On extensions of wavelet systems to dual pairs of frames

0.17 MB·

by  Ole Christensen

On extensions of wavelet systems to dual pairs of
frames
4
1 Ole Christensen∗, Hong Oh Kim†, Rae Young Kim‡
0
2
January 7, 2014
n
a
J

5

] Abstract
A
F It is an open problem whether any pair of Bessel sequences with wavelet structure
h. can be extended to a pair of dual frames by adding a pair of singly generated wavelet
t systems. We consider the particular case where the given wavelet systems are gener-
a
m ated by the multiscale setup with trigonometric masks and provide a positive answer
[ under extra assumptions. We also identify a number of conditions that are necessary

for the extension to dual multiscale wavelet frames with any number of generators,
1
v and show that they imply that an extension with two pairs of wavelet systems is
2 possible. Along the way we provide examples showing that extensions to dual frame
6
pairs are attractive because they often allow better properties than the more popular
8
0 extensions to tight frames.
.
1
Keywords: Bessel sequences, Dual frame pairs, Wavelet systems
0
4 2010 Mathematics Subject Classification: 42C15, 42C40
1
:
v
i
X 1 Introduction
r
a
Extension problems exist in a large variety in the frame literature. In its basic version the
question is whether a given sequence of elements in a Hilbert space can be extended to a
frame with prescribed properties. For example, it is natural to ask for extensions such that
the resulting frame is computationally convenient, e.g., a tight frame or a frame for which
a dual frame can be found easily. A natural generalization of this idea is to start with two
sequences and ask for extension of these sequences to dual frame pairs.
It is known that any pair of Bessel sequences in a separable Hilbert space can be
extended to a pair of dual frames by adding appropriate collections of vectors. But if we
∗DepartmentofAppliedMathematicsandComputerScience,TechnicalUniversityofDenmark,Building
303, 2800 Lyngby, Denmark ([email protected])
†Department of Mathematical Sciences, KAIST, 373-1, Guseong-dong, Yuseong-gu, Daejeon, 305-701,
Republic of Korea ([email protected])
‡Department of Mathematics, Yeungnam University, 214-1, Dae-dong, Gyeongsan-si, Gyeongsangbuk-
do, 712-749,Republic of Korea ([email protected])
1
require the added sequences to have a special structure or to satisfy certain constraints,
many open problems appear. A key question is whether any given pair of Bessel sequences
with wavelet structure can be extended to a pair of dual frames by adding a pair of wavelet
systems, each with a single generator (see the paper [16] by D. Han for a discussion and
conjecture about the corresponding tight case).
In this paper we will analyze this problem under the extra assumption that the given
wavelet systems are generated by the MRA-setup as considered, e.g., in [18, 9, 12]. Given
two scaling functions, we will consider the associated wavelet systems generated by letting
the masks be trigonometric polynomials, and ask for extensions to dual pairs of frames by
addingwavelet systems ofthesametype. Wewillfirstidentifyaconditionontherefinement
masks that is necessary for this extension to be possible, and then show that this condition
is also sufficient for the possibility to extend to dual pairs using two generators. A stronger
condition characterizes the possibility to extend the given wavelet systems to a pair of dual
wavelet frames by adding a pair of wavelet systems, each with a single generator.
Note that Daubechies and B. Han already in [11] showed that for any given pair of
scaling functions one can construct dual pairs of wavelet frames, each with two generators.
Our setup is different from the one in [11]: in our extension of Bessel systems to dual pairs
with two generators, we consider one of the pairs of Bessel sequences to be given, i.e., we
only have freedom with respect to one pair of wavelet systems.
The rest of this introduction gives a short introduction to the key ingredients of the
paper and connects our results to the literature. In Section 2 we present a few preliminary
results onBessel sequences andthe mixed extension principle. The results aboutextensions
with singly generated systems are stated in Section 3, while the extension by systems with
two generators is treated in Section 4.
Asequence{f } inaseparableHilbertspaceHiscalledaframeifthereexistconstants
i i∈I
A,B > 0 such that
A||f||2 ≤ |hf,f i|2 ≤ B||f||2, ∀f ∈ H.
i
i∈I
X
A frame is tight if we can choose A = B; and if at least the upper frame condition is
satisfied, {f } is called a Bessel sequence. For any frame {f } , there exist at least one
i i∈I i i∈I
dual frame, i.e., a frame {g } such that f = hf,g if , ∀f ∈ H. A tight frame with
i i∈I i∈I k k
A = B = 1 leads to an expansion of arbitrary elements f ∈ H of exactly the same type as
P
we know for orthonormal bases, i.e., f = hf,f if , ∀f ∈ H. For more information on
i∈I k k
frames we refer to the books [10, 4].
In this paper we will exclusively consPider systems of functions in L2(R) with wavelet
structure, that is, collections of functions of the type {2j/2ψ(2jx − k)}j,k∈Z for a fixed
function ψ. Considering the operators on L2(R) given by T f(x) := f(x−k) and Df(x) :=
k
21/2f(2x), the wavelet system can be written as {DjTkψ}j,k∈Z.
Let T denote the unit circle which will be identified with [−1/2,1/2]. Also, for f ∈
L1(R)∩L2(R) we denote the Fourier transform by Ff(γ) = fˆ(γ) = ∞ f(x)e−2πixγdx. As
−∞
usual, the Fourier transform is extended to a unitary operator on L2(R).
R
In the entire paper we will use the following setup that appeared, e.g., in [12], except
that we restrict our attention to trigonometric masks.
2
General setup: Consider a function ϕ ∈ L2(R) such that
(i) ϕ is continuous at the origin and ϕˆ(0) = 1;
(ii) There exists a 1-periodic trigonometric polynomial m (called a refinement mask)
0
b
such that
ϕ(2γ) = m (γ)ϕ(γ), a.e. γ ∈ R. (1.1)
0
Given 1-periodic trigonometrbic polynomialsbm ,m ,...,m , consider the functions ψ ∈
1 2 n ℓ
L2(R) defined by
ψ (2γ) = m (γ)ϕ(γ), ℓ = 1,...,n. (1.2)
ℓ ℓ
b
Note that the technical condition b
|ϕ(γ +k)|2 ≤ K < ∞,
k∈Z
X
b
which is used in [12], automatically is satisfied in our setting. In fact, it is well known
that ϕ has compact support whenever the scaling equation (1.1) holds for a trigonometric
polynomial m . Thus,
0
|ϕ(γ +k)|2 = hϕ,ϕ(·−k)ie−2πikγ
k∈Z k∈Z
X X
b
is a trigonometric polynomial.
We will base the analysis on the mixed extension principle (MEP) by Ron and Shen [18]
˜
which is formulated in terms of the (n+1)×2 matrix-valued functions M and M defined
by
m (γ) m γ + 1 m (γ) m γ + 1
0 0 2 0 0 2
m (γ) m γ + 1 m (γ) m γ + 1
M(γ) =  1. 1(cid:0) . 2(cid:1), M(γ) =  1. 1(cid:0) . 2(cid:1). (1.3)
.. .. e .. e ..
(cid:0) (cid:1) (cid:0) (cid:1)
   
m (γ) m γ + 1  me (γ) me γ + 1 
 n n 2  f  n n 2 
   
Formulated for trigonometric(cid:0)masks,(cid:1)the MEP reads as follows: (cid:0) (cid:1)
e e
Proposition 1.1 Assume that ϕ,ϕ ∈ L2(R) satisfy the conditions in the general setup,
with associated masks m ,m . For each ℓ = 1,··· ,n, let m ,m be trigonometric polynomi-
0 0 ℓ ℓ
alsand defineψℓ,ψℓ ∈ L2(R)by (1.2e). Assume that{DjTkψi}i=1,···,n;j,k∈Z and{DjTkψi}l=1,···,n;j,k∈Z
are Bessel sequences. If theecorresponding matrix-valued funcetions M and M satisfy
e e
M(γ)∗M(γ) = I, γ ∈ T, f (1.4)
then {DjTkψi}i=1,···,n;j,k∈Z and {DfjTkψi}l=1,···,n;j,k∈Z form dual frames for L2(R).
e
3
The MEP was later extended to the mixed oblique extension principle in [9, 12]; these
papers also contain several explicit examples. Other papers about the MEP include [1, 13,
15].
Most of the concrete wavelet frame constructions in the literature are obtained via
the related unitary extension principle [18] and its variants, which lead to tight frame
constructions (see, e.g., the papers [19, 17, 14, 3, 6], just to mention a few out of many).
But it is already noted in, e.g., [12, 9] that the extra flexibility in the MEP frequently leads
to moreattractive constructions, a claim that is also supported by some ofthe results inthe
current paper. For example, we consider a case where the extension of a wavelet system to
a tight frame introduces a wavelet generator without compact support, while the extension
to a dual pair of frames is possible with compactly supported generators.
Note that the analysis in the current paper is complementary to the one in [5]. In [5] we
formulated the general question whether any pair of Bessel sequences {DjTkψ1}j,k∈Z and
{DjTkψ1}j,k∈Z can be extended to a pair of dual frames by adding a wavelet system to each
of the given Bessel sequences. A sufficient condition for a positive answer turned out to be
that ψeis compactly supported on [−1,1]. In contrast, the extension principle applied in
1
the current paper involves functions that are compactly supported in time.
Ficnally, for the sake of the non-specialist, we note that it is known that the dual frames
of a wavelet frame not necessarily have wavelet structure: there are cases where no dual
wavelet frame exist at all (see, e.g., the books by Chui [7] and Daubechies [10]), and
there are cases where some duals have wavelet structure and some do not (see the paper
by Bownik and Weber [2]). This issue is one of the key motivations behind the various
extension principles, which construct, simultaneously, a frame and a dual with wavelet
structure.
2 Preliminaries on Bessel sequences and the MEP
Intheentire paper we assume thatwe have given trigonometricpolynomials m ,m ,m ,m
0 1 0 1
as described in the general setup. We will search for trigonometric polynomials m ,··· ,m
2 n
and m ,··· ,m for the cases n = 2 and n = 3 satisfying the condition (1.4), i.e.,
2 n e e
M(γ)∗M(γ) = I, γ ∈ T. (2.1)
e e
Note that (2.1) is equivalent to the two conditions
f
n
m (γ)m (γ) = 1, γ ∈ T, (2.2)
ℓ ℓ
ℓ=0
X
n
e
m (γ)m (γ +1/2) = 0, γ ∈ T. (2.3)
ℓ ℓ
ℓ=0
X
In Proposition 1.1 we need that {eDjTkψi}i=1,···,n;j,k∈Z and {DjTkψi}l=1,···,n;j,k∈Z are
Bessel sequences. Since these systems are finite union of wavelet systems, it is sufficient
that each of these form a Bessel sequence. The following lemma proviedes necessary and
sufficient conditions for this:
4
Lemma 2.1 Let ϕ ∈ L2(R) be a scaling function, with a refinement mask m satisfying
0
the conditions in the general setup. Let m be a trigonometric polynomial and define ψ by
ψ(2γ) = m(γ)ϕ(γ), ℓ = 1,...,n. (2.4)
Then the following are equivalent:
b
b
(a) {DjTkψ}j,k∈Z is a Bessel sequence;
(b) m(0) = 0.
Proof. (a) ⇒ (b) : Assume that {DjTkψ}j,k∈Z is a Bessel sequence with bound B. Using
2
[8, Theorem 1] (or see Proposition 11.2.2 in [4]), we have ψ(2jγ) ≤ B. By the
j∈Z
conditions in the general setup, ψ is continuous at the originP, so it(cid:12)(cid:12) follows(cid:12)(cid:12) that ψ(0) = 0.
b
Hence, via (2.4) and the assumption ϕ(0) = 1 we conclude that m((cid:12)0) = 0.(cid:12)
(b) ⇒ (a) : See [11, Lemma 2.1].b b (cid:3)
b
Example 2.2 Let B , N ∈ N, denote the Nth order B-spline, defined recursively by
N
B := χ , B := B ∗B .
1 [0,1] N+1 1 N
It is well known that ϕ := B satisfies the conditions in the general setup with m (γ) =
N 0
(e−πiγcos(πγ))N . Thus Lemma 2.1 shows that {DjTkBN}j,k∈Z is not a Bessel sequence in
L2(R). On the other hand, a finite linear combination
K
ψ (x) = c B (2x−k)
1 k N
k=−K
X
generates a Bessel sequences {DjTkψ1}j,k∈Z if and only if Kk=−Kck = 0. (cid:3)
P
ThefollowingexampledemonstratesthatthematrixconditioninTheorem1.1isnotsuf-
ficientforduality, i.e., theassumptionofthesequences{DjTkψi}i=1,···,n;j,k∈Zand{DjTkψi}l=1,···,n;j,k∈Z
being Bessel sequences is essential:
e
Example 2.3 Consider the scaling functions ϕ(x) = ϕ(x) = B (x) and
2
1 1
ψ (x) = ψ (x) = − B (2x)+B (2x−1)− B (2x−2),
1 1 2 2 2 e 2 2
1 1 1
ψ (x) = −e B (2x+2)+ B (2x)− B (2x−2), ψ (x) = 2B (2x),
2 2 2 2 2 2
4 2 4
with the associated masks
e
m (γ) = m (γ) = e−2πiγcos2(πγ),
0 0
m (γ) = m (γ) = e−2πiγsin2(πγ), m (γ) = 2cos2(πγ)sin2(πγ), m (γ) = 1.
1 1 2 2
e
Then the MEP-condition (1.4) with n = 2 in Proposition 1.1 is satisfied. But by Lemma
2.1 we know that e{DjTkψ2}j,k∈Z is not a Bessel sequence. Hence {DjTkψei}i=1,2;j,k∈Z and
{DjTkψi}l=1,2;j,k∈Z does not form dual frames for L2(R). (cid:3)
e
e
5
In Lemma 2.5 we will state three necessary conditions for the existence of MEP-type
wavelet systems {DjTkψi}i=1,···,n;j,k∈Z and{DjTkψi}l=1,···,n;j,k∈Z. Wefirst need thefollowing
factorizations:
e
Lemma 2.4
(i) Letf bea 1-periodictrigonometric polynomialwith f(0) = 0.Then f(γ) = e−πiγsin(πγ)Λ (γ)
1
for a 1-periodic trigonometric polynomial Λ ;
1
(ii) Letg be a1-periodictrigonometricpolynomialwith g(1/2) = 0.Theng(γ) = e−πiγcos(πγ)Λ (γ)
2
for a 1-periodic trigonometric polynomial Λ .
2
Proof. For the proof of (i), write f as f(γ) = c e−2πikγ. Since f(0) = c = 0,
k∈Z k k∈Z k
we have f(γ) = c (e−2πikγ −1). Define f and f by
k6=0 k +P − P
P
f (γ) := c (e−2πikγ −1), f (γ) := c (e2πkiγ −1).
+ k − −k
k∈N k∈N
X X
Then we see that
k−1 k−1
f (γ) = c (e−2πiγ −1) e−2πiℓγ = e−πiγsin(πγ) −2i c e−2πiℓγ
+ k k
!
k∈N ℓ=0 k∈N ℓ=0
X X X X
=: e−πiγsin(πγ)Λ (γ).
+
Similarly,
k
f (γ) = e−πiγsin(πγ) 2i c e2πiℓγ =: e−πiγsin(πγ)Λ (γ).
− −k −
!
k∈N ℓ=1
X X
Then we have f(γ) = f (γ)+f (γ) = e−πiγsin(πγ)Λ (γ), where Λ (γ) := Λ (γ)+Λ (γ)
+ − 1 1 + −
is a 1-periodic trigonometric polynomial. This proves (i).
For the proof of (ii), let g˜(γ) := g(γ + 1/2). Since g˜(0) = 0, there exists a 1-periodic
trigonometric polynomial Λ such that g˜(γ) = e−πiγsin(πγ)Λ(γ). Then we have g(γ) =
g˜(γ −1/2) = e−πiγcos(πγ)Λ (γ), where Λ (γ) := −iΛ(γ −1/2). This proves (ii). (cid:3)
2 2
Lemma 2.5 Under the hypothesis of Proposition 1.1, the following hold:
(a) m (0) = m (0) = 0, ℓ = 1,2,··· ,n;
ℓ ℓ
(b) m (1/2) = m (1/2) = 0;
0 0
e
(c) 1−m (γ)m (γ) = sin2(πγ)Λ(γ) for some 1-periodic trigonometric polynomial Λ.
0 0
e
e
6
Proof. (a) : This follows by Lemma 2.1.
(b) : Note that m (0) = m (0) = 1 by the assumptions in the general setup. This together
0 0
with (a) and (2.3) imples m (1/2) = m (1/2) = 0.
0 0
(c) : By (a) and Lemma 2.4, the functions m ,m , ℓ = 1,2··· ,n, can be factorized as
e ℓ ℓ
e
m (γ) = e−πiγsin(πγ)Λ (γ), m (γ) = e−πiγsin(πγ)Λ (γ),
ℓ ℓ eℓ ℓ
for some 1-periodic trigonometric polynomialseΛℓ,Λℓ. Combining tehis with (2.2) leads to
1−m (γ)m (γ) = sin2(πγ)Λ(γ), where Λ(γ) = n Λ (γ)Λ (γ). (cid:3)
0 0 ℓ=1 ℓ ℓ
e
In Secteion 4 we will show that, on the other hPand, the aessumptions (a), (b) and (c) in
Lemma2.5impliesthatthewavelet systems {DjTkψ1}j,k∈Z, {DjTkψ1}j,k∈Z canbeextended
to pairs of dual wavelet frames by adding two wavelet systems.
f
3 Extension with one pair of generators
In the rest of the paper we will consider scaling functions ϕ,ϕ ∈ L2(R) as in the general
setup, with associated trigonometric polynomial masks m ,m . Assuming that we have
0 0
given trigonometric polynomials m ,m and defined the associated functions ψ ,ψ by
1 1 e 1 1
(1.2), our goal is to extend the Bessel sequences {DjTkψ1}j,k∈fZ, {DjTkψ1}j,k∈Z to pairs of
dual wavelet frames. It turns out to bfe convenient to consider the functions M andfM ,
α β
defined by f
f f
M (γ) := 1−m (γ)m (γ)−m (γ)m (γ); (3.1)
α 0 0 1 1
M (γ) := −m (γ)m (γ +1/2)−m (γ)m (γ +1/2). (3.2)
β 0 0 1 1
f e e
Lemma 3.1 If thfe conditions (a), (b) and (c) in Lemma 2.5 are satisfied, then M and
e e α
M can be factorized as
β
f
f M (γ) = sin2(πγ)Λ (γ), M (γ) = −isin(πγ)cos(πγ)Λ (γ) (3.3)
α α β β
for some 1-perifodic trigonometric polynomfials Λα and Λβ.
Proof. By Lemma 2.4, and (a) and (b) of Lemma 2.5, m ,m , ℓ = 0,1 can be factorized
ℓ ℓ
as
e
m (γ) = e−πiγcos(πγ)Λ (γ), m (γ) = e−πiγcos(πγ)Λ (γ),
0 0 0 0
m (γ) = e−πiγsin(πγ)Λ (γ), m (γ) = e−πiγsin(πγ)Λ (γ),
1 1 1 1
e e
for some 1-periodic trigonometric polynomialseΛℓ,Λℓ for ℓ = 0,1. eTogether with (c) of
Lemma 2.5 this implies
e
M (γ) = sin2(πγ) Λ(γ)−Λ (γ)Λ (γ) =: sin2(πγ)Λ (γ),
α 1 1 α
(cid:16) (cid:17)
f e
7
and
M (γ) = −isin(πγ)cos(πγ) Λ (γ)Λ (γ +π/2)−Λ (γ)Λ (γ +π/2)
β 0 0 1 1
=: −isin(πγ)cos(πγ)Λ(cid:16) (γ), (cid:17)
β
f e e
(cid:3)
as desired.
We are now ready to state a condition for extension of MRA-type wavelet systems
{DjTkψ1}j,k∈Z,{DjTkψ1}j,k∈Z to dual frames {DjTkψℓ}ℓ=1,2;j,k∈Z, {DjTkψℓ}ℓ=1,2;j,k∈Z. Note
thatinTheorem3.2below,thecondition(i)in(I)meansthat{DjTkψ2}j,k∈Zand{DjTkψ2}j,k∈Z
are Bessel sequences, wehile (ii) simply is the MEP-condition. We also noete that the proof
shows how to choose the corresponding masks m ,m . With this information we canefind
2 2
the functions ψ ,ψ explicitly: if, e.g., m (γ) = d e2πikγ, then ψ = 2 d ϕ(2x+k).
2 2 2 k 2 k
f
Theorem 3.2 Left ϕ,ϕ ∈ L2(R) be as in the Pgeneral setup, with trigoPnometric polyno-
mial masks m ,m , respectively. Let m ,m be trigonometric polynomials, and define
0 0 1 1
ψ ,ψ ∈ L2(R) by (1.2). Assume that the conditions (a), (b) and (c) in Lemma 2.5 for
1 1 e
m ,m ,m ,m are satisfied. Then the following are equivalent:
0 0 1 1 e e
e
(I) There exist 1-periodic trigonometric polynomials m ,m such that
2 2
e e
(i) m (0) = m (0) = 0;
2 2
e
(ii) the matrix-valued functions M,M in (1.3) with n = 2 satisfy
e
M(γf)∗M(γ) = I, γ ∈ T,
(II) M (γ)M (γ +1/2) = M (γ)M (fγ +1/2), γ ∈ T.
α α β β
Inthefaffirmaftivecase, the mfulti-wafveletsystems{DjTkψℓ}ℓ=1,2;j,k∈Z and{DjTkψℓ}ℓ=1,2;j,k∈Z,
with ψ ,ψ defined by (1.2), form dual frames for L2(R).
2 2
e
Proof. (I)⇒(II): This follows from (2.2) and (2.3):
e
M (γ)M (γ +1/2) = m (γ)m (γ)m (γ +1/2)m (γ +1/2)
α α 2 2 2 2
= M (γ)M (γ +1/2).
β β
f f
e e
(II)⇒(I):LetM ,M befactorizedas(3f.3)forf1-periodictrigonometricpolynomialsΛ ,Λ .
α β α β
Then Lemma 2.5 (b) implies
f f
Λ (γ)Λ (γ +1/2) = Λ (γ)Λ (γ +1/2). (3.4)
α α β β
Let Γ be the common factor of Λ and Λ , that is,
α β
Λ (γ) = Γ(γ)Γ (γ), Λ (γ) = Γ(γ)Γ (γ), (3.5)
α α β β
8
for some 1-periodic trigonometric polynomials Γ ,Γ with no common factors. This to-
α β
gether with (3.4) implies
Γ (γ)Γ (γ +1/2) = Γ (γ)Γ (γ +1/2).
α α β β
Since Γ and Γ have no common factor, we have
α β
Γ (γ) = Γ (γ +1/2) (3.6)
β α
up to constant. Define m and m by
2 2
m (γ) := e−πiγsin(πγ)Γ(γ), m (γ) := e−πiγsin(πγ)Γ (γ).
2 e 2 α
Then (i) is trivial. By (3.5) and (3.6), we have
e
m (γ)m (γ) = sin2(πγ)Γ(γ)Γ (γ) = M (γ),
2 2 α α
and e f
m (γ)m (γ +1/2) = −isin(πγ)cos(πγ)Γ(γ)Γ (γ +1/2) = M (γ).
2 2 α β
These lead to (iie) by (2.2) and (2.3) with n = 2. f (cid:3)
For the case where ϕ = ϕ = B , we can characterize the possible trigonometric poly-
2
nomials m ,m with at most three terms that satisfy the conditions in Theorem 3.2. Our
1 1
main reason for stating this is that we can use the result to identify concrete candidates
e
for pairs of wavelet systems that can not be extended to a pair of dual wavelet frames by
f
adding a single pair of wavelet systems, see Example 3.5.
Corollary 3.3 Let d ,d ,d ,d ∈ C. Define ψ and ψ by
0 1 0 1 1 1
ψ1(x) := de0Be2(2x)+(d1 −d0)B2(2xe−1)−d1B2(2x−2); (3.7)
ψ (x) := d B (2x)+(d −d )B (2x−1)−d B (2x−2). (3.8)
1 0 2 1 0 2 1 2
Then the followeing are equievalent: e e e
(a) There exist 1-periodic trigonometric polynomials m and m such that (I) in Theorem
2 2
3.2 holds;
e
(b) 3d d +3d d −d d −d d = 2.
0 0 1 1 1 0 0 1
Proof. Let ϕ := ϕ := B with the associated masks m (γ) := m (γ) := (e−πiγcos(πγ))2.
e e e2 e 0 0
From (3.7) and (3.8), m and m are defined by
1 1
e e
d d −d d d d −d d
m (γ) = 0 + 1 0e−2πiγ −e 1e−4πiγ, m (γ) = 0 + 1 0e−2πiγ − 1e−4πiγ. (3.9)
1 1
2 2 2 2 2 2
e e e e
e
9
Trivially, (a), (b) and (c) in Lemma 2.5 for m ,m ,m ,m are satisfied. We now check the
0 0 1 1
condition (II) in Theorem 3.2. Note that
e e
M (γ) = 1−cos4(πγ)−m (γ)m (γ), M (γ) = cos2(πγ)sin2(πγ)−m (γ)m (γ +1/2).
α 1 1 β 1 1
Tfhen we have e f e
M (γ)M (γ +1/2) = 1−cos4(πγ) 1−sin4(πγ) − 1−cos4(πγ) m (γ +1/2)m (γ +1/2)
α α 1 1
− 1−sin4(πγ) m (γ)m (γ)+m (γ)m (γ +1/2)m (γ)m (γ +1/2)
(cid:0) (cid:1)(cid:0) 1 1 (cid:1) (cid:0)1 1 (cid:1) 1 1
f f e
(cid:0) (cid:1)
and
e e e
M (γ)M (γ +1/2)
β β
= cos4(πγ)sin4(πγ)−sin2(πγ)cos2(πγ)m (γ +1/2)m (γ)
1 1
f f
−cos2(πγ)sin2(πγ)m (γ)m (γ +1/2)+m (γ)m (γ +1/2)m (γ)m (γ +1/2).
1 1 1 1 1 1
e
Using the identities
e e e
1−cos4(πγ) = sin2(πγ) 1+cos2(πγ) , 1−sin4(πγ) = cos2(πγ) 1+sin2(πγ) ,
cos2(πγ)−cos4(πγ) = sin2(πγ)−sin4(πγ) = cos2(πγ)sin2(πγ),
(cid:0) (cid:1) (cid:0) (cid:1)
the condition (II) in Theorem 3.2 is equivalent to
sin2(πγ)cos2(πγ) 2− m (γ)−m (γ +1/2) (m (γ)−m (γ +1/2))
1 1 1 1
= sin2(πγ)m (γ +(cid:16)1/2)(cid:16)m (γ +1/2)+cos2(πγ(cid:17))m (γ)m (γ). (cid:17) (3.10)
1 1 1 1
e e
From (3.9), m and m can be factorized as
1 1 e e
m (γ) = e−πiγsin(πγ)Λ (γ), m (γ) = e−πiγ sin(πγ)Λ (γ),
1 e 1 1 1
where e e
Λ (γ) := i(d +d e−2πiγ), Λ (γ) := i(d +d e−2πiγ). (3.11)
1 0 1 1 0 1
Since m (γ)m (γ) = sin2(πγ)Λ (γ)Λ (γ), (3.10) is equivalent with
1 1 1 1 e e e
2− m1(γ)−e m1(γ +1/2) (m1(γ)e−m1(γ +1/2)) = Λ1(γ +1/2)Λ1(γ +1/2)+Λ1(γ)Λ1(γ),
(cid:16) (cid:17)
This together with (3.9) and (e3.11) leaeds to 2− d −d d −d e= 2(d d +d d ), tehat
1 0 1 0 0 0 1 1
is, 3d0d0 +3d1d1 −d1d0 −d0d1 = 2, as desired. (cid:0) (cid:1)(cid:16) (cid:17) (cid:3)
e e e e
Leteus conseider a ceoncreteecase and find the functions ψ2,ψ2 explicitly:
f
10