2 K. ENTACHER seies called Es classes, Koobov[4] developed the theoy of good lattice points. Recently, in a seies of papes, Lache et al. [6, 7, 8, 9]

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1 BIT 37:(4) (997), 845{860. QUASI-MONTE CARLO METHODS FOR NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES KARL ENTACHER Depatment of Mathematics, Univesity of Salzbug, Hellbunnest. 34 A-5020 Salzbug, Austia. Abstact. In the pesent pape we study quasi-monte Calo methods to integate functions epesentable by genealized Haa seies in high dimensions. Using (t; m; s)-nets to calculate the quasi-monte Calo appoximation, we get best possible estimates of the integation eo fo pactically elevant classes of functions. The local stuctue of the Haa functions yields inteesting new aspects in poofs and esults. The esults ae supplemented by concete compute calculations. AMS subject classication: Pimay 65D30, 42C0. Key wods: Numeical integation, genealized Haa functions, low-discepancy point sets, quasi-monte Calo methods. Intoduction Quasi-Monte Calo methods ae the most eective appoach fo the appoximate calculation of high dimensional integals. We efe the eade to the compehensive monogaph of Niedeeite [2]. The basic tools fo these methods ae special low-discepancy point sets P = (x n ) N? n=0 ; N 2 N, in the s-dimensional unit cube [0; [s, s 2. The quasi-monte Calo appoximation of an integal of a function f : [0; [ s?! R is given by Z [0;[ s f(x) dx N Optimal estimates of the integation eo (.) R N (f; P) = Z N? n=0 [0;[ s f(x) dx? N f(x n ): N? n=0 f(x n ) ae obtained fo classes of functions epesentable by special othogonal seies and fo suitable point sets. Using special classes of apidly conveging Fouie Reseach suppoted by the Austian Science Foundation (FWF), poject no. P9285, P43-MAT, and by CEI-PACT, WP

2 2 K. ENTACHER seies called Es classes, Koobov[4] developed the theoy of good lattice points. Recently, in a seies of papes, Lache et al. [6, 7, 8, 9] studied Koobov's appoach using genealized Walsh seies. The appopiate point sets in this study ae Niedeeite's (t; m; s)-nets. The latte autho has given a detailed theoy on these low-discepancy point sets and ecient constuction methods (see [2]). Histoically, the denition of (t; m; s)-nets in base 2 is due to Sobol'[3, 4], who called them P -nets. It was Sobol's goal to study quasi-monte Calo integation in tems of the classical Haa system. He mainly consideed classes of functions which satisfy the Holde condition, called H classes and special Haa seies with in a cetain sense bounded sum of the Haa coecients (S p classes). Consideing this bacgound, the following questions natually aise: If we study Koobov's appoach using (t; m; s)-nets in tems of genealized Haa function systems, how ecient is this appoach in compaison to Sobol's estimates and the estimates in the Walsh case, and how does the local natue of the Haa functions inuence poofs and esults? This pape is devoted to an elaboate study of these two questions. Ou appoach yields best possible integation eo estimates fo (t; m; s)-nets as in the Walsh case. The integation eos fo ou E ~ classes show the same ode of magnitude as Sobol' deived fo his H classes, which ae subclasses of ou classes. Analogous esults to those of Lache [7, 9] can be poven in an entiely dieent and sometimes easie way (compae ou Theoem 3.2 and Theoem 3 in [7]) due to the local denition of the Haa functions. Contay to the Walsh case it can be shown that pactically elevant classes of functions satisfy the conditions equied fo the eo estimates. The esults of this pape, which ae pat of the autho's PhD thesis[] ae obtained by multiesolution popeties of the Haa functions only. We conjectue that ou concept may be extended to othe othogonal wavelet systems on compact intevals. 2 Denitions 2. The genealized Haa function system In this section we pesent the denition of the Haa function system elative to an abitay intege base b 2. The notations and the denitions ae taen fom Helleale [3]. In the following, we shall identify the s-dimensional tous (R=Z) s ; s, with the half-open unitcube [0; [ s. The nomalized Haa measue on (R=Z) s, espectively the Lebesgue measue on [0; [ s, will be denoted by s. P Let b 2 be a xed intege. Fo a nonnegative intege, let = j=0 jb j, j 2 f0; ; : : : ; b?g, be the unique b-adic expansion of in base b. Evey numbe P x 2 [0; [ has a unique b-adic expansion x = j=0 x jb?j?, x j 2 f0; ; : : : ; b?g, unde the condition that x j 6= b? fo innitely many j. In the following, this uniqueness condition is assumed P without futhe notice. P g? Fo g 2 N, we dene (g) := j=0 jb j g? and x(g) := j=0 x jb?j?. Futhe

3 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 3 let (0) := 0 and x(0) := 0: Note that (g) 2 f0; ; : : : ; b g? g and x(g) 2 f0; =b g ; : : : ; (b g? )=b g g. Definition 2.. () Let g be a nonnegative intege. We dene b (g) := f 2 N : b g < b g+ g. Futhe, let b (?) := f0g and the sets N 0 := N [ f0g, and N = N 0 [ f?g: Q (2) If g = (g ; : : : ; g s ) 2 N s s, fo an intege s 2, then b (g) := i= b(g i ). Rema 2.. Thoughout this pape, we will use the sets b (g). The eade should note that, if 2 b (g); g 0, then = 0 + b+: : :+ g b g = (g)+ g b g, with g 2 f; : : : ; b? g. If g 0, then ] b (g) = (b? ) b g, whee ]M denotes the numbe of elements of a given set M. Definition 2.2. Let e b : Z b! K, whee Z b = f0; ; : : : ; b? g is the least esidue system modulo b and K := fz 2 C : jzj = g, denote the function e b (a) := e 2i a b ; a 2 Z b : The -th Haa function h (b) ; 0, to the base b is dened as follows: If = 0, then h (b) 0 (x) := 8x 2 [0; [. If 2 b(g); g 0, then h (b) with the elementay b-adic intevals b? (x) := b g 2 e b (a g ) (b) D (a)(x); a=0 D (b) (g) (a) := + a (g) ; + a + b g bg+ b g b g+ : The -th nomalized Haa function H (b) on [0; [ is dened as H (b) 0 := h (b) 0 ; and, if 2 b (g); g 0, then H (b) := b? g 2 h (b) : Definition 2.3. The fundamental domain D (b) of the -th Haa function h (b) is dened as the following elementay b-adic inteval: If = 0, then D (b) 0 := [0; [; if 2 b (g); g 0, then [ b? D (b) := a=0 D (b) (g) (a) = b ; (g) + : g b g Rema 2.2. The notation \fundamental domain" is not common usage. We wite h, H, (g) and D if it is clea fom the context which base b is meant o if we pesent popeties of h (b) and H (b) which ae valid fo all bases b 2. This notation will be used in the multidimensional case dened below. Definition 2.4. Let H b := fh (b) : := ( ; 2 ; : : : ; s ) 2 N s 0g denote the Haa function system to the base b on the s-dimensional Q tous [0; [ s ; s. The -th Haa function h (b) is dened as h(b) (x) := s i= h(b) i (x i ); x = (x ; : : : ; x s ) 2 [0; [ s : The nomalized vesion H (b) is dened in the same way and the -th s- dimensional fundamental domain denotes D (b) := Q s i= D(b) i :

4 4 K. ENTACHER Rema 2.3. The main popeties of the Haa functions ae given in [3, Rema 2., 2.2]. The genealization of popety (4) in Rema 2. of the latte pape is the following: thee ae exactly (b? ) Haa functions h (b), 2 b (g); g = (g ; : : : ; g s ) 2 N s, whee s? is the numbe of indices i such that g i =?, that have the same fundamental domain. In this case we get H (b) = b? h (b) ; with = 2 s i= g i 6=? g i : 2.2 The function class b s (C) We dene ou function classes b E ~ s (C) slightly dieent fom Lache. Since the suppots of the Haa functions h ; 2 (g); g 0; ae elementay b-adic intevals of length b?g, we dene the classes with espect to the esolution b?g. Fo f 2 L ([0; [ s ; s ), let S f denote the Haa seies of f, S f (x) := ^f() h (x); x 2 [0; [ s ; with ^f() := f h d s : Z[0;[ s 2N s 0 Definition 2.5. Fo a given intege base b 2 and fo > 0 and C > 0, let b s (C) be the class of all functions f 2 L ([0; [ s ; s ) with f S f on [0; [ s whee the Haa coecients ^f have the following popety (2.) with C j ^f()j b () ; 8 2 (g); and g = (g ; : : : ; g s ) 2 N s ; b () := sy i= b ( i ) and b ( i ) := : i = 0 b gi : i 2 (g i ); g i 0: Rema 2.4. If f 2 b s (C) and > =2, then the Haa seies S f is absolutely convegent (see []). Example below indicates a Haa seies which is divegent fo all x 2 [0; [ s if = = Examples of functions and thei classes In this section, we pesent some examples of functions fo dieent classes 2 s (C). The poofs that the functions belong to these classes ae given in Entache[]. This pape is also available in the intenet on the Wold-Wide-Web seve: (also accessible via ftp).

5 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 5 () Ou test function to calculate the integation eo in high dimensions is the function F : [0; [ s?! R with (2.2) F (x) := sy i= f(x i ); x = (x ; : : : ; x s ) 2 [0; [ s ; and f(x) := 2(+ 2 )? 2 (2 (? 2 )? ) 2 g(? 2 ) fo x 2? 2 g ;? 2 g+ ; g 0: F belongs to the class 2 s (C) with > =2 and C := maxf; (2? p 2)?s g. (2) Conside the class H (L), 0 < of functions f : [0; [?! R, whee 8 x; y 2 [0; [ : jf(x)? f(y)j Ljx? yj. These function classes and thei genealization to dimension s 2 have been studied by Sobol' [4]. It is easy to show that H (L) (L=2). (3) Examples of the classes 2 3 2s (C). Fom (2) we obtain that the function f : [0; [?! R, f(x) := x n, n 2 N, belongs to the class ( n 2 ). Theefoe (ja 0 j + 2 P n i= i ja ij) is the class of the polynomial f(x) = a 0 + a x + : : : + a n x n ; n 2 N 0 ; a i 2 R; and 2 3 2s ( 2 P s i= i ja ij) is the class of the function F (x ; : : : ; x s ) := a x + a 2 x : : : + a s x s s ; (x ; : : : ; x s ) 2 [0; [ s : Conside F : [0; [ s?! R with (2.3) F (x ; : : : ; x s ) := (x + : : : + x s ) n ; n 2 N: This function belongs to the class 2 3 2s (C) with C = n n ;:::;ns=0 n +:::+ns=n n! 2 (s?]fni=0g) Q s i= (n i? )! whee (?)! := : 2.4 (t; m; s)-nets In analogy to Lache's method of numeical integation of Walsh seies, we get optimal integation eos by using (t; m; s)-nets. Fo ecient constuction methods of (t; m; s)-nets we efe to [5, 0,, 2]. The esults of the integation eo will be compaed to the estimates deived by using the unifom lattice. Special constuction methods of (t; m; s)-nets, the so called digital nets, play an outstanding ole in the Walsh case. Fist esults concening the use of digital nets in ou famewo ae given in [, 2]. The denition of (t; m; s)-nets and the basic popeties ae given in [2, p. 48]. Definition 2.6. Let n 2 N. The unifom lattice is dened as the following point set P consisting of N = n s points P := f( n ; : : : ; s n ) : 0 i n? ; i sg:

6 6 K. ENTACHER 3 The esults 3. Estimates of the integation eo Theoem 3.. Fo b 2, let f 2 b E ~ s (C) with > 2. Futhe let P = fx 0 ; : : : ; x N? g be a point set in [0; [ s and R N (f; P) the integation eo (.). (a) If P is the unifom lattice with N = b ns ; n ; then R N (f; P) C A b (? 2 ) b s N ; =? 2 s ; A := i= i s? b ( 2?)i : (b) The esult in (a) is, apat of the constant C, best possible since thee exists a function f 2 b s (), with R N (f; P) = (b? )b(? 2 ) b (? 2 )? N ; =?=2 s : (c) If P is a (t; m; s)-net to the base b, then R N (f; P) C A ( + 2s? )b (+ 2 )s (log b) s? b (+ 2 )t (log N)s? N (? 2 ) : (d) The esult in (c) is, apat of the constant, best possible, since fo evey > 2 and C > 0 thee exists a (0; m; 2)-net P to the base 2 and a function f 2 2 E ~ 2 (C) with R N (f; P) C 2 (? 2 ) log N N (? 2 ) : Rema 3.. An analogue to Theoem 3. fo the Walsh case is given in Lache and Taunfellne [9, Theoem and Theoem 2]. Recent impovements of these esults can be found in [8, Theoem and Theoem 2]. The (0; m; 2)- net in pat (d) is an example of a digital (t; m; 2)-net. Despite the fact that the eo estimates above suggest to use (0; m; s)-nets to calculate the quasi- Monte Calo appoximation of a function, we can give examples of 2-dimensional functions belonging to ou classes whee, fo example, the integation eo using a (; m; 2)-net to calculate the quasi-monte Calo appoximation is smalle than the eo obtained by a (0; m; 2)-net (see [, 2]). Using (t; m; s)-nets to base 2 (oiginally called P -nets), Sobol' deived an integation eo estimate fo his H classes (compae Sect. 2.3 (2)) of the same ode of magnitude as we obtained fo ou classes, see [4, p. 239]. Sobol's moe geneal S p classes exhibit highe eo bounds. Theoem 3. yields the best integation eo estimate by using (0; m; s)-nets. But fo m 2, a (0; m; s)-net in base b can only exist fo dimensions s b+ (see [2, Coollay 4. 2]). Fo applications on binay computes, ecient calculations ae done with base 2, and fo this eason, (0; m; s)-nets can only exist up to dimension s = 3. We bidge this gap by the following theoem which shows that a function f 2 b E ~ s (C); > =2, belongs to a class b L E ~ s (D); L 2. This guaantees a

7 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 7 calculation of the quasi-monte Calo appoximation with (0; m; s)-nets to the base b L in highe dimensions. Theoem 3.2. Let f 2 b E ~ s (C) with > 2. Then we have with f 2 B s (C C(; b) s ) whee C(; b) = C(b) B b (+ 2 ) B = b L ; L 2; and C(b) = b b? sin a : a= b b (+ 2 )? Rema 3.2. The analogue to Theoem 3.2 fo the Walsh case (see Lache et al. [7, Theoem 3]) is poven fo the case b = 2. In compaison to ou esult, Lache's theoem is only valid fo > + L, 0:25 L < 0:5, and the appopiate constant D contains B, wheeas ou constant D contains only B. 3.2 Numeical esults Hee, we pesent some numeical esults of the integation eo (.) fo the test function 2.2 descibed in Section 2.3. Numeical esults fo function (2.3) ae given in [2]. Futhe numeical compaisons fo special Walsh seies and the function f(x ; : : : ; x s ) := (x + : : : + x s ) =2, using (t; m; s)-nets, good lattice points and Halton sequences ae given R in [6]. We nomalized f, hence we get [0;[ s F (x) dx = in all dimensions. Using = + 2 ; 2 N, we can calculate the quasi-monte Calo appoximation fo this function in intege aithmetic except of a nal division. Thus, ound o eos ae avoided. The (0; m; s)-nets, 2 s 9, ae geneated by the constuction method published in [0]. = 3=2 b = 4 = 7=2 b = 4 N n s e e-05 2.e-04.2e e-0 7.9e e-06.9e e-06 6.e e e e-2 4.7e-0 3.6e e e e e-06 5.e-05.9e-4.3e- 2.9e- 2.e e e e-08.e e-4 6.0e-3 2.4e e-0 6.0e-0.e-08 4.e e-5 4.7e-2 = 3=2 b = 8 = 7=2 b = 8 N n s e e-04.e e e-08.5e-05.7e e e e-06.e e-03 2.e-0.4e e e e-08 4.e-07.e e e-3 3.3e-2 7.8e e e-09.7e e-08.9e e-6 7.2e-3 7.2e-2 3.0e e-2.3e-0.2e-09 3.e e-4 7.3e-05 The following table pesents some esults using the unifom lattice in dimensions 3 s 8.

8 8 K. ENTACHER = 3=2 b = 4 = 7=2 b = N R N N R N N R N N R N N R N N R N e e e e e e-05 = 3=2 b = 8 = 7=2 b = e e e e e e-03 4 The poofs Let f 2 b s (C), > =2, and P = fx 0 ; : : : ; x N? g be a point set in [0; [ s. Since f = S f on [0; [ s, we easily get with R N (f; P) = S N (h ; P) := N Fom Denition 2.5, it follows (4.) R N (f; P) C 6=0 N? n=0 6=0 ^f() S N (h ; P) h (x n ); 2 N s 0 : b () js N(h ; P)j: In the following, we estimate the \Weyl sums" S N (h ; P) fo the dieent point sets. 4. Poof of Theoem 3. Fo a given g = (g ; : : : ; g s ) 2 N s, let ; s denote the numbe of i with g i 0, and let the emaining g i =? (i.e. i = 0). Fo the calculations below, the ode in (g ; : : : ; g s ) does not matte. We only have to note that thee ae possibilities to aange the s? numbes? in g = (g ; : : : ; g s ). Theefoe? s let, w. l. o. g., g 2 N s with (4.2) g i 0 fo i and g i =? fo + i s: Pat (a) Let P be the unifom lattice with N = b ns points. Lemma 4.. () S N (h ; P) = 0 fo all 2 (g) and g with at least one g j < n fo j. (2) Fo all g with g j n; j, we have S N (H ; P) = 8 < : b?n fo (b? ) b n vectos 2 (g) 0 othewise:

9 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 9 Poof. () W. l. o. g., we conside the case 0 g ; : : : ; g l < n and g l+ ; : : : ; g n; l : The fundamental domain D of h ; 2 (g) has the fom Y i (g i ) D = ; i(g i ) + (4.3) [0; [ s? =: b gi b gi i= Two cases ae possible: Case I: D \ P = ;, then S N (h ; P) = 0. Case II: ](D \ P) = b (s?)n ly i= b n?gi ; In the second case, each of the intevals I i ; i l, has b n?gi possibilities fo the i-th coodinate, and the intevals I i ; l + i, contain exactly one coodinate x i of a lattice point x 2 P. Futhe, H i (x i ) = as the coodinate x i necessaily belongs to D i (0). Finally, each of the I i ; + i s, has b n possibilities fo the i-th coodinate x (j) i ; 0 j b n?, whee H 0 (x (j) i ) =. The intevals I ; : : : ; I l ae patitioned into b elementay b-adic subintevals b i (g i ) + a i D i (a i ) = b gi+ ; b i(g i ) + a i + b gi+ ; 0 a i b? ; i l: Each of these subintevals contains b n?gi? coodinates. Hence we get S N (H ; P) = N b(s?)n ly j= b n?gj? ly b? i= a i=0 sy i= I i : e b (a i ( i ) gi ) : {z } (2) Simila to pat (), if D \ P = ;, then S N (h ; P) = 0. Othewise, we have, ](D \ P) = b n(s?). Since, fom (I i ) b?n ; i, it follows that the l st coodinates ( l b n ; : : : ; b n ) of a point p 2 D \ P ae xed. The emaining coodinates vay in f0; =b n ; : : : ; (b n? )=b n g. Theefoe S N (H ; P) = N H l ( l b n ) H ( = b n Y i= H i ( l i b n ): b n ) b n? l +;:::;l s=0 =0 H 0 ( l+ b n ) H 0 ( ls b n ) Fo any numbe x = l i =b n, we obtain that the digit x gi = 0, because of g i n; i. Fom this, it follows that H i (x) = e b (x gi ( i ) gi ) =. Thee ae b n possible D with ](D \ P) = b n(s?). Fo a given D, thee ae (b? ) dieent 2 (g) to get this paticula fundamental domain. We

10 0 K. ENTACHER only have to vay the digit ( i ) gi 2 f; : : : ; b? g fo i. Thus we get S N (H ; P) = b?n fo (b? ) b n points 2 (g), and S N (H ; P) = 0 othewise. We continue with Pat (a) of the poof of Theoem 3.. Fo a given g 2 N s with popety (4.2), let S g := 2(g) b () js N(h ; P)j: Using Lemma 4., we only have to conside g with g ; : : : ; g n. In this case we obtain S g (b? ) b ( 2?)(g+:::+g) : By patitioning the sum on the ight side of (4.) into aeas (g), we get R N (f; P) C s? s (b? ) = i=n g ;:::;g =n g +:::+g=i b ( 2?)i :? i?n+? Thee ae? (i? n + )? solutions of g + : : : + g = i with g j n; j (see [, Sect..]). This yields R N (f; P) C s? s (b? ) b ( 2?)(n?) = C A b (? 2 ) b s N??=2 s : Pat (b) We conside the function f : [0; [ s?! R, f(x) := ^f() h (x); 2N s 0 i s? b ( 2?)i i= {z } with ^f( ; 0; : : : ; 0) = b?g fo 2 (g), g 0, and ^f() = 0 othewise. This denition yields f 2 b E ~ s (), and with g := (g ;?; : : : ;?), g 0, R N (f; P) = g =0 2(g) b g S N(h ; P) : Using S N (h ; P) = b g 2 SN (H ; P) and Lemma 4. fo =, we get R N (f; P) = b? g =n b g(? 2 =:A (b? )b (? 2 ) = ) b (? 2 )? N?=2 s Pat (c) Let P be a (t; m; s)-net in base b and g 2 N s with popety (4.2). :

11 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES Lemma 4.2. P s () S N (h ; P) = 0 fo all 2 (g) and g with i= (g i + ) m? t. (2) js N (H ; P)j b?(g+:::+g) fo all 2 (g), and g with m? t < s i= (g i + ) < m? t + : Note that P this case is only possible fo 2 s. s (3) If i= (g i + ) m? t +, then thee ae at most (b? ) b m vectos 2 (g) with js N (H ; P)j 6= 0. In this case we have js N (H ; P)j b t?m. Poof. () The fundamental domain D of h ; 2 (g), is patitioned into b elementay b-adic subintevals of the fom I = Y i= D i (a i ) [0; [ s? ; whee the a i vay in f0; : : : ; b? g. P The function h is obviously constant on s each inteval I. The equiement i= (g i + ) m? t yields s (I) b t?m so that each I contains exactly s (I) b m points of P. Hence, we have S N (h ; P) = b? b (g+:::+g)+ a =0 b? Y a = i= b g i 2 e b (a i gi ) = 0: (2) The inequality m?t? < g +: : :+g < m?t yields b t?m < (D ) < b +t?m. Theefoe ](D \ P) = b m (D ) and thus we get js N (H ; P)j (D ) = b?(g+:::+g). (3) In this case we have (D ) b t?m. Hence, D contains at most b t elements of P. The esult follows easily fom Rema 2.3. We continue with the poof of Pat (c) of Theoem 3.. Again, we conside S g := 2(g) b () js N(h ; P)j: Fo the case (2) of the lemma above, we get S g (b? ) b ( 2?)(g+:::+g), and fo the case (3) S g (b? ) b t b ( 2?)(g+:::+g). This and pat () of the lemma above yield R N (f; P) C s? s (b? ) =2 s? + C b t s (b? ) = m?t? g ;:::;g=0 m?t?<g +:::+g<m?t g ;:::;g=0 g +:::+gm?t b ( 2?)(g+:::+g) b ( 2?)(g+:::+g) :

12 2 K. ENTACHER The numbe of solutions of the equation g + : : : + g = i; g i 2 N 0, equals (i + )?. This yields? i+?? R N (f; P) C s? s? (b? ) =2 = i= s? + Cb t s (b? ) (m? t? + + i) s? b ( 2?)(m?t?+i) i= (m? t? + + i) s? b ( 2?)(m?t?+i) : Fo all i 2 N we have m?t?++i mi fo 2, and m?t?++i (m+)i fo. Hence R N (f; P) C ms? b (? 2 )(t+s) b (? 2 )m s? s (b? ) =2 + C ms? 2 s? b t b (? 2 )(t+s) b (? 2 )m = i= i s? b ( 2?)i s? s (b? ) Again let A := P i= is? b ( 2?)i, then we get the esult R N (f; P) C A ( + 2s? )b (+ 2 )(t+s) (log b) s? Pat (d) is poven in Entache[2]. Poof of Theoem 3.2 i= (log N)s? N? : 2 i s? b ( 2?)i : Let B = b L, L 2, fo a given base b 2. We shall pove the esult by induction fo dimension s. Case I: dimension s = We stat with the examination of the Haa seies S (B) ; h 2 H b, with espect to the system H B. To avoid too many indices, we shall denote the Haa coecients of a given function f with espect to the system H B by ^f(n), hence with agument n. Futhe we will use the agument to signify that ^f() denotes the Haa coecient of f with espect to the Haa system H b. In the next lemma, we shall pove that the -th Haa function h 2 H b is a Haa polynomial with espect to the system H B. Lemma 4.3. Let 2 B (j), j 0. Then the n-th Haa coecient c h (n) with espect to the system H B of the -th Haa function h 2 H b is equal to zeo fo all n with n < B j o n B j+. If = 0, then c h (n) = 0 fo all n. Poof. We have ch (n) = Z D \D (B) n h h (x)h (B) n (x) dx: The case = 0 is easily veied. Let j 0 and n B j+. Then we have eithe D \ D n (B) = ; o D n (B) D. In the second case, the Haa function h, is

13 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 3 constant on D \ D n (B) = D n (B) and theefoe h c(n) = 0. The inequality n < B j is teated similaly. Coollay 4.4. Let B j < B j+, j 2 N 0. The Haa function h 2 H b has a nite Haa seies with espect to the system H B, and h (x) = n2 B(j) ch (n)h (B) n (x): Let f 2 b (C). Ou goal is to estimate the Haa coecients ^f(n); n 2 N. The following lemma shows a epesentation of the numbes ^f(n) in tems of the Haa coecients ^f() with egad to H b and the Haa coecients c h (n) with espect to the function system H B. Lemma 4.5. If f 2 b (C), > =2, then ^f(n) = L(g+)? g=lg 2(g) ^f()c h (n) fo n 2 B (g); g 0: This lemma will yield the ode of ^f(n) if we ae able to estimate c h (n). Poof. We have S f (x) = ^f(0) + g=0 (g+)l? g=gl 2(g) ^f()h (x): Since 2 (g) with gl g < (g + )L if and only if B g < B g+, Coollay 4.4 yields S f (x) = ^f(0) + = ^f(0) + g=0 (g+)l? g=gl g=0 n2 B(g) 2(g) ^f() (g+)l? g=gl n2 B(g) 2(g) ch (n)h (B) n (x) ^f()c h (n) A h (B) n (x): This yields the esult. Let us conside the integes ; n, with B g ; n < B g+ ; g 0, since fom Lemma 4.3 it follows that only in these cases c h (n) 6= 0 is possible. In the following, let n 2 B (g); g 0, and 2 (g) with Lg g < L(g + ), moe pecisely (4.4) n 2 B (g) and 2 (g) with g = Lg + j; 0 j < L: Futhe let us dene E a := D n (B) (a) = [ (a) ; (a+) [ with (a) := n(g) B g + a ; 0 a < B: Bg+

14 4 K. ENTACHER Since we get (4.5) B? h (B) n (x) = B g 2 e B (a n g ) Ea (x); ch (n) = B g 2 a=0 B? a=0 e B (a n g ) d Ea (): Fom Helleale [3, Lemma 3.2], we conclude that we may have d Ea () 6= 0 only if (g) 2 fb g (a) (g); b g (a+) (g)g, and in this case the given estimation of jd Ea ()j does not depend on the explicit value of (g). Theefoe we have to examine the set A(; g) := fa 2 f0; : : : ; B? g : (g) = b g (a) (g)g: The case (g) = b g (B) (g) will be teated below. Let n(g) = n 0 +n B+: : :+n g? B g? be the B-adic expansion of n(g). Changing to base b yields n(g) = n 0 + n b + : : : + n Lg? b Lg?, n i 2 f0; : : : ; b? g, and fom this we get fo a = a 0 + a b + : : : + a L? b L?, n(g) B + a = a 0 + a b + : : : + a L? b L? + n 0 b L + : : : + n Lg? b L(g+)? : Because of g = Lg + j, j 2 f0; : : : ; L? g, it follows and (a) g (a) (g) = ( = a L?j?. Thus we have 0:n Lg? n Lg?2 : : : n 0 a L? : : : a L?j if j 0:n Lg? : : : n 0 if j = 0; b g (a) (g) = a L?j + a L?j+ b + : : : + a L? b j? + n 0 b j + : : : + n Lg? b g? : The equiement (g) = b g (a) (g) is equivalent to 0 = a L?j ; : : : ; j? = a L? ; j = n 0 ; : : : ; g? = n Lg?. Hence, the numbes a have the fom a = a 0 + a b + : : : + a L?j? b L?j? + 0 b L?j + : : : + j? b L? : If we vay a 0 ; : : : ; a L?j?, we get A(g; ) = fb L?j (j) + i : i 2 f0; : : : ; b L?j? gg and ]A(g; ) = b L?j : Note that if a = b L?j (j), then a 0 = 0; : : : ; a L?j? = 0, and theefoe (4.6) (a) g = 0 and (a)? (a) (g + ) = 0:

15 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 5 Finally, we have to conside the case (B) = (n(g) + )=B g, since the case (g) = b g (B) (g) is not included above. Hee we distinguish between (B) = and (B) <. Let a := B?. The st case yields Ea d() =?c J (); J := [0; (a) [, and theefoe we have Ea d() 6= 0 only if a 2 A(g; ) due to the above poof. If (B) <, one can easily show that g (B) = 0 and (B)? (B) (g +) = 0. Fom Lemma 3. in [3], we obseve that Ea d() =?c J ; J := [0; (a) [, and we ae in the same situation as befoe. Let us etun to the examination of h c(n) in (4.5). Lemma 4.6. If n 2 B (g); g 0, and 2 (g) with g = Lg + j, 0 j < L, then thee ae exactly (b? ) b j integes 2 (g) with jc h (n)j 6= 0. In this case we have jc h (n)j B g 2 b L?j b? g 2? sin g b Poof. The consideations above yield A(g; ) = fb L?j (j) + i : i 2 f0; : : : ; b L?j? gg: Let (j) 6= 0. We conside a = b L?j (j)?. Then a 62 A(g; ), but a + = b L?j (j) 2 A(g; ). In this case, (4.6) implies that (a+) g = 0 and (a+)? (a+) (g+) = 0. Hence Lemma 3., Pat 2 of Helleale [3] implies d Ea () = 0. If (j) = 0, the situation above is not possible, since A(g; ) = f0; : : : ; b L?j?g. Hence fo both cases, (j) 6= 0 and (j) = 0, we obtain, by equation (4.5), ch (n) = B g b L?j? 2 i=0 e B (a i n g ) d Eai () with a i := b L?j (j) + i: The esult follows fom Lemma 3.2 in [3] and fom the fact that thee ae b j dieent possibilities fo (j) and b? possibilities fo g. Now we ae able to estimate ^f(n). Lemma 4.5 yields j ^f(n)j C L(g+)? g=lg 2(g) Using j = g? Lg and the lemma above, we get 2(g) b g j c h (n)j b g j c h (n)j: g b g(+ 2 ) B( 2 +) b : b? sin a : a= b {z } =: C(b) These ae exactly the numbes with g 2 f; : : : ; b? g, and (g) vaies abitaily in f0; ; : : : ; b j? g.

16 6 K. ENTACHER Finally we obseve that j ^f(n)j b (+ 2 ) < C C(b) B b (+ 2 )? B g : {z } =: C(; b) Case II: dimension s 2 The poof of this case is obtained by induction, in same way as in [7, p. 709]. In the latte pape a simila theoem is poved fo the Walsh case. Acnowledgement. I want to than my supeviso Pete Helleale fo his suppot duing the wo on my PhD thesis. Futhe, I want to than Wolfgang Ch. Schmid fo his assistance with compute pogams and numeical esults. REFERENCES. K. Entache. Genealized Haa function systems in the theoy of unifom distibutions of sequences modulo one. PhD thesis, Univesity of Salzbug, K. Entache. Genealized Haa function systems, digital nets and quasi-monte Calo integation. In H.H. Szu, edito, Wavelet Applications III, Poc. SPIE 2762, 996. Available on the intenet at 3. P. Helleale. Geneal discepancy estimates III: The Edos-Tuan-Kosma Inequality fo the Haa function system. Mh. Math., 20:25{45, N.M. Koobov. Numbe-Theoetic Methods in Appoximate Analysis. Fizmatgiz, Moscow, 963. (In Russian). 5. G. Lache, A. Lau, H. Niedeeite, and W.Ch. Schmid. Optimal polynomials fo (t; m; s)-nets and numeical integation of Walsh seies. SIAM J. Nume. Analysis, 33, 996 pp. 2239{ G. Lache and W.Ch. Schmid. Multivaiate Walsh seies, digital nets and quasi-monte Calo integation. In H. Niedeeite and P. Jau-Shyong Shiue, editos, Monte Calo and Quasi-Monte Calo Methods in Scientic Computing, volume 06 of Lectue Notes in Statistics, pages 252{262. Spinge, G. Lache, W.Ch. Schmid, and R. Wolf. Repesentation of functions as Walsh seies to dieent bases and an application to the numeical integation of high-dimensional Walsh seies. Math. Comp., 63:70{76, G. Lache, W.Ch. Schmid, and R. Wolf. Quasi-Monte Calo methods fo the numeical integation of multivaiate Walsh seies. Mathl. Comput. Modelling, 23(8/9):55{67, 996.

17 NUMERICAL INTEGRATION OF MULTIVARIATE HAAR SERIES 7 9. G. Lache and C. Taunfellne. On the numeical integation of Walsh seies by numbe-theoetic methods. Math. Comp., 63:277{29, H. Niedeeite. Point sets and sequences with small discepancy. Mh. Math., 04:273{337, H. Niedeeite. Low-discepancy and low-dispesion sequences. J. Numbe Theoy, 30:5{70, H. Niedeeite. Random Numbe Geneation and Quasi-Monte Calo Methods. SIAM, Philadelphia, USA, I. M. Sobol'. The distibution of points in a cube and the appoximate evaluation of integals. Zh. Vycisl. Mat. i Mat Fiz., 7:784{802, 967. (In Russian). 4. I. M. Sobol'. Multidimensional Quadatue Fomulas and Haa Functions. Naua., Moscow, 969. (In Russian).