# Generalization of the interaction between the Haar approximation and polynomial operators to higher order methods

(1)

(2)

## to higher order methods

(3)

k=

k=0

(k)

k

(4)

2

δ

2

k

k∈Z

k

(5)

δ

δ

k

k

δ

δ

n

n

n

i

j

δ

δ

δ

δ

δ

2

δ

L2(R)

1

δ

k

2

2

k

k

δ

δ

δ

(6)

1

1

|s−t|≤µ

|s−t|≤µ

δ

δ

δ

δ

δ

i

i

N+1

δ

δ

δ

N+1

### sup

(k,l)∈IN,|s−t|≤(µ+M)δ

(k)

(l)

N

i

j

i+j

(7)

[0,1)

(8)

N,δ

(9)

δ

N,δ

δ

k∈Z

l=N

l=0

l

[k,k+1)

l

1

N,δ

N

δ

δ

k∈Z

k

[k,k+1)

N

N,δ

N

δ

N,δ

δ

δ

δ1

k

δ1

k

k∈Z

δ

δ

i

j

i+j

N

N,δ

N,δ

N,δ

N

1

(10)

N

N+1

N

N+1

N

N

N+1

N+1

N

z

z

N,δ

δ

N,δ

δ

δ

N

N,δ

(11)

δ

N,δ

δ

δ

δ1

δ

N

q

δ

δ

δ

N+1

N+1

p

k∈Z

p

k∈Z

δ

δ

N,δ

δ

i

N,δ

N,δ

N

(12)

N

N,I

N

N,I

N,I

N,I

(j)k

(j)˜k

2

2

(13)

N,I

N,I

p

k∈I

I

N,I

p

k∈I

N,I

I

N,I

N,I

p

(j)

p

(j)

11

p

p

k∈Z

N,I

l

l

k∈I

0

I

N,I

1

I

N,I

n def

n−1

1

I

N,I

N,I

1

I

I

I

n

1

n

n

I

0

1

I−1

N,I

I−1

I

0

1

I−1

I

I

N,I

I

(14)

k

k

k∈Z

k∈Z

i=n

i=0

(i)k

i

(i)k

i

i

[k,k+1)

i

i(j)

i,j

i(j)

i

i

### (t) > 0 if 0 < t < 1.

0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5

0 0 . 0 0 5

0 . 0 1 0 . 0 1 5 0 . 0 2

p1(t)=t3

L1(t)=0 R2(t)=0

R1(t)=

δ3+3δ2t L2(t)=

δ3-3δ2t

p2(t)=(δ-t)3

δ

1

3

2

3

1

1

3

2

2

3

2

2

3

i

i

1

2

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

k

i+j≤n

i

j

i+j

n

(15)

k

k

k

k

k

k+1

k

k+1

k

k+1

k+1

k

j

j

j

k

I

k

N

I

N

I

I

I

k

k

k

k

k

k

k+1

k

k+1

k

k+1

k

j

I

I

I

(16)

I

I

I

I

N

I

I

I

I

I

I

I

I

I

I

I

### x is the decomposition of a pair of vectors into the symmetric and their antisymmetric parts. � Theorem 5 yields a polynomial formula which is similar to (1):

0 0 . 5

0 0 . 0 1 0 . 0 2

Smooth part 0 0 . 5

- 0 . 0 1 - 0 . 0 0 5

0

Singular part

L1(t)=0 L1(t)=0

R1(t) = δ3= L2(t)

R2(t)=0 R2(t)=0

R1(t) = 3δ2t = -L2(t)

N

I

k=

k=0

(2k)

2k

I

I

3

3

N

(17)

I

I

N,δ

N,δ

k

k

k

k

k

n

n

i=n

i=0

i

i

n

i=n

i=0

i

i

(2k)

I

I

I

I

(18)

0

n

r

k

0

r

0

r

Mr+r

Mr

Mr+r

0

Mr+r

r

r

r

r

k

0

r

r

r

r

0

r

0

r

0

r

2

r

r

r

0

i

(19)

j

n

i

j

i

j

N

i

j

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

δ

i

i

δ

(k,l)∈In

(k)

(l)

i

δ

j

(20)

N+1

N

0

N

N+1

i=N

i=0

i

i def

0

N

N+1

N+1

N+1

0

N

N+p

N+p−1

N+p−1

N+p−1

0

N

N+p−1

N

N+p

N+1

i

i

N

iN+p1

N+p

i

iN+p1

i

N

i

N+1

i

iN+p

iN+1

Ni +1

Ni +p

N+p(k)

i

(k)

i

N+1(k)

i

N+p−1

N+1

(k)

i

N+1(k)

i

N+p−1

(k)

i

N+p−1(k)

i

i

N(k−1)+p−1

i

N+p1

(k)

i

i

N+p1

(k−1)

i

N+p

(k)

i

(21)

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## References

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