y j = (cid:26) α j if j ∈ u , j u . Remark 1.
Let F be the space of functions with finite norm k f k F := X u ⊆ [ s ] γ u Z [0 , | u | (cid:12)(cid:12)(cid:12)(cid:12) ∂ | u | f∂ x u ( x u , ) (cid:12)(cid:12)(cid:12)(cid:12) d x u , where for u = ∅ we put ∂ | u | f∂ x u ( x u , ) = f (1 , , . . . , P s is an upper bound for the worst-caseerror of the QMC rule Q N based on P s , that is, e ( Q N , F ) ≤ D ∗ N, γ ( P s ) . It is well known that there is a close connection between discrepancy and charactersums. In discrepancy theory such relations are known under the name “Erd˝os-Tur´an-Koksma inequalities” . One particular instance of an Erd˝os-Tur´an-Koksma inequalityis given in the following lemma which is perfectly suited for our applications. Beforestating the result, we introduce the following auxiliary function. For q = p k , we define T ( q, s ) = ((cid:0) k + 1 (cid:1) s if p = 2 , (cid:0) π log p + (cid:1) s k s if p > . (5)The result is the following: 6 emma 3. For q = p k and z , . . . , z N − ∈ F sq , let P s = { x , . . . , x N − } be the N -element point set defined by (1) and (2) . Then we have D ∗ N ( P s ) ≤ sq + T ( q, s ) N max w ∈ F sq \{ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =0 ψ ( w · z n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where T ( q, s ) is defined as in (5) .Proof. For a non-zero s × k matrix H = ( h ij ) with entries h ij ∈ ( − p/ , p/ ∩ Z wedefine the exponential sum S N ( H ) = N − X n =0 exp π i p s X i =1 k X j =1 h ij c ( j ) n,i ! , where the c ( j ) n,i ∈ F p are defined by (1) and where i = √−
1. By a general discrepancybound taken from [18, Theorem 3.12 and Lemma 3.13] we get D ∗ N ( P s ) ≤ sq + T ( q, s ) N max H =0 | S N ( H ) | , where the maximum is extended over all non-zero matrices H with entries h ij ∈ ( − p/ , p/ ∩ Z . Let { δ , . . . , δ k } be the dual basis of the given ordered basis { β , . . . , β k } of F q over F p , that is, Tr( δ j β i ) = ( i = j, i = j, where Tr denotes the trace function from F q to F p . For any basis, there exists a dualbasis and this basis is unique, see [16, p. 55] for a proof. Then we have c ( j ) n,i = Tr( δ j z n,i ) for 1 ≤ j ≤ k, ≤ i ≤ s, and n ∈ N , where z n = ( z n, , . . . , z n,s ). Therefore S N ( H ) = N − X n =0 exp π i p s X i =1 k X j =1 h ij Tr( δ j z n,i ) ! = N − X n =0 exp π i p Tr s X i =1 k X j =1 h ij δ j z n,i !! = N − X n =0 ψ s X i =1 k X j =1 h ij δ j z n,i ! . Put w = ( w , . . . , w s ) ∈ F sq with w i = k X j =1 h ij δ j for i = 1 , . . . , s. Since H is not the zero matrix and { δ , . . . , δ k } is a basis of F q over F p , it follows that w is not the zero vector. This fact finishes the proof.7ow we extend the star discrepancy estimate from Lemma 3 to the weighted stardiscrepancy: Lemma 4.
For z , . . . , z N − ∈ F sq let P s = { x , . . . , x N − } be the N -element point setdefined by (1) and (2) . Then we have D ∗ N, γ ( P s ) ≤ max ∅6 = u ⊆ [ s ] γ u | u | q + T ( q, | u | ) N max w ∈ F | u | q \{ } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =0 ψ ( w · z n, u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! . where z n, u ∈ F | u | q is the projection of z n to the coordinates indexed by u .Proof. The result follows immediately from Lemma 3 together with the fact that D ∗ N, γ ( P s ) ≤ max ∅6 = u ⊆ [ s ] γ u D ∗ N ( P u ) , where P u consists of the points from P s projected onto the components whose indicesbelong to u .The previous lemma gives us our first main result. Theorem 1.
For the point set P s defined as in Definition 1 the following bound holds: D ∗ q, γ ( P s ) ≤ max ∅6 = u ⊆ [ s ] γ u | u | (cid:18) q + 3 T ( q, | u | ) q / (cid:19) . For the point set P s defined as in Definition 2 the following bound holds: D ∗ T, γ ( P s ) ≤ max ∅6 = u ⊆ [ s ] γ u | u | (cid:18) q + 3 T ( q, | u | ) q / T (cid:19) . Proof.
The result follows from Lemma 4 and Lemmas 1 and 2.We point out that the discrepancy estimate from Theorem 1 holds for every choice ofweights. Also, it is important to remark that point sets defined by (4) are more flexiblein terms of size. It is easy to check that for any prime p and T not divisible by p , thereexists q = p k , such that T divides q −
1. This k is the multiplicative order of p modulo T . Since the multiplicative group of F q is cyclic there is an element θ ∈ F q of order T . Tractability.
For a recent overview of results concerning tractability properties of theweighted star discrepancy we refer to [7, 8].
Important Remark 2.
Unfortunately the proof of [8, Theorem 3.2 (ii)] (also [7, The-orem 7(2)]) is not correct and hence this part of the theorem must be discarded. Allother parts of these papers are correct.We now restrict ourselves to product weights and present a condition on the weightsfor strong polynomial tractability. 8 heorem 2.
Let P s be the point set from Definition 1. Assume that for an orderedsequence of weights γ = ( γ j ) j ≥ with γ ≥ γ ≥ . . . , there is a ≤ δ < / such that lim sup j →∞ jγ j < δ . (6) Then there is a constant c γ ,δ > , which depends only on γ and δ but not on s such thatfor all ≤ s ≤ q we have D ∗ q, γ ( P s ) ≤ c γ ,δ q / − δ . If lim sup j →∞ jγ j = 0 , then the result holds for all δ > .Proof. We show the result for p > r ≤ r for r ≥ γ we have D ∗ q, γ ( P s ) ≤ C (1) max r =1 ,...,s r Q rj =1 (cid:0) γ j k (cid:0) π log p + (cid:1)(cid:1) q / ≤ C (1) max r =1 ,...,s Q rj =1 (cid:0) γ j k (cid:0) π log p + (cid:1)(cid:1) q / . Notice that 2 k ( π log p + ) can also be bounded by c log q for some 0 < c <
3. Now, let ℓ be the largest integer such that cγ ℓ log q >
1. Then we have D ∗ q, γ ( P s ) ≤ C (1) Q ℓj =1 ( cγ j log q ) q / . The condition lim sup j →∞ jγ j < δ/ L > jγ j <δ/ j ≥ L . Without loss of generality we may assume that ℓ ≥ L . (Otherwise, if ℓ < L , consider a new weight sequence γ ′ = ( γ ′ j ) j ≥ with γ ′ j = γ j for all j ∈ { , . . . , ℓ } ∪{ L, L + 1 , . . . } and γ ′ j = γ ℓ for j ∈ { ℓ + 1 , . . . , L − } , and hence γ j ≤ γ ′ j for all j ≥ r ∈ N let c r = r Y j =1 ( cγ j log q ) , so we have c ℓ c ℓ − = cγ ℓ log q < cδ ℓ log q. By the definition of ℓ we have c ℓ − < c ℓ , hence1 < cδ ℓ log q, which implies ℓ < cδ (log q ) /
3, or ℓ ≤ ⌊ cδ (log q ) / ⌋ .Therefore, there is a constant C (2) γ > ℓ Y j =1 ( cγ j log q ) = L − Y j =1 ( cγ j log q ) ℓ Y j = L ( cγ j log q ) ≤ C (2) γ ( c log q ) L − ⌊ cδ (log q ) / ⌋ Y j = L cδ log q j . x := cδ (log q ) /
3. Then ℓ Y j =1 ( cγ j log q ) ≤ C (2) γ (cid:18) c log qx (cid:19) L − (( L − x ⌊ x ⌋ ⌊ x ⌋ ! ≤ C (2) γ (cid:18) δ (cid:19) L − (( L − x = C (3) γ ,δ q cδ/ Note that P ∞ j =1 γ j < ∞ , together with the monotonicity γ ≥ γ ≥ . . . implies lim sup j →∞ jγ j = 0. To see this let ε > 0. From P ∞ j =1 γ j < ∞ it follows withthe Cauchy condensation test that also P ∞ k =0 k γ k < ∞ . In particular, 2 k γ k → k → ∞ . This means that γ k ≤ ε/ k +1 for k large enough. Thus, for large enough j with 2 k ≤ j < k +1 we obtain γ j ≤ γ k ≤ ε k +1 < εj . In particular, for j large enough we have jγ j < ε . This implies thatlim sup j →∞ jγ j = 0 . Of course the converse is not true in general (for example, γ j = 1 / ( j log j )). Corollary 1. With the notation and conditions as in Theorem 2, in particular lim sup j →∞ γ j <δ/ , the weighted star discrepancy (or, equivalently, integration in F ) is strongly poly-nomially tractable with ε -exponent at most / (1 − δ ) .Proof. For ε > M := ⌈ ( c γ ,δ ε − ) / (1 − δ ) ⌉ . Let q be the smallest prime power whichis greater or equal to M . According to the Postulate of Bertrand we have q < M .Then we have D ∗ q, γ ( P s ) ≤ ε and hence the information complexity satisfies N ( ε, s ) ≤ q ≤ M = 2 ⌈ ( c γ ,δ ε − ) / (1 − δ ) ⌉ . This means that we have strong polynomial tractability.For the proof of Corollary 1 it is enough to use the construction of Definition 1 witha prime q . However, from a practical point of view the construction of Definition 1with any prime power q and the construction of Definition 2 provide more flexibility. Inparticular the case q = 2 r can be efficiently implemented using (optimal) normal basesand the Itoh-Tsujii inversion algorithm, see [14, Chapter 3] and [13], respectively.10 Integration of H¨older continuous, absolutely con-vergent Fourier series and cosine series Absolutely convergent Fourier series. For f ∈ L ([0 , s ) and h ∈ Z s we definethe h th Fourier coefficient of f as b f ( h ) = R [0 , s f ( x )e − π i h · x d x . Then we can associateto f its Fourier series f ( x ) ∼ X h ∈ Z s b f ( h )e π i h · x . (7)Let α ∈ (0 , 1] and t ∈ [1 , ∞ ]. Similarly to [4] we consider the norm k f k K α,t = X u ⊆ [ s ] | u | X k u ∈ Z | u |∗ | b f ( k u , ) | + | f | H α,t , where Z ∗ = Z \ { } and where | f | H α,t = sup x , x + h ∈ [0 , s | f ( x + h ) − f ( x ) |k h k αℓ t , is the H¨older semi-norm where k · k ℓ t denotes the norm in ℓ t .We define the following sub-class of the Wiener algebra K α,t := { f ∈ L ([0 , s ) : f is one-periodic and k f k K α,t < ∞} . The choice of t will influence the dependence on the dimension of the worst-case errorupper bound. As in [4] we remark that for any f ∈ K α,t the Fourier series (7) of f converges to f at every point x ∈ [0 , s . This follows directly from [25, Corollary 1.8,p. 249], using that f is continuous since it satisfies a H¨older condition, i.e. | f | H α,t < ∞ .More information on K α,t can be found in [4]. Theorem 3. Let P s be the point set from Definition 1 with k = 1 and q = p = N andlet Q N be the QMC rule based on P s . Then for α ∈ (0 , and t ∈ [1 , ∞ ] we have e ( Q N , K α,t ) ≤ max (cid:18) √ N , s α/t N α (cid:19) . In particular, if t = ∞ we have e ( Q N , K α, ∞ ) ≤ N min( α, / . Proof. For f ∈ K α,t we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N − X n =0 f ( x n ) − Z [0 , s f ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X k ∈ Z s \{ } b f ( k ) 1 N N − X n =0 e π i k · x n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ X k ∈ Z sN ∤ k | b f ( k ) | N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =0 e π i p k · c n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X k ∈ Z s \{ } N | k | b f ( k ) | = X ∅6 = u ⊆ [ s ] X k u ∈ Z | u |∗ N ∤ k u | b f (( k u , )) | N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N − X n =0 e π i p k u · c n, u (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X k ∈ Z s \{ } | b f ( N k ) | , N | k if all coordinates of k are divisible by N and N ∤ k otherwise. Now weapply Lemma 1 to the first sum and [4, Lemma 1] to the second sum and obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N N − X n =0 f ( x n ) − Z [0 , s f ( x ) d x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ √ N X ∅6 = u ⊆ [ s ] | u | X k u ∈ Z | u |∗ N ∤ k u | b f (( k u , )) | + s α/t N α | f | H α,t ≤ max (cid:18) √ N , s α/t N α (cid:19) k f k K α,t . The result follows. Corollary 2. Integration in K α, ∞ is strongly polynomially tractable with ε -exponent atmost max( α , .Proof. The proof is similar to the one of Corollary 1. Absolutely convergent cosine series. So far we required that the functions areperiodic. Now we show how we can get rid of this assumption. Let us consider cosineseries instead of classical Fourier series.The cosine system { cos( kπx ) : k ∈ N } forms a complete orthogonal basis of L ([0 , σ k ( x ) := (cid:26) k = 0 , √ kπx ) if k ∈ N , then the system { σ k : k ∈ N } forms an ONB of L ([0 , k = ( k , . . . , k s ) ∈ N s and x = ( x , . . . , x s ) ∈ [0 , s define σ k ( x ) = s Y j =1 σ k j ( x j ) . The σ k with k ∈ N s constitute an ONB of L ([0 , s ).To an L -function g we can associate the cosine series g ( x ) ∼ X k ∈ N s e g ( k ) σ k ( x )with cosine coefficients e g ( k ) = R [0 , s g ( x ) σ k ( x ) d x .In order to apply the results for Fourier series to cosine series we need the tenttransformation φ : [0 , → [0 , 1] given by φ ( x ) = 1 − | x − | . For vectors x the tent transformed point φ ( x ) is understood component wise. The tenttransformation is a Lebesgue measure preserving map and we have Z [0 , s g ( x ) d x = Z [0 , s g ( φ ( x )) d x . k g k C α,t = X u ⊆ [ s ] | u | | u | / X k u ∈ N | u | | e g (( k u , )) | + 2 α | g | H α,t and let C α,t = { g ∈ L ([0 , s ) : k g k C α,t < ∞} . For g ∈ L ([0 , s ) we have that the function f = g ◦ φ is one-periodic and k f k K α,t = k g k C α,t . The cosine series of g converges point-wise and absolute to g for all points in [0 , s .Now we consider integration of functions from C α,t : For a point set P s = { x , . . . , x N − } let Q = { φ ( x ) , . . . , φ ( x N − ) } be the tent transformed version of P s . Let Q N be a QMCrule based on P s . Then we denote by Q φN the QMC rule based on Q .As in [4, Proof of Theorem 2] we have the identity of worst-case errors in K α,t and C α,t when we switch from a QMC rule to the tent transformed version of this rule,namely e ( Q N , K α,t ) = e ( Q φN , C α,t ) . With this identity we can transfer the results for periodic functions to not necessarilyperiodic ones. Corollary 3. Let P s be the point set from Definition 1 with k = 1 and q = p = N andlet Q φN be the tent transformed version of the QMC rule based on P s . Then for α ∈ (0 , and t ∈ [1 , ∞ ] we have e ( Q φN , C α,t ) ≤ max (cid:18) √ N , s α/t N α (cid:19) . In particular, if t = ∞ we have e ( Q φN , C α, ∞ ) ≤ N min( α, / . Corollary 4. Integration in C α, ∞ is strongly polynomially tractable with ε -exponent atmost max( α , . References [1] Ch. Aistleitner, Covering numbers, dyadic chaining and discrepancy. J. Complexity27 (2011), 531–540.[2] Ch. Aistleitner and M. Hofer, Probabilistic discrepancy bound for Monte Carlopoint sets. Math. Comp. 83 (2014), 1373–1381.[3] Z. Chen, Finite binary sequences constructed by explicit inversive methods. FiniteFields Appl. 14 (2008), 579–592.[4] J. 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Email: [email protected] Domingo Gomez-Perez, Faculty of Sciences, University of Cantabria, E-39071 Santander,Spain. Email: [email protected] Friedrich Pillichshammer, Department for Financial Mathematics and Applied NumberTheory, Johannes Kepler University Linz, Altenbergerstr. 69, 4040 Linz, Austria. Email: [email protected] Arne Winterhof, Johann Radon Institute for Computational and Applied Mathemat-ics, Austrian Academy of Sciences, Altenbergerstr. 69, 4040 Linz, Austria. Email: [email protected]@oeaw.ac.at