Tractability of $\mathbb{L}_2$-approximation in hybrid function spaces

0.19 MB·

by  Peter Kritzer

L
Tractability of -approximation in hybrid function spaces
2
Peter Kritzer , Helene Laimer , Friedrich Pillichshammer
∗ † ‡
7 Abstract
1
0 WeconsidermultivariateL -approximation inreproducingkernelHilbertspaceswhich
2
2
are tensor products of weighted Walsh spaces and weighted Korobov spaces. We study

n the minimal worst-case error eL2 app,Λ(N,d) of all algorithms that use N information
−
a
J evaluations from the class Λ in the d-dimensional case. The two classes Λ considered in
1 thispaperaretheclass Λall consistingofalllinear functionalsandtheclass Λstd consisting
1 only of function evaluations.
The focus lies on the dependence of eL2 app,Λ(N,d) on the dimension d. The main
] −
A results are conditions for weak, polynomial, and strong polynomial tractability.
N
. Keywords: Multivariate approximation; Walsh space; Korobov space; hybrid function space.
h
t 2010 MSC: 41A25, 41A63, 65D15, 65Y20.
a
m
[
1 Introduction

1
v
0 We consider L2-approximation of functions in certain reproducing kernel Hilbert spaces (K),
H
1 which are embedded into L ([0,1]d), where K denotes the reproducing kernel. To be more
9 2
2 precise, we approximate the embedding operator
0
1. EMB : (K) L ([0,1]d), EMB (f) = f,
d 2 d
0 H →
7 and measure the approximation error in the L -norm. Since (K) is a reproducing kernel
1 2
H
: Hilbert space it is known (cf. [16, 20]) that there is no loss of generality when we restrict
v
Xi ourselves to linear approximation algorithms of the form AN,d(f) = Nk=1akLk(f) with coeffi-
cients a L ([0,1]d) and continuous linear functionals L on (K) from a permissible class
r k ∈ 2 k H P
a of information Λ. Here N is the number of information evaluations.
Westudytheproblemintheso-calledworst-case setting, i.e., wemeasure theapproximation
error of an algorithm A by means of the worst-case error,
N,d
eL2−app(AN,d) = sup kEMBd(f)−AN,d(f)kL2([0,1]d).
f (K)
∈H
kfkH(K)≤1
∗P. Kritzer is supported by the Austrian Science Fund (FWF): Project F5506-N26.
†H. Laimer is supported by the Austrian Science Fund (FWF): Projects F5506-N26 and F5508-N26.
‡F. Pillichshammer is supported by the Austrian Science Fund (FWF): Project F5509-N26.
AllprojectsarepartsoftheSpecialResearchProgram”Quasi-MonteCarloMethods: TheoryandApplications”.
1
The Nth minimal worst-case error is given by
eL2 app,Λ(N,d) = inf eapp(A ),
− N,d
AN,d
where the infimum is extended over all linear algorithms A using N information evaluations
N,d
fromtheclassΛ. WeareparticularlyinterestedinthedependenceoftheNth minimalworst-case
error on the dimension d. To study this dependence systematically we consider the information
complexity NL2 app,Λ(ε,d), which is the minimal number N for which there exists an algorithm
−
using N information evaluations from the class Λ Λall,Λstd with an error of at most ε.
We would like to avoid cases where the inform∈a{tion comp}lexity NL2 app,Λ(ε,d) grows ex-
−
ponentially or even faster with the dimension d or with ε 1. To quantify the behavior of
−
the information complexity we use different notions of tractability, namely strong polynomial
tractability, polynomial tractability, and weak tractability (we refer to Section 3 for the precise
definitions).
The current state of the art of tractability theory is summarized in the three volumes of the
book of Novak and Wo´zniakowski [16, 17, 18] which we refer to for extensive information on
this subject and further references.
In previous papers, several authors have studied similar approximation problems in various
reproducing kernel Hilbert spaces, see, e.g., [2, 3, 4, 12, 15, 22]. These investigations have
in common that the reproducing kernel Hilbert spaces considered are tensor products of one-
dimensional spaces whose kernels are all of the same type (but maybe equipped with different
weights). In the current paper we consider the case where the reproducing kernel is a product
of kernels of different type. We call such spaces hybrid spaces. Some results on tractability
in general hybrid spaces can be found in the literature. For example, in [17] multivariate
integration is studied for arbitrary reproducing kernels K without relation to K . Here we
d d+1
consider as a special instance the tensor product of Walsh and Korobov spaces. The problem
of numerical integration in such spaces was recently considered in [11]. The study of a hybrid
of Korobov and Walsh spaces could be important in view of functions which are periodic
with respect to some of the components and, for example, piece-wise constant with respect to
the remaining components. Moreover, it has been pointed out by several scientists (see, e.g.,
[10, 13]) that hybrid problems may be relevant for certain applications. Indeed, communication
with the authors of [10] and [13] have motivated our idea for considering function spaces where
we may have very different properties of the elements with respect to different components, as
for example regarding smoothness.
From the analytical point of view, it is very challenging to deal with hybrid spaces. The
reason for this is the rather complex interplay between the different analytic and algebraic
structures of the kernel functions. In the present study we are concerned with Fourier analysis
carried out simultaneously with respect to the Walsh and the trigonometric function systems.
The problem is also closely related to the study of hybrid point sets which received much
attention in recent years (see, for example, [8, 9]). Hence we also have considerable theoretical
interest in studying this problem.
The paper is organized as follows. In Section 2 we introduce the Hilbert space under
consideration. In Section 3 we specify the problem setting and state our main result. The
proofs are presented in Section 4.
2
2 The hybrid function space
We study a specific reproducing kernel Hilbert space, namely the tensor product of a Korobov
space and a Walsh space, that was introduced in [11]. See [1] for general information about
reproducing kernel Hilbert spaces.
Fix a prime number b and let i = √ 1. For k N with b-adic expansion k = κ ba+ +
0 a
− ∈ ···
κ b+κ with κ 0,...,b 1 we define the kth Walsh function wal : [0,1) C by
1 0 j k
∈ { − } →
ξ κ + +ξ κ
wal (x) = exp 2πi 1 0 ··· a+1 a ,
k
b
(cid:18) (cid:19)
for x [0,1) with b-adic expansion x = ξ1 + ξ2 + (unique in the sense that infinitely many
∈ b b2 ···
of the ξ are different from b 1). Note that a = log k .
i − ⌊ b ⌋
For k = (k ,...,k ) Ns and x = (x ,...,x ) [0,1)s the kth s-variate Walsh function
1 s ∈ 0 1 s ∈
wal : [0,1)s C is given by wal (x) = s wal (x ).
k → k j=1 kj j
Further, for l Zt we define the t-variate lth trigonometric function e : [0,1)t C as
∈ Q l →
e (y) = exp(2πil y),
l
·
where denotes the usual Euclidean inner product.
Let·now s,t N, α,β > 1 and let γ(1),γ(2) be two non-increasing sequences γ(i) = (γ(i))
∈ j j≥1
(i)
for i 1,2 , where 0 < γ 1. We define two functions ρ and r as follows: For
∈ { } j ≤ α,γ(1) β,γ(2)
k = (k ,...,k ) Ns and l = (l ,...,l ) Zt let
1 s ∈ 0 1 t ∈
s t
ρ (k) = ρ (k ) and r (l) = r (l ),
α,γ(1) α,γ(1) j β,γ(2) β,γ(2) j
j j
j=1 j=1
Y Y
where
1 if k = 0,
j
ρ (k ) =
α,γj(1) j (γj(1)b−α⌊logb(kj)⌋ if kj 6= 0,
and
1 if l = 0,
j
r (l ) =
β,γj(2) j (γj(2)|lj|−β if lj 6= 0.
With the help of these functions one can define so-called Walsh spaces [5, 7] and Korobov
spaces [6, 14, 17].
Here we define a hybrid function space as the tensor product of the Walsh and Korobov
spaces. The hybrid space (K ), where γ = (γ(1),γ(2)), is the reproducing kernel Hilbert
s,t,α,β,γ
H
space with kernel function given by K : [0,1)s+t [0,1)s+t C,
s,t,α,β,γ
× →
K ((x,y),(x,y )) = ρ (k)r (l)wal (x)wal (x)e (y)e (y )
s,t,α,β,γ ′ ′ α,γ(1) β,γ(2) k k ′ l l ′
k Nsl Zt
X∈ 0X∈
and inner product
1 1
f,g = f(k,l)g(k,l),
h is,t,α,β,γ ρ (k)r (l)
k Nsl Zt α,γ(1) β,γ(2)
X∈ 0X∈
b
b
3
with
f(k,l) = f(x,y)wal (x)e (y)dxdy.
k l
Z[0,1]sZ[0,1]t
b
The space (K ) is the tensor product of a Walsh space and a Korobov space. If
s,t,α,β,γ
H
s = 0, then we obtain the Korobov space, if t = 0, then we obtain the Walsh space.
Remark 1. For convenience we will inthefollowing use the notation f(x,y)dxdy, where
[0,1]d
d = s+t, by which we mean f(x,y)dxdy.
[0,1]s [0,1]t R
R R
The hybrid space (K ) is the space of all absolutely convergent series f of the form
s,t,α,β,γ
H
f(x,y) = f(k,l)wal (x)e (y) with f < .
k l d,α,β,γ
k k ∞
(k,l) Ns Zt
X∈ 0×
b
For further information on the space (K ) we refer to [11, Section 2.2].
s,t,α,β,γ
H
L
3 -approximation
2
Our goal is now to approximate the embedding from the hybrid space (K ) to the space
s,t,α,β,γ
H
L ([0,1]s+t), i.e.,
2
EMB : (K ) L ([0,1]s+t), EMB (f) = f.
s,t s,t,α,β,γ 2 s,t
H →
As already mentioned, it is enough to consider linear algorithms A of the form
N,s,t
N
A (f) = a L (f), (1)
N,s,t k k
k=1
X
with a L ([0,1]s+t) and continuous linear functionals L on (K ) from a permissible
k 2 k s,t,α,β,γ
∈ H
class of information Λ. We consider two classes:
Λ = Λall , the class of all continuous linear functionals defined on (K ). Since
s,t,α,β,γ
• H
(K ) is a Hilbert space, for every L Λall there exists a function f from
s,t,α,β,γ k k
H ∈
(K ) such that L (f) = f,f for all f (K ).
s,t,α,β,γ k k d,α,β,γ s,t,α,β,γ
H h i ∈ H
Λ = Λstd, the class of standard information consisting only of function evaluations. That
•
is, L Λstd if there exists (x ,y ) [0,1]s+t such that L (f) = f(x ,y ) for all
k ∈ k k ∈ k k k
f (K ).
s,t,α,β,γ
∈ H
Since (K ) is a reproducing kernel Hilbert space, function evaluations arecontinuous
s,t,α,β,γ
H
linear functionals, and therefore Λstd Λall. More precisely,
⊆
L (f) = f(x ,y ) = f,K ( ,(x ,y ))
k k k h s,t,α,β,γ · k k is,t,α,β,γ
and
L = K = K1/2 ((x ,y ),(x ,y )).
k kk k s,t,α,β,γks,t,α,β,γ s,t,α,β,γ k k k k
4
The worst-case error in (K ) of a linear algorithm as in (1) is
s,t,α,β,γ
H
eL2−app(AN,s,t) = sup kEMBs,t(f)−AN,s,t(f)kL2([0,1]s+t).
f∈H(Ks,t,α,β,γ)
kfkH(Ks,t,α,β,γ)≤1
The Nth minimal worst-case error is given by
eL2 app,Λ(N,s,t) = inf eapp(A ),
− N,s,t
AN,s,t
where the infimum is extended over all linear algorithms A using information from the class
N,s,t
Λ. The information complexity is given as
NL2 app,Λ(ε,s,t) := min N : eL2 app,Λ(N,s,t) ε .
− −
{ ≤ }
Since Λstd Λall, it follows that NL2 app,Λall(ε,s,t) NL2 std,Λstd(ε,s,t).
− −
⊆ ≤
We say that the L -approximation problem EMB = (EMB ) is:
2 s,t s,t 1
≥
weakly tractable, if
•
logNL2 app,Λ(ε,s,t)
−
lim = 0;
s+t+ε−1 s+t+ε−1
→∞
polynomially tractable, if we can find constants C,τ ,τ 0 such that
1 2
• ≥
NL2 app,Λ(ε,s,t) Cε τ1(s+t)τ2 for all ε (0,1) and all s,t N;
− −
≤ ∈ ∈
strongly polynomially tractable, if we can find constants C,τ 0 such that
1
• ≥
NL2 app,Λ(ε,s,t) Cε τ1 for all ε (0,1) and all s,t N. (2)
− −
≤ ∈ ∈
The infimum τ (Λ) of the real numbers τ such that (2) holds is called the ε-exponent of
∗ 1
strong polynomial tractability.
For γ = (γ(1),γ(2)) we define the sum exponent
∞ ∞
s = inf κ > 0 : (γ(1))κ < and (γ(2))κ < (3)
γ j ∞ j ∞
( )
j=1 j=1
X X
with the convention that inf = .
∅ ∞
Our main goal in this paper is to show the following theorem.
Theorem 1. Consider the approximation problem EMB. Then we have:
1. Strong polynomial tractability and polynomial tractability in the class Λall are equivalent,
and they hold if and only if s < , where s is defined in (3). In this case the exponent
γ γ
∞
of strong polynomial tractability is τ (Λall) = 2max(s , 1, 1).
∗ γ α β
2. The problem is weakly tractable in the class Λall if and only if
s γ(1) + t γ(2)
j=1 j j=1 j
lim = 0. (4)
s+t s+t
→∞ P P
5
3. The problem is strongly polynomially tractable in the class Λstd if
∞ ∞
(1) (2)
γ < and γ < .
j ∞ j ∞
j=1 j=1
X X
The exponent of strong polynomial tractability in the class Λstd satisfies
τ (Λstd) [2max(1, 1,s ),4+2max(1, 1,s )].
∗ ∈ α β γ α β γ
4. The problem is polynomially tractable in the class Λstd if
s γ(1) t γ(2)
j=1 j j=1 j
limsup < and limsup < .
logs ∞ logt ∞
s P t P
→∞ →∞
5. The problem is weakly tractable in the class Λstd if and only if
s γ(1) + t γ(2)
j=1 j j=1 j
lim = 0.
s+t s+t
→∞ P P
Remark 2. Sinceit caneasilybeverified that integrationin (K ) isnotharder thanap-
s,t,α,β,γ
H
proximation, the last item in Theorem 1 implies that the sufficient condition for weak tractabil-
ity of integration shown in [11] is also necessary.
4 Proof of Theorem 1
We recall that strong polynomial tractability implies polynomial tractability which in turn
implies weak tractability. Furthermore, all sufficient conditions for the class Λstd are also
sufficient for the class Λall with τ (Λall) τ (Λstd) in the case of strong polynomial tractability.
∗ ∗
≤
All necessary conditions for the class Λall are also necessary for the class Λstd.
4.1 Proof of Item 1
In order to give a necessary and sufficient condition for strong polynomial tractability for Λall
we use a criterion from [16, Section 5.1]. Let us consider the self-adjoint operator W :=
s,t
EMB EMB : (K ) (K ), which in our case is given by
∗s,t s,t H s,t,α,β,γ → H s,t,α,β,γ
W f = ρ (k)r (l)f(k,l)wal (x)e (y).
s,t α,γ(1) β,γ(2) k l
(k,l) Ns Zt
X∈ 0×
b
The eigenvalues are then given by the collection of the numbers
ρ (k)r (l) for (k,l) Ns Zt.
α,γ(1) β,γ(2) ∈ 0 ×
Furthermore, the largest eigenvalue is ρ (0)r (0) = 1.
α,γ(1) β,γ(2)
6
From [16, Theorem 5.2] we find that the problem EMB is polynomially tractable for Λall if
and only if there exist ν > 0 and q 0 such that
≥
1/ν
sup (ρ (k)r (l))ν (s+t) q < . (5)
 α,γ(1) β,γ(2)  − ∞
s,t
(k,l) Ns Zt
X∈ 0×
 
Furthermore, we have strong polynomial tractability if and only if (5) holds with q = 0.
It is easy to check that we require ν > max(1, 1) in order for (5) to hold with q = 0. Let
α β
us now assume that ν is indeed bigger than max(1, 1). For the sum in (5) we have
α β
s t
(ρ (k)r (l))ν = 1+(γ(1))νµ(αν) 1+(γ(2))ν2ζ(βν) , (6)
α,γ(1) β,γ(2) j j
(k,l)X∈Ns0×Zt Yj=1(cid:16) (cid:17)Yj=1(cid:16) (cid:17)
where µ(x) = bx(b−1) for x > 1 and ζ(x) is the Riemann zeta function.
bx b
Now, using ar−guments outlined in [19] (see also [14, Section 4.5]), it is easy to see that the
existence of some ν > max(1, 1) with
α β
∞ ∞
(γ(1))ν < and (γ(2))ν <
j ∞ j ∞
j=1 j=1
X X
is a necessary and sufficient condition for (5) with q = 0 and therefore for strong polynomial
tractability of the problem EMB.
Again according to [16, Theorem 5.2], the exponent of strong polynomial tractability is
2max(1, 1,s ), where s is defined in (3).
α β γ γ
It remains to show the equivalence of strong polynomial and polynomial tractability. Of
course, it suffices to show that polynomial tractability implies strong polynomial tractability.
So assume that the problem EMB is polynomially tractable for the class Λall. Then we obtain
polynomial tractability also for the embedding problem in the pure Walsh space (K )
s,0,α,β,γ
H
and in the pure Korobov space (K ). According to [21, Theorem 2] this is equivalent to
0,t,α,β,γ
H
strong polynomial tractability for the embedding problem in the pure Walsh space (K )
s,0,α,β,γ
H
and in the pure Korobov space (K ). According to [3] and [12] this implies the existence
0,t,α,β,γ
H
of ν > 0 such that (γ(1))ν1 < and the existence of ν > 0 such that (γ(2))ν2 < .
1 j 1 j ∞ 2 j 1 j ∞
Hence we have s < ≥and this in turn implies strong polynomial tractability fo≥r the class Λall,
γ
P∞ P
as shown above. This completes the proof of Item 1.
4.2 Proof of Item 2
Sufficiency ofCondition (4) followsbyItem 5oftheTheorem which weshow inthenext section.
For showing necessity of Condition (4), we use [16, Theorem 5.3] in the following. To keep
notation simple, we shall frequently write again d instead of s+t. Theorem 5.3 in [16] states
that our approximation problem is weakly tractable for Λall if and only if
lim λ log2j = 0 for all d N and
d,j
• j ∈
→∞
7
there exists some function f: (0, 1] N such that
• 2 →
1
sup sup sup λ log2j < , (7)
η2 d,j ∞
η∈(0,12] d≥f(η)j≥⌈exp(d√η)⌉+1
where λ = λ denotes the jth eigenvalue of W in non-increasing order.
d,j s+t,j s,t
Let us now assume that the approximation problem is weakly tractable for Λall. This then
in particular implies that
lim λ log2j = 0 for all d N. (8)
d,j
j ∈
→∞
We are now going to show that (8) implies (4). To this end, recall that the eigenvalues of W
s,t
are of the form
ρ (k)r (l) for (k,l) Ns Zt.
α,γ(1) β,γ(2) ∈ 0 ×
Note that we have λ = 1; furthermore, note that ρ (1) = γ(1) for any j N, and
d,1 α,γ(1) j ∈
j
r (1) = γ(2) for any i N. Hence, by choosing all components of (k,l) Ns Zt but one
β,γi(2) i ∈ ∈ 0 ×
equal to zero, and the remaining equal to one, we see that
γ(1),...,γ(1) and γ(2),...,γ(2)
1 s 1 t
are eigenvalues of W . Consequently,
s,t
s t d
(1) (2)
γ + γ λ ,
j j ≤ d,j
j=1 j=1 j=1
X X X
and hence
s γ(1) + t γ(2) d λ
j=1 j j=1 j j=1 d,j
lim lim .
s+t s+t ≤ d d
→∞ P P →∞ P
However, due to (8), it follows that the latter limit is 0, which shows that indeed (4) holds.
4.3 Proof of Items 3–5
Any f (K ) can be displayed as
s,t,α,β,γ
∈ H
f(x,y) = f(k,l)wal (x)e (y).
k l
(k,l) Ns Zt
X∈ 0×
b
In order to approximate f(k,l), we are going to use quasi-Monte Carlo algorithms based
on classical and on polynomial lattice point sets.
b
Classical lattice point sets. For N N and z = (z ,...,z ) Zt , where Z := z
∈ 1 t ∈ N N { ∈
1,...,N 1 : gcd(z,N) = 1 , the lattice point set q N 1 with generating vector z is
{ − } } { v}v=−0
defined by
vz vz
q = 1 ,..., t for all 0 v N 1.
v N N ≤ ≤ −
(cid:16)n o n o(cid:17)
Here denotes the fractional part of a real number.
{·}
8
Polynomial lattice point sets. Let F be the finite field of prime order b, F [x] be the set of
b b
polynomials over F , and let F ((x 1)) be the field of formal Laurent series over F . The latter
b b − b
contains the field of rational functions as a subfield. Given m N, set G := a F [x] :
b,m b
∈ { ∈
deg(a) < m and define a mapping ν : F ((x 1)) [0,1) by
m b −
} →
m
∞
ν t x l := t b l.
m l − l −
!
l=z l=max(1,z)
X X
Let f F [x] with deg(f) = m and g = (g ,...,g ) F [x]s. The polynomial lattice point set
b 1 s b
∈ ∈
(p ) with generating vector g is defined by
v v∈Gb,m
v(x)g (x) v(x)g (x)
p := ν 1 ,...,ν s for all v G .
v m f(x) m f(x) ∈ b,m
(cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)
Note that we can associate the polynomial v(x) = mr=−01vrxr ∈ Gb,m with the integer v =
mr=−01vrbr, where, with a slight abuse of notation, thPe element vr ∈ Fb is associated with the
integer v 0,1...,b 1 . In this way we can index the points of a polynomial lattice point
r
P ∈ { − }
set by integers ranging from 0 to bm 1.
−
NowsupposethatN isoftheformbm forsomem N, andletPL = p ,...,p [0,1)s
be a polynomial lattice point set and L = q ,...,q∈ [0,1)t be{a 0lattice pNo−i1n}t ⊆set. We
consider the point set (PL,L) = (p,q) ={(p0,q ): vN=−10},⊆...,N 1 .
{ v v v − }
For M 1 define the set
≥
= (k,l) Ns Zt : (ρ (k)) 1(r (l)) 1 M . (9)
AM { ∈ 0 × α,γ(1) − β,γ(2) − ≤ }
In order to approximate the embedding EMB (f) = f for f (K ) we use the
s,t s,t,α,β,γ
∈ H
algorithm
N 1
1 −
A (f)(x,y) = f((p,q) )wal (p )e (q ) wal (x)e (y). (10)
N,s,t,M N v k v l v k l
!
(k,Xl)∈AM Xv=0
It can easily be checked that A is a linear algorithm of the form (1) with
N,s,t,M
1
a (x,y) = wal (x p )e (y q ) and L (f) = f((p,q) ), 0 v N 1.
v N k ⊖ v l − v v v ≤ ≤ −
(k,Xl)∈AM
The error of approximation for given f (K ) is then
s,t,α,β,γ
∈ H
(f A (f))(x,y) = f(k,l)wal (x)e (y)
N,s,t,M k l
−
(k,Xl)∈/AM
b N 1
1 −
+ f(k,l) f((p,q) )wal (p )e (q ) wal (x)e (y).
− N v k v l v k l
!
(k,Xl)∈AM Xv=0
b (11)
We use (11) and Parseval’s identity to obtain
EMB (f) A (f) 2 = S +S ,
k s,t − N,s,t,M kL2([0,1]s+t) 1 2
9
where
S := f(k,l) 2,
1
| |
(k,Xl)∈/AM
b
and
2
N 1
1 −
S := f (x,y)dxdy f ((p,q) ) ,
2 (k,l) (k,l) v
(k,Xl)∈AM (cid:12)(cid:12)(cid:12)Z[0,1]s+t − N Xv=0 (cid:12)(cid:12)(cid:12)
with (cid:12) (cid:12)
(cid:12) (cid:12)
f (x,y) := f(x,y)wal (x)e (y).
(k,l) k l
From the definition of it follows easily that
M
A
1
S < f 2 .
1 Mk kH(Ks,t,α,β,γ)
Letusnowconsider S . Thetermin-between theabsolutevaluesignsinS istheintegration
2 2
error of the QMC rule using the nodes (PL,L) for the function f (x,y). Since the product
(k,l)
of two Walsh functions is again a Walsh function, and the analogue is true for trigonometric
functions, it can easily be verified that f (K ). Hence we can bound S by
(k,l) s,t,α,β,γ 2
∈ H
S (eint(PL,L))2 f 2 ,
2 ≤ k (k,l)kH(Ks,t,α,β,γ)
(k,Xl)∈AM
where eint(PL,L) is the worst-case integration error in (K ) of the QMC rule based on
s,t,α,β,γ
H
the nodes (PL,L), i.e.,
N 1
1 −
eint(PL,L) = sup f(x,y)dxdy f((p,q) ) .
v
− N
kffk∈HH(K(Ks,st,,αt,,αβ,,βγ,)γ≤)1(cid:12)(cid:12)(cid:12)Z[0,1]s+t Xv=0 (cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
From [11, Theorem 3] it then follows that
s t
2
S (1+γ(1)2µ(α)) (1+γ(2)4ζ(β)) f 2 . (12)
2 ≤ N j ! j ! k (k,l)kH(Ks,t,α,β,γ)
Yj=1 Yj=1 (k,Xl)∈AM
Next we find an estimate for f 2 for (k,l) . From the easily seen fact
k (k,l)kH(Ks,t,α,β,γ) ∈ AM
that f (h,m) = f(k h,l+m) we obtain
(k,l)
⊕
f(k h,l+m) 2
b b
f 2 = | ⊕ |
k (k,l)kH(Ks,t,α,β,γ) h Nsm Zt ρα,γ(1)(h)rβ,γ(2)(m)
X∈ 0 X∈ b
f(k h,l+m) 2 ρ (k h)r (l+m)
= | ⊕ | α,γ(1) ⊕ β,γ(2) .
ρ (k h)r (l+m) ρ (h)r (m)
h Nsm Zt α,γ(1) ⊕ β,γ(2) α,γ(1) β,γ(2)
X∈ 0 X∈ b
Combining results from [3] and [12] we find
ρ (k h)r (l+m) 1 t t
α,γ(1) ⊕ β,γ(2) max(1,2βγ(2)) M max(1,2βγ(2)),
ρ (h)r (m) ≤ ρ (k)r (l) j ≤ j
α,γ(1) β,γ(2) α,γ(1) β,γ(2) j=1 j=1
Y Y
10