Weighted integration over a hyperrectangle based on digital nets and sequences

Quasi-Monte Carlo (QMC) methods are equal weight quadrature rules to approximate integrals over the unit cube with respect to the uniform measure. In this paper we discuss QMC integration with respect to general product measures defined on an arbitrary hyperrectangle. We only require that the cumulative distribution function is invertible. We develop a worst-case error bound and study the dependence of the error on the number of points and the dimension for digital nets and sequences as well as polynomial lattice point sets, which are mapped to the domain using the inverse cumulative distribution function. We do not require any smoothness properties of the probability density function and the worst-case error does not depend on the particular choice of density function and its smoothness. The component-by-component construction of polynomial lattice rules is based on a criterion which depends only on the size of the hyperrectangle but is otherwise independent of the product measure.

Usually, quasi-Monte Carlo (QMC) rules are used for uniform integration over the unit-cube, i.e., when a i = 0, b i = 1 and ϕ i ≡ 1 for all i ∈ [s].In this context there exists a multitude of literature.For introductory texts and surveys we refer to [1][2][3][4].However, often the integration problem is not defined on the unit cube and/or with respect to the uniform measure.The standard approach in this case is to perform a transformation to standardize the problem to a setting where QMC can be applied directly.However, such a transformation can have a big influence on the integration problem and the performance of QMC methods.For instance, [5] applied QMC to a problem from statistics, [6] applied QMC to a partial differential equations with random coefficients, [7] used a tensor approximation to transform QMC points to other distributions appearing in statistics.It is known from [8] that well distributed point sets exist with respect to very general measures, but it is difficult to obtain good explicit constructions (see for instance [9] for a successful transformation of lattice point sets to R s ).Here we restrict ourselves to product measures.The more general case of mixture distributions and beyond will be studied in a forthcoming paper.
In this paper we aim at applying QMC for weighted integrals of the form (1). We will use QMC rules of the form 1 where y 0 , y 1 , . . ., y N−1 are N points from the s-dimensional unit-cube [0, 1] s .In particular we study digital nets and sequences as well as polynomial lattice point sets.See [2,4,10] for introductions to these topics.
Let R + be the set of positive real numbers and let γ := {γ u ∈ R + : u ⊆ [s]} be a given set of positive so-called coordinate weights.These coordinate-weights model in some sense the importance of variables or groups of variables of integrands for the integration problem.This point of view has been introduced by Sloan and Woźniakowski [11] in order to explain the effectiveness of QMC also in high dimensions.In fact, in [11] the authors used a special type of weights which are so-called product weights of the form with a weight sequence (γ i ) i≥1 in R + .
Another type of weights that attracted a lot of attention during the past years are product order dependent (POD) weights which are of the form where (Γ ℓ ) ℓ≥0 and (γ j ) j≥1 are sequences in R + .POD weights are very important in the context of PDEs with random coefficients, see, e.g., [12,13].
We consider functions with bounded mixed partial derivatives up to order one in the sup-norm and define, for p ∈ [1, ∞], the ''weighted'' norm ∥F ∥ p,s,γ := ( with the obvious modifications if p = ∞.Here, for u = {u 1 , u 2 , . . ., The integration error for weighted integration using a QMC-rule (2) with underlying set P = {y 0 , y 1 , . . ., y N−1 } of integration nodes is defined by err(F ; ϕ; P) := In particular, for every F with ∥F ∥ p,s,γ < ∞ we have |err(F ; ϕ; P)| ≤ ∥F ∥ p,s,γ wce(P; p, s, γ, ϕ).Notice that the norm of the function does not depend on the integration weight ϕ.In this paper we study the decay of the worst-case error wce(P; p, s, γ, ϕ) as the number of QMC points N increases.Besides the convergence rate of the worst-case error for N → ∞, we are also interested in the dependence of the integration problem on the dimension s.

This question is related to tractability studies.
A somewhat surprising result of this paper is that the worst-case error can be bounded independent of ϕ (see Theorem 1).The method itself requires that Φ −1 exists and the points Φ −1 (y n ) can be generated, but the decay of the error does otherwise not depend on properties of ϕ or Φ −1 .This starkly differs from the straight forward approach where one considers the composition F • Φ −1 as the integrand, in which case one ends up with the function norm ∥F • Φ −1 ∥ for some norm ∥ • ∥.In this case the smoothness properties of Φ −1 are important.Our proof uses Haar functions which are orthogonal with respect to the weight ϕ, and we then study the decay of the Haar coefficients with respect to these ϕ-orthogonal Haar functions.
The paper is organized as follows: In Section 2 we introduce the basic analytic tools which are Haar and Walsh functions.The main result of this section is a bound on the Haar coefficients of functions F with bounded norm ∥F ∥ p,s,γ (see Lemma 1).In Section 3 we present a bound on the integration error which is in many cases easy to handle (see Theorem 1).This upper bound on the integration error will then be studied for digital nets and sequences as well as for polynomial lattice point sets.The definitions and some basics are recalled in Section 4. The error analysis for these node sets follows in Sections 5-7.The paper concludes with a discussion of the dependence of the worst-case error bounds on the dimension.We will provide sufficient conditions on the weights γ which guarantee that the obtained error bounds hold uniformly in s; the technical term for this property is strong polynomial tractability; see Section 8. Section 9 concludes the paper with a numerical experiment.

Haar and Walsh functions
In the following we recall the definition of Haar functions, which form our basic analytic tool, estimate the Haar coefficients of functions F with ∥F ∥ p,s,γ < ∞ and show a representation of Haar functions in terms of Walsh functions.
Let F : [a, b] → R with ∥F ∥ p,s,γ < ∞.We expand F in its Haar series where, due to the L 2 orthonormality of h (ϕ) j,m , we have In order to estimate the integration error we need to know the decay rate of F (ϕ) (j, m).For a proper statement of this estimate we use the following notation: For u ⊆ [s] and j ∈ N s −1 set j u := (j i ) i∈u and let (j u , −1) ∈ N s −1 be the vector whose ith component is j i if i ∈ u and −1 otherwise.We use a similar notation for (m u , 0). where Proof.To simplify the notation we write a, b, ϕ, Φ and Φ −1 instead of a i , b i , ϕ i , Φ i and Φ −1 From this we directly obtain for (j, m) ̸ = (−1, 0) that and further ) ] .
Using integration by parts in each coordinate i ∈ u we obtain This implies that where λ j i ,m i is as in (8).□ The bound on the Haar coefficients F (ϕ) (j, m) in Lemma 1 depends on Φ −1 .When estimating the worst-case error later on we can remove this dependence since there we only need to estimate sums of the λ j i ,m i and then it is enough to use the property that ϕ i is a probability density function.More precisely, we have: Proof.We have Note that in Lemma 2 we need the assumption that −∞ < a i < b i < ∞.
Walsh functions.Later, when we analyze the integration error, it will be convenient to represent the Haar functions in terms of Walsh functions.In the following we introduce Walsh functions and show a well known connection to Haar functions.
Let the real number x ∈ [0, 1) have dyadic expansion For any dyadic expansion we assume that infinitely many digits are different from 1, which makes the expansion unique.For k ∈ N 0 with dyadic expansion k = κ 0 + κ 1 2 + • • • + κ r−1 2 r−1 and κ 0 , . . ., κ r−1 ∈ {0, 1}, define the kth (dyadic) Walsh function by In dimensions s > 1 we use products of the Walsh functions.Let Then we define the kth (dyadic) Walsh function by We have the following well known representation of Haar functions in terms of Walsh functions.
Here ⊖ denotes the digit-wise dyadic subtraction, i.e., for Proof.For completeness we provide a short proof.Let binary expansions of m and x, respectively.Then we have 1 ) .
This observation will be used later on (in multivariate form).

General error analysis
In this section we present a general error analysis for functions F with bounded norm ∥F ∥ p,s,γ .We obtain an upper bound on the integration error that can be used later on for digital nets and sequences as well as for polynomial lattice point sets.The error analysis requires the use of projection regular point sets with N = 2 m elements in [0, 1) s .

Definition 1 (Projection Regular Point Set). Let m ∈ N and let
be the one-dimensional projections of P. We call P projection regular, if for every The error bound involves a function µ on N 0 which measures the length of the binary digit expansion of a natural number.
Now we can state the announced error estimate.
Theorem 1.Let m ∈ N and let P = {y 0 , . . ., , with the obvious modifications if q = ∞, and where Here, for u ⊆ [s], y n,u denotes the projection of the point y n to the components which belong to u and likewise k u = (k i ) i∈u .
Proof.Conveniently we have Using this property and the Haar series expansion (7) the integration error can be rewritten as err(F ; ϕ For u ⊆ [s] let P u be the set of projections of the points from P to the components which belong to u, i.e., Now we rearrange the sum over all j ∈ N s −1 \ {−1} in (10) according to the sets of components of j's which differ from −1.We have where we used Lemma 1 and Hölder's inequality, which is justified because 1 Denote the summand, that is raised to the power of q by S u , i.e., In order to estimate S u we distinguish two cases.We write where and S (2)   u := S u − S (1)  u .
We consider S (1)   u where the summation is restricted to all j u ∈ {0, 1, . . ., m − 1} |u| .We separate the sum over m u into two parts using , where we used Lemma 2 in the last step.Further, according to Lemma 3, where the notation mu 2 j u has to be interpreted as the vector ( Therefore max We use (13) to estimate S (1)  u and obtain this way .
We illustrate the last step for the univariate case: .
The same argumentation works for general projections u.With the definition of E(P, u) in ( 9) we obtain ) Now we estimate the sum S (2)   u , where the summation is over all We partition the range of summation in the following way: For j u ∈ N |u| 0 \ {0, 1, . . ., m − 1} |u| and any m u ∈ D j u , the set P u ∩ I j u ,mu has at most 1 element, since P, and therefore also Then Using Lemma 2 again we get Combining the estimates ( 14) and (15) give ) .
Inserting this estimate into (11) we obtain |err(F ; ϕ; This yields the desired result.□ Remark 1.In this remark we assume that a i = 0 and b i = 1 for all i ∈ [s].With some small modifications, the bound from Theorem 1 also bounds the star-discrepancy of a point set (cf. [2, Theorem 5.36]).This is not surprising, since for any function F with ∥F ∥ p,s,γ < ∞ and inverse cumulative distribution function Φ −1 as above, we have that the Hardy-Krause variation of F • Φ −1 is also finite.
In the following sections we will study the upper bound on the worst-case error from Theorem 1 for digital nets, digital sequences and also for polynomial lattice point sets.The basic definitions are recalled in the next section.

Digital nets and sequences, and polynomial lattice point sets
In this section we recall the definition and basic results about the node sets in use.Readers who are already acquainted with the digital construction schemes can jump directly to Section 5.
Digital ((t u ) u , m, s)-nets.Let F 2 be the finite field of order 2. We identify F 2 with the set {0, 1} equipped with arithmetic operations modulo 2. Consider the following construction principle for point sets consisting of 2 m points in [0, 1) and multiply for every i ∈ [s] the matrix C i with the vector ⃗ n = (n 0 , . . ., n m−1 ) ⊤ of digits of n in F 2 , Now we set and The point set {x 0 , . . ., x 2 m −1 } is called a digital (t, m, s)-net over F 2 and the matrices C 1 , . . ., C s are called the generating matrices of the digital net.
We remark that explicit constructions for digital (t, m, s)-nets are known with some restrictions on the parameter t (the so-called quality parameter t is independent of m but depends on s), see for instance [2,4] for more information.
It is clear that every projection of a digital (t, m, s)-net over F 2 to coordinates from a set u ⊆ [s], u ̸ = ∅, forms a digital (t, m, |u|)-net over F 2 , see, e.g.[2,Section 4.4.3].However, it may happen that the quality parameter t of a projection is smaller than the overall quality parameter t of the full projection.In order to include this possibility into the definition of digital nets we can define a more general form of the quality parameter t in the following way.
A variant of digital nets are shifted digital nets.Here one chooses ( ⃗ with all but finitely many components different from zero and replaces (17) by 1 , . . ., y (i) m ) are given by (16).
Digital ((t u ) u , s)-sequences.Digital sequences are infinite versions of digital nets.
) k,ℓ∈N we assume that for each ℓ ∈ N there exists a K (ℓ) ∈ N such that c i,k,ℓ = 0 for all k > K (ℓ).Assume that for every m ≥ t the upper left m × m submatrices Consider the following construction principle for infinite sequences of points in [0, 1) ∈ N 0 , and define the infinite binary digit vector of n by where the matrix vector product is evaluated over F 2 .Now set The infinite sequence (x n ) n≥0 is called a digital (t, s)-sequence over F 2 with generating matrices C 1 , . . ., C s .
In the same way as above for digital nets we can regard a digital (t, s)-sequence over For general properties of digital nets and sequences we refer to the books [2,4].

Polynomial lattices. Let
Polynomial lattice point sets have been first introduced by Niederreiter [14] and can be viewed as polynomial analogs of lattice point sets.They form special instances of digital nets; see [2, Chapter 10] or [4] for further information.
a polynomial lattice point set P(g , f ) is given by the points QMC rules that use polynomial lattice point sets as underlying nodes are called polynomial lattice rules.The polynomial f is called the modulus and g the generating vector of the polynomial lattice point set.Note that |P(g, It is well-known that P(g , f ) is projection regular whenever gcd(g i , f ) = 1 for all i ∈ [s] (see, e.g., [2,Remark 10.3]).

Definition 7.
The dual net of the polynomial lattice point set P(g , f ) from Definition 6 is defined as An important property of polynomial lattice point sets is that (see [2, Lemmas 4.75 and 10.6]) where we identify integers k ∈ N 0 with polynomials over F 2 in the natural way: .

Error bound for digital nets
In this section we study the error bound from Theorem 1 for digital nets.The main result of this section is: Theorem 2. For every digital ((t u ) ∅̸ =u⊆[s] , m, s)-net P over F 2 with generating matrices of full row rank we have wce(P; p, s, γ, ϕ) , where 1 ≤ p, q ≤ ∞ with 1 p + 1 q = 1, and with the obvious modifications if q = ∞.
The proof of this result relies on Theorem 1 and an estimate of the quantities E(P, u) for digital nets.To this end, let In all other cases the above Walsh sum equals zero.Hence the quantity E(P, u) from ( 9) boils down to

Now we estimate E(P, u).
Lemma 4. Let ∅ ̸ = u ⊆ [s] and let C i , i ∈ u, be the generating matrices of a digital (t u , m, s)-net P over F 2 .Then we have Proof.To simplify the notation we show the result only for u = [s].The other cases follow by the same arguments.We have . . .
we know from [16, Proof of Lemma 7] that Σ(j 1 , . . ., then Together with condition (21) we obtain Now we have to estimate the sums Σ 1 and Σ 2 .First we have ) , where we used the fact that for fixed l the number of non-negative integer solutions of we obtain and hence We have Now we estimate Σ 1 .If m − t ≥ s − 1 we proceed similar to above and obtain Here we used the estimate for integers b > 1 and k, t 0 > 0 (see, e.g., [16]).
For this case (m − t ≥ s − 1) we obtain where we used that for s > 1.It can be easily checked that the bound (24) holds true also for s = 1.Now we consider the case where m − t < s − 1.We have Thus we obtain This finishes the proof.□ Theorem 2 follows from combining Theorem 1 and Lemma 4.

Error bound for digital sequences
In this section we study the error bound from Theorem 1 for digital sequences.
. ., y N−1 } be the initial segment of a digital sequence and let, for j ∈ [r], From this we immediately obtain where C (m j ×m j ) i is the upper-left m j × m j sub-matrix of C i , 0 (N×m j ) denotes the N × m j zero matrix.Furthermore, also the matrix Hence the dyadic digit vector of k is given by ⃗ k = (a 0 , a 1 , . . ., a m j −1 , l 0 , l 1 , l 2 , . ..) ⊤ =: ) , where a 0 , . . ., a m j −1 are the binary digits of a and l 0 , l 1 , l 2 , . . .are the dyadic digits of ℓ.Note that the elements of the sequence l 0 , l 1 , l 2 , . . .become eventually zero.With this notation and with the above decomposition of the matrix C i we have For the point set Q j under consideration, the vector is constant and its components become eventually zero (i.e., only a finite number of components is nonzero).Furthermore, This means that the point set Q j is a digitally shifted ((t u ) ∅̸ =u⊆[s] , m j , s)-net over F 2 with generating matrices and hence the claim is proven.Using Theorems 1 and 2 (the result also holds for digitally shifted digital nets) we obtain wce(Q j ; p, s, γ, ϕ) and therefore wce(P N ; p, s, γ, ϕ) , where we estimated 2/log 2 = 2.8853 . . .< 3 in order to make the result a bit easier to state.□

Error bound for polynomial lattice point sets
In this section we analyze the worst-case error of polynomial lattice points P(g , f ) as introduced in Definition 6.We restrict our analysis to the case p = ∞ and hence q = 1.Then Theorem 1 yields wce(P; ∞, s, γ, ϕ) For a polynomial lattice point set P(g , f ) we use as figure-of-merit.
According to (19) we have 1 Hence Existence results.For given irreducible polynomial f ∈ F 2 [x] with deg(f ) = m we average B γ (g , f ) over all possible generating vectors g ∈ G s m , where Furthermore, for every c ≥ 1 we have ) . ( In particular, there exists a Proof.The proof uses standard techniques.Note that |G s m | = 2 sm .We have where we used that )|u| This yields 1 as desired.
The estimate (28) follows from an application of Markov's inequality.The existence of g * satisfying (29) follows from choosing c = 1 in (28).□ Combining the above existence result with (26) we obtain the following corollary: Fast component-by-component construction.Initially developed for lattice rules (see, e.g., [17][18][19][20]), nowadays the fast component-by-component construction is also used to construct polynomial lattice rules (see, e.g., [21,22]).This construction method is very efficient and yields polynomial lattice rules which guarantee the almost optimal order of error convergence in the number of employed nodes and even the currently best results in terms of the dependence of the error on the dimension s. Here Proof.The proof uses standard arguments and is based on induction on d.
For d = 1 we have B γ ((1), f ) = 0 and hence the desired bound holds true trivially.
In the induction hypothesis we assume that for some d ∈ {2, . . ., s} we have already constructed g * d−1 := Now we consider B γ ((g Separating the summation over all ∅ ̸ = u ⊆ [d] into summation over all ∅ ̸ = u ⊆ [d − 1] and summation over all u ⊆ [d] that contain the element d we obtain where Note that the only dependence of B γ ((g * follows that the linear polynomial congruence for some δ ∈ (0, 1] and absolute constant C > 0 (independent of m and s, but which may depend on δ and on maybe other parameters).Then we have strong polynomial tractability with exponent τ * at most 1/(1 − δ).
where ld denotes the logarithm in base 2. Then This implies that . Hence we achieve strong polynomial tractability with exponent τ * at most 1/(1 − δ).□ In the following we will look at several constructions of digital nets and check how the weights have to be chosen in order to satisfy the condition in Lemma 5 and to achieve strong polynomial tractability.

Niederreiter and Sobol' digital sequences
A number of explicit constructions of digital nets are known.For a survey we refer to [2, Chapter 8].In the following we study the dependence of the worst-case error on the dimension for two well established explicit constructions of digital nets for general weights.
There are explicit constructions of digital ((t u ) u , m, s)-nets over F 2 due to Sobol' [26] and Niederreiter [27].For nets obtained from a Niederreiter sequence the quality parameter t u satisfies Using this bound, we obtain from Theorem 2 wce(P; p, s, γ, ϕ) where 1 ≤ p, q ≤ ∞ and 1 p + 1 q = 1, with the obvious modifications for q = ∞.
Theorem 6.Let P be a digital net obtained from a Niederreiter sequence.Assume there exists a δ ∈ (0, 1) such that with the obvious modifications for q = ∞.Then we have wce(P; p, s, γ, ϕ) where 1 ≤ p, q ≤ ∞ and 1 p + 1 q = 1.For strictly positive weights γ u , if where in the above γ ∅ :=
This proves (38).Now we prove the assertion for strictly positive weights.Assume that all weights γ u are strictly positive.Using Jensen's inequality, (36) can be re-written as wce(P; p, s, γ, ϕ) Now we use [29, Lemma 4, Eq. ( 15)] which states that This implies wce(P; p, s, γ, ϕ) ≤ Hence we require the weight condition In .
This proves the desired result.□ Remark 3.For product weights γ and hence condition (39) is equivalent to For example, if Γ t = (t!) λ with some λ > 0, then max v∈[i−1] Γ v+1 Γv = i λ and then condition (39) is equivalent to For nets obtained from a Sobol' sequence, the quality parameter t u satisfies (see [30,Eq. 15]) t u ≤ ∑ i∈u (ld (i) + ld ld (i + 1) + ld ld ld (i + 3) + c) , for some constant c > 0 independent of i, u and s.Hence, in the same way as above, we obtain the following theorem.

Polynomial lattices
We also consider polynomial lattices.Again we restrict ourselves to the case p = ∞ and hence q = 1.] .
Then for arbitrary δ as in the statement of the theorem, the proof follows in the same way as the proof of Theorem 6. □ Like in Remark 3 we can re-write weight condition (44) for product-and for POD weights.
Remark 4. For product weights γ u = ∏ i∈u γ i condition (44) is equivalent to For POD weights condition (44) is equivalent to In particular, for Γ t = (t!) λ with some λ > 0, condition (44) is equivalent to

Numerical experiment
Let a i = 0 and b i = 1.We consider the probability density function ϕ(x) = 2x.The cumulative distribution function is then Φ(x) = x 2 and its inverse is given by Φ −1 (x) = √ x.We consider the integrand We use Sobol' points generated in Matlab R2020a to approximate the value of the integral.Fig. 1 shows the convergence of the integration error as the number of points increases.

Definition 3 .
Let s ≥ 1, m ≥ 1 and 0 ≤ t ≤ m be integers.Choose m × m matrices C 1 , . . ., C s over F 2 with the following property: For any non-negative integers d 1 , . . ., d s with d 1 + • • • + d s = m − t the system of the first d 1 rows of C 1 , together with the first d 2 rows of C 2 , together with the . . .first d s−1 rows of C s−1 , together with the first d s rows of C s is linearly independent over F 2 .

Definition 4 .
Let C 1 , . . ., C s be m × m matrices over F 2 .Then the digital net with generating matrices C 1 , .

F 2 [
x] be the set of all polynomials over F 2 and let F 2 ((x −1 )) be the field of formal Laurent series consisting of elements g = ∞ ∑ k=w a k x −k with a k ∈ F 2 and w ∈ Z with a w ̸ = 0.For g ∈ F 2 ((x −1 )) and m ∈ N ∪ {∞} we define the ''fractional part'' function F 2 ((x −1 )) → [0, 1) by

d− 1 ,
g d ), f ) on g d is via Θ(g d ) and hence a minimizer of B γ ((g * d−1 , g d ), p) is also a minimizer of Θ(g d ) and vice versa.Now we use an averaging argument.Since the minimum never exceeds the average we have

Fig. 1 .
Fig. 1.The absolute integration error is shown on the y-axis using 2 m Sobol' points generated in Matlab.The x-axis shows m.The two lines illustrate a convergence rate of order 1/N and 1/ √ N for comparison.The dimension s = 8.In this experiment, the estimated rate of convergence is ≈ N −0.9897 .
). □ {y 0 , y 1 , . . ., y 2 m −1 } be a digital ((t u ) u , m, s)-net over F 2 with generating matrices C 1 , . . ., C s .Hence N = |P| = 2 m .Throughout we assume that the generating matrices C i , i ∈ [s], have full row rank.This is no big restriction but it guarantees that the digital net P is projection regular and therefore Theorem 1 applies.
n=0 wal ku (y n,u ) = 1 if k u = (k i ) i∈u , where u ⊆ [s], belongs to the dual net, i.e., if Now we claim that the point setsQ j , j ∈ [r], are shifted digital ((t u ) ∅̸ =u⊆[s] , m j , s)-nets over F 2 .Indeed, let C 1 , . .., C s ∈ F N×N 2be the generating matrices of the digital sequence.According to our standing assumption these matrices are non-singular upper triangular matrices.For i ∈ [s] the matrix C i is of the form wce(P N ; p, s, γ, ϕ) ≤ we propose a component-by-component algorithm that is based on the figure-of-merit B γ .Given a polynomial f ∈ F 2 [x], with deg(f ) = m, and weights γ = {γ u ∈ R + : u ⊆ [s]}.Construct a vector For d = 2, 3, . .., s: if g * 1 , . .., g * Note that the figure-of-merit B γ depends only on the size of the cube but is otherwise independent of the product measure.Hence, the polynomial lattice rule constructed by Algorithm 1 is universal in the sense that it works for any product ϕ of PDFs.In the following we restrict ourselves again to irreducible polynomials f ∈ F 2 [x] and show that in this case the resulting generating vector g * is of good quality with respect to the figure-of-merit B γ .The resulting polynomial lattice point set P(g * , f ) is projection regular, because f is irreducible and g * ∈ G m and hence gcd(g * i , f ) = 1 for every i ∈ [s].Let f ∈ F 2 [x] be irreducible with deg(f ) = m.Suppose that g * = (g * 1 , . . ., g * d−1 are already chosen, then find g * d ∈ G m by minimizing B γ ((g * 1 , . . ., g * d−1 , g d ), f ) as a function of g d over G m .Remark 2.
Consider a generic sequence of integration problems (I s : H s → R) s≥1 , where H s is a normed space of s-variate functions, with worst-case error wce.Assume we have for every s, m ∈ N a 2 m -element point set P s,m in [0, 1] s such that wce(P s,m ) ≤ 1, then for any this case we have for any 1, then for any