Journal of Inequalities in Pure and
Applied Mathematics
PRICE AND HAAR TYPE FUNCTIONS AND UNIFORM
DISTRIBUTION OF SEQUENCES
volume 5, issue 2, article 26,
2004.
V. GROZDANOV AND S. STOILOVA
Department of Mathematics,
South West University
66 Ivan Mihailov str.
2700 Blagoevgrad, Bulgaria.
Received 08 August, 2003;
accepted 10 December, 2003.
Communicated by: N. Elezović
EMail: vassgrozdanov@yahoo.com
EMail: stanislavast@yahoo.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
138-03
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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 LeVeque 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.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
Contents
Title Page
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Price Functional System, Haar Type Functional System and
Analogues of the Criterion of Weyl . . . . . . . . . . . . . . . . . . . . . . 6
3
Price and Haar Type Integrals and u.d. of Sequences . . . . . . . 9
4
Preliminary Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5
Proofs of the Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
References
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1.
Introduction
Let ξ = (xi )i≥0 be a sequence in the unit interval [0, 1). We define A(ξ; J; N ) =
{i : 0 ≤ i ≤ N − 1, xi ∈ J} for an arbitrary integer N ≥ 1 and an arbitrary
subinterval J ⊆ [0, 1). The sequence ξ is 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 where µ(J) is the length of J.
Let ξN = {x0 , . . . , xN −1 } be an arbitrary net of real numbers in [0, 1). The
extreme and quadratical discrepancies D(ξN ) and T (ξN ) of the net ξN are defined respectively as
D(ξN ) = sup |N −1 A(ξN ; J; N ) − µ(J)|,
V. Grozdanov and S. Stoilova
J⊆[0,1)
Z
T (ξN ) =
1
|N −1 A(ξN ; [0, x); N ) − x|2 dx
Price and Haar Type Functions
and Uniform Distribution of
Sequences
21
.
0
The discrepancy DN (ξ) of the sequence ξ is defined as DN (ξ) = D(ξN ), for
each integer N ≥ 1, and ξN 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 →∞ DN (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
Z 1
N −1
1 X
f (xi ) =
f (x)dx
(1.1)
lim
N →∞ N
0
i=0
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holds.
The theory of uniformly distributed sequences is divided into quantitative
and qualitative parts. Quantitative theory considers measures, showing the deviation 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 sequence
PN −1ξ = (xi )i≥0
1
is uniformly distributed if and only if the equality limN →∞ N i=0 ek (xi ) = 0
holds for each integer k 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 Section 2 we 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.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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In Section 3 we 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 Section 4 we prove some preliminary statements, which are used to prove
the main results. The proofs of the main results are given in Section 5. In Section 6 we 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.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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2.
Price Functional System, Haar Type Functional
System and Analogues of the Criterion of Weyl
Let B = {b1 , b2 , . . . , bj , . . . : bj ≥ 2, j≥ 1}
be an arbitrary fixed sequence of
2πi
integer numbers. We define ωj = exp bj for each integer j ≥ 1. We define
the set of the generalized powers {Bj }∞
j=0 as: B0 = 1 and for each integer
Qj
j ≥ 1, Bj = s=1 bs .
Definition 2.1.
P
−1
(i) For real x ∈ [0, 1) in the B−adic form x = ∞
i=1 xi Bi , where for i ≥ 1
xi ∈ {0, 1, . . . , bi − 1} and each integer j ≥ 0, Price [10] defines the
xj+1
functions χBj (x) = ωj+1
.
P
(ii) For each integer k ≥ 0 in the B−adic form k = nj=0 kj+1 Bj , where for
1 ≤ j ≤ n + 1, kj ∈ {0, 1, . . . , bj − 1}, kn+1 6= 0 Q
and real x ∈ [0, 1), the
k-th function of Price χk (x) is defined as χk (x) = nj=0 (χBj (x))kj+1 .
The system χ(B) = {χk }∞
k=0 is called the Price functional system. This
system is a complete orthonormal system in L2 [0, 1).
Let bj = b in the sequence B for each j ≥ 1. Then, the system {χ0 } ∪
(b) ∞
{χbk }∞
k=0 is the Rademacher [11] system {φk }k=0 of order b. The system of
(b)
Chrestenson [2] {ψk }∞
k=0 of order b is obtained from the system χ(B). If for
each j ≥ 1 bj = 2, then the original system of Walsh [17] is obtained.
In 1947 Vilenkin [15] introduced the system χ(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
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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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) the k th Haar type function h0k (x), k ≥ 0 to the
base B is defined as follows: If k = 0, then h00 (x) = 1, ∀x ∈ [0, 1). If k ≥ 1 is
an arbitrary integer and
(2.1)
k = Bn + p(bn+1 − 1) + s − 1,
where for some integer n ≥ 0, 0 ≤ p ≤ Bn − 1 and s ∈ {1, . . . , bn+1 − 1}, then
 √
+a
+a+1
sa
, if pbBn+1
≤ x < pbn+1
and a = 0, 1, . . . , bn+1 − 1,
 Bn ωn+1
Bn+1
n+1
0
hk (x) =

0,
otherwise.
We will consider another one:
Definition 2.3. For x ∈ [0, 1) the k th Haar type function h00k (x), k ≥ 0 to the
base B is defined as follows: If k = 0, then h000 (x) = 1, ∀x ∈ [0, 1). If k ≥ 1 is
an arbitrary integer and
(2.2)
k = kn Bn + p,
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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where for some integer n ≥ 0, 0 ≤ p ≤ Bn − 1 and kn ∈ {1, . . . , bn+1 − 1},
then
 √
+a
+a+1
kn a
, if pbBn+1
≤ x < pbn+1
and a = 0, 1, . . . , bn+1 − 1,
 Bn ωn+1
Bn+1
n+1
h00k (x) =

0,
otherwise.
00 ∞
It can be easily seen that the systems {h0k }∞
k=0 and {hk }k=0 are complete
orthonormal systems in L2 [0, 1). In the case when for each j ≥ 1 bj = 2 the
00 ∞
original system of Haar is obtained from the systems {h0k }∞
k=0 and {hk }k=0 .
Theorem 2.1 (Analogues of the criterion of Weyl). The sequence (xi )i≥0 of
[0, 1) is u. d. if and only if:
N −1
1 X
lim
χk (xi ) = 0, for each k ≥ 1,
N →∞ N
i=0
N −1
1 X 0
lim
hk (xi ) = 0, for each k ≥ 1,
N →∞ N
i=0
and
Price and Haar Type Functions
and Uniform Distribution of
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V. Grozdanov and S. Stoilova
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N −1
1 X 00
lim
hk (xi ) = 0, for each k ≥ 1.
N →∞ N
i=0
Close
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
Rx
We consider
the
integrals
of
Price
and
Haar
type
functions
J
(x)
=
χk (t)dt,
k
0
Rx 0
R x 00
0
00
Ψk (x) = 0 hk (t)dt and Ψk (x) = 0 hk (t)dt for each integer k ≥ 1 and
x ∈ [0, 1).
For an arbitrary integer k ≥ 1 we define the integer n ≥ 0 by the condition
Bn ≤ k < Bn+1 . We define the modified integrals of Price function as
(3.1)
Jn,q,k (x) = Jk (x) +
1
1
· q
δq.Bn ,k ,
Bn+1 ωn+1
−1
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
for all x ∈ [0, 1) and each q = 1, 2, . . . , bn+1 − 1, and for arbitrary integers
i, j ≥ 0, δi,j is the Kronecker’s symbol.
If k is an integer of the kind (2.1), we define
(3.2)
Ψ0n,s,k (x)
=
Ψ0k (x)
+
1
bn+1
−3
Bn 2
1
s
ωn+1
−1
Contents
,
for all x ∈ [0, 1) and each s = 1, 2, . . . , bn+1 − 1.
If k is an integer of the kind (2.2), we define
(3.3)
Ψ00n,kn ,k (x) = Ψ00k (x) +
1
bn+1
−3
Bn 2
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1
kn
ωn+1
−1
,
for all x ∈ [0, 1) and each kn = 1, 2, . . . , bn+1 − 1.
We will call the integrals Ψ0n,s,k (x) and Ψ00n,kn ,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). Let ξN = {x0 , x1 . . . ,
xN −1 } be an arbitrary net, composed of N ≥ 1 points of [0, 1). The discrepancy
D(ξN ) of the net ξN satisfies the inequalities:

2  13
−1 (q+1)Bn −1 N −1
∞ bn+1
X
X
X
X
12

D(ξN ) ≤
Jn,q,k (xm )  ,
2
N
n=0
q=1
k=qBn
m=0
2  13
−1 (s+1)Bn −1 N −1
∞ bn+1
X
X
X
X
12
Ψ0n,s,k (xm )  ,
D(ξN ) ≤  2
N n=0 s=1 k=sB m=0

n
2  13
−1 (kn +1)Bn −1 N −1
∞ bn+1
X
X
X
X
12
D(ξN ) ≤  2
Ψ00n,kn ,k (xm )  .
N n=0 k =1 k=k B m=0

n
n
n
Theorem 3.2 (Integral analogues of the criterion of Weyl). Let an absolute
constant B exist, such as for each j ≥ 1 bj ≤ B. Let k ≥ 1 be an arbitrary
integer and Bn ≤ k < Bn+1 . The sequence ξ = (xi )i≥0 of [0, 1) is u. d. if and
only if:
(i)
N −1
1 X
lim
Jn,q,k (xi ) = 0 for each q = 1, 2, . . . , bn+1 − 1,
N →∞ N
i=0
(ii) If k ≥ 1 is of the kind (2.1) then
1
N →∞ N
lim
N
−1
X
i=0
Ψ0n,s,k (xi ) = 0 for each s = 1, 2, . . . , bn+1 − 1,
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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(iii) If k ≥ 1 is of the kind (2.2) then
N −1
1 X 00
Ψn,kn ,k (xi ) = 0 for each kn = 1, 2, . . . , bn+1 − 1.
N →∞ N
i=0
lim
Theorem 3.3 (An analogue of the inequality of Erdös-Turan). Let an absolute constant B exist, such that bj ≤ B for each j ≥ 1 and we signify
b = min{bj : j ≥ 1}. Let ξN = {x0 , . . . , xN −1 } be an arbitrary net, composed of N ≥ 1 points of [0, 1). For an arbitrary integer H > 0 we define the
integers M ≥ 0 and q ∈ {1, 2, . . . , bM +1 − 1} as qBM ≤ H < (q + 1)BM .
Then the following inequality holds

2
N
−1
X
1

Jn,s,k (xm )
D(ξN ) ≤ 12
N m=0
n=0 s=1
k=sBn
2
q−1 (s+1)BM −1 −1
X
X 1 N
X
+ 12
J
(x
)
M,s,k m N
s=1 k=sB
m=0
M
−1 bn+1
X
X−1 (s+1)B
Xn −1 M
 13
2
H
−1
1 N
π 2
X
X
3B(1
+
2b
sin
)
1 
B
+12
JM,q,k (xm ) +
.
2
π
N m=0
(b − 1)b sin B BM
k=qB
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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Theorem 3.4 (Analogues of the formula of Koksma). Let ξN = {x0 , x1 . . . ,
xN −1 } be an arbitrary net, composed of N ≥ 1 points of [0, 1). The quadratical
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discrepancy T (ξN ) of the net ξN satisfies the equalities
(N T (ξN ))2 =
N
−1 X
m=0
(N T (ξN ))2 =
1
xm −
2
N
−1 X
m=0
1
xm −
2
!2
2
∞ bn+1
−1
X
X−1 (q+1).B
Xn −1 N
X
+
Jn,q,k (xm ) ,
n=0
!2
q=1
k=q.Bn
m=0
2
∞ bn+1
−1
X
X−1 (s+1)B
Xn −1 N
X
+
Ψ0n,s,k (xm ) ,
n=0 s=1
m=0
k=sBn
Price and Haar Type Functions
and Uniform Distribution of
Sequences
and
(N T (ξN ))2 =
N
−1 X
m=0
1
xm −
2
2
!2 X
∞ bn+1
−1
X−1 (kn +1)B
Xn −1 N
X
00
+
Ψ
(x
)
n,kn ,k m .
n=0
m=0
kn =1
V. Grozdanov and S. Stoilova
k=kn Bn
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4.
Preliminary Statements
P∞
−1
Let x ∈ [0, 1) have the B−adic representation x =
j=0 xj+1 Bj+1 , where
for j ≥ 0, xj+1 ∈ {0, 1, . . . , bj+1 − 1}. For each integer j ≥ 0 we have that
b
xj+1 = j+1
arg χBj (x). Hence, we obtain the representation
2π
∞
1 X 1
x=
arg χBj (x).
2π j=0 Bj
(4.1)
Lemma 4.1. Let k ≥ 1 be an arbitrary integer and k = βn+1 Bn + k 0 , where
βn+1 ∈ {1, . . . , bn+1 −1} and 0 ≤ k 0 < Bn . For x ∈ [0, 1) the k th Price integral
satisfies the following equality
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
(4.2) Jk (x) =
βn+1 bn+1
2π
1 − ωn+1
1
Bn+1
arg χBn (x)
χk0 (x)
β
n+1
1 − ωn+1
∞
1 X −r
+
bn+1 arg χBn brn+1 x χk (x).
2πBn r=1
Proof. Let b ≥ 2 be a fixed integer and ω = exp 2πi
. For an arbitrary integer
b
β, 1 ≤ β ≤ b − 1 and real x ∈ [0, 1) let
Z x
(b)
(b)
Jβ (x) =
ψβ (t)dt.
0
(4.3)
(b)
Jβ (x) =
(b)
arg φ0 (x)
11−ω
b
1 − ωβ
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We will prove the following equality
βb
2π
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∞
1 X −r
(b)
(b)
+
b arg ψbr (x)ψβ (x).
2π r=1
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We put s = [bx], where [bx] denotes the integer part of bx and we have that
Z x
(b)
(b)
(4.4)
Jβ (x) =
[φ0 (t)]β dt
0
=
s−1 Z
X
(h+1)/b
hβ
x
ω dt +
h/b
h=0
Z
(b)
[φ0 (t)]β dt
s/b
1 1 − ω β[bx]
[bx]
(b)
=
+ ψβ (x) x −
β
b 1−ω
b
From (4.1) and (4.4) we obtain (4.3).
If
Z
(bn+1 )
Jβn+1 ·Bn (x) =
.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
x
χβn+1 Bn (t)dt,
0
then
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Jk (x) =
(4.5)
(bn+1 )
χk0 (x)Jβn+1
Bn (x).
We have the equalities
(bn+1 )
Jβn+1
·Bn (x)
Z
=
x
β
χBn+1
(t)dt
n
Z0 x h
iβn+1
(bn+1 )
=
φ0
(Bn t)
dt
0
Z Bn x
1 (bn+1 )
1
(bn+1 )
(t)dt =
(Bn x),
=
ψβn+1
J
Bn 0
Bn βn+1
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so that
1 (bn+1 )
(Bn x).
J
Bn βn+1
From the last equality and (4.5) we obtain that
(b
)
n+1
Jβn+1
·Bn (x) =
Jk (x) =
(4.6)
1
(bn+1 )
χk0 (x) · Jβn+1
(Bn x).
Bn
From (4.3) we obtain
(b
)
n+1
(4.7) Jβn+1
(Bn x) =
1
bn+1
·
1−ω
βn+1 bn+1
2π
Price and Haar Type Functions
and Uniform Distribution of
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arg χBn (x)
β
n+1
1 − ωn+1
∞
1 X −r
bn+1 arg χBn (brn+1 x)χβn+1 Bn (x).
+
2π r=1
For every integer n ≥ 1 we consider the set B(n) = {b1 , b2 , . . . , bn }. We
e
e1 = bn , B
e2 =
define the “reverse” set B(n)
= {bn , bn−1 , . . . , b1 }, so that B
en = bn · · · b1 . For an arbitrary integer p, 0 ≤ p < Bn and for a
bn bn−1 , . . . , B
p
p
B-adic rational Bn let (p)B(n) , (p)B(n)
, Bn
and Bpn
be the corree
B(n)
p
Bn
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From (4.6) and (4.7) we obtain (4.2).
sponding representations of p and
V. Grozdanov and S. Stoilova
e
B(n)
e
to the systems B(n) and B(n).
Lemma 4.2. (Relationships between the Price and the Haar type functions)
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(i) Let k ≥ 1 be an arbitrary integer of the kind (2.1). Then for all x ∈ [0, 1)
h0k (x) =
×
1
√
bn+1 Bn
Bn+1 −1 bn+1 −1
X
X
α=0
a=0
Ψ0n,s,k (x) =
a.s
χ(pbn+1 +a)B(n+1)
ωn+1
e
t=0
χα (x);
Bn+1
e
B(n+1)
1
√
Price and Haar Type Functions
and Uniform Distribution of
Sequences
bn+1 Bn
n bj+1
X
X−1 (t+1)B
Xj −1 bn+1
X−1
a.s
×
ωn+1
χ(pbn+1 +a)B(n+1)
e
j=0
!
α
α=tBj
Bn+1
a=0
!
α
Jj,t,α (x).
(ii) Let k ≥ 1 be an arbitrary integer of the kind (2.2). Then for all x ∈ [0, 1)
×
(4.9)
bn+1 Bn
Bn+1 −1 bn+1 −1
X X
a.kn
ωn+1
χ(pbn+1 +a)B(n+1)
e
α=0
a=0
Ψ00n,kn ,k (x)
=
t=0
α=tBj
!
α
χα (x);
Bn+1
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bn+1 Bn
a=0
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e
B(n+1)
1
√
n bj+1
X
X−1 (t+1)B
Xj −1 bn+1
X−1
a.kn
×
ωn+1
χ(pbn+1 +a)B(n+1)
e
j=0
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1
√
(4.8) h00k (x) =
V. Grozdanov and S. Stoilova
e
B(n+1)
α
Bn+1
Page 16 of 38
!
Jj,t,α (x).
e
B(n+1)
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Proof. For an arbitrary integer p, 0 ≤ p < Bn and x ∈ [0, 1), following Kremer
[6] we define the function


p+1
p

, Bn
1, if x ∈

Bn


B(n)
B(n)
q(Bn ; (p)B(n)
;
x)
=
e



p+1
p

, Bn
.
 0 if x 6∈
Bn
B(n)
B(n)
Price and Haar Type Functions
and Uniform Distribution of
Sequences
The equality
Bn −1
1 X
q(Bn ; (p)B(n)
; x) =
χ e
e
Bn α=0 (p)B(n)
α
Bn
!
χα (x)
holds. Let us use the significations: For an arbitrary integer p, 0 ≤ p <
Bn , (p)B(n)
= (p̃1 p̃2 . . . p̃n )B(n)
, for an arbitrary integer α, 0 ≤ α < Bn ,
e
e
α
= (0·αn αn−1 . . . α1 )B(n)
, for real x ∈ [0, 1), x = (0·x1 . . . xn+1 . . .)B .
e
Bn
e
B(n)
Then, we obtain the equalities
(4.10)
χ(p)B(n)
e
α
Bn
!
=
αn−1 p̃n−1
ωnαn p̃n ωn−1
e
B(n)
V. Grozdanov and S. Stoilova
e
B(n)
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. . . ω1α1 p̃1
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(4.11)
χα (x) =
ω1α1 x1 ω2α2 x2
. . . ωnαn xn .
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If x ∈
p
Bn
,
B(n)
p+1
Bn
, then for all j = 1, . . . , n, xj = p̃j . From
B(n)
(4.10) and (4.11) we obtain
α
Bn
χ(p)B(n)
e
If x 6∈
p
Bn
,
B(n)
p+1
Bn
!
χα (x) = 1.
e
B(n)
, then, some δ, 1 ≤ δ ≤ n exists, so that
B(n)
xδ 6= p̃δ . Then, we have that
bX
δ −1
α (xδ −p̃δ )
ωδ δ
V. Grozdanov and S. Stoilova
= 0.
αδ =0
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From (4.10) and (4.11), we obtain
BX
n −1
χ(p)B(n)
e
α=0
α
Bn
Contents
!
χα (x) =
e
B(n)
j −1
n bX
Y
α (xj −p̃j )
ωj j
= 0.
j=1 αj =0
Now let k ≥ 1 be an integer of the kind (2.2). In order to prove (4.8) we note
that for all x ∈ [0, 1)
bn+1 −1
h00k (x) =
p
Bn
X
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a.kn
ωn+1
q Bn+1 ; (pbn+1 + a)B(n+1)
;x .
e
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a=0
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We will prove (4.9). Using the proved formula for h00k (x), we have that
Z x
00
(4.12)
Ψk (x) =
h00k (t)dt
0
=
1
√
Bn+1 −1 bn+1 −1
bn+1 Bn
X
X
α=0
a=0
a.kn
ωn+1
× χ(pbn+1 +a)B(n+1)
e
=
1
√
bn+1 Bn
!Z
α
χα (t)dt
Bn+1
t=0
× χ(pbn+1 +a)B(n+1)
e
× Jj,t,α (x) −
a.kn
ωn+1
V. Grozdanov and S. Stoilova
!
α
Bn+1
1
Price and Haar Type Functions
and Uniform Distribution of
Sequences
a=0
α=tBj
0
e
B(n+1)
n bj+1
X
X−1 (t+1)B
Xj −1 bn+1
X−1
j=0
x
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e
B(n+1)
1
t
Bj+1 ωj+1
−1
Contents
δtBj ,α .
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It is not difficult to prove that
(4.13)
n
X
1
j=0
Bj+1
bj+1 −1
X
1
t=0
t
ωj+1
−1
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(t+1)Bj −1 bn+1 −1
X
X
α=tBj
a=0
× χ(pbn+1 +a)B(n+1)
e
From (4.12) and (4.13), we obtain (4.9).
α
Bn+1
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a.kn
ωn+1
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!
δtBj ,α
e
B(n+1)
B −1
= kn n
.
ωn+1 − 1
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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, . . . , bn+1 − 1} and 0 ≤ p ≤ Bn − 1. Then, for all x ∈ [0, 1) the
equalities
Bn −1
1 X
(n)
χk (x) = √
α h00
(x);
Bn j=0 p,j̃ sBn +j̃
(4.14)
Bn −1
1 X
(n)
Jn,s,k (x) = √
α Ψ00
(x);
Bn j=0 p,j̃ n,s,sBn +j̃
(4.15)
Bn −1
1 X
(n)
χk (x) = √
αp,j̃ h0Bn +j̃(bn+1 −1)+s−1 (x)
Bn j=0
and
1
Jn,s,k (x) = √
Bn
(n)
BX
n −1
(n)
αp,j̃ Ψ0n,s,Bn +j̃(bn+1 −1)+s−1 (x),
PBn −1
j=0
(n)
αp,j̃ = Bn · δsBn ,k .
Proof. Let x ∈ [0, 1) be fixed. We define t̃ = (t)B(n)
, 0 ≤ t̃ < Bn as
he
(n)
t̃
t̃+1
t̃ t̃+1
≤
x
<
.
We
denote
∆
=
,
. It is obvious
Bn
Bn
Bn Bn
t̃
B(n)
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j=0
hold, where αp,j̃ are complex mumbers, so that
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B(n)
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(n)
that ∆t̃
=
Sbn+1 −1
a=0
(n+1)
∆t̃bn+1 +a . There is some a, 0 ≤ a ≤ bn+1 − 1, so that
(n+1)
x ∈ ∆t̃bn+1 +a . We have the equalities
!
p
t̃
a.s
a.s
χk (x) = χp
ωn+1
and h00s·Bn +t̃ (x) = Bn ωn+1
.
Bn B(n)
Hence, we obtain
1
χk (x) = √ χp
Bn
t̃
Bn
!
h00s·Bn +t̃ (x).
Price and Haar Type Functions
and Uniform Distribution of
Sequences
B(n)
(n)
Let t̃0 be an arbitrary integer, so that 0 ≤ t̃0 < Bn , and t̃0 6= t̃. For x ∈ ∆t̃ we
have that h00s·Bn +t̃0 (x) = 0. Hence, we obtain the equality
!
Bn −1
j̃
1 X
h00s·Bn +j̃ (x).
χk (x) = √
χp
Bn B(n)
Bn j=0
(n)
j̃
Let for integers 0 ≤ p < Bn and 0 ≤ j < Bn we signify αp,j̃ = χp
.
Bn
B(n)
We use the representations p = (pn pn−1 . . . p1 )B(n) and
j̃
= (0 · j1 j2 . . . jn )B(n) ,
Bn B(n)
where for 1 ≤ τ ≤ n pτ , jτ ∈ {0, 1, . . . , bτ − 1}. Then, we obtain the equality
(4.16)
BX
n −1
j=0
(n)
αp,j̃ =
n bX
τ −1
Y
τ =1 jτ =0
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ωτpτ jτ .
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P n −1 (n)
If p = 0, then, for 1 ≤ τ ≤ n, pτ = 0 and from (4.16) we obtain B
j=0 αp,j̃ =
Bn . If pP6= 0 then, some δ, 1 ≤ δ ≤ n exists, so that pδ 6= 0. From (4.16), we
(n)
n −1
obtain B
j=0 αp,j̃ = 0.
We will prove (4.15). From (4.14) for all x ∈ [0, 1) we have
Bn −1
1 X
1
1
− 32
(n)
00
Jk (x) = √
α
ΨsBn +j̃ (x) +
Bn s
bn+1
ωn+1 − 1
Bn j=0 p,j̃
−
1
bn+1
Bn−2
1
BX
n −1
s
ωn+1
−1
j=0
(n)
αp,j̃ .
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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 ξN = {x0 , . . . , xN −1 }, composed of N ≥ 1 points of
[0, 1) and x ∈ [0, 1) we signify R(ξN ; x) = A(ξN ; [0, x); N ) − N x. Then, the
next lemma holds:
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Lemma 4.4.
(i) The Fourier-Price coefficiens of R(ξN ; x) satisfy the equalities:
(χ)
a0
=−
N
−1 X
m=0
1
xm −
2
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and for each integer k ≥ 1, k = kn Bn + p, kn ∈ {1, 2, . . . , bn+1 − 1},
0 ≤ p < Bn
N
−1
X
(χ)
ak = −
Jn,kn ,k (xm ).
m=0
(ii) The Fourier-Haar type coefficiens of R(ξN ; x) satisfy the equalities:
(h0 )
a0
=−
N
−1 X
m=0
1
xm −
2
,
(h00 )
a0
=−
N
−1 X
m=0
1
xm −
2
;
Let k ≥ 1 be an arbitrary integer of kind (2.1). Then,
(h0 )
ak
=−
N
−1
X
Ψ0n,s,k (xm ).
m=0
(4.17)
=−
N
−1
X
Ψ00n,kn ,k (xm ).
m=0
Proof. Let for 0 ≤ m ≤ N − 1, cm (x) be the characteristic function of the
interval (xm , 1). Then,
R(ξN ; x) =
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Let k ≥ 1 be an arbitrary integer of kind (2.2). Then,
(h00 )
ak
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and Uniform Distribution of
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N
−1
X
m=0
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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 integer k ≥ 1 of the kind (2.2) we have:
Z 1
(h00 )
(4.18)
ak =
R(ξN ; x)h00k (x)dx
=
0
N
−1 Z 1
X
m=0
cm (x)h00k (x)dx
Z
−N
0
1
xh00k (x)dx.
0
Price and Haar Type Functions
and Uniform Distribution of
Sequences
The equalities
N
−1 Z 1
X
(4.19)
m=0
cm (x)h00k (x)dx
=
0
N
−1 Z 1
X
m=0
N
−1
X
=−
h00k (x)dx
xm
Z
h00k (x)dx
−
0
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Ψ00k (xm )
Contents
m=0
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hold. From (4.1) and (4.8) we obtain
Z 1
(4.20) xh00k (x)dx
=
Bn+1 −1 bn+1 −1
√
X
X
a·kn
ωn+1
χ(pbn+1 +a)B(n+1)
e
2πbn+1 Bn α=0 a=0
Z 1
∞
X
1
×
arg χBr (x)χα (x)dx
Br 0
r=0
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0
1
V. Grozdanov and S. Stoilova
0
α
Bn+1
!
e
B(n+1)
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=
BX
n −1
1
α
! "bn+1 −1
X
√
χ(pbn+1 )B(n+1)
e
Bn+1
2πbn+1 Bn α=0
Z 1
∞
X
1
×
arg χBr (x)χα (x)dx
Br 0
r=0
e
B(n+1)
bn+1 −1 (t+1)Bn −1
1
X
α
√
χ(pbn+1 )B(n+1)
e
Bn+1
2πbn+1 Bn t=1 α=tB
n
"bn+1 −1
# ∞
Z 1
X (t−k )a X
1
n
×
ωn+1
arg χBr (x)χα (x)dx
Br 0
a=0
r=0
!
(kn +1)Bn −1
X
α
1
χ(pbn+1 )B(n+1)
= √
e
Bn+1 B(n+1)
2π Bn α=k B
e
n n
Z
∞
1
X
1
×
arg χBr (x)χα (x)dx.
B
r
0
r=0
+
a·kn
ωn+1
a=0
X
#
!
e
B(n+1)
The following equality can be proved: Let n ≥ 0 and q ∈ {1, 2, . . . , bn+1 − 1}
be fixed integers. Then, for each integer α ≥ 0
Z 1
1
1
1
(4.21)
arg χBn (x)χα (x)dx =
· q
δq·Bn ,α .
2π 0
bn+1 ωn+1 − 1
From (4.20) and (4.21) we obtain
Z 1
(4.22)
xh00k (x)dx =
0
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and Uniform Distribution of
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1
bn+1
−3
Bn 2
1
kn
ωn+1
−1
.
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From (4.18), (4.19), (4.22) and (3.3) we obtain (4.17).
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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5.
Proofs of the Main Results
Proof of Theorem 3.1. For an arbitrary net ξN = {x0 , . . . , xN −1 }, composed
of N ≥ 1 P
points of [0, 1), following Kuipers and Niederreiter [8] we denote
N −1
1
1
S(ξN ) =
m=0 xm − 2 and for x ∈ [0, 1), Q(ξN ; x) = N (R(ξN ; x) +
S(ξN )). The inequality
Z 1
3
D (ξN ) ≤ 12
Q2 (ξN ; x)dx.
0
is proved. Hence, we obtain
12
D (ξN ) ≤ 2
N
3
(5.1)
Z
1
R (ξN ; x)dx − S (ξN ) .
2
Price and Haar Type Functions
and Uniform Distribution of
Sequences
2
0
V. Grozdanov and S. Stoilova
We obtain
2  13
−1 (kn +1)Bn −1 N −1
∞ bn+1
X
X
X
X
12
Jn,kn ,k (xm ) 
D(ξN ) ≤  2
N n=0 k =1 k=k B m=0

n
n
n
from Lemma 4.4 (i) and Parseval’s equality.
The other inequalities of Theorem 3.1 follow from (5.1) and Lemma 4.4
(ii).
Proof of Theorem 3.2.
Necessity of (i): We assume that the sequence is u. d. We will prove the equality
(5.2)
lim
N →∞
N
−1
X
i=0
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Jn,q,k (xi ) = 0.
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We use the representation k = q · Bn + p, where n ≥ 0, q ∈ {1, 2, . . . , bn+1 − 1}
and 0 ≤ p < Bn . From (3.1) and Lemma 4.1, we obtain
(5.3) Jn,q,k (x) = −
1
χp (x)
q
Bn+1 ωn+1 − 1
·
+
1
Bn+1
qbn+1
+
1
Bn+1
arg χ
·
1
q
ωn+1
−1
δq.Bn ,k
(x)
Bn
2π
ωn+1
·
χp (x)
q
ωn+1
−1
∞
1 X −r
bn+1 arg χBn brn+1 x χk (x).
+
2πBn r=1
Firstly, we assume that k = q · Bn , hence, p = 0 and δq·Bn ,k = 1. From (5.3) for
Jn,q,k (x), we have
Price and Haar Type Functions
and Uniform Distribution of
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V. Grozdanov and S. Stoilova
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Jn,q,k (x) =
1
Bn+1
qbn+1
2π
arg χ
Bn
ωn+1
·
q
ωn+1
−1
(x)
∞
1 X −r
+
bn+1 arg χBn brn+1 x χk (x).
2πBn r=1
We will prove the equalities
N −1
n+1
1 X qb2π
arg χBn (xi )
lim
ωn+1
=0
N →∞ N
i=0
and
N −1 ∞
1 X X −r
lim
bn+1 arg χBn (brn+1 xi )χk (xi ) = 0.
N →∞ N
i=0 r=1
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From (1.1), it is sufficient to prove that
Z 1 qbn+1
arg χBn (x)
2π
ωn+1
dx = 0
0
and
Z
∞
1X
0
r
b−r
n+1 arg χBn bn+1 x χk (x)dx = 0.
r=1
We have the following equalities
Z
1
qbn+1
2π
ωn+1
arg χBn (x)
dx =
BX
n −1 bn+1
X−1 Z
0
j=0
=
1
bn+1
s=0
jbn+1 +s+1
Bn+1
jbn+1 +s
Bn+1
Price and Haar Type Functions
and Uniform Distribution of
Sequences
qbn+1
2π
ωn+1
arg χBn (x)
dx
V. Grozdanov and S. Stoilova
bn+1 −1
X
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sq
ωn+1
= 0.
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s=0
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Since k = q·Bn from (4.21), we have the equalities
#
Z 1 "X
∞
r
b−r
n+1 arg χBn bn+1 x χk (x) dx
0
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r=1
=
=
∞
X
r=1
∞
X
r=1
b−r
n+1
Z
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I
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1
arg χBn
brn+1 x
χk (x)dx
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0
b−r
n+1
Z
0
1
(b
)
arg φ0 n+1 (brn+1 Bn x)
h
iq
(b
)
φ0 n+1 (Bn x)
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=
∞
X
b−r
n+1
Bn
h
iq
(bn+1 )
n+1 )
arg φ(b
(t)
φ
(t)
dt
0
r
0
r=1
= Bn
Z
∞
X
r=1
b−r
n+1
Z
0
1
(b
)
n+1
arg ψbrn+1
(t)ψq(bn+1 ) (t)dt = 0.
If k 6= q·Bn , then, from (5.3), we will obtain a useful formula for Jn,q,k (x) and
by analogy, we can prove the equality (5.2).
Sufficiency of (i): We assume that the sequence ξ = (xi )i≥0 is not u. d. Then,
limN →∞ DN (ξ) = D > 0.
Let M > 0 be a fixed integer. From Theorem 3.1 we have
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
2
−1
M
−1 bn+1
X
X−1 (q+1)B
Xn −1 1 N
X
3
(5.4) DN (ξ) ≤ 12
Jn,q,k (xm )
N m=0
n=0 q=1
k=qBn
2
∞ bn+1
−1
X−1 (q+1)B
Xn −1 1 N
X
X
+ 12
Jn,q,k (xm ) .
N m=0
n=M q=1
k=qB
n
For arbitrary integers n ≥ 0, q ∈ {1, 2, . . . , bn+1 −1} and qBn ≤ k < (q +1)Bn
we can prove
−1
1 N
X
1
1
Jn,q,k (xm ) ≤ B +
(5.5)
.
π
N
2
sin
B
n+1
B
m=0
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Let b = min{bn : n ≥ 1}. Then, using the inequality (5.5), we have
2
∞ bn+1
−1
X
X−1 (q+1)B
Xn −1 1 N
X
(5.6)
Jn,q,k (xm )
N m=0
n=M q=1
k=q.Bn
2 X
∞
1
1
< B+
π
2 sin B
Bn+1
n=M
2 X
∞
1
1
≤ B+
π
n+1
b
2 sin B
n=M
2
1
1
1
.
=
B+
π
bM
b−1
2 sin B
Now we choose M , so that
(5.7)
12
b−1
1
B+
2 sin Bπ
2
V. Grozdanov and S. Stoilova
Title Page
1
1
< D3 .
M
b
2
Let ε > 0 be arbitrary. Then, an integer N0 exists, so that for each N ≥
3
N0 , D 3 − ε < D N
(ξ). From (5.4), (5.6), (5.7) and the last inequality, we obtain
2
M
−1 bn+1
−1
X
X−1 (q+1)B
Xn −1 1 N
X
1 3
D − ε < 12
Jn,q,k (xm ) .
N m=0
2
n=0 q=1
k=qBn
We choose an integer ν > 0, so that 12 D3 − ε > ν1 D3 and we obtain
2
M
−1 bn+1
−1
X
X−1 (q+1)B
Xn −1 1 N
X
1 3
0 < D < 12
Jn,q,k (xm ) .
ν
N
n=0 q=1
m=0
k=qB
n
Price and Haar Type Functions
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Finally, (n, q, k) exists, such that
N
−1
X
1
lim
J
(x
)
n,q,k m > 0.
N →∞ N m=0
The demonstration of (ii) and (iii) of the theorem is a consequence of the formulae obtained in Lemma 4.2, Lemma 4.3 and (i) of Theorem 3.2.
Proof of Theorem 3.3.
(5.8)
2
−1
M
−1 bn+1
X
X−1 (s+1)B
Xn −1 1 N
X
3
D (ξN ) ≤ 12
Jn,s,k (xm )
N
n=0 s=1
m=0
k=sBn
q−1 (s+1)BM −1 −1
2
X
X 1 N
X
+ 12
JM,s,k (xm )
N
s=1 k=sBM
m=0


2
(q+1)BM −1 H
N −1
 X
X
1 X
+
JM,q,k (xm )
+ 12

 N
m=0
k=qBM
k=H+1
2
bM +1 −1 (s+1)BM −1 −1
X
X 1 N
X
+ 12
J
(x
)
M,s,k m N
s=q+1
m=0
k=sBM
2  31
bn+1 −1 (s+1)Bn −1 ∞
−1
X
X
X 1 N
X
+ 12
Jn,s,k (xm )  .
N
n=M +1
s=1
k=sBn
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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We have the following inequality
2
(q+1)BM −1 −1
X 1 N
X
(5.9)
Σ=
JM,q,k (xm )
N m=0
k=H+1
2
N −1
1 X
+
JM,s,k (xm )
N m=0
s=q+1
k=sBM
2
bn+1 −1 (s+1)Bn −1 −1
∞
X
X
X 1 N
X
+
Jn,s,k (xm )
N
m=0
n=M +1 s=1
k=sBn
2
−1
∞ bn+1
X
X−1 (s+1)B
Xn −1 1 N
X
Jn,s,k (xm ) .
≤
N
bM +1 −1 (s+1)BM −1 X
n=M
s=1
X
m=0
k=sBn
The following inequalities
N −1
2
1 X
(5.10)
Jn,s,k (xm ) ≤
N
m=0
N
−1
X
1
|Jn,s,k (xm )|
N
m=0
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|Jn,s,k (x)| ≤ |Jk (x)| +
1
1
, ∀x ∈ [0, 1)
π ·
2b sin B Bn
hold. It is not difficult to prove that for each k, Bn ≤ k < Bn+1
(5.12)
V. Grozdanov and S. Stoilova
Contents
!2
and
(5.11)
Price and Haar Type Functions
and Uniform Distribution of
Sequences
|Jk (x)| <
1
, ∀x ∈ [0, 1).
Bn
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From (5.11) and (5.12), we obtain
(5.13)
|Jn,s,k (x)| ≤ 1 +
1
2b sin Bπ
1
, ∀x ∈ [0, 1).
Bn
From (5.9), (5.10) and (5.13) the following inequalities
Σ≤
∞ bn+1
X
X−1 (s+1)B
Xn −1 n=M
s=1
h=sBn
1
1+
2b sin Bπ
2
B(1 + 2b sin Bπ )2 1
1
<
Bn2
4(b − 1)b sin2 Bπ BM
hold. The statement of the theorem holds from the last inequality and (5.8).
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
Theorem 3.4 is a direct consequence of Lemma 4.4 and Parseval’s equality.
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6.
Conclusion
In conclusion, the authors will present possible variants to extend this study.
The obtained results show that Price functions and Haar type functions can
be used as a means of examining uniformly distributed sequences. The proved
results raise the issue of their generalization. The most natural generalization is
to obtain results in the s−dimensional cube [0, 1)s , s ≥ 2.
This is not difficult to do when Price functions and Haar type functions are
used (see Kuipers and Niederreiter [8, Corollaries 1.1 and 1.2]).
The obtaining of results, in which the modified integrals from the corresponding s−dimensional functions are in use, is connected with great technical
difficulties. In the one-dimensional case the proof of sufficiency of Theorem
3.2 is based on the analog of the LeVeque inequality, exposed in Theorem 3.1.
A multidimensional variant of the inequality of Le Veque is not known in this
form to the authors. In this case, the proof of sufficiency of the multidimensional variant of Theorem 3.2 is connected with proving the multidimensional
variant of LeVeque’s inequality.
Another direction to generalize the obtained results is the possibility to use
arbitrary orthonormal bases in L2 [0, 1) as a means of examining uniformity distributed sequences. The obtained results in such a study would have more general nature. The definition and the use of the modified integrals of the functions
of an arbitrary system will be difficult in practice because these integrals depend
on the concrete values of the corresponding functions.
Regarding the applications of uniformly distributed sequences, the inequality
of Koksma-Hlawka gives an estimation of the error of s−dimensional quadrature formula in the terms of discrepancy of the used net. In this sense, the
Price and Haar Type Functions
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V. Grozdanov and S. Stoilova
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problem of obtaining quantitative estimations of discrepancy is interesting, as
the functions presented in this paper can be used as a means of solving the above
problem.
The shown generalizations are a part of the problems to solve in connection to the study of uniformity distributed sequences by orthonormal bases in
L2 [0, 1).
The methods to prove the theorems in this paper, by suitable adaptation may
be used to solve some of the exposed problems.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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References
[1] G.N. AGAEV, N. JA. VILENKIN, G.M. DZAFARLY AND A.I. RUBINSTEIN, Multiplicative systems and harmonic analysis on zerodimensional groups, ELM, Baku, 1981.
[2] H. E. CHRESTENSON, A class of generalized Walsh functions, Pacific J.
Math., 5 (1955), 17–31.
[3] V.S. GROZDANOV AND S.S. STOILOVA, Multiplicative system and uniform distribution of sequences, Comp. Rendus Akad. Bulgare Sci., 54(5)
(2001), 9–14.
[4] V.S. GROZDANOV AND S.S. STOILOVA, Haar type functions and uniform distribution of sequences, Comp. Rendus Akad. Bulgare Sci. 54, 12
(2001) 23-28.
[5] A. HAAR, Zur Theorie der Orthogonalen Funktionensysteme, Math. Ann.,
69 (1910), 331–371.
[6] H. KREMER, Algorithms for Haar functions and the fast Haar transform,
Symposium: Theory and Applications of Walsh functions, Hatfield Polytechnic, England.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
V. Grozdanov and S. Stoilova
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[7] L. KUIPERS, Simple proof of a theorem of J. F. Koksma, Nieuw. Tijdschr.
voor, 55 (1967), 108–111.
[8] L. KUIPERS AND H. NIEDERREITER, Uniform distribution of sequences, Wiley, New York, 1974.
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[9] W.J. LEVEQUE, An inequality connected with Weyl’s criterion for uniform, Proc. Symp. Pure Math., v. VIII, Providence: Amer. Math. Soc.,
(1965), 20–30.
[10] J.J. PRICE, Certain groups of orthonormal step functions, Canad. J. Math.,
9 (1957), 413–425.
[11] H.A. RADEMACHER, Einige Sätze Über Reihen von allgemeinen Orthogonalenfunktionen, Math. Annalen, 87 (1922), 112–138.
[12] F. SCHIPP, W.R. WADE AND P. SIMON, An Introduction to Dyadic Harmonical Analysis, Adam Hilger, Bristol, 1990.
Price and Haar Type Functions
and Uniform Distribution of
Sequences
[13] B.G. SLOSS AND W.F. BLYTH, Walsh functions and uniform distribution
mod 1, Tohoku Math. J., 45 (1993), 555–563.
V. Grozdanov and S. Stoilova
[14] I.M. SOBOL’, Multidimensional quadrature formulae and Haar functions,
Nauka Moscow, 1969.
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[15] N. YA. VILENKIN, A class of complete orthonormal series, Izv. Akad.
Nauk SSSR, Ser. Mat., 11 (1947), 363–400.
[16] N. YA. VILENKIN, On the theory of orthogonal system with geps, Izv.
Akad. Nauk SSSR, Ser. Mat., 13 (1949), 245–252.
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[17] J.L. WALSH, A closed set of normal orthogonal function, Amer. J. Math.,
45 (1923), 5–24.
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[18] H. WEYL, Uber die Gleichverteilung von Zahlen mod. Eins., Math. Ann.,
77 (1916), 313–352.
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