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volume 5, issue 2, article 26, 2004.

Received 08 August, 2003;

accepted 10 December, 2003.

Communicated by:N. Elezovi´c

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Journal of Inequalities in Pure and Applied Mathematics

PRICE AND HAAR TYPE FUNCTIONS AND UNIFORM DISTRIBUTION OF SEQUENCES

V. GROZDANOV AND S. STOILOVA

Department of Mathematics, South West University 66 Ivan Mihailov str.

2700 Blagoevgrad, Bulgaria.

EMail:vassgrozdanov@yahoo.com EMail:stanislavast@yahoo.com

c

2000Victoria University ISSN (electronic): 1443-5756 138-03

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Price and Haar Type Functions and Uniform Distribution of

Sequences V. Grozdanov and S. Stoilova

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J. Ineq. Pure and Appl. Math. 5(2) Art. 26, 2004

http://jipam.vu.edu.au

Abstract

The Weyl criterion is shown in the terms of Price functions and Haar type functions. We define the so-called modified integrals of Price and Haar type functions and obtain the analogues of the criterion of Weyl, the inequalities of LeV- eque and Erdös-Turan and the formula of Koksma in the terms of the modified integrals of Price and Haar type functions.

2000 Mathematics Subject Classification:65C05; 11K38; 11K06.

Key words: Uniform distribution of sequences; Price and Haar type functions; Weyl criterion; Inequalities of LeVeque and Erdös-Turan; Formula of Koksma.

1. Introduction

Letξ = (x_i)i≥0be a sequence in the unit interval[0,1).We defineA(ξ;J;N) = {i : 0 ≤ i ≤ N −1, x_i ∈ J}for an arbitrary integerN ≥ 1and an arbitrary subintervalJ ⊆[0,1).The sequenceξis called uniformly distributed (abbrevi- ated u. d.) if for every subinterval J of [0,1)the equality limN→∞ A(ξ;J;N)

N =

µ(J)holds, and whereµ(J)is the length ofJ.

Letξ_N = {x₀, . . . , x_N₋₁}be an arbitrary net of real numbers in[0,1). The extreme and quadratical discrepanciesD(ξ_N)andT(ξ_N)of the netξ_N are defined respectively as

D(ξN) = sup

J⊆[0,1)

|N⁻¹A(ξN;J;N)−µ(J)|,

T(ξ_N) = Z 1

0

|N⁻¹A(ξ_N; [0, x);N)−x|²dx ¹2

.

The discrepancy D_N(ξ) of the sequenceξ is defined asD_N(ξ) = D(ξ_N),for each integer N ≥ 1, and ξ_N is the net, composed of the first N elements of the sequence ξ. It is well-known that the sequence ξ is u. d. if and only if limN→∞D_N(ξ) = 0.

According to Kuipers and Niederreiter [8, Corollaries 1.1 and 1.2], the sequence ξ is u. d. if and only if for each complex-valued and integrable in the sense of Riemann function f, defined on R and periodical with period 1, the following equality

(1.1) lim

N→∞

1 N

N−1

X

i=0

f(x_i) = Z 1

0

f(x)dx

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holds.

The theory of uniformly distributed sequences is divided into quantitative and qualitative parts. Quantitative theory considers measures, showing the de- viation of the distribution of a concrete sequence from an ideal distribution.

Qualitative theory main idea of uniformly distributed sequences is to find necessary and sufficient conditions for uniformity of the distribution of sequences.

Weyl [18] obtains such a condition (the so-called Weyl criterion) which is based on the use of the trigonometric functional system T = {ek(x) = exp(2πikx), k∈Z, x∈R}.The criterion of Weyl is: The sequenceξ = (x_i)i≥0

is uniformly distributed if and only if the equalitylimN→∞ 1 N

PN−1

i=0 e_k(x_i) = 0 holds for each integerk 6= 0.

The Walsh functional system has been recently used as an appropriate means of studying the uniformity of the distribution of sequences. Sloss and Blyth [13] use this system to obtain future necessary and sufficient conditions for a sequence to be u. d.

The link, which is realized for studying sequences in[0,1),constructed in a generalized number system and some orthonormal functional systems on[0,1), constructed in the same system, is quite natural. The purpose of our paper is to reveal the possibility some other classes of orthonormal functional system, as the Price functional system and two systems of Haar type functions to be used as a means of obtaining new necessary and sufficient conditions for uniform distribution of sequences.

In Section2we obtain new necessary and sufficient conditions for uniform distribution of sequences, which are analogues of the classical criterion of Weyl.

These conditions are based on the functions of Price and Haar type functions.

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In Section3we introduce the so-called modified integrals of the Price functions and Haar type functions. Integral analogues of the Weyl criterion are obtained in terms of these integrals. Analogues of the classical inequalities of LeVeque [9] and Erdös-Turan (see Kuipers and Niederreiter [8]), and the formula of Koksma, (see Kuipers [7]) are obtained.

In Section4we prove some preliminary statements, which are used to prove the main results. The proofs of the main results are given in Section5. In Sec- tion 6we give a conclusion, where we announce some open problems, having to do with the problems, solved in our paper.

The results of this paper were announced in Grozdanov and Stoilova [3] and [4]. Here we explain the full proofs of them. The results which are based on the Price functions generalize the ones of Sloss and Blyth [13]. The results which are based on the Haar type functions are new.

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2. Price Functional System, Haar Type Functional System and Analogues of the Criterion of Weyl

Let B = {b₁, b₂, . . . , b_j, . . . : b_j ≥ 2, j ≥ 1}be an arbitrary fixed sequence of integer numbers. We defineω_j = exp

2πi bj

for each integerj ≥ 1.We define the set of the generalized powers {Bj}^∞_j=0 as: B0 = 1 and for each integer j ≥1, B_j =Qj

s=1b_s. Definition 2.1.

(i) For realx ∈[0,1)in theB−adic formx=P∞

i=1x_iB_i⁻¹,where fori≥ 1 x_i ∈ {0,1, . . . , b_i − 1} and each integer j ≥ 0, Price [10] defines the functionsχ_B_j(x) =ω_j+1^x^j+1.

(ii) For each integerk ≥0in theB−adic formk = Pn

j=0k_j+1B_j,where for 1≤j ≤n+ 1, k_j ∈ {0,1, . . . , b_j−1}, k_n+1 6= 0and realx∈[0,1),the k-th function of Priceχ_k(x)is defined asχ_k(x) = Qn

j=0(χ_B_j(x))^k^j+1. The system χ(B) = {χ_k}^∞_k=0 is called the Price functional system. This system is a complete orthonormal system inL₂[0,1).

Let b_j = b in the sequence B for each j ≥ 1. Then, the system {χ₀} ∪ {χ_b^k}^∞_k=0 is the Rademacher [11] system{φ^(b)_k }^∞_k=0 of orderb. The system of Chrestenson [2]{ψ_k^(b)}^∞_k=0 of orderb is obtained from the system χ(B).If for eachj ≥1b_j = 2,then the original system of Walsh [17] is obtained.

In 1947 Vilenkin [15] introduced the systemχ(B)and Price [10] defined it independently of him in 1957. Some names are used about the system χ(B)in special literature: both Price system (see Agaev, Vilenkin, Dzafarly, Rubinstein

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[1]) and Vilenkin system (see Schipp, Wade, Simon [12]). We use the name Price functional system in this paper.

We will consider two kinds of the so-called Haar type functions. Starting from the original Haar [5] system, Vilenkin [16] proposes a new system of functions, which is called a Haar type system, (see Schipp, Wide, Simon [12]). This definition is:

Definition 2.2. For x ∈ [0,1)thek^th Haar type functionh⁰_k(x), k ≥ 0to the baseBis defined as follows: Ifk = 0,thenh⁰₀(x) = 1,∀x∈ [0,1).Ifk ≥ 1is an arbitrary integer and

(2.1) k =B_n+p(b_n+1−1) +s−1,

where for some integern≥0,0≤p≤B_n−1ands∈ {1, . . . , b_n+1−1},then

h⁰_k(x) =







√Bnω^sa_n+1, if ^pb_Bⁿ⁺¹^+a

n+1 ≤x < ^pbⁿ⁺¹_B ^+a+1

n+1 and a= 0,1, . . . , bn+1−1,

0, otherwise.

We will consider another one:

Definition 2.3. For x ∈ [0,1)thek^th Haar type functionh⁰⁰_k(x), k ≥ 0to the baseBis defined as follows: Ifk = 0,thenh⁰⁰₀(x) = 1,∀x∈ [0,1).Ifk ≥ 1is an arbitrary integer and

(2.2) k =knBn+p,

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where for some integern ≥ 0,0 ≤ p ≤ B_n−1andk_n ∈ {1, . . . , b_n+1−1}, then

h⁰⁰_k(x) =







√B_nω^k_n+1ⁿ^a, if ^pb_Bⁿ⁺¹^+a

n+1 ≤x < ^pbⁿ⁺¹_B ^+a+1

n+1 and a= 0,1, . . . , b_n+1−1,

0, otherwise.

It can be easily seen that the systems {h⁰_k}^∞_k=0 and {h⁰⁰_k}^∞_k=0 are complete orthonormal systems in L₂[0,1). In the case when for eachj ≥ 1b_j = 2the original system of Haar is obtained from the systems{h⁰_k}^∞_k=0 and{h⁰⁰_k}^∞_k=0. Theorem 2.1 (Analogues of the criterion of Weyl). The sequence (x_i)i≥0 of [0,1)is u. d. if and only if:

Nlim→∞

1 N

N−1

X

i=0

χ_k(x_i) = 0, for each k ≥1,

N→∞lim 1 N

N−1

X

i=0

h⁰_k(x_i) = 0, for each k≥1,

and

N→∞lim 1 N

N−1

X

i=0

h⁰⁰_k(x_i) = 0, for each k≥1.

The proof of this theorem is based on the equality (1.1) and the properties of Price and Haar type functional systems.

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3. Price and Haar Type Integrals and u.d. of Sequences

We consider the integrals of Price and Haar type functionsJ_k(x) = Rx

0 χ_k(t)dt, Ψ⁰_k(x) = Rx

0 h⁰_k(t)dt and Ψ⁰⁰_k(x) = Rx

0 h⁰⁰_k(t)dt for each integer k ≥ 1 and x∈[0,1).

For an arbitrary integerk ≥ 1we define the integern ≥0by the condition B_n≤k < B_n+1.We define the modified integrals of Price function as

(3.1) J_n,q,k(x) = J_k(x) + 1 Bn+1

· 1

ω^q_n+1−1δ_q.B_n_,k,

for all x ∈ [0,1) and each q = 1,2, . . . , b_n+1 −1, and for arbitrary integers i, j ≥0, δ_i,j is the Kronecker’s symbol.

Ifkis an integer of the kind (2.1), we define (3.2) Ψ⁰_n,s,k(x) = Ψ⁰_k(x) + 1

b_n+1B⁻

3

n2

1 ω_n+1^s −1, for allx∈[0,1)and eachs = 1,2, . . . , b_n+1−1.

Ifkis an integer of the kind (2.2), we define (3.3) Ψ⁰⁰_n,k_n_,k(x) = Ψ⁰⁰_k(x) + 1

bn+1

B⁻

3

n2

1 ω_n+1^kⁿ −1, for allx∈[0,1)and eachk_n= 1,2, . . . , b_n+1−1.

We will call the integralsΨ⁰_n,s,k(x)andΨ⁰⁰_n,k_n_,k(x)modified integrals of Haar type functions. The next theorems hold:

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Theorem 3.1 (Analogues of the inequality of LeVeque). Letξ_N ={x₀, x₁. . . , xN−1}be an arbitrary net, composed ofN ≥1points of[0,1). The discrepancy D(ξ_N)of the netξ_N satisfies the inequalities:

D(ξ_N)≤



 12 N²

∞

X

n=0 bn+1−1

X

q=1

(q+1)Bn−1

X

k=qBn

N−1

X

m=0

J_n,q,k(x_m)

2



1 3

,

D(ξ_N)≤



 12 N²

∞

X

n=0 bn+1−1

X

s=1

(s+1)Bn−1

X

k=sBn

N−1

X

m=0

Ψ⁰_n,s,k(x_m)

2



1 3

,

D(ξ_N)≤



 12 N²

∞

X

n=0 bn+1−1

X

kn=1

(kn+1)Bn−1

X

k=knBn

N−1

X

m=0

Ψ⁰⁰_n,k

n,k(x_m)

2



1 3

.

Theorem 3.2 (Integral analogues of the criterion of Weyl). Let an absolute constant B exist, such as for each j ≥ 1 b_j ≤ B. Let k ≥ 1 be an arbitrary integer andB_n ≤ k < B_n+1.The sequenceξ = (x_i)i≥0 of[0,1)is u. d. if and only if:

(i)

N→∞lim 1 N

N−1

X

i=0

J_n,q,k(x_i) = 0 for each q = 1,2, . . . , b_n+1−1,

(ii) Ifk ≥1is of the kind (2.1) then

N→∞lim 1 N

N−1

X

i=0

Ψ⁰_n,s,k(x_i) = 0 for each s= 1,2, . . . , b_n+1−1,

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(iii) Ifk ≥1is of the kind (2.2) then

N→∞lim 1 N

N−1

X

i=0

Ψ⁰⁰_n,k

n,k(x_i) = 0 for each k_n= 1,2, . . . , b_n+1−1.

Theorem 3.3 (An analogue of the inequality of Erdös-Turan). Let an ab- solute constant B exist, such that b_j ≤ B for each j ≥ 1 and we signify b = min{bj : j ≥ 1}. Let ξN = {x0, . . . , xN−1} be an arbitrary net, com- posed ofN ≥ 1points of[0,1).For an arbitrary integerH > 0we define the integers M ≥ 0and q ∈ {1,2, . . . , b_M₊₁−1} asqB_M ≤ H < (q+ 1)B_M. Then the following inequality holds

D(ξ_N)≤



12

M−1

X

n=0 bn+1−1

X

s=1

(s+1)Bn−1

X

k=sBn

1 N

N−1

X

m=0

J_n,s,k(x_m)

2

+ 12

q−1

X

s=1

(s+1)BM−1

X

k=sBM

1 N

N−1

X

m=0

J_M,s,k(x_m)

2

+12

H

X

k=qBM

1 N

N−1

X

m=0

JM,q,k(xm)

2

+3B(1 + 2bsin_B^π)² (b−1)bsin² _B^π

1 B_M





1 3

.

Theorem 3.4 (Analogues of the formula of Koksma). Let ξ_N = {x₀, x₁. . . , x_N−1}be an arbitrary net, composed ofN ≥1points of[0,1).The quadratical

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discrepancyT(ξ_N)of the netξ_N satisfies the equalities

(N T(ξ_N))² =

N−1

X

m=0

x_m−1 2

!2

+

∞

X

n=0 bn+1−1

X

q=1

(q+1).Bn−1

X

k=q.Bn

N−1

X

m=0

J_n,q,k(x_m)

2

,

(N T(ξ_N))² =

N−1

X

m=0

x_m−1 2

!2

+

∞

X

n=0 bn+1−1

X

s=1

(s+1)Bn−1

X

k=sBn

N−1

X

m=0

Ψ⁰_n,s,k(x_m)

2

, and

(N T(ξ_N))² =

N−1

X

m=0

x_m− 1 2

!2

+

∞

X

n=0 bn+1−1

X

kn=1

(kn+1)Bn−1

X

k=knBn

N−1

X

m=0

Ψ⁰⁰_n,k_n_,k(x_m)

2

.

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4. Preliminary Statements

Let x ∈ [0,1) have the B−adic representation x = P∞

j=0xj+1B_j+1⁻¹ , where for j ≥ 0, x_j+1 ∈ {0,1, . . . , b_j+1 −1}. For each integerj ≥ 0 we have that x_j+1 = ^b^j+1_2π argχ_B_j(x).Hence, we obtain the representation

(4.1) x= 1

2π

∞

X

j=0

1

B_j argχ_B_j(x).

Lemma 4.1. Let k ≥ 1be an arbitrary integer and k = β_n+1B_n +k⁰,where βn+1 ∈ {1, . . . , bn+1−1}and0≤k⁰ < Bn.Forx∈[0,1)thek^thPrice integral satisfies the following equality

(4.2) J_k(x) = 1 Bn+1

1−ω

βn+1bn+1

2π argχ_Bn(x) n+1

1−ω^β_n+1ⁿ⁺¹ χ_k⁰(x)

+ 1

2πB_n

∞

X

r=1

b^−r_n+1argχ_B_n b^r_n+1x

χ_k(x).

Proof. Letb≥2be a fixed integer andω= exp ^2πi_b

.For an arbitrary integer β,1≤β ≤b−1and realx∈[0,1)let

J_β^(b)(x) = Z x

0

ψ^(b)_β (t)dt.

We will prove the following equality (4.3) J_β^(b)(x) = 1

b

1−ω^βb^2π^arg^φ^(b)⁰ ^(x) 1−ω^β + 1

2π

∞

X

r=1

b^−rargψ_b^(b)r (x)ψ_β^(b)(x).

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We puts= [bx],where[bx]denotes the integer part ofbxand we have that J_β^(b)(x) =

Z x 0

[φ^(b)₀ (t)]^βdt (4.4)

=

s−1

X

h=0

Z (h+1)/b h/b

ω^hβdt+ Z x

s/b

[φ^(b)₀ (t)]^βdt

= 1 b

1−ω^β[bx]

1−ω^β +ψ_β^(b)(x)

x−[bx]

b

. From (4.1) and (4.4) we obtain (4.3).

If

J_β^(bⁿ⁺¹⁾

n+1·Bn(x) = Z x

0

χ_β_n+1_B_n(t)dt,

then

(4.5) J_k(x) = χ_k⁰(x)J_β^(bⁿ⁺¹⁾

n+1Bn(x).

We have the equalities J_β^(bⁿ⁺¹⁾

n+1·Bn(x) = Z x

0

χ^β_Bⁿ⁺¹

n (t)dt

= Z x

0

h

φ^(b₀ⁿ⁺¹⁾(B_nt)iβn+1

dt

= 1 Bn

Z Bnx 0

ψ_β^(bⁿ⁺¹⁾

n+1 (t)dt= 1 Bn

J_β^(bⁿ⁺¹⁾

n+1 (B_nx),

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so that

J_β^(bⁿ⁺¹⁾

n+1·Bn(x) = 1 Bn

J_β^(bⁿ⁺¹⁾

n+1 (B_nx).

From the last equality and (4.5) we obtain that

(4.6) Jk(x) = 1

B_nχk⁰(x)·J_β^(b_n+1ⁿ⁺¹⁾(Bnx).

From (4.3) we obtain

(4.7) J_β^(bⁿ⁺¹⁾

n+1 (B_nx) = 1

b_n+1 · 1−ω^βn+1bn+1^2π ^arg^χ^Bn^(x) 1−ω_n+1^βⁿ⁺¹ + 1

2π

∞

X

r=1

b^−r_n+1argχ_B_n(b^r_n+1x)χ_β_n+1_B_n(x).

From (4.6) and (4.7) we obtain (4.2).

For every integer n ≥ 1we consider the set B(n) = {b1, b2, . . . , bn}. We define the “reverse” set B(n) =e {b_n, bn−1, . . . , b₁}, so that Be₁ = b_n, Be₂ = b_nb_n−1, . . . ,Be_n = b_n· · ·b₁.For an arbitrary integer p,0 ≤ p < B_n and for a B-adic rational _B^p

n let (p)_B(n),(p)

B(n)e ,

p Bn

B(n) and p

Bn

B(n)e be the corre- sponding representations ofpand _B^p

n to the systemsB(n)andBe(n).

Lemma 4.2. (Relationships between the Price and the Haar type functions)

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(i) Letk ≥1be an arbitrary integer of the kind (2.1). Then for allx∈[0,1)

h⁰_k(x) = 1 b_n+1√

B_n

×

Bn+1−1

X

α=0

bn+1−1

X

a=0

ω^a.s_n+1χ_(pb_n+1_+a)

B(n+1)e

α B_n+1

B(n+1)e

!

χ_α(x);

Ψ⁰_n,s,k(x) = 1 b_n+1√

B_n

×

n

X

j=0 bj+1−1

X

t=0

(t+1)Bj−1

X

α=tBj

bn+1−1

X

a=0

ω_n+1^a.s χ_(pb_n+1_+a)

B(n+1)e

α B_n+1

B(n+1)e

!

J_j,t,α(x).

(ii) Letk ≥1be an arbitrary integer of the kind (2.2). Then for allx∈[0,1)

(4.8) h⁰⁰_k(x) = 1 b_n+1√

B_n

×

Bn+1−1

X

α=0

bn+1−1

X

a=0

ω^a.k_n+1ⁿχ_(pb

n+1+a)

B(n+1)e

α B_n+1

B(n+1)e

!

χ_α(x);

(4.9) Ψ⁰⁰_n,k_n_,k(x) = 1 b_n+1√

B_n

×

n

X

j=0 bj+1−1

X

t=0

(t+1)Bj−1

X

α=tBj

bn+1−1

X

a=0

ω_n+1^a.kⁿχ_(pb_n+1_+a)

B(n+1)e

α B_n+1

B(n+1)e

!

J_j,t,α(x).

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Proof. For an arbitrary integerp,0≤p < B_nandx∈[0,1),following Kremer [6] we define the function

q(B_n; (p)

B(n)e ;x) =











1, if x∈

p Bn

B(n),

p+1 Bn

B(n)

0 if x6∈

p Bn

B(n), p+1

Bn

B(n)

.

The equality

q(B_n; (p)_B(n)_e ;x) = 1 B_n

Bn−1

X

α=0

χ_(p)

B(n)e

α B_n

B(n)e

! χ_α(x)

holds. Let us use the significations: For an arbitrary integer p, 0 ≤ p <

Bn, (p)_B(n)_e = (˜p1p˜2. . .p˜n)_B(n)_e , for an arbitrary integer α, 0 ≤ α < Bn, α

Bn

B(n)e

= (0·α_nαn−1. . . α₁)

B(n)e ,for realx∈[0,1), x= (0·x₁. . . x_n+1. . .)B. Then, we obtain the equalities

(4.10) χ_(p)

B(n)e

α B_n

B(n)e

!

=ω^α_nⁿ^p^˜ⁿω_n−1^αⁿ⁻¹^p^˜ⁿ⁻¹. . . ω₁^α¹^p^˜¹

and

(4.11) χ_α(x) =ω₁^α¹^x¹ω₂^α²^x². . . ω_n^αⁿ^xⁿ.

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If x ∈

p Bn

B(n)

,

p+1 Bn

B(n)

, then for all j = 1, . . . , n, x_j = ˜p_j. From (4.10) and (4.11) we obtain

χ_(p)

B(n)e

α B_n

B(n)e

!

χα(x) = 1.

If x 6∈

p Bn

B(n), p+1

Bn

B(n)

, then, some δ, 1 ≤ δ ≤ n exists, so that x_δ 6= ˜p_δ.Then, we have that

bδ−1

X

αδ=0

ω^α_δ^δ^(x^δ^−˜^p^δ⁾ = 0.

From (4.10) and (4.11), we obtain

Bn−1

X

α=0

χ_(p)

B(n)e

α B_n

B(n)e

!

χα(x) =

n

Y

j=1 bj−1

X

αj=0

ω^α_j^j^(x^j^−˜^p^j⁾ = 0.

Now letk ≥ 1be an integer of the kind (2.2). In order to prove (4.8) we note that for allx∈[0,1)

h⁰⁰_k(x) =p B_n

bn+1−1

X

a=0

ω_n+1^a.kⁿq

B_n+1; (pb_n+1+a)_B(n+1)_e ;x .

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We will prove (4.9). Using the proved formula forh⁰⁰_k(x),we have that Ψ⁰⁰_k(x) =

Z x 0

h⁰⁰_k(t)dt (4.12)

= 1

b_n+1√ B_n

Bn+1−1

X

α=0

bn+1−1

X

a=0

ω_n+1^a.kⁿ

×χ_(pb_n+1_+a)

B(n+1)e

α B_n+1

B(n+1)e

!Z x 0

χ_α(t)dt

= 1

bn+1

√Bn n

X

j=0 bj+1−1

X

t=0

(t+1)Bj−1

X

α=tBj

bn+1−1

X

a=0

ω^a.k_n+1ⁿ

×χ_(pb_n+1_+a)

B(n+1)e

α B_n+1

B(n+1)e

!

×

J_j,t,α(x)− 1 B_j+1

1

ω_j+1^t −1δ_tB_j_,α

.

It is not difficult to prove that (4.13)

n

X

j=0

1 B_j+1

bj+1−1

X

t=0

1 ω_j+1^t −1

(t+1)Bj−1

X

α=tBj

bn+1−1

X

a=0

ω_n+1^a.kⁿ

×χ_(pb_n+1_+a)

B(n+1)e

α Bn+1

B(n+1)e

!

δ_tB_j_,α= B_n⁻¹ ω_n+1^kⁿ −1. From (4.12) and (4.13), we obtain (4.9).

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In the following lemma the relationships in the opposite direction are proved.

Lemma 4.3. Let k ≥ 1 be an arbitrary integer and k = sBn +p, where s ∈ {1, . . . , b_n+1 −1} and 0 ≤ p ≤ B_n −1. Then, for all x ∈ [0,1) the equalities

(4.14) χ_k(x) = 1

√Bn Bn−1

X

j=0

α⁽ⁿ⁾

p,˜jh⁰⁰_sB

n+˜j(x);

(4.15) J_n,s,k(x) = 1

√B_n

Bn−1

X

j=0

α⁽ⁿ⁾

p,˜jΨ⁰⁰_n,s,sB

n+˜j(x);

χ_k(x) = 1

√B_n

Bn−1

X

j=0

α⁽ⁿ⁾

p,˜jh⁰_B

n+˜j(bn+1−1)+s−1(x)

and

J_n,s,k(x) = 1

√B_n

Bn−1

X

j=0

α⁽ⁿ⁾

p,˜jΨ⁰_n,s,B

n+˜j(bn+1−1)+s−1(x), hold, whereα⁽ⁿ⁾_p,_˜_j are complex mumbers, so thatPBn−1

j=0 α⁽ⁿ⁾_p,_˜_j =Bn·δsBn,k. Proof. Let x ∈ [0,1) be fixed. We define t˜ = (t)_B(n)_e , 0 ≤ t < B˜ n as _˜

t Bn

B(n)

≤ x < _˜

t+1 Bn

B(n)

. We denote ∆⁽ⁿ⁾_˜_t = h _˜

t Bn,^˜^t+1_B

n

. It is obvious

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that ∆⁽ⁿ⁾_˜_t = Sbn+1−1

a=0 ∆⁽ⁿ⁺¹⁾_˜_tb

n+1+a. There is some a, 0 ≤ a ≤ b_n+1 − 1, so that x∈∆⁽ⁿ⁺¹⁾_˜_tb

n+1+a.We have the equalities χ_k(x) = χ_p

˜t B_n

B(n)

!

ω_n+1^a.s and h⁰⁰_s·B

n+˜t(x) =p

B_nω^a.s_n+1.

Hence, we obtain

χ_k(x) = 1

√B_nχ_p

t˜ B_n

B(n)

! h⁰⁰_s·B

n+˜t(x).

Let˜t⁰ be an arbitrary integer, so that0≤t˜⁰ < Bn,andt˜⁰ 6= ˜t.Forx∈ ∆⁽ⁿ⁾_t_˜ we have thath⁰⁰_s·B

n+˜t⁰(x) = 0.Hence, we obtain the equality χ_k(x) = 1

√B_n

Bn−1

X

j=0

χ_p ˜j

B_n

B(n)

! h⁰⁰_s·B

n+˜j(x).

Let for integers0≤p < B_nand0≤j < B_nwe signifyα⁽ⁿ⁾

p,˜j =χ_p _˜

j Bn

B(n)

. We use the representationsp= (p_npn−1. . . p₁)_B(n) and

˜j B_n

B(n)

= (0·j1j2. . . jn)B(n),

where for1≤τ ≤n p_τ, j_τ ∈ {0,1, . . . , b_τ−1}.Then, we obtain the equality (4.16)

Bn−1

X

j=0

α⁽ⁿ⁾

p,˜j =

n

Y

τ=1 bτ−1

X

jτ=0

ω_τ^p^τ^j^τ.

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Ifp= 0,then, for1≤τ ≤n, p_τ = 0and from (4.16) we obtainPBn−1 j=0 α⁽ⁿ⁾

p,˜j = B_n.Ifp6= 0then, someδ, 1≤ δ ≤ nexists, so thatp_δ 6= 0.From (4.16), we obtainPBn−1

j=0 α⁽ⁿ⁾

p,˜j = 0.

We will prove (4.15). From (4.14) for allx∈[0,1)we have

J_k(x) = 1

√B_n

Bn−1

X

j=0

α⁽ⁿ⁾

p,˜j

Ψ⁰⁰_sB

n+˜j(x) + 1 b_n+1B⁻

3

n2

1 ω^s_n+1−1

− 1

b_n+1B_n⁻² 1 ω_n+1^s −1

Bn−1

X

j=0

α⁽ⁿ⁾_p,_˜_j.

From the last equality and (3.3) we obtain (4.15).

Sobol [14] proved a similar result, giving the relationship between the original Haar and the Walsh functions.

For an arbitrary net ξ_N = {x₀, . . . , xN−1}, composed ofN ≥ 1 points of [0,1)andx ∈ [0,1)we signifyR(ξ_N;x) = A(ξ_N; [0, x);N)−N x.Then, the next lemma holds:

Lemma 4.4.

(i) The Fourier-Price coefficiens ofR(ξ_N;x)satisfy the equalities:

a^(χ)₀ =−

N−1

X

m=0

x_m− 1 2

;

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and for each integerk ≥ 1, k = k_nB_n +p, k_n ∈ {1,2, . . . , b_n+1 −1}, 0≤p < Bn

a^(χ)_k =−

N−1

X

m=0

J_n,k_n_,k(x_m).

(ii) The Fourier-Haar type coefficiens ofR(ξ_N;x)satisfy the equalities:

a^(h₀ ⁰⁾=−

N−1

X

m=0

x_m−1 2

, a^(h₀ ⁰⁰⁾=−

N−1

X

m=0

x_m−1 2

;

Letk ≥1be an arbitrary integer of kind (2.1). Then,

a^(h_k⁰⁾ =−

N−1

X

m=0

Ψ⁰_n,s,k(x_m).

Letk ≥1be an arbitrary integer of kind (2.2). Then,

(4.17) a^(h_k⁰⁰⁾ =−

N−1

X

m=0

Ψ⁰⁰_n,k_n_,k(x_m).

Proof. Let for 0 ≤ m ≤ N −1, cm(x) be the characteristic function of the interval(x_m,1).Then,

R(ξN;x) =

N−1

X

m=0

cm(x)−N x.

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We will prove only (4.17). The proof of the remaining equalities of the lemma is similar. For an arbitrary integerk ≥1of the kind (2.2) we have:

a^(h_k⁰⁰⁾ = Z 1

0

R(ξ_N;x)h⁰⁰_k(x)dx (4.18)

=

N−1

X

m=0

Z 1 0

c_m(x)h⁰⁰_k(x)dx−N Z 1

0

xh⁰⁰_k(x)dx.

The equalities

N−1

X

m=0

Z 1 0

c_m(x)h⁰⁰_k(x)dx=

N−1

X

m=0

Z 1 0

h⁰⁰_k(x)dx− Z xm

0

h⁰⁰_k(x)dx (4.19)

=−

N−1

X

m=0

Ψ⁰⁰_k(x_m)

hold. From (4.1) and (4.8) we obtain Z 1

0

xh⁰⁰_k(x)dx (4.20)

= 1

2πbn+1

√Bn Bn+1−1

X

α=0

bn+1−1

X

a=0

ω_n+1^a·kⁿχ_(pb_n+1_+a)

B(n+1)e

α Bn+1

B(n+1)e

!

×

∞

X

r=0

1 B_r

Z 1 0

argχ_B_r(x)χ_α(x)dx