Dyadic diaphony of digital sequences

Transcription

1 Dyadic diaphony of digital sequences Friedrich Pillichshammer Abstract The dyadic diaphony is a quantitative measure for the irregularity of distribution of a sequence in the unit cube In this paper we give formulae for the dyadic diaphony of digital, s)-sequences over Z, s, These formulae show that for fixed s {, }, the dyadic diaphony has the same values for any digital, s)- sequence For s, it follows that the dyadic diaphony and the diaphony of special digital, )-sequences are up to a constant the same We give the exact asymptotic order of the dyadic diaphony of digital, s)-sequences and show that for s it satisfies a central limit theorem Introduction The diaphony F see [3] or [4, Definition 9] or [8, Exercise 57, p 6]) of the first elements of a sequence ω x n ) n in [, ) s is given by / F ω) e πi k,xn ρk), k Z s k where for k k,, k s ) Z s it is ρk) s i max, k i ) and, denotes the usual inner product in R s It is well known that the diaphony is a quantitative measure for the irregularity of distribution of the first points of a sequence In fact, a sequence ω is uniformly distributed modulo if and only if lim F ω) Throughout this paper we will call the diaphony the classical diaphony In [7] Hellekalek and Leeb introduced the notion of dyadic diaphony which is similar to the classical diaphony but with the trigonometric functions replaced by Walsh functions Before we give the exact definition of the dyadic diaphony recall that Walsh-functions in base can be defined as follows: for a non-negative integer k with base representation k κ m m + + κ + κ and a real x with canonical) base representation x x + x + the k-th Walsh function in base is defined as wal k x) : ) x κ +x κ + +x m+ κ m For dimension s, x,, x s [, ) and k,, k s we define s wal k,,k s x,, x s ) : wal kj x j ) The author is supported by the Austrian Science Foundation FWF), Project S969, that is part of the Austrian ational Research etwork Analytic Combinatorics and Probabilistic umber Theory j

2 For vectors k k,, k s ) s and x x,, x s ) R s we write wal k x) : wal k,,k s x,, x s ) ow we can give the definition of the dyadic diaphony see Hellekalek and Leeb [7]) Definition The dyadic diaphony F, of the first elements of a sequence ω x n ) n in [, ) s is defined by F, ω) 3 s k s k ψk) wal k x n ) where for k k,, k s ) it is ψk) s i ψk i) and for k, { if k, ψk) r if r k < r+ with r Throughout the paper we will write rk) r if r is the unique determined integer such that r k < r+ It is shown in [7, Theorem 3] that the dyadic diaphony is a quantitative measure for the irregularity of distribution of the first points of a sequence: a sequence ω is uniformly distributed modulo if and only if lim F, ω) Further it was shown in [] that the dyadic diaphony is up to a factor depending only on s the worst-case error for quasi-monte Carlo integration of functions from a certain Hilbert space We consider the dyadic diaphony of a special class of sequences in [, ) s, namely of so-called digital, s)-sequences over Z for s, Digital, s)-sequences or more generally digital t, s)-sequences were introduced by iederreiter [, ] and they provide at the moment the most efficient method to generate sequences with excellent distribution properties We remark that a digital, s)-sequence over Z only exists if s or s For higher dimensions s 3 the concept of digital t, s)-sequence over Z with t > has to be stressed see [] or []) Before we give the definition of digital, s)-sequences we introduce some notation: for a vector c γ, γ, ) Z and for m we denote the vector in Z m consisting of the first m components of c by cm), ie, cm) γ,, γ m ) Further for an matrix C over Z and for m we denote by Cm) the left upper m m submatrix of C Definition For s {, }, choose s matrices C,, C s over Z with the following property: for every m and every n m the vectors c ) m),, c n ) s) m), c m),, c m nm) s) are linearly independent in Z m j) Here c i is the i-th row vector of the matrix C j In particular for any m the matrix C j m) has full rank over Z for all j {,, s}) For n let n n + n + n + be the base representation of n For j {,, s} multiply the vector n n, n, ) with the matrix C j, C j n : x j n ), xj n ), ) Z /,

3 and set Finally set x n : x ) n,, x s) n ) x n j) : xj n ) + xj n ) + Every sequence x n ) n constructed in this way is called digital, s)-sequence over Z The matrices C,, C s are called the generator matrices of the sequence To guarantee that the points x n belong to [, ) s and not just to [, ] s ) and also for the analysis of the sequence we need the condition that for each n and j s, we have x j ni) for infinitely many i This condition is always satisfied if we assume that for each j s and r we have c j i,r for all sufficiently large i, where cj i,r are the entries of the matrix C j Throughout this paper we assume that the generator matrices fulfill this condition see [, p7] where this condition is called S6)) For example if s and if we choose as generator matrix the identity matrix, then the resulting digital, )-sequence over Z is the well known van der Corput sequence in base Hence the concept of digital, )-sequences over Z is a generalization of the construction principle of the van der Corput sequence For the classical diaphony it was proved by Faure [5] that F ω)) π u u, ) if ω is a digital, )-sequence over Z whose generator matrix C is a non-singular upper triangular matrix Faure and we shall do so as well) called these sequences UT-sequences Here denotes the distance to the nearest integer function, ie, x : minx x, x x )) See also [, 6, ] for further results concerning the classical diaphony of special -dimensional sequences The aim of this paper is to prove a similar formula for the dyadic diaphony of digital, s)-sequences over Z for s {, } see Theorems and ) These formulae show that for fixed s the dyadic diaphony is invariant for digital, s)-sequences over Z Further we find that the dyadic diaphony and the classical diaphony of UT-sequences s ) only differ by a multiplicative constant Corollary ) We obtain the exact asymptotic order of the dyadic diaphony of digital, s)-sequences over Z Corollary and 4) Moreover it follows from our formula that the squared dyadic diaphony of digital, )-sequences over Z satisfies a central limit theorem Corollary 3) For digital, )-sequences we will obtain a similar, but weaker result Corollary 5) The Results for s First we give the formula for the dyadic diaphony of digital, )-sequences over Z This formula shows that the dyadic diaphony is invariant for digital, )-sequences over Z Theorem Let ω be a digital, )-sequence over Z Then for any we have F, ω)) 3 u u 3

4 We defer the proof of this formula to Section 4 Remark In Theorem we have an infinite sum for the dyadic diaphony of a digital, )-sequence over Z This formula can easily be made computable since for m we have / u / u for u m + Therefore we have m ) F, ω)) 3 + u From Theorem we find the surprising result that the classical diaphony and the dyadic diaphony of a UT-sequence are essentially the same Corollary Let ω be a digital UT-sequence over Z Then for any we have u m F, ω) 3 π F ω) Proof This follows from Theorem together with Faures formula ) From Remark one can see immediately that the dyadic diaphony of a digital, )- sequence over Z is of order F, ω) O log /) But we can even be much more precise From a thorough analysis of the sum in Remark we obtain the exact dependence of the dyadic diaphony of digital, )-sequences over Z on log / Corollary Let ω be a digital, )-sequence over Z For m we have and F, ω)) m )m 9 m 9 m lim sup F, ω)) log 3 log The proof of this result will be given in Section 5 We just remark that the result for the lim sup follows also from a result of Chaix and Faure [, Théoréme 43] for the classical diaphony of the van der Corput sequence together with Corollary and Theorem In [3] the authors showed that the star discrepancy and all L p -discrepancies of the van der Corput sequence in base satisfy a central limit theorem The same arguments as in their proofs can now be used to obtain the subsequent result Corollary 3 Let ω be a digital, )-sequence over Z Then for every real y we have { M # < M : F, ω)) 4 log + y } 4 log Φy) + o), 3 where Φy) π y e t dt denotes the normal distribution function and log denotes the logarithm to the base Ie, the squared dyadic diaphony of a digital, )-sequence over Z satisfies a central limit theorem 4

5 Remark Together with Corollary we also obtain a central limit theorem for the square of the classical diaphony of UT-sequences Proof As already mentioned, the proof follows exactly the lines of the proof of [3, Theorem ] One only has to compute the expectation and the variance of the random variable S m m X w, w where X is uniformly distributed on [, ) By tedious but straightforward calculations we obtain ES m m/ and VarS m m/ m )/6 3 The Results for s We give the formula for the dyadic diaphony of digital, )-sequences over Z which shows that the dyadic diaphony is invariant for digital, )-sequences as well Theorem Let ω be a digital, )-sequence over Z Then for any we have F, ω)) 9 4 u u u We defer the proof of this formula to Section 4 Remark 3 In Theorem we have an infinite sum for the dyadic diaphony of a digital, )-sequence over Z Again this formula can easily be made computable compare with Remark ) For m we have F, ω)) 9 m ) 4 + 3m 4 u u + m 4 u From Remark 3 one can see immediately that the dyadic diaphony of a digital, )- sequence over Z is of order F, ω) Olog /) Also here we obtain from a thorough analysis of the sum in Remark 3 the exact dependence of the dyadic diaphony of digital, )-sequences over Z on log / Corollary 4 Let ω be a digital, )-sequence over Z Then for any m we have and F, ω)) m 8 + 7m O m m ) lim sup F, ω)) log ) 8log ) The proof of this result will be given in Section 5 Following this proof the Om/ m )- term in the above bound can easily be made explicit Unfortunately we could not show that the squared dyadic diaphony of a digital, )- sequence over Z satisfies a central limit theorem However, we were able to prove the following result 5

6 Corollary 5 Let ω be a digital, )-sequence over Z Then for any ε > we have { ) } lim < m 3 : m 3 ε < F, ω) < 3 log 3 + ε Proof By tedious but straightforward calculations using Theorem we obtain and F, ω)) 3 3 m m + Om m ) m m From this the result immediately follows F, ω)) m4 m + Om 3 m ) 4 The Proofs of Theorems and For the proofs of Theorems and we need the subsequent lemma This result was implicitly proved in [3] For the sake of completeness we provide the short proof Lemma Let the non-negative integer U have binary expansion U U + U + + U m m For any non-negative integer n U let n n + n + + n m m be the binary representation of n For p m let Up) : U + + U p p Let b, b,, b m be arbitrary elements of Z, not all zero Then U ) b n + +b m n m ) b w+u w+ + +b m U m w+ U, where w is minimal such that b w Proof From splitting up the sum we obtain U ) b n + +b m n m w+ U w+ + +U m m w ) + Uw) ) nw ) b w+u w+ + +b m U m ) nw ) b w+n w+ + +b m n m Uw) + ) b w+u w+ + +b m U m ) nw w+ { ) b w+u w+ + +b m U m Uw) if Uw) < w, w+ Uw) if Uw) w, { ) b w+u w+ + +b m U m Uw)/ w+ w+ if Uw)/ w+ < /, Uw)/ w+ if Uw)/ w+ /, ) b w+u w+ + +b m U m w+ Uw) w+ 6

7 Since Uw) U w+ the result follows w+ ow we can give the Proof of Theorem Let r k < r+ Then k k + k + + k r r with k i {, }, i < r and k r Let, denote the usual inner product in Z and let c i Z be the i-th row vector of the generator matrix C of the digital, )-sequence for short we write C instead of C here) Since the i-th digit x n i) of the point x n, i, n, is given by c i, n see Definition ) we have wal k x n ) ) k c, n + +k r c r+, n ) k c + +k r c r+, n ) Let C c i,j ) i,j For k, k k + k + + k r r, k i {, }, i < r and k r define uk) : min{l : k c,l + +k r c r+,l } ote that since C generates a digital, )-sequence over Z we obviously have uk) r + For fixed k, r k < r+ let b b, b, ) : k c + + k r c r+ Let m m If uk) m we obtain from ) together with Lemma, wal k x n ) ) b, n ) n b + +n m b m ) n uk) + ) uk)b uk) + uk) uk) But if uk) > m the above equality is trivially true Therefore we have F, ω)) k r r uk) rk) r+ uk) r k r r uk) r+ u u u u u u ru uk) ) r+ k r uk)u r+ r k r uk)u 7

8 ow we have to evaluate the sum r+ for r u and u This is the number of k r uk)u vectors k,, k r ) Z r such that Cr + ) k k r x u+ x r+ Z r+ 3) for arbitrary x u+,, x r+ Z Recall that for an integer m we denote by Cm) the left upper m m submatrix of the matrix C, see Section ) We consider two cases: Assume that r u Then system 3) becomes k Cr + ) k r Since the r + ) r + ) matrix Cr + ) is regular over Z it is clear that there exists a vector k k,, k r ) Z r+ such that Cr + ) k,,, ) Assume that k r, then we have Cr) k,, k r ),, ) Again we know that Cr) is regular over Z and therefore we obtain k k r Hence k, the zero vector in Z r+ This is now a contradiction since k is a solution of the system Cr + ) k,,, ) Therefore we have u k u uk)u Assume that r u Since Cr) is regular over Z it is clear that Dr) : Cr) ) is regular over Z Hence for any vector k Z r there is a vector l Z r such that k Dr) l Therefore system 3) can be rewritten as Cr + ) Dr) l ) x u+ x r+ 8

9 with l Z r ow we use the definition of the matrix Dr) and find that the above system is equivalent to the system l + c r+ r + ), l r x u+ d d d r d r x r+ where d,, d r ) : c,r+,, c r,r+ )Dr) ow one can easily see that for arbitrary x u+,, x r there exists exactly one solution l l,, l r ) Z r such that the first r lines of the above system are fulfilled Further there is exactly one possible choice of x r+ Z such that this vector l is a solution of the above system Therefore we obtain r+ k r uk)u r u ow we have F, ω)) The result follows u u ) u + u ) r u 6 r ru u u Proof of Theorem Let ω x n ) n be a digital, )-sequence over Z Let x n x n, y n ) for n Clearly the sequences x n ) n and y n ) n are digital, )-sequences over Z We have F, ω)) 8 k k ψk) wal k x n ) wal 8 rk) k x n ) + wal 8 rl) l y n ) k l + wal 8 rk)+rl) k x n )wal l y n ) k,l 3 u + wal 8 rk)+rl) k x n )wal l y n ), 4) u k,l where for the last equality we used Theorem We have to consider Σ : wal rk)+rl) k x n )wal l y n ) k,l 9

10 Assume that r k < r+ and t l < t+ Then k k + k + + k r r with k i {, }, i < r and k r and l l + l + + l t t with l j {, }, j < t and l t Let c i Z be the i-th row vector of the generator matrix C and let d i Z be the i-th row vector of the generator matrix C, i Since the i-th digit x n i) of x n is given by c i, n and the i-th digit y n i) of y n is given by d i, n see Definition ) we have wal k x n )wal l y n ) ) k c, n + +k r c r+, n +l d, n + +l t d t+, n ) k c + +k r c r+ +l d + +l tdt+, n Let C c i,j ) i,j and C d i,j ) i,j Define uk, l) : min{j : k c,j + + k r c r+,j + l d,j + + l s d s+,j } Since C, C generate a digital, )-sequence over Z we obviously have uk, l) r+s+ As in the proof of Theorem we now apply Lemma and obtain wal k x n )wal l y n ) uk,l) Therefore we have Σ k,l r,t r,t rk)+rl) uk,l) r+t We have to evaluate the double-sum r+ k r u t+ uk,l) l t uk,l) r+t+ u r+t u r+ k r t+ l t } {{ } uk,l)u number of k,, k r, l,, l t Z such that c, c r+, d, d t+, c, c r+, d, d t+, c,r+t+ c r+,r+t+ d,r+t+ d t+,r+t+ uk,l) r+ k r uk,l) t+ l t } {{ } uk,l)u for u r + t + This is the k k r l l t x u+ x r+t+ 5)

11 for arbitrary x u+,, x r+t+ Z ote that the above matrix we denote it by Cr, t) is an r + t + ) r + t + ) matrix Since C, C generate a digital, )-sequence over Z, it follows that Cr, t) is regular ow we consider three cases: Assume that u r + t + Then system 5) becomes Cr, t) h 6) Since Cr, t) is regular there exists a vector h k,, k r, l,, l t ) Z r+z+, h, such that Cr, t) h,,, ) Assume that l t Then But then c, c r+, d, d t, c, c r+, d, d t, c,r+t+ c r+,r+t+ d,r+t+ d t,r+t+ c, c r+, d, d t, c, c r+, d, d t, c,r+t+ c r+,r+t+ d,r+t+ d t,r+t+ k k r l l t k k r l l t Since the above matrix is again regular we obtain that k,, k r, l,, l t ),, ) and therefore h, a contradiction Hence l t and in the same way one can show that k r We have shown that system 6) has exactly one solution and therefore we have r+ k r s+ l s } {{ } uk,l)u Assume that u r + t + Let x Z Since Cr, t) is regular there exists exactly one vector h Z r+t+ such that Cr, t) h,,,, x) Assume that h is of the form h k,, k r,, l,, l t, ) particular we have h c, c r+, d, d t+, c, c r+, d, d t+, c,r+t c r+,r+t d,r+t d t+,r+t Z r+t+ In 7)

12 Since c, c r, d, d t, c, c r, d, d t, c,r+t c r,r+t d,r+t d t,r+t is regular we find that h is the unique solution of 7) Hence h is exactly the same vector as in the first case where u r + t + But then h cannot be a solution of Cr, t) h,,,, x) Therefore we obtain r+ k r s+ l s } {{ } uk,l)u 3 Assume that u r + t We rewrite system 5) in the form c, c r, d, d t, c, c r, d, d t, c,r+t c r,r+t d,r+t d t,r+t c,r+t+ c r,r+t+ d,r+t+ d t,r+t+ c,r+t+ c r,r+t+ d,r+t+ d t,r+t+ k k r l l t x u+ x r+t x r+t+ x r+t+ + y r,t where y r,t c r+, + d t+,,, c r+,r+t+ + d t+,r+t+ ) Z r+t+ Since the upper r + t) r + t) sub-matrix of the above matrix is regular we find for arbitrary x u+,, x r+t exactly one solution of the first r + t rows of the above system This solution can be made a solution of the whole system by an adequate choice of x r+t+ and x r+t+ Therefore we have r+ k r s+ l s } {{ } uk,l)u r+t u

13 ow we have Σ r+t u r+t u r,t r,t r+t u r+t u u u u u r,t r+t u u u u wu u u + 4) + 6 8u ) u u u u + 6 r+t u + r+t+) r,t r+t+ + 6 r+t u w + w + 6 u r,t r+tu u u ) u u ) u The result follows by inserting this expression into 4) u r+t+ ) 5 The Proofs of Corollaries and 4 We will say that a real β in [, ) is m-bit if β b + + bm m with b i {, } Ie, an m-bit number is of the form k/ m with k {,,, m } The essential technical tool for the proof of Corollary is provided by Lemma Assume that β, b b this here and in the following always means base representation) has two equal consecutive digits b i b i+ with i m and let i be minimal with this property, ie, Replace β by Then m u u γ m u u β + β, b i+ or β, b i+ or β, b i+ or β, b i+ γ, b i+ resp γ, b i+ resp γ, b i+ resp γ, b i+ 9 9 ) )i τ) in the first two cases, i ) )i τ in the last two cases, i 3

14 where τ :, b i+ b i+3 Remark 4 In any case we have m u u γ m u u β with equality iff τ in the first two cases and iff τ in the last two cases Proof of Lemma This is simple calculation We just handle the first case here m i u γ u β ) γ i β + u γ) u β ) 8) u Here γ 3 u γ) 3 u ) + τ and i β τ Further, for even u we have i i+ ) ) u+ u+ i + τ i u and u β 3 i + τ i+ u, and for odd u we have u γ) ) + u+ τ and u β ) + u+ τ 3 i i u 3 i i+ u Inserting this into 8) we obtain m u γ u β ) 9 u The other cases are calculated in the same way + i ) τ) From Lemma we obtain the subsequent result concerning the maximum of m u u β over all β We remark that in [9] the authors considered the same problem without the square at the -function Lemma 3 Consider β R with the canonical base representation ie, with infinitely many digits equal to zero) Then there exists m max β u u β m )m 7 m 7 m and it is attained if and only if β is of the form β with β ) ) m+ or β ) m ) 3 3 Remark 5 ote that 3 ) ) m+ {, if m is odd,, if m is even, and ) m ) {, if m is odd, 3, if m is even 4

15 Proof of Lemma 3 For any γ, c c c m c m+ with fixed c,, c m the sum m u u γ obviously becomes maximal if c m and c m+ c m+, or if c m and c m+ c m+ Hence by Lemma the u β m sup β u only can be attained, respectively approached by if m is even, and by if m is odd ow we check easily that m u for k,, 6 and the result follows β, or b m is the last zero) β, or β 3, b m is the last one) β 4, or b m is the last zero) β 5, or β 6, b m is the last one) u β k m )m 7 m 7 m We give the Proof of Corollary We have max m m u u m max u β m β m bit )m 7 m 7 m by Lemma 3 The result follows now together with Remark u For the proof of Corollary 4 we can in principle proceed as for the proof of Corollary However, in this case the detailed computations are by far more involved than above First we have Lemma 4 Assume that β, b b has two equal consecutive digits b i b i+ with i m and let i be minimal with this property, ie, β, b i+ or β, b i+ or β, b i+ or β, b i+ 5

16 Replace β by γ, b i+ resp γ, b i+ resp γ, b i+ resp γ, b i+ Then m u u γ m u) + m 9 m 9 where τ :, b i+ b i+3 m u u β m u) ) )i i + 4 ) i)) τ) in the first two cases, i 9 7 i i ) )i i + 4 ) i)) τ in the last two cases, i 9 7 i i Remark 6 In any case, for m > 3, we have m u u γ m u) m u u β m u) Proof of Lemma 4 We have m u γ u β )m u) u i m γ m i) i β + u γ) m u ) u β m u) ) u The result now follows as in the proof of Lemma by some tedious but straightforward algebra With Lemma 4 we obtain Lemma 5 We have m max β m bit u { m u β m u) + m ) ) m 7 m + m 3 8 m + m m + ) )) m 7 7 m + 4 m 3 For even m the maximum is attained if and only if {, β 3 + ), 3 m m ) For odd m the maximum is attained if and only if { ), β 3 + m ), 3 m+ 8 m ) and and if m is even, if m is odd 6

17 Proof For short we write f m β) : m u u β m u) Let m be even It follows from Lemma 4 that the m-bit number β which maximizes our sum has to be of the form β, b m b m or β, b m b m First we deal with β, b m b m ow we consider four cases corresponding to the possible choices for b m and b m If b m, b m ), ), then f m β ) m 8 + m If b m, b m ), ), then f m β ) m 8 + m If b m, b m ), ), then f m β ) m 8 + m If b m, b m ), ), then f m β ) m 8 + m m 6 m 6 m 7 m 8 64 m 8 m m 5 m + m 7 m 8 m 8 m m 4 m m m 8 6 m 8 m m m m 7 4 m 8 4 m 8 m Therefore we find that the choice b m b m ), ) gives the maximal value, ie, β, For β, b m b m we find in the same way that b m, b m ), ) gives the maximal value, ie, β, Since f m β ) f m β ) m 8 + m m 4 7 m ) m + 3 ) 8 ) 8 m the result follows for even m For odd m 3 the result can be proved analogously We give the Proof of Corollary 4 We have max m m u u m u max u β m u) m β m bit 8 + m m ) 8 + O m u by Lemma 5 The result follows now together with Remark 3 Acknowledgement The author wishes to thank Peter Kritzer for his many comments and suggestions 7

18 References [] Chaix, H, Faure, H, Discrépance et diaphonie en dimension un Acta Arith 63: 3-4, 993 [] Dick, J, Pillichshammer, F, Diaphony, discrepancy, spectral test and worst-case error Math Comput Simulation 7: 59 7, 5 [3] Drmota, M, Larcher, G, Pillichshammer, F, Precise distribution properties of the van der Corput sequence and related sequences Manuscripta Math 8: 4, 5 [4] Drmota, M, Tichy, RF, Sequences, Discrepancies and Applications Lecture otes in Mathematics 65, Springer-Verlag, Berlin, 997 [5] Faure, H, Discrepancy and Diaphony of digital, )-sequences in prime base Acta Arith 7: 5 48, 4 [6] Grozdanov, V, On the diaphony of one class of one-dimensional sequences Internat J Math Math Sci 9: 5 4, 996 [7] Hellekalek, P, Leeb, H, Dyadic diaphony Acta Arith 8: 87 96, 997 [8] Kuipers, L, iederreiter, H, Uniform Distribution of Sequences John Wiley, ew York, 974 [9] Larcher, G, Pillichshammer, F, Sums of distances to the nearest integer and the discrepancy of digital nets Acta Arith 6: , 3 [] iederreiter, H, Point sets and sequences with small discrepancy Monatsh Math 4: , 987 [] iederreiter, H, Random umber Generation and Quasi-Monte Carlo Methods o 63 in CBMS-SF Series in Applied Mathematics SIAM, Philadelphia, 99 [] Proinov, PD, Grozdanov, VS, On the diaphony of the van der Corput-Halton sequence J umber Theory 3: 94 4, 988 [3] Zinterhof, P, Über einige Abschätzungen bei der Approximation von Funktionen mit Gleichverteilungsmethoden Sitzungsber Österr Akkad Wiss Math-atur Kl II 85: 3, 976 Author s Address: Friedrich Pillichshammer, Institut für Finanzmathematik, Universität Linz, Altenbergstraße 69, A-44 Linz, Austria friedrichpillichshammer@jkuat 8