On lower bounds for integration of multivariate permutation-invariant functions

Transcription

1 arxiv: v1 [math.na] 15 Oct 2013 On lower bouns for integration of multivariate permutation-invariant functions Markus Weimar October 16, 2013 Abstract In this note we stuy multivariate integration for permutation-invariant functions fromacertain Banach spacee,α of Korobov typeinthe worst case setting. We present a lower error boun which particularly implies that in imension every cubature rule which reuces the initial error necessarily uses at least + 1 function values. Since this hols inepenently of the number of permutation-invariant coorinates, this shows that the integration problem can never be strongly polynomially tractable in this setting. Our assertions generalize results ue to Sloan an Woźniakowski [SW97]. Moreover, for large smoothness parameters α our boun can not be improve. Finally, we exten our results to the case of permutation-invariant functions from Korobov-type spaces equippe with prouct weights. 1 Introuction an main result Consier the integration problem Int = (Int ) N, Int : E,α C, Int (f) = f(x)x, [0,1] for perioic, complex-value functions in the Korobov class { E,α := f L 1 ([0,1] ) f := f E,α := sup f(k) ( } ) α k 1... k < k Z Philipps-University Marburg, Faculty of Mathematics an Computer Science, Hans-Meerwein-Straße, Lahnberge, Marburg, Germany. weimar@mathematik.uni-marburg.e This work has been supporte by Deutsche Forschungsgemeinschaft DFG (DA 360/19-1). 1

2 where N an α > 1. Here Z enotes the set of integers, N := {1,2,...}, an we set k m := max{1, k m }. Moreover, for f L 1 ([0,1] ) f(k) := f,e 2πik L 2 := f(x)e 2πikx x, k = (k 1,...,k ) Z, [0,1] enotes its kth Fourier coefficient, where kx = m=1 k m x m, an i = 1. To approximate Int (f), withoutloss ofgenerality, weconsier algorithmsfromtheclassofalllinear cubature rules A(f) := A N, (f) := N w n f ( t (n)), N N 0 := {0} N, (1) n=1 that use at most N values of the input function f at some points t (n) [0,1], n = 1...,N. The weights w n can be arbitrary complex numbers. Clearly, every function f E,α has a 1-perioic extension since their Fourier series are absolutely convergent: 2πik ( ) α f(k)e f k1... k = f (1+2ζ(α)) <. k Z k Z As usual, ζ(s) = m=1 m s is the Riemann zeta function evaluate at s > 1. In [SW97] Sloan an Woźniakowski showe that for every N the Nth minimal worst case error of Int = (Int ) N, e(n,;int,e,α ) := inf A N, sup f E,α 1 Int (f) A N, (f), equals the initial error e(0,;int,e,α ) = 1 provie that N < 2. In other wors, the integration problem on the full spaces (E,α ) N suffers from the curse of imensionality, since for every fixe ε (0,1) its information complexity grows exponentially with the imension : n(ε,) := n(ε,;int,e,α ) := min{n N 0 e(n,;int,e,α ) ε} 2, N. We generalize this result to the case of permutation-invariant 1 subspaces in the sense of [Wei12]. To this en, for N let I {1,...,} be some subset of coorinates an consier the integration problem Int = (Int ) N restricte to the subspace S I (E,α ) of all 1 In [Wei12] we use the name symmetric what cause some confusion. 2

3 I -permutation-invariant functions f E,α. That is, in imension we restrict ourselves to functions f that satisfy f(x) = f(σ(x)) for all x [0,1] (2) an any permutation σ from { S I := σ: {1,...,} {1,...,} σ bijective an σ } {1,...,}\I = i (3) that leaves the elements in the complement of I fixe. For the ease of presentation we shall use the same notation for permutations σ S I an for the corresponing permutations σ : R R of -imensional vectors, given by x = (x 1,...,x ) σ (x) := (x σ(1),...,x σ() ). Observe that in the case I = we clearly have S I (E,α ) = E,α. One motivation to stuy the integration problem restricte to those subspaces is relate to approximate solutions of partial ifferential equations. Many approaches to obtain such solutions lea us to the problem of calculating high-imensional integrals, e.g., to obtain certain wavelet coefficients. Obviously, it is of interest whether this can be one efficiently since taking into account a large number of coefficients woul lea to better approximations to the exact solution. Therefore it is important to incorporate as many structural properties (such as permutation-invariance of the integrans uner consieration) as possible in orer to reuce the effort for every single calculation. In information-base complexity (IBC) this effort is measure by the behavior of the information complexity n(ε,) which can be formalize by several notions of tractability. We remin the reaer that a problem is calle polynomially tractable if its information complexity n(ε,) is boune from above by some polynomial in an ε 1, i.e., n(ε,) Cε p q for some C,p > 0, q 0 an all ε (0,1], N. If the last inequality remains vali even for q = 0 then we say that the problem is strongly polynomially tractable. Apart from that, several weaker notions of tractability were introuce recently; for etails see, e.g., Sielecki [Sie13]. If for some fixe s, t (0, 1] the information complexity satisfies lnn(ε,) lim = 0, (4) ε 1 + ε s + t then we have (s, t)-weak tractability. This generalizes the well-establishe notion of weakly tractable problems (which is inclue as the special case s = t = 1). It is use to escribe the 3

4 case of at most subexponentially growing information complexities such as, e.g., n(ε, ) = exp( ). Finally, a given problem is sai to be uniformly weakly tractable if the limit conition (4) hols 2 for every s,t (0,1]. Our main result states that the integration problem for permutation-invariant functions in the above sense can never be strongly polynomially tractable, inepenent of the size of the sets I which escribes the number of impose permutation-invariance conitions. The assertion reas as follows: Theorem 1. Let Then, for every N < N, an N := N (,I ) := (#I +1) 2 #I, N. (5) e(n,;int,s I (E,α )) = 1 (6) e(n,;int,s I (E,α )) for all N an α > 1. Consequently, for all N. lim α e(n,;int,s I (E,α )) = 0 ( 1+ ζ(α) ) 1 (7) 2 α 1 Before we give the proof of this theorem in Section 3 we a some comments on this result in the next section. 2 Discussion an further results Note that, in particular, (6) yiels that for every imension the initial error of the integration problem uner consieration equals 1. Thus the problem is well-scale an we o not nee to istinguish between the absolute an the normalize error criteria. We stress that ue to results of Smolyak an Bakhvalov the choice of linear, nonaaptive cubature rules A in (1) is inee without loss of generality. For etails an further references see, e.g., [SW97, Remark 1] an Novak an Woźniakowski [NW08, Section 4.2.2]. 2 Note that it clearly suffices to check (4) for every s = t (0,1] in orer to conclue uniform weak tractability. 4

5 Moreover, observe that ue to (7) the assertion state in (6) can not be extene to N N provie that the smoothness α is sufficiently large. That means that at least in this case our result is sharp. Note that (6) can be reformulate equivalently in terms of the information complexity: n(ε,;int,s I (E,α )) (#I +1) 2 #I for all N an every ε (0,1). Therefore our Theorem 1 inee generalizes [SW97, Theorem 1] since the latter is containe as the special case I = for every N. Furthermore, observe that in any case the right-han sie of the last inequality is lower boune by +1. Hence, even for the fully permutation-invariant problem, where we have I = {1,...,} for all N, the information complexity grows at least linearly with imension provie that ε < 1. Together with some obvious estimates this proves the following corollary. Corollary 1. Assume α > 1 an let b := #I. We stuy Int = (Int ) N for the sequence of I -permutation-invariant subspaces (S I (E,α )) N in the worst case setting: If the problem is polynomially tractable with the constants C, p, q then q 1 an (b ) N O(ln). In particular, the problem is never strongly polynomially tractable. If the problem is uniformly weakly tractable then (b ) N o( t ) for all t (0,1]. If the problem is (s,t)-weakly tractable for some s,t (0,1] then (b ) N o( t ). In particular, weak tractability implies (b ) N o(). If (b ) N / o() then we have the curse of imensionality. In turn, alreay the absence of the curse implies (b ) N o(). For some applications it might be useful to impose permutation-invariance conitions with respect to finitely many isjoint subsets I (1),...,I(R) {1,...,} of the coorinates, R > 1; see [Wei12] for etails. In this case the respective subspace S (1) I,...,I(R) (E,α ) := R r=1 S (r) I (E,α ) E,α consists of all f E,α which satisfy (2) for all σ R r=1 S, where S I (r) (r) I an (E,α ) are efine as before. It turns out that Theorem 1 remains vali for this S I (r) 5

6 case when we replace the efinition of N given in (5) by ( ( ) R ( ) ) 2 N := N,I (1),...,I(R) := #I (r) +1 R r=1 #I(r). Without going into etails we mention that our proof given in Section 3 below can be transferre almost literally to this case. Consequently, an analogue of Corollary 1 remains vali if we set b := R r=1 #I(r) for N. During the last two ecaes many numerical problems such as (high-imensional) integration were proven to be computationally har. Fortunately, it turne out that the introuction of weights to the norm of the unerlying source spaces can ramatically reuce the complexity of those problems such that they can be hanle efficiently; see, e.g., [NW08]. In [Woź09] Woźniakowski consiere the integration problem efine above for weighte Korobov-type spaces E γ,α := { f L 1 ([0,1] ) f γ := f E γ r=1 where N an α > 1, as well as γ := {γ,u(k) γ,u(k) := m u(k) γ,m are prouct weights with,α ( ) α } k1... k := sup f(k) <, k Z γ,u(k) (8) k Z }. Here the quantities u(k) := {m {1,...,} k m 0} an 1 = γ, γ,1... γ, 0 for N an k = (k 1,...,k ) Z. This generalizes [SW97] since if γ,m 1 for all m = 1,..., an N then E γ,α = E,α. Theorem 1 in [Woź09] states that η,n e(n,;int,e γ,α ) 1 for all α > 1, N, an N < 2, where η,n enotes the (N+1)st largest weight in the sequence (γ,u ) u {1,...,} [0,1]. So what about weights in our setting? First observe that the subset of I -permutation-invariant functions in E γ,α again forms a linear subspace. In accorance with our previous notation we enote it by S I (E γ,α ). Furthermore, we note that the Fourier coefficients of an I -permutation-invariant function f are I -permutation-invariant again, i.e., they satisfy f(k) = f(σ(k)) for every k Z an all σ S I. Since clearly also ( ) α k 1... k is I -permutation-invariant, f E γ,α belongs to the unit ball of the subspace S I (E γ,α ) if an only if γ,u(σ(k)) µ,u(k) f(k) min ( ) α =: ( ) α, k Z, k1... k k1... k σ S I 6

7 where the new weights µ,u are efine as a prouct of an orer-epenent part (w.r.t. coorinates from I ) an a part consisting of usual prouct weights. For the ease of notation, in what follows we assume permutation-invariance w.r.t. the first #I coorinates, i.e., I = {1,2,...,#I }. Then µ,u(k) = min σ S I m u I (σ(k)) γ,m m u I c (k) γ,m = #u I (k) i=1 γ,#i i+1 m u I c (k) γ,m. (9) Here we set u I (k) := u(k) I an u I c (k) := u(k)\i to enote the support of k Z w.r.t. the sets I an I c = {1,...,}\I, respectively. Thus we have S I (E γ,α ) = S I (E µ,α ), where E µ,α is efine by (8) with γ,u(k) replace by µ,u(k) an S I once more enotes the restriction to I -permutation-invariant functions. Again the lower boun for the error of numerical integration epens on the (N+1)th largest weight µ,u(k) that can appear. To formalize this point let ψ: {0,1,...,# 1} enote a rearrangement of the multi-inices k from the set := { k = (m,l) {0,1} #I {0,1} ( #I ) m 1... m #I } {0,1} Z (10) such that the corresponing weights µ,u(k) possess a non-increasing orering, i.e. ν,n := µ,u(ψ(n)) µ,u(ψ(n+1)) = ν,n+1 for all n = 0,1,...,# 2. (11) Then the weighte analogue of Theorem 1 reas as follows: Theorem 2. Assume α > 1, for every N let I {1,...,} be given, an consier N (,I ) efine as in (5). Then we have for all N an every N < N (,I ). We illustrate Theorem 2 by two examples. ν,n e ( N,;Int,S I (E µ,α )) 1 (12) 7

8 Example 1. Consier the fully permutation-invariant problem, i.e., assume I = {1,...,} for every N. For simplicity let γ,m := 1 m for every m {1,...,} an all N. Then = {k {0,1} k 1... k } an # = + 1. Moreover, we conclue µ,u(k) = #u(k) i=1 1/( i+1), k, an consequently ν,0 = 1, ν,1 = 1/, ν,2 = 1/( ( 1)),. ν, = 1/(!). Hence, for α > 1 an N we obtain e(0,;int ;S I (E µ,α )) = 1 an e(n,;int ;S I (E µ,α )) 1 ( 1)... ( N +1), if N {1,...,}. Our secon, more sophisticate example shows that permutation-invariance is not as powerful as aitional knowlege moele by weights. It generalizes an assertion ue to Woźniakowski [Woź09, p. 648]. Example 2. For N let #I {0,1,...,} be arbitrary an assume permutationinvariance w.r.t. the first #I coorinates in imension. Furthermore, let γ,1 :=... := γ,#i := 1 an 1 γ,#i γ, > 0. That is, we assume the weights only act on coorinates without permutation-invariance conitions an leave the remaining coorinates unweighte. Thus we ask how much permutation-invariance can relax (by now well-establishe) necessary conitions on the sequences (γ,m ) m=1, N, for (strong) polynomial tractability: Assuming polynomial tractability with constants C 1, p > 0 an q 0 it is easy to check that e(n,;int,s I (E µ,α )) (2C)1/p q/p N 1/p, 8

9 see [Woź09] for etails. For every N an all κ > p this implies e(n,;int,s I (E µ,α ))κ 1+(2C) κ/p qκ/p ζ(κ/p) C qκ/p N=0 with some finite constant C > 0 which only epens on C,κ, an p. On the other han, Theorem 2 provies the lower boun e(n,;int ;S I (E µ,α ))κ N=0 In conclusion lim sup = N (,I ) 1 N=0 m {0,1} #I m 1... m #I = (#I +1) ν κ,n = 1 # 1 N=0 l {0,1} #I #I j=1 µ κ,u(ψ(n)) = k µ κ,u(k) j u(l) ( ) 1+γ κ,#i +j. γ,#i +j (#I ( +1) #I j=1 1+γ κ,#i +j) < (13) qκ/p is a necessary conition for (strong) polynomial tractability. In particular, this implies that we can have strong polynomial tractability (q = 0) only if the number of permutation-invariant coorinates #I is uniformly boune. If so, then (13) yiels that lim sup j=1 γ κ,j < for all κ > p which resembles the assertion state in [Woź09, Cor. 1]. If we assume polynomial tractability with q > 0 then we fin that for all κ > p #I #I O( qκ/p j=1 γ,#i κ ) an limsup +j <. ln Note that if #I is boune then the latter conition again coincies with the known conition given by Woźniakowski. Otherwise, if #I grows like say β for some β (0,1], then our conition turns out to be less restrictive, i.e., we may have polynomial tractability although polynomially many coorinates are unweighte (but permutationinvariant). 9 κ

10 3 Proofs In orer to prove Theorem 1 we basically combine the ieas state in [SW97, Woź09] with the technique evelope in [Wei12]. Since the proof is a little bit technical we ivie it into three steps which are organize as follows. InafirststepweshowthatforanygivenintegrationrulethatusesN < N functionvalues there exists a certain fooling function which shows that e(n,;int,s I (E,α )) 1. Afterwars, in Step 2, we notice that this lower boun is sharp, because e(0,;int,s I (E,α )) 1 for every N an all α > 1. Finally, we present a cubature rule that uses at most N = N function values, whereas its worst case error is no larger than the boun state in (7). Proof (Theorem 1). Step 1. Following Sloan an Woźniakowski [SW97] we fix α > 1, as well as N, an consier an arbitrary linear cubature rule A := A N,, given by (1), with N < N (,I ) := (#I + 1) 2 #I. Without loss of generality let us assume that I = {1,...,#I }. We will show that there exists a function f N in the unit ball of S I (E,α ) such that A(f N ) = 0, whereas the integral of f N equals 1. Fork Z let e k := exp(2πik ). Following thelines of [Wei12] we efine alinear operator S I : E,α E,α calle symmetrizer by (S I e k )(x) := 1 #S I σ S I e k (σ(x)), x [0,1], ancontinuousextension. ThereinS I isgivenin(3). ThenS I e k = (1/#S I ) λ S I e λ 1 (k)( ) is I -permutation-invariant in the sense of (2), i.e., (S I e k )(σ(x)) = (S I e k )(x) for all x [0,1] an any σ S I. Moreover, for k Z let M I (k) be efine as in [Wei12]. That is, M I (k)! enotes the number of ifferent permutations σ S I such that σ(k) = λ(k) for any fixe λ S I. To prove the claim we consier the set introuce in (10). Observing that # = N (,I ) = (#I +1) 2 #I we choose a bijection ψ: {0,1,...,N (,I ) 1}. (14) Furthermore, we consier the homogeneous linear system N n=0 a n (S I e ψ(n) )(t (i) ) M I (ψ(n))! = 0, i = 1,...,N, which consists of N < N (,I ) linear equations in N + 1 complex variables a n, n = 0,1,...,N. Here the points t (i) [0,1], i = 1,...,N, enote the integration noes use by 10

11 thecubatureruleaapplietothefunction0 S I (E,α ). Clearly, wecanselect anon-trivial solution a = (a n ) N n=0 CN+1 of this system, scale such that for some n {0,1,...,N} a n = 1 max n=0,1,...,n a n. Next we efine the function f N : [0,1] C by ( N ) f N (x) := #S I (S I e ψ(n ))(x) a n (S I e ψ(n) )(x), M I (ψ(n))! x [0,1]. (15) n=0 Observe that then f N (t (i) ) = 0 for all i = 1,...,N an thus we have A(f N ) = 0. Moreover, weseethatf N isi -permutation-invariantasaprouctofi -permutation-invariantfunctions. It remains to show that f N is a suitable fooling function for our integration problem. That is, we show that f N is an element of the unit ball of S I (E,α ), i.e., its Fourier coefficients satisfy f N (k) 1 ( k1... k ) α for every k = (k 1,...,k ) Z, (16) an that its integral is as large as possible, i.e., Int (f N ) = f N (0) = 1. For k Z we calculate f N (k) = f N,e k L2 = #S I 1 #S I = 1 #S I σ S I λ S I n=0 σ S I e σ 1 ( ψ(n )) N N n=0 a n M I (ψ(n))! a n e σ(ψ(n M I (ψ(n))! ))e λ(ψ(n)),e k L 2 1 #S I λ S I e λ 1 (ψ(n)),e k L 2 an e σ(ψ(n ))e λ(ψ(n)),e k ( 2πi ( λ(ψ(n)) k σ(ψ(n )) ) ) x x L 2 = exp [0,1] = δ [λ(ψ(n)) k σ(ψ(n ))=0], whereδ [C] equals1iftheconitionc isfulfillean0otherwise. Inparticularfrom f N (k) 0 it follows that there exist λ an σ in S I, as well as n {0,1,...,N}, such that k = λ(ψ(n)) σ(ψ(n ) { 1,0,1}, (17) 11

12 since ψ(n) {0,1} for all n an for every permutation λ S I the multi-inex λ(ψ(n)) is an element of {0,1}, too. Hence we arrive at f N (k) = 0 if k Z \{ 1,0,1} an f N (k) = 1 #S I σ S I δ [ k+σ(ψ(n )) {0,1} ] λ S I ( N n=0 a n M I (ψ(n))! δ [λ(ψ(n))=k+σ(ψ(n ))] if k { 1,0,1}. We will show that the summation within the brackets can be reuce to at most one non-vanishing term. Therefore assume σ S I to be fixe an consier h := k+σ(ψ(n )) {0,1}. For this h there exists one (an only one) multi-inex j such that (j m ) m I c = (h m ) m I c an #{m I j m = 1} = #{m I h m = 1}, because λ an σ in S I leave the coorinates in I c = {#I + 1,...,} fixe. Since the mapping ψ was assume to be bijective, j uniquely efines n(k,n,σ) := ψ 1 (j) {0,1,...,# 1}. Hence, there can be at most one n {0,1,...,N} with λ(ψ(n)) = h. If so, then there exist exactly M I (ψ(n(k,n,σ)))! ifferent permutations λ S I such that λ(ψ(n(k,n,σ))) = h = k+σ(ψ(n )). If k = 0 Z then h = σ(ψ(n )) which implies j = ψ(n ) an thus n(0,n,σ) = ψ 1 (j) = n for all σ S I. Consequently, we obtain f N (k) = 1 #S I which particularly yiels σ S I δ [k+σ(ψ(n )) {0,1} ] δ [n(k,n,σ) N] a n(k,n,σ), k { 1,0,1}, (18) Int (f N ) = f N (0) = 1 #S I σ S I a n = a n = 1. ) 12

13 Furthermore, formula (18) implies f N (k) ( ) α k 1... k = fn (k) 1 #S I σ S I 1 #S I δ [ k+σ(ψ(n )) {0,1} ] σ S I a n = 1, δ[n(k,n an(k,n,σ) N],σ) for k { 1,0,1}. Together with f N (k) ( ) α k 1... k = 0 for k Z \ { 1,0,1} this finally proves (16) an completes this step. Step 2. Clearly, we have e(0,;int,s I (E,α )) 1 for all N an any I {1,...,} since the worst case error of the zero algorithm A 0, 0 is given by wor (A 0, ;Int,S I (E,α )) = sup f(0) 1. f S I (E,α ) 1 Step 3. To show (7) we once more follow the arguments given in [SW97]. There it has been shown that the 2 -point prouct-rectangle rule R 2,: E,α C, f R 2,(f) := 1 ( j1 f 2 2, j 2 2,..., j ), 2 j {0,1} is a suitable algorithm for Int on E,α. For N an α > 1 its worst case error was foun to be ( wor (R 2,;Int,E,α ) = 1+ ζ(α) ) 1. (19) 2 α 1 Since S I (E,α ) is a linear subspace of E,α, where S I (E,α ) = E,α, we can restrict the algorithm R 2, to I -permutation-invariant functions in E,α an its worst case error remains boune from above by (19). On the other han, it is easy to see that A N,(f) := k = 1 2 k #S I 2 M I (k)! f ( k 2 f(σ(k)/2) M I (k)! σ S I ) = 1 2 j {0,1} f ( ) j = R 2 2,(f) (20) 13

14 on S I (E,α ). Hence the restriction of R 2,, i.e., the algorithm A N, efine by (20), is a cubature rule in the sense of (1) that uses not more than # = N = N (,I ) function values in imension ; see (10). Consequently, e(n,;int,s I (E,α )) wor (A N,;Int,S I (E,α )) ( wor (R 2,;Int,E,α ) = 1+ ζ(α) ) 1 2 α 1 which completes the proof. The proof of Theorem 2 can be erive using only one aitional argument. As in the previous proof, we construct a suitable fooling function g N that lies in the unit ball of S I (E µ,α ) an for which Int (g N ) is as large as possible while A(g N ) = 0. Proof (Theorem 2). Since we now eal with the weighte case we nee to show that for every k = (k 1,...,k ) Z we have µ,u(k) ĝ N (k) ( ) α k1... k instea of (16). We start by selecting a bijection ψ: {0,1,...,N (,I ) 1} which provies a non-increasing orering of the weights; see (14) an (11). Setting g N (x) := µ,u(ψ(n)) µ,u(ψ(n )) f N (x), x [0,1], it now remains to show that for every k Z with ĝ N (k) 0 we can estimate µ,u(ψ(n)) µ,u(ψ(n )) µ,u(k). (21) Recall that here the relation of k an n is given by k = λ(ψ(n)) σ(ψ(n )) for some λ,σ S I an a certain n {0,1,...,N}; see (17). To prove (21) we first note that the latter representation of k yiels #u I (k) min{#u I (ψ(n))+#u I (ψ(n )),#I }, since λ an σ o not change the size of the support of the respective multi-inices ψ(n) an 14

15 ψ(n ) in. Consequently we obtain #u I (k) i=1 γ,#i i+1 = #u I (ψ(n)) i=1 #u I (ψ(n)) i=1 #u I (ψ(n)) i=1 γ,#i i+1 γ,#i i+1 γ,#i i+1 min{#u I (ψ(n))+#u I (ψ(n )),#I } i=#u I (ψ(n))+1 min{#u I (ψ(n )),#I #u I (ψ(n))} #u I (ψ(n )) i=1 i=1 γ,#i i+1 γ,#i #u I (ψ(n)) i+1 γ,#i i+1, (22) by exploiting the general assumption 1 γ,1... γ,#i 0. Moreover, for the coorinates relate to the prouct weight part we conclue which immeiately implies that γ,m m u I c (k) u I c (k) u I c (ψ(n)) u I c (ψ(n )) m u I c (ψ(n)) γ,m m u I c (ψ(n )) γ,m. (23) Together with the representation of µ state in (9) the estimates (22) an (23) imply µ,u(ψ(n)) µ,u(ψ(n )) µ,u(k). This leas us to the claime boun (21) using the impose orering µ,u(ψ(n)) µ,u(ψ(n)) for n = 0,1,...,N. Finally, since f N (0) = 1, the integral of g N is lower boune by Int (g N ) = ĝ N (0) = µ,u(ψ(n)) µ,u(ψ(n )) f N (0) µ,u(ψ(n)) = ν,n. The fact that A(g N ) = µ,u(ψ(n)) µ,u(ψ(n )) A(f N ) = 0 completes the proof. Acknowlegements I want to thank Henryk Woźniakowski for pointing out the existence of the reference [Woź09] to me. Moreover, I like to thank Dirk Nuyens an Gowri Suryanarayana for motivating me to stuy integration problems of permutation-invariant functions, as well as for their kin hospitality uring my stays in Leuven in October 12 an in March

16 References [NW08] E. Novak an H. Woźniakowski - Tractability of Multivariate Problems. Vol. I: Linear Information. EMS Tracts in Mathematics 6. European Mathematical Society (EMS), Zürich [Sie13] P. Sielecki - Uniform weak tractability. J. Complexity 29, 2013, pp [SW97] I. Sloan an H. Woźniakowski - An intractability result for multiple integration. Mathematics of Computation 66, 1997, pp [Wei12] M. Weimar - The complexity of linear tensor prouct problems in (anti)symmetric Hilbert spaces. J. Approx. Theory 164(10), 2012, pp [Woź09] H. Woźniakowski - Tractability of multivariate integration for weighte Korobov spaces: my 15 year partnership with Ian Sloan, in: P. L Ecuyer an A.B. Owen (Es.) - Monte Carlo an Quasi-Monte Carlo Methos Springer, Berlin. 2009, pp