# ON THE EXISTENCE OF A COMPACTLY SUPPORTED L

(1)

p

p

k∈Zd

k

k

d

d

α1d

k∈Zd

k

ihβk,ξi

d

1

1

∞ m=1

m

k

p

[325]

(2)

1

k

+

∞ m=1

m

p

k

2

0≤i,j≤N

(i,j)

0≤i,j≤N

(i,j)

1

2

2

p

p

k

N

N

2

2

2

2

2

N

N

i,j

K

2

K

(3)

2

i,j

2

(k,l)

N

N

N

N

i(k,l)1,i2;j1,j2

## = c

(2i1−j1+k,2i2−j2+l)

1

2

1

2

(i,j)

2

N

N

(k,l)

i1,i2

j1,j2

i(k,l)1,i2;j1,j2

j1,j2

(0,0)

(0,1)

(1,0)

(1,1)

(i,j)

2

k,l∈{0,1}

(k,l)

−1(k,l)

(0,0)

2

(0,1)

(1,0)

(1,1)

2

1

k

l

1

k

J

j1

jk

J

J

J

J

j1

jk

J

i,j∈{0,1}

(J,(i,j))

(J,J1)

J

1

0≤i,j≤N

(i,j)

(4)

2

p

p

N

N

(i,j)

(i,j)

N

N

i,j

t

k,l

0≤i,j≤N

(i,j)

2

N

N

i,j

[i,i+1]×[j,j+1]

Q

m(Q)1

T

Q

p

2

2

(k,l)

i,j

## = f

[k/2,(k+1)/2)×[l/2,(l+1)/2)+(i,j)

K+(i,j)

[0,1/2)2+(i,j)

## + ~ f

[0,1/2)×[1/2,1)+(i,j)

## + ~ f

[1/2,1)×[0,1/2)+(i,j)

[1/2,1)2+(i,j)

(5)

N

N

0

k+1

k

k

J

J

p

k

i,j

KJ+(i,j)

J

k

p

0

1

1

1

1

1

J

−1(0,0)

J1

k+1

k

(0,0)

k

(0,0)

J1

((0,0),J1)

J

−1J

J

J

p

p

N

N

i,j

|J|=n

Ji,j

KJ

n

n

i,j

KJ+(i,j)

0

n

0

(6)

pLp

pLp

0≤i,j≤N −1

|J|=n

\

KJ

i,j

p

n

pLp

n

0≤i,j≤N −1

|J|=n

KJ+(i,j)

p

n

KJ+(i,j)

p

\

KJ

i,j

p

n

pLp

pLp

n

p

k

\

K

k

\

K

k+1

\

K

k

\

[0,1/2]×[0,1/2]

(0,0)

k

\

[0,1/2]×[1/2,1]

(0,1)

k

\

[1/2,1]×[0,1/2]

(1,0)

k

\

[1/2,1]×[1/2,1]

(1,1)

k

14

(0,0)

(0,1)

(1,0)

(1,1)

\

K

k

14

(0,0)

(0,1)

(1,0)

(1,1)

(1,1)

(0,0)

(0,1)

(1,1)

(7)

(i,j)

(i,j)

(1,1)

(0,0)

(0,1)

(1,1)

N

N

J

(0,0)

J

(0,1)

J

(1,0)

p

2

n → ∞

n

|J|=n

J

i,j

p

l

|J|=l

J

p

0

k+1

k

n

n+1

n

n+1

0

0

n

n

J

(0,0)

(J,(0,0))

J

(0,1)

(J,(0,1))

J

(1,0)

(J,(1,0))

J

(1,1)

(J,(1,1))

n

pLp

\

K

n

p

|J|=n+1

\

KJ

n

p

|J|=n

\

K(J,(0,0))

\

K(J,(0,1))

\

K(J,(1,0))

\

K(J,(1,1))

n

p

n

|J|=n

J

(0,0)

p

J

(0,1)

p

J

(1,0)

p

J

(1,1)

p

p

n

(8)

p

n

pLp

Jl

k

(i,j)

kl

l

kl

Jl

k

(i,j)

n

|J|=n

J

p

kl

n+kl

|J|=n+kl

J

(i,j)

p

l

l

nl

|J|=nl+kl

J

(i,j)

p

(d−1)(p−1)

1≤l≤d

l

l

1

N

N

d

l=1

l

Jl

k

(i,j)

p1

d

l=1

l

p

1

n

|J|=n

J

p1

n

|J|=n

d l=1

l

J

Jl

k

(i,j)

p 1

(d−1)(p−1)

d

l=1

l

p

n

|J|=n

J

Jl

k

(i,j)

p1

(d−1)(p−1)

d

l=1

l

p

n

|J|=n+kl

J

(i,j)

p1

(d−1)(p−1)

d

l=1

l

p

(d−1)(p−1)

p1

l

|J|=l

J

p

p

l

|J|=l

J

J1

(i,j)

p

J1

(i,j)

p

(9)

l+n

|J|=l+n

J

(i,j)

p

l+n

|J|=l

|J1|=n

J

J1

(i,j)

p

n

|J|=n

J1

(i,j)

p

n+l

pLp

n

pLp

n+kl

p

k=0

n

n→∞

n

l

|J|=l

J

i

p

1

k

2

p

n

(0,0)

(0,1)

(1,0)

(1,1)

n

p

n

|J|=n

J

p

n

|J|=n

J

p

n

|J|=n

J

p

k

p

p

x

\

R

y

\

R

(10)

x

N j=0

xj

x

xj

N i=0

(i,j)

y

N

i=0

yi

y

yi

N j=0

(i,j)

x

y

x

y

q

q

x

y

p

p