modulation spaces mp,q for 0

JOURNAL OF c© 2006, Scientific Horizon

FUNCTION SPACES AND APPLICATIONS http://www.jfsa.net

Volume 4, Number 3 (2006), 329-341

Modulation spaces Mp,q for 0 < p, q �� ∞

Masaharu Kobayashi

(Communicated by Hans Triebel)

2000 Mathematics Subject Classification. 42B35, 46A16, 46E30, 46E35.

Keywords and phrases. Modulation spaces, quasi-Banach spaces, Lp estimates

for convolutions, density of the rapidly decreasing functions.

Abstract. The purpose of this paper is to construct modulation spacesMp,q(Rd) for general 0 < p, q � ∞ , which coincide with the usual modulationspaces when 1 � p, q � ∞ , and study their basic properties including their

completeness. Given any g ∈ S(Rd) such that supp �g ⊂ {ξ | |ξ| � 1} and�k∈Zd

�g(ξ − αk) ≡ 1, our modulation space consists of all tempered distributions

f such that the (quasi)-norm

‖f‖Mp,q[g]

:=

��Rd

��Rd

��f ∗ �Mωg�(x)

��pdx

� qp

dω

� 1q

is finite.

1. Introduction

The purpose of this paper is to construct modulation spaces Mp,q(Rd)for general 0 < p, q � ∞ , which coincide with the usual modulationspaces when 1 � p, q � ∞ , and study their basic properties.

330 Modulation spaces

By the usual modulation space Mp,q(Rd), we mean the space of alltempered distributions f ∈ S′(Rd) for which the norm

‖f‖Mp,q =( ∫

Rd

( ∫Rd

|Vgf(x, ω)|pdx) q

p

dω

) 1q

is finite, where Vgf(x, ω) is the Short-Time Fourier Transform of f withrespect to a general window g ∈ S(Rd) defined by

Vgf(x, ω) = 〈f,MωTxg〉 =∫Rd

f(t)g(t− x)e−2πiω·tdt.

Most of the researches on modulation spaces have been restricted to thecase 1 � p, q � ∞. (See Grochenig [3].) The reason is that in the casep < 1, the estimate of the Lp -norm of the convolution is not easy.

To estimate Lp -norm of convolution of two functions, Y. Galperin and S.Samarah [2] use the fact that if f is a function on Rd that has polynomialgrowth, |f(t)| = O(|t|N ), then Bargmann transform Bf(z)

Bf(z) = 2d/4

∫Rd

f(t)e2πt·z−πt2−π2 z2

dt

is an entire function on Cd . For such f , they develop a theory of modulationspaces Mp,q(Rd) for the case 0 < p, q � ∞ . But for general f ∈ S′(Rd),whether or not Bf(z) is an entire function is non-trivial and is not provedin the paper, and completeness of the space Mp,q(Rd) is not proved. InH. Triebel [6] modulation spaces are studied along the lines of the theoryof Besov spaces, but f is restricted to those functions in S(Rd) that thesupport of f is compact. The completeness is not proved, either.

In this article we use only a simple, but key, lemma (see Lemma 2.6)to estimate convolutions, and define modulation spaces Mp,q(Rd) for0 < p, q � ∞ and prove its completeness, in particular.

2. Basic definition

Let S(Rd) be the Schwartz space of all complex-valued rapidly decreasinginfinitely differentiable functions on Rd with the topology defined by thesemi-norms

pM (ϕ) = supx∈Rd

(1 + |x|)M∑

|α|�M

|∂αϕ(x)|, M = 1, 2, · · ·

M. Kobayashi 331

for ϕ ∈ S(Rd). And let S′(Rd) be the topological dual of S(Rd).The Fourier transform is f(ω) =

∫f(t)e−2πit·ωdt , and the inverse Fourier

transform is f(t) = f(−t). We define for 0 < p <∞

‖f‖Lp =( ∫

Rd

|f(x)|pdx) 1

p

and ‖f‖L∞ = ess. supx∈Rd |f(x)| . We use 〈f, g〉 to denote the extension toS′(Rd) × S(Rd) of the inner product 〈f, g〉 =

∫f(t)g(t)dt on L2(Rd).

Definition 2.1. If f ∈ S′(Rd) and g ∈ S(Rd), we define the convolutionf ∗ g by

f ∗ g(x) = 〈f, g(x− ·) 〉 =∫Rd

f(t)g(x− t)dt, x ∈ Rd.

We define the translation and the modulation operators by

Txf(t) = f(t− x), and Mωf(t) = e2πiω·tf(t) (x, ω ∈ Rd),

respectively. The following Lemmas are useful.

Lemma 2.2. For g ∈ S(Rd) , k ∈ Zd and x ∈ Rd we have(Tkg

)∨(x) =(Mkg

)(x).

Lemma 2.3. Let Γ be a compact subset of Rd and f ∈ S′(Rd) . Ifsupp f ⊂ Γ then for any ξ0 ∈ Rd , we have

(1) supp(Mξ0f

) ⊂ ξ0 + Γ.

This is employed to prove that the constant C in the estimate (3) belowdepends only on Γ.

Definition 2.4. Let 0 < p � ∞ , and Γ be a compact subset of Rd .Then Lp

Γ is defined by

(2) LpΓ = {f ∈ S′(Rd) | ∃ξ0 ∈ Rd, supp f ⊂ ξ0 + Γ, ‖f‖Lp <∞}.

By the Paley-Wiener-Schwartz theorem LpΓ consists of entire analytic

functions.

Theorem 2.5. Let Γ be a compact subset of Rd and let 0 < p � q � ∞.

Then there exists a positive constant C (which depends only on the diameterof Γ and p) such that

(3) ‖f‖Lq � C‖f‖Lp

holds for all f ∈ LpΓ .


Proof. If supp f ⊂ Γ, (3) is just the famous Nikol’skij-Triebel inequality(see [5] Theorem 1.4.1). Let f ∈ Lp

Γ . Then we can find ξ0 ∈ Rd such thatsupp f ⊂ ξ0 + Γ. By lemma 2.3,

supp(M−ξ0f

)⊂ Γ and ‖f‖Lp = ‖M−ξ0f‖Lp .

So we have

‖f‖Lq = ‖M−ξ0f‖Lq � C‖M−ξ0f‖Lp = C‖f‖Lp. �From Theorem 2.5 we have the following key Lemma.

Lemma 2.6. Let 0 < p � 1 and Γ,Γ′ be compact subsets of Rd . Thenthere exists a positive constant C (which depends only on the diameters ofΓ,Γ′ and p ) such that

(4) ‖ |f | ∗ |g| ‖Lp � C‖f‖Lp‖g‖Lp

holds for all f ∈ LpΓ and all g ∈ Lp

Γ′ .

Proof. Step 1. First we assume f ∈ LpΓ, g ∈ Lp

Γ′ be such that

supp f ⊂ Γ, supp g ⊂ Γ′.

Then for a.e. x ∈ Rd , f(y)g(x− y) ∈ Lp (as a function of y ) and

supp(f(·)g(x− ·))⊂ Γ − Γ′.

Here we used the notation −A = {x | − x ∈ A} . By Theorem 2.5

|f | ∗ |g|(x) =( ∫

Rd

|f(y)g(x− y)|dy)

� C

( ∫Rd

|f(y)g(x− y)|pdy) 1

p

,

where C depends only on Γ,Γ′ and p . Taking the Lp -norm of both sides,we have (4).

Step 2. Let f ∈ LpΓ, g ∈ Lp

Γ′ . Then there exist ξ0, ξ′0 ∈ Rd such that

supp f ⊂ ξ0 + Γ and supp g ⊂ ξ′0 + Γ′. Applying Step.1 to M−ξ0f and

M−ξ′0g instead of f and g respectively, we have (4). �

M. Kobayashi 333

3. Modulation spaces

Definition 3.1. For α > 0 we define Φα(Rd) as follows:

Φα(Rd)

:={g ∈ S(Rd)

∣∣∣∣ supp g ⊂ {ξ | |ξ| � 1}, and∑

k∈Zd

g(ξ−αk) ≡ 1, ∀ξ ∈ Rd

}.

In the following, we choose a sufficiently small α > 0 so that the functionspace Φα(Rd) is not empty.

Definition 3.2. Given a g ∈ Φα(Rd), and 0 < p, q � ∞ , we define themodulation space Mp,q

[g] (Rd) to be the space of all tempered distributionsf ∈ S′(Rd) such that the (quasi)-norm

(5) ‖f‖Mp,q[g]

:=( ∫

Rd

(∫Rd

∣∣f ∗ (Mωg

)(x)

∣∣pdx) qp

dω

) 1q

is finite.

Our definition is close to the original definition of modulation spaces byFeichtinger [1].

Remark. Since f ∈ S′(Rd) and g ∈ S(Rd), we have Vgf(x, ω) =e−2πix·ωf ∗ (Mω g)(x), where g(x) = g(−x). Since the usual modulationspace Mp,q(Rd) (1 � p, q � ∞) is independent of the choice of a windowg ∈ S(Rd)�{0} (see [3, Proposition 11.3.2]), our modulation space coincideswith the usual one if 1 � p, q � ∞ .

Theorem 3.3. Let 0 < p � ∞ , 0 < q � ∞ , then

(6)( ∑

k∈Zd

( ∫Rd

∣∣f ∗ (Mαkg

)(x)

∣∣pdx) qp) 1

q

is an equivalent quasi-norm on Mp,q[g] (Rd) , with modifications if p or

q = ∞ .

Proof. Step 1. We first prove that quasi-norm (5) can be estimatedfrom above by the quasi-norm (6). For this purpose fix an ω ∈ Rd . Thenthere exists k = (k1, · · · , kd) ∈ Zd such that

(7) ω ∈ [αk1, α(k1 + 1)] × · · · × [αkd, α(kd + 1)] = [αk, α(k + 1)].


Then there exists a positive constant N which depends only on the size ofsupp g , α > 0 and the dimension d , such that

f ∗ (Mωg) = f ∗ (Tω g)∨

= f ∗( ∑

|r|�N

Tα(k+r)g · Tωg

)∨

= f ∗( ∑

|r|�N

Tα(k+r)g

)∨∗ (Tω g

)∨

=∑

|r|�N

(f ∗ (Mα(k+r)g)

)∗ (Mωg).

Taking the Lp -norm on both sides and p-th power, we obtain

‖f ∗ (Mωg)‖pLp =

∫Rd

∣∣∣∣ ∑|r|�N

(f ∗ (Mα(k+r)g)

) ∗ (Mωg)(x)∣∣∣∣p

dx

� C∑

|r|�N

‖(f ∗ (Mα(k+r)g)) ∗ (Mωg)‖p

Lp

� C′ ∑|r|�N

‖f ∗ (Mα(k+r)g)‖pLp .

(8)

Here, for the last inequality we used Young’s theorem for p � 1

‖F ∗G‖Lp � ‖F‖Lp‖G‖L1

and Lemma 2.6 for p � 1. We take the q/p-th power of (8), integrate over[αk, α(k + 1)], and sum over k ∈ Zd , and obtain

∫Rd

‖f ∗ (Mωg)‖qLpdω � C′ ∑

k∈Zd

∫[αk,α(k+1)]

( ∑|r|�N

‖f ∗ (Mα(k+r)g)‖pLp

) qp

dω

� C′′ ∑k∈Zd

‖f ∗ (Mαkg)‖qLp .

Step 2. Fix a k ∈ Zd . Then for all ω ∈ [αk, α(k + 1)] we have

f ∗ (Mαkg) = f ∗ (∑

|r|�N

Tω+αrg · Tαkg)∨

=∑

|r|�N

f ∗ (Mω+αrg) ∗ (Mαkg).

M. Kobayashi 335

Taking p-th power and integrating over Rd , we obtain

‖f ∗ (Mαkg)‖pLp � C

∑|r|�N

‖f ∗ (Mω+αrg)‖pLp .

Similarly we have

‖f ∗ (Mαkg)‖qLp � C

( ∑|r|�N

‖f ∗ (Mω+αrg)‖pLp

) qp

� C′∫

[αk,α(k+1)]

( ∑|r|�N

‖f ∗ (Mω+αrg)‖pLp

) qp

dω

� C′′ ∑|r|�N

∫[αk,α(k+1)]

‖f ∗ (Mω+αrg)‖qLpdω.

Taking sum over k ∈ Zd , we get

∑k∈Zd

‖f ∗ (Mαkg)‖qLp � C′

∫Rd

‖f ∗ (Mωg)‖qLpdω.

So the proof is complete. �In the sequel, we shall not distinguish between equivalent quasi-norms of

a given quasi-normed space.

Corollary 3.4. Let 0 < p0 � p1 � ∞ and 0 < q0 � q1 � ∞ . Then

Mp0,q0[g] (Rd) ⊂Mp1,q1

[g] (Rd).

Proof. This follows form the monotonicity of the lp -spaces and Theorem2.5. �

Lemma 3.5. If 0 < p � ∞ , 0 < q � ∞ and g1, g2 ∈ Φα(Rd) , then‖f‖Mp,q

[g1]and ‖f‖Mp,q

[g2]are equivalent quasi-norms on Mp,q(Rd) .

Proof. It is easy to see that both ‖f‖Mp,q[g1]

and ‖f‖Mp,q[g2]

are quasi-norms.In order to prove the equivalence of these two quasi-norms we apply Young’sinequality (or Lemma 2.6 if p � 1). Since Tαkg1 =

∑|r|�N

Tα(k+r)g2 · Tαkg1

we have

f ∗ (Mαkg1) =(f · Tαkg1 ·

∑|r|�N

Tα(k+r)g2

)∨


= f ∗( ∑

|r|�N

Mα(k+r)g2

)∗ (Mαkg1).

Hence there exists a positive constant c such that

‖f ∗ (Mαkg1)‖Lp � c

∥∥∥∥f ∗( ∑

|r|�N

Mα(k+r)g2

)∥∥∥∥Lp

.

This proves that ‖f‖Mp,q[g1]

can be estimated from above by c‖f‖Mp,q[g2]

. Hencethese two quasi-norms are equivalent to each other. �

Theorem 3.6. We have the continuous imbeddings

(9) S(Rd) ⊂Mp,q(Rd) ⊂ S′(Rd)

for 0 < p � ∞ and 0 < q � ∞.

Proof. To prove the left-hand side of (9), let f ∈ S(Rd) and denote

Δ =d∑

j=1

(∂2/∂2j ). Then for any positive integer L we have

(1 + 4π2|ω|2)L|f ∗ (Mωg)(x)|

=∣∣∣∣∫Rd

f(x− t)g(t)(1 + 4π2|ω|2)Le2πiω·tdt∣∣∣∣

=∣∣∣∣∫Rd

f(x− t)g(t)(1 − Δ)Le2πiω·tdt∣∣∣∣

=∣∣∣∣∫Rd

∑|α+β|�2L

Cα,β∂αf(x− t)∂βg(t)e2πiω·tdt

∣∣∣∣.Therefore

|f ∗ (Mωg)(x)|(1 + |x|2)M (1 + 4π2|ω|2)L � C p2(L+M)(f).

Taking L,M sufficiently large, we have

‖f‖Mp,q � Cp2(L+M)(f).

Next we prove the right-hand side of (9) with q = ∞ . First note thatthere exists a constant N (depending only on the size of supp g , α > 0and dimension d) such that Tαkg =

∑|r|�N

Tα(k+r)g · Tαkg for all k ∈ Zd . If

M. Kobayashi 337

f ∈Mp,∞(Rd) and ψ ∈ S(Rd), then

|〈f, ψ〉| =∣∣∣∣ ∑

k∈Zd

〈(Tαkg · f)∨,∑

|r|�N

(Mα(k+r)g) ∗ ψ〉∣∣∣∣

�∑

k∈Zd

‖f ∗ (Mαkg)‖L∞

∥∥∥∥ ∑|r|�N

(Mα(k+r)g) ∗ ψ∥∥∥∥

L1

.

Recall that both f ∗(Mαkg) and∑

|r|�N

(Mα(k+r)g)∗ψ are analytic functions.

Hence, the last estimate makes sense. By Theorem 2.5 we have

‖f ∗ (Mαkg)‖L∞ � C‖f ∗ (Mαkg)‖Lp.

Hence,

|〈f, ψ〉| � C′‖f‖Mp,∞∑

k∈Zd

∥∥∥∥ ∑|r|�N

(Mα(k+r)g) ∗ ψ∥∥∥∥

L1

.

The last sum can be estimated from above by C‖ψ‖M1,1 . Consequently, ifM is a sufficiently large natural number, we have

|〈f, ψ〉| � C‖f‖Mp,∞pM (ψ)

for any ψ ∈ S(Rd). This proves that Mp,∞(Rd) is continuously embeddedin S′(Rd). �

Theorem 3.7. Mp,q(Rd) is a quasi-Banach space if 0 < p � ∞ and0 < q � ∞ (Banach space if 1 � p � ∞ and 1 � q � ∞ ) .

To prove the theorem we prepare the following lemma.

Definition 3.8. For 0 < p, q � ∞ we define Lp,q(Rd × Zd) as follows:

Lp,q(Rd × Zd)

:={f : Rd × Zd → C is measurable and ‖f‖Lp,q := ‖ ‖f‖Lp‖lq <∞

}.

This is a quasi-normed space and hence there is an 0 < r <∞ such that

(10)∥∥∥∥

n∑l=1

fl

∥∥∥∥r

Lp,q

� 2n∑

l=1

‖fl‖rLp,q .

(See Kothe [4] p.162.)


Lemma 3.9. Let 0 < p, q � ∞ . If {fn(x, k)}∞n=1 is a sequence inLp,q(Rd × Zd) such that

(11)∑n∈N

‖fn‖rLp,q <∞.

Then the series

(12) f(x, k) =∑n∈N

fn(x, k)

converges absolutely for a.e. x ∈ Rd and for all k ∈ Zd and is inLp,q(Rd × Zd) and

(13) ‖f1 + · · · + fn − f‖Lp,q → 0, as n→ ∞.

In particular, Lp,q(Rd × Zd) is complete.

Proof. When either p or q = ∞ , the following inequalities give the proofwith r = min{q, 1} or min{p, 1} .∣∣∣∣

∣∣∣∣∣∣∣∣∣∣∣∣ ∑

n∈N

|fn(x, k)|∣∣∣∣∣∣∣∣L∞

∣∣∣∣∣∣∣∣r

lq�

∣∣∣∣∣∣∣∣ ∑

n∈N

‖fn(x, k)‖L∞

∣∣∣∣∣∣∣∣r

lq

�∑n∈N

‖fn(x, k)‖rLp,q , if q � p = ∞,

∣∣∣∣∣∣∣∣ ∑

n∈N

|fn(x, k)|∣∣∣∣∣∣∣∣r

Lp

�∑n∈N

‖fn(x, k)‖rLp

�∑n∈N

(sup

k‖fn(x, k)‖Lp

)r, if p < q = ∞.

Let 0 < p, q <∞ and define

gn(x, k) :=n∑

l=1

|fl(x, k)| and g(x, k) := limn→∞ gn(x, k) =

∞∑n=1

|fn(x, k)|.

By (11) and by the Beppo-Levi theorem

‖g‖rLp,q = lim

n→∞ ‖gn‖rLp,q � lim

n→∞ 2n∑

l=1

‖fl‖rLp,q = 2

∞∑n=1

‖fn‖rLp,q <∞.

This proves that g ∈ Lp,q . Consequently, for a.e. x and for all k , theseries (12) converges absolutely. Since |f(x, k)|p � g(x, k)p ∈ L1(Rd) and‖f(x, k)‖q

Lp � ‖g(x, k)‖qLp ∈ l1(Zd), we conclude that f ∈ Lp,q(Rd × Zd).

M. Kobayashi 339

Similarly, since |f1 + · · ·+ fn − f | �∞∑

l=n+1

|fn| � g ∈ Lp,q , we have (13) by

a repeated application of the Lebesgue dominant convergence theorem.To prove the completeness of Lp,q let {hn}∞n=1 be a Cauchy sequence in

Lp,q . Then we can choose a subsequence {hnk}∞k=1 so that

‖hnk+1 − hnk‖r

Lp,q � 2−(k+1) for all k ∈ N.

From above argument, {hnk}∞k=1 converges to f = limk→∞ hnk

in Lp,q .Since {hn}∞n=1 is a Cauchy sequence, {hnk

} also converges to f in Lp,q . �

Proof of Theorem 3.7. Let {fn}∞n=1 be a sequence in Mp,q(Rd) such that∑n∈N

‖fn‖rMp,q <∞.

Then by Lemma 3.9, the series∑n∈N

(fn ∗ (Mαkg)(x)

)

converges absolutely for a.e. x ∈ Rd and for all k ∈ Zd and is inLp,q(Rd × Zd). And by Theorem 3.6, we have f =

∑n∈N

fn converges in

S′(Rd). Since for all x ∈ Rd and k ∈ Zd

f ∗ (Mαkg)(x) =⟨ ∑

n∈N

fn,Mαkg(x− ·)⟩

=∑n∈N

〈fn,Mαkg(x− ·)〉

=∑n∈N

(fn ∗ (Mαkg)(x)

),

we have f ∈Mp,q(Rd). So Mp,q(Rd) is complete. �

Theorem 3.10. If 0 < p <∞ and 0 < q <∞ , then S(Rd) is dense inMp,q(Rd) .

Proof. Let f ∈Mp,q(Rd) and set

fn =∑|k|�n

f ∗ (Mαkg), n ∈ N.


Of course, fn ∈Mp,q(Rd). Then, it follows that

(f − fn) ∗ (Mαlg)(x) =∑|k|>n

f ∗ (Mαkg) ∗ (Mαlg)(x)

=∑

r∈Λl,n

f ∗ (Mα(l+r)g) ∗ (Mαlg)(x)

where

Λl,n :={r ∈ Zd

∣∣∣∣ |r + l| > n and supp(Tα(l+r)g) ∩ supp(Tαlg) �= ∅}.

Note that

|Λl,n| � ∃N, (∀l ∈ Zd, ∀n ∈ N).

Moreover∣∣∣∣∣∣∣∣ ∑

r∈Λl,n

f ∗ (Mα(l+r)g) ∗ (Mαlg)(x)∣∣∣∣∣∣∣∣Lp

� C

∣∣∣∣∣∣∣∣ ∑

r∈Λl,n

f ∗ (Mα(l+r)g)∣∣∣∣∣∣∣∣Lp

‖Mαlg‖Lmin{1,p}

and thus

‖f − fn‖qMp,q =

∑l∈Zd

∣∣∣∣∣∣∣∣ ∑

r∈Λl,n

f ∗ (Mα(l+r)g)∣∣∣∣∣∣∣∣q

Lp

� C′ ∑|r|>n−N

∣∣∣∣∣∣∣∣f ∗ (Mαrg)

∣∣∣∣∣∣∣∣q

Lp

→ 0, (n→ ∞).

Hence fn approximates f in Mp,q(Rd). Let ϕ ∈ S(Rd) with ϕ(0) = 1and supp ϕ ⊂ {ξ | |ξ| � 1} . Let (fn)δ(x) = ϕ(δx)fn(x) with 0 < δ < 1then (fn)δ ∈ S(Rd) and this is an approximation of fn in Mp,q(Rd). Thisproves that S(Rd) is dense in Mp,q(Rd). �

Lemma 3.11. Let Γ be a compact subset of Rd . If β > 0 is sufficientlysmall (depending on Γ) , then we have equivalency

( ∑k∈Zd

|f(βk)|p) 1

p

∼ ‖f‖Lp, f ∈ LpΓ, 0 < p � ∞.

Proof. See for instance [5] Theorem 1.4.1. �

M. Kobayashi 341

Combining Theorem 3.3 and Lemma 3.11, we have the followingCorollary.

Corollary 3.12. Let 0 < p � ∞ , 0 < q � ∞ . If β > 0 is sufficientlysmall then

(14)( ∑

k∈Zd

( ∑l∈Zd

∣∣f ∗ (Mαkg

)(βl)

∣∣p) qp) 1

q

is an equivalent quasi-norm on Mp,q(Rd) .

References

[1] H. G. Feichtinger, A new class of function spaces, In : Proc. Conf.“Constructive Function Theory”, Kiew, 1983, (1984).

[2] Y. Galperin and S. Samarah, Time-frequency analysis on modulationspaces Mp,q

m , 0 < p, q ≤ ∞ , Appl. Comput. Harmon. Anal., 16 (1)(2004), 1–18.

[3] K. Grochenig, Foundations of Time-Frequency Analysis, Birkhauser,2001.

[4] G. Kothe, Topologische Lineare Raume I, Zweite Auflage, Springer,1966.

[5] H. Triebel, Theory of Function Spaces, Birkhauser, Boston, 1983.

[6] H. Triebel, Modulation spaces on the euclidean n-space, Z. Anal.Anwendungen, 2(5) (1983), 443–457.

Department of MathematicsTokyo University of ScienceKagurazaka 1-3Shinjuku-kuTokyo 162-8601Japan(E-mail : [email protected])

(Received : September 2005 )

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