A Study of Weighted Polynomial Approximations with Several Variables (I)

(1)

http://www.scirp.org/journal/am ISSN Online: 2152-7393

ISSN Print: 2152-7385

DOI: 10.4236/am.2017.89095 Sep. 19, 2017 1267 Applied Mathematics

A Study of Weighted Polynomial

Approximations with Several Variables (I)

Ryozi Sakai

Department of Mathematics, Meijo University, Nagoya, Japan

Abstract

In this paper, we investigate the weighted polynomial approximations with

several variables. Our study relates to the approximation for p

( )

s

Wf∈L 

by weighted polynomials. Then we will estimate the degree of approximation.

Keywords

Weighted Polynomial Approximations with Several Variables, the Degree of Approximations

1. Introduction

Let s:= × × ×    (

s

times, s≥1 integer) be the direct product space,

and let W x x

(

1, 2,,xs

)

:=w x w x1

( ) ( )

1 2 2 w xs

( )

s , where w xi

( )

i ≥0 are even

weight functions. We suppose that for every nonnegative integer n,

( )

0 d , 0,1, 2, , 1, 2, , .

n i

x w x x n i s

∞

< ∞ = =

∫

 

In this paper, we will study to approximate the real-valued weighted function

( )(

Wf x x1, 2,,xs

)

by weighted polynomials

( )(

WP x x1, 2,,xs

)

, where

(

1, 2, ,

)

, , ,

( )

s

s n n n

P x x  x ∈ _  . Here, _{, , ,}

( )

s

(

:

_;

( )

s

)

n nn



=

n s





means a class of

all polynomials with at most n-degree for each variable x ii, 1, 2,= ,s. We

need to define the norms. Let 0< ≤ ∞p , and let : s

f  → be measurable.

Then we define

( )

( )(

)

(1 )

( )(

)

1

1 1

1 , ,

, , d d , if 0 ;

:

sup , , , if .

p s

s s

p p

s s

L

s x x

Wf x x x x p

Wf

Wf x x p

∞ ∞

−∞ −∞

∈

  _{< < ∞}

 

 

= 

= ∞ 



∫



  





How to cite this paper: Sakai, R. (2017) A Study of Weighted Polynomial Approxima-tions with Several Variables (I). Applied Mathematics, 8, 1267-1306.

https://doi.org/10.4236/am.2017.89095

Received: August 3, 2017 Accepted: September 16, 2017 Published: September 19, 2017

(2)

DOI: 10.4236/am.2017.89095 1268 Applied Mathematics

We assume that for 0< ≤ ∞p the integral is independent of the order of

integration with respect to each x ii, 1, 2,= ,s. When

Wf

_Lp

( )

_s

< ∞

, we write

( )

p s

Wf∈L  . If p= ∞, we require that f is continuous and

( )( )

lim_X_→∞ Wf X =0, where X =

(

x₁,,x_s

)

=max x_i ;i=1, 2,,s.

Our purpose in this paper is to approximate the weighted function

( )

p s

Wf∈L  by weighted polynomials WP P; ∈_{n s}_;

( )

s . The paper is

arranged as the following. In Section 2, we give the definition of the weights which are treated in this paper. In Section 3, we consider the approximation for the functions in _Lp

( )

s . In Section 4, we consider a property of higher order derivatives. In Section 5, we estimate the degree of approximations. In Section 6, we consider the approximation for the functions with bounded variation. In Section 7, we consider the approximation of the Lipschitz-type functions. In Section 8, we treat the functions with higher order derivatives.

2. Class of Weight Functions and Preliminaries

Throughout the paper C C C, 1, 2, denote positive constants independent of

, ,

n x t or polynomials P x

( )

. The same symbol does not necessarily denote the

same constant in different occurrences. Let f x

( )

~g x

( )

mean that there

exists a constant C>0 such that C f x−1

( )

≤g x

( )

≤Cf x

( )

holds for all x∈I, where I⊂ is a subset.

We say that f : 0,→

[

∞

)

is quasi-increasing if there exists C>0 such

that f x

( )

≤Cf y

( )

for 0< <x y. Hereafter we consider following weights.

Definition 2.1. Let Q: 0,→

[

∞

)

be a continuous and even function, and

satisfy the following properties:

(a) Q x′

( )

is continuous in , with Q

( )

0 =0.

(b) Q′′

( )

x exists and is positive in \ 0

{ }

.

(c) limx→∞Q x

( )

= ∞.

(d) The function

( )

: xQ x

( )

, 0

T x x

Q x

′

= ≠

is quasi-increasing in

(

0,∞

)

, with

( )

1, \ 0 .

{ }

T x ≥ Λ > x∈

(e) There exists C1>0 such that

( )

1

( )

, . . .

Q x Q x

C a e x

Q x Q x

′ ′′

≤ ∈

′ 

Then we write

( )

2

exp

w= −Q ∈ C .

Moreover, if there also exists a compact subinterval J

( )

0 of , and

2 0

C > such that

( )

2

( )

, . . \ ,

Q x Q x

C a e x J

Q x Q x

′ ′′

≥ ∈

(3)

then we write _w₌_exp

( )

₋_Q _∈_

(

_C2₊

)

. If _{T x}

( )

is bounded, then the weight

( )

(

2

)

exp

w= −Q ∈ C + is called a Freud-type weight, and if T x

( )

is

unbounded, then w is called an Erdös-type weight.

For

( )

(

( )

)

(

2

)

3

(

{ }

)

exp , \ 0

w x = −Q x ∈ C + Q∈C  , if there exists K>0

such that for x ≥K,

( )

,

Q x Q x

C

Q x Q x

′′′ ′′

≤

′′ ′ (2.1)

and there exist

λ

, 0

C

>

such that for 0 3 2 λ < < _,

( )

, ( )

Q x C Q x λ

′

≤ _(2.2)

then we write

(

3

)

w∈λ C + . Furthermore, if

( )

4

3 ~

Q x Q x Q x

C

Q x Q x Q x

′′′ ′′

≤

′′ ′ (2.3)

and the inequality (2.2) with 0 4

3

λ

< < _{hold, then we write} _w

(

_C4

)

λ

∈ + .

We have some examples satisfying Definition 2.1.

Example 2.2 (cf. [1][2]). (1) If an exponential Q x

( )

satisfies

( )

(

)

( )

1 2

1 xQ x ,

Q x

′ ′

< Λ ≤ ≤ Λ

′

where Λi, 1, 2i= are constants, then we call w=exp

(

−Q x

( )

)

the Freud

weight. The class

(

2

)

C +

 contains the Freud weights.

(2) For α >1,l≥1 we define

( )

l;

( )

expl

( )

exp 0 ,l

( )

Q x =Q_α x = xα −

where exp_l

( )

x =exp exp exp

(

expx

)



)

times

(

l

)

. Moreover, we define

( )

{

( )

*

( )

}

; , exp exp 0 , 1, 0, 0,

m

l m l l

Q_α x = x xα −α α+ >m m≥ α≥

where *

0

α = _if

α

=0, and otherwise *

1

α = _{. We note that}Q_l_;0,_m gives a

Freud-type weight, that is, T x

( )

is bounded..

(3) We define

( )

(

1

)

x 1, 1.

Q x x

α

α = + − α>

(4) Let

( )

(

2

)

exp

w= −Q ∈ C + , and let us define

( )

: lim sup , : lim inf .

x x

Q x Q x Q x Q x

Q x Q x Q x Q x

µ

+

µ

− _→∞

→∞

′′ ′ ′′ ′

= _′ = _′

If µ₊ =µ₋_{, then we say that the weight}_w_{is regular. All weights in examples}

(4)

(5) More generally we can give the examples of weights

(

3

)

w∈λ C + . If the

weight w is regular and if _Q∈_C3

(

_{\ 0}

{ }

)

satisfies (2.1), then for the regular

weights we have

(

3

)

w∈λ C + (see [3], Corollary 5.5 (5.8)).

The following fact is very important for our study.

Proposition 2.3 ([3], Theorem 4.1 and (4.11)). Let 0< <λ 3 2 and

α

∈.

Then for _w _exp

( )

_Q

(

_C3

)

λ

= − ∈ + , we can construct a new weight

(

2

)

wα∈ C + such that

( ) ( )

~

( )

on ,

w

T xαw x w_α x 

and for some C≥1,

( )

/ and ~ ,

n C n Cn w w

a w_α ≤a w ≤a w_α T_α x T x =T x

where a wn

( )

α and an

( )

w are MRS-numbers for the weight wα and

w

,

respectively, and Tw_α and Tw are correspond for wα or

w

, respectively.

Let

{ }

pn be orthonormal polynomials with respect to a weight w, that is, n

p is the polynomial of degree n such that

( ) ( ) ( )

2

(

)

d the Kronecker delta .

n m mn

p x p x w x x

δ

∞

−∞ =

∫

For 1≤ ≤ ∞p , we denote by Lp

( )

 the usual p

L space on  (here for p= ∞, if wf ∈L∞

( )

 , then we require f to be continuous, and wf to have

limit 0 at ±∞). For p

( )

wf ∈L  , we set

(

)

( ) ( )

( )

( ) ( ) ( )

1

0

2

, : ,

where d

n

n k k

k

k k

s f x b f p x

b f f t p t w t t −

= ∞ −∞

=

∑

∫

(2.4)

for n∈ (the partial sum of Fourier-type series). The de la Vallée Poussin

mean of order n is defined by

(

)

2

(

)

1

, : , .

n

n j

j n

v f x s f x

= +

=

∑

_(2.5)

Let _w_∈_

(

_C2₊

)

. We need the Mhaskar-Rakhmanov-Saff numbers

(MRS-numbers) ax;

( )

(

)

1

1 2

0 2

2

d , 0.

π ₁

x x

a uQ a u

x u x

u

′

= >

−

∫

We easily see

0

lim _x and lim _x 0

x→∞a = ∞ x→+ a =

and

0

lim x 0 and lim x .

x x

a a

x x

→∞ = →+ = ∞

(5)

is defined by

(

)

(

)

_{( )}

, ; : inf p .

n

n p _L

P

E w f w f P

∈

= − _



3. Approximations for

L

p

-Functions

In this section, we treat the function such as p

( )

s

Wf∈L  , where 1≤ ≤ ∞p ),

and if p= ∞, then we suppose that Wf is continuous and

( )( )

lim_X_→∞ Wf X =0 . For any multivariate point

(

1, ,

)

s s

X = x  x ∈ , we

consider the weights;

( )

:

1

( )

1

exp

(

( )

)

.

s s

j j j j

j j

W X

=

∏

₌

w x

=

∏

₌

−

Q x

As shown under, we will also use X u

( ) (

:= u u1, 2,,us

)

. Let

(

)

(

3

)

exp

i i

w = −Q ∈λ C + , 0< <λ 3 2,

i

=

1, 2,



,

s

. From Proposition 2.3 we

see 1 4

(

2

)

,1 4

~ , 1, 2, ,

i i i

T w w ∈ C + i=  s. Then we admit to write

(

)

1 4 2

i i

T w∈ C + . For the weight W we construct the modulus of continuity

of f . It involves the function

( )

(

( )

)

,

1

: 1 i , 1, 2, , ,

t i i

i _i _i

x

x i s

t _T _t

σ _σ

Φ = − + = _

where

σ

i

( )

t is defined by

( )

( ) ( )

: inf : , 0,

i

i n

a

t a t t

n

σ = _ ≤ _ >

 

 

where ( )i n

a is the MRS-number for the weight w xi

( )

. If

( )i n

a n=t, then we

have

( )

( )i

i t an

σ = _{. In the sequel, if}1≤ ≤j s is an integer, then f _{p j}_; will

denote the p

L norm of f taken with respect to the j-th variable. This is a

function of the remaining

(

s−1

)

variables. For each fixed

(

)

1

1 1 1

ˆ : , , , , , s

j j j s j

X = x  x₋ x₊  x ∈− , we write

( )

(

)

ˆ_j

:

1

,

j1

, ,

j1

,

s

,

1, 2,

, .

X

f

x

=

f x



x

₋

x x

₊



x

j

=



s

(3.1)

Using

( )

ˆ : ˆ ˆ ,

2 2

j j j

h X X X

h h

f x f x  f x 

∆ = _ + _− _ − _

   

we define the modulus of continuity. For the Freud-type weight, we define

(

)

( )

(

( )

)

( )

(

( )

)

( )

₍ _{( )}₎ ˆ

,

1 ˆ

0

2

ˆ

constant ₄

, ; 1

: d

inf .

j

j p

j

j p

j _j

p j X j

p p

t

j h X

L x t

j j

X

c _L _x _t

f w t

w x f x h

t

f x c w x

σ

ω

≤

 

=_ ∆ _

 

+ −

∫

(6)

(

)

( )

(

( )

)

( )

(

( )

)

( )

₍ _{( )}₎

,

ˆ ,

1 ˆ

0

2

ˆ

constant ₄

, ; 1

: d

inf .

j

t j j p

j

j p

j _j

p j X j

p p

t

j h x X

L x t

j j

X

c _L _x _t

f w t

w x f x h

t

f x c w x

σ

ω

Φ

≤

 

=_ ∆ _

 

+ −

∫

We remark that if T xj

( )

is bounded, then we see Φt j,

( )

x ~ 1, so we do not

need the definition for the Freud-type weight.

Let vn be the de la Vallée Poussin mean opetator, and let vn j,

( )

f ,j=1, 2,,s

denote the operation to f with respect to j-th co-ordinate, and [ ]j n

v will denote

the operator vn applied to f with respect to each of the first j co-ordinates.

Clearly,

[ ]1

( )

[ ]

( )

[ ]1

(

( )

)

,1 , , , 2, 3, , .

j j

n n n n n j

v f =v f v f =v − v f j=  s (3.2)

Let ( )j n

a be the MRS-number for the weight w_j=exp

( )

−Q_j .

First, we consider the following Proposition.

Proposition 3.1 ([4], Theorem 3.14). For 1≤ < ∞p , Cc

( )

s is dence in

( )

p s

L  , where

( )

s

c

C  is a set of all continuous functions with a compact

support on s_.

From this proposition, for any

ε

>0 there exist a constant K>0 and a

continuous function fK with a compact support

[

,

]

s

K K

− _{such that}

( ) ( )

(

K

( )

)

p₍ ₎ .

L X K

W X f X f X

ε

≤

− < _(3.3)

Then we give the following assumption:

Assumption 3.2. In (3.3) we suppose that for every co-ordinate x_j,j=1, 2,,s

( )

(

( ) ( ) ( )

)

( )

ˆ ˆ

j j p

j X K X

L x K

w x f x f x

ε

≤

− < _(3.4)

holds.

We define a new class of functions *

( )

, 1

s p

L  ≤ ≤ ∞p as follows:

( )

{

( )

}

*

:

holds 3.4 ,

p s s

W p

L



=

f Wf

∈

L



(3.5)

where if p= ∞, then LWp*

( )

s =LWp

( )

s and we suppose that f is continuous and

( ) ( )

lim 0

X→∞W X f X =

(we write this fact as 0

( )

s

Wf∈C  ). We state the theorem in this section.

Theorem 3.3. (1) We suppose

( )

(

3

)

(

)

exp 0 3 2 , 1, 2, ,

j j

w = −Q ∈λ C + < <λ j=  s, and let

( )

2 3

, 1, 2, , .

j

j n j

n n

T a c j s

a

 

≤ _ _ =

   (3.6)

(7)

[ ]

( )

(

)

( )

(

)

(

)

( )

1

1 1 4 1 4

ˆ ,

1 , 1

1

, ; ,

p s

j

p s

j

s n

L

j s

j _n

j i s i j i k k p j X j j j j

L W f v f

a

C w T f T w c

n

ω

−

≤ ≤ ≠ =

=

−

 

≤ _ _

 

∑

∏



(3.7)

where

(

)

{

}

1

1 1 1

:

,

s

j

x

j

x

j

x

s −

− +

=





(3.8)

and 0 1 4

1 k 1

k=T =

∏

. Especially *

( )

s

p s

T W

f ∈L  , then we have

(

)

(

)

( )

1

1 1 4 1 4

ˆ ,

1 , 1

1

, ; 0

as .

j

p s

j

j s

j _n

j i s i j i k k p j X j j j j

L a

C w T f T w c

n

ω

−

≤ ≤ ≠ =

=

 

→

 

 

→ ∞

∑

∏

 (3.9)

(2) We suppose _exp

( )

(

2

)

_{, 1, 2,} _,

j j

w = −Q ∈ C + j=  s, and let (3.6) holds.

Let n≥1, 1≤ ≤ ∞p . Then we have

[ ]

( )

(

)

( )

(

)

( )

1

ˆ , 1 ,

1 4

1 1

, ; .

j

p s

p s j

s

j s

n

j i p j j j

s i s i j X

j k

k L

L

W f v f _a

C w f w c

n

T ω −

≤ ≤ ≠ =

=

−  

≤ _ _

 

∑

∏

_ 

(3.10)

Especially p*

( )

s

W

f ∈L  , then we have

(

)

( )

1

ˆ , 1 ,

1

, ; 0 as .

j

p s

j

j s

n j i s i j i p j X j j j

L a

C w f w c n

n

ω

−

≤ ≤ ≠ =

 

→ → ∞

 

 

∑

∏



(3.11)

First we will show (3.7). We need some preliminaries. Proposition 3.4 ([5], Theorem 1). Let 1≤ ≤ ∞p .

(1) We assume that _w_∈_

(

_C2₊

)

satisfies

( )

(

)

2 3

n n

T a ≤C n a . Then there

exists a constant C>0 such that when wg∈Lp

( )

 , then

( )

( ) ( )

1 4 p ,

p

n

L L

wv g

C wg

T _ ≤ 

and so,

( )

_{( )}

( )

1 4 _{( )}

. p p

n L n L

wv g _ ≤CT a wg _

(2) We assume that

(

3

)

(

)

0 3 2

w∈_λ C + < <λ satisfies T a

( )

_n ≤C n a

(

_n

)

2 3.

Then there exists a constant C>0 such that if T wg1 4 ∈Lp

( )

 , then

( )

_{( )} 1 4 _{( )}

.

p p

n _L

L wv g _ ≤C T wg



Proposition 3.5 ([5], Corollary 6.2 (6.5)). Let 1≤ ≤ ∞p .

(1) Let

(

2

)

w∈ C + , and n≥1 be an integer. Then

( )

(

)

( )

(

)

,

1 4 ; ,

p

n

n p L

w g v g

CE w g T

−

≤

(8)

where C do not depend on g and n.

(2) Let _w

(

_C3

)

₀

(

_{3 2}

)

λ λ

∈ + < < , and n≥1 be an integer. Then

( )

(

)

_{( )}

(

1 4

)

, ; ,

p

n _L n p

w g−v g ≤CE T w g



where C do not depend on g and n.

Proposition 3.6. ([6]) Let

( )

(

2

)

exp

w= −Q ∈ C + , and let 0< ≤ ∞p . Then

there exist n0∈ and positive constants C1, C2 such that for every

( )

p wg∈L 

(and for p= ∞, we require g to be continuous, and wf to vanish at ±∞)

and every n≥n0,

(

)

, ; 1 , ; 2 ,

n

n p p

a E g w C g w C

n

ω  

≤ _ _

 

where n0 and C C1, 2 do not depend on g and

n

, and

(

)

*

, ,

p g w t

ω

_{will be}

defined in Section 6. We set

( )

{

}

1 4 1

:

j

,

:

,

j

i j j

i

T

=

∏

₌

T



=

x

∈



(

)

{

}

1

{

(

)

1

}

1

:

,

:

,

j j s j s j

j

x

j j

x

j

x

s

− + − +

≤

=



∈

≤

=



∈



(

)

{

}

1 1

1 1 1

:

,

s s

j

x

j

x

j

x

s

− −

− +

=



∈



1 1,

:

s _i

,

_j

:

s _i

,

1, 2,

, .

i i i j

W

=

∏

₌

w W

=

∏

_{= ≠}

w

j

=



s

We need the infinite-finite inequality.

Theorem 3.7 (Infinite-finite inequality). Let 0< ≤ ∞p , 0L> , and let

( )

, ,

( )

(

: ;

( )

)

s s

n n n s

P X ∈_  =   . Then

( ) ( )

p

( )

s

( ) ( )

p

(

( )i

(

₁ ( )i

)

_{, 1,2, ,}

)

.

i n n

L L x a L i s

W X P X ≤C W X P X _≤ ₋_δ ₌



 (3.12)

If r>1, then there exists

ε

>0 such that

( ) ( )

( )

( ) ( )

( )

,

exp ,

p s p s

i r

L L

W X P X ≤C −nε W X P X

  (3.13)

where

{

( )

}

1

,

:

;

i

s s

i r

x x

i i

a

rn i −

=

≥

×



.

To prove Theorem 3.7 we use the following proposition with s=1.

Proposition 3.8 ([2], Theorem 1.9). Let 0< ≤ ∞p , 0L> , and let P x

( )

∈n

( )

 .

Then

( ) ( )

p_{( )}

( ) ( )

p₍ ₍₁ ₎₎.

n n

L L x a L

w x P x _ ≤C w x P x _≤ ₋_δ (3.14)

If r>1, then there exists

ε

>0 such that

( ) ( )

p₍ ₎ exp

( )

( ) ( )

p₍ ₎.

rn n

L a x L x a

w x P x _≤ ≤C −nε w x P x _≤ (3.15)

(9)

3.8 (3.14), we have

( )

{

( ) (

)

}

( )

(

( )

)

( )

(

( )

)

( ) (

)

( )

(

( )

)

( )

(

( )

)

( )

( ) (

)

( )

(

( )

)

( )

(

( )

)

1 1 1 1 1 1

2 2 1 1 1 1 2

1

1 2 2 1 1 1 1

1

1 ₁ 1 1 2 2 1 2 1

1

2 ₁

, , d d d

, , d d

, , d d d

n n

p p

p

s s s s

a L

p p

s s s s

a L

a L _p p

s s s s

a L

A w x w x w x P x x x x x

C w x w x w x P x x x x

C w x w x w x P x x x x x

C

δ δ

∞ ∞ ∞

−∞ −∞ −∞

−

∞ ∞

−∞ −∞ − −

− ∞ ∞

−∞ −∞

− −

−

− −

= ×

≤

=

≤

∫

   

( )

(

( )

)

( )

(

( )

)

( ) ( )

( )

( ) (

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( )

( ) (

)

( ) ( )

(

( )

(

( )

)

2 2 2 2

1 1 1 1

1

1 1 2 2 1

3 2 1 3 1 2

1 1

1 1 1 1

1 1

1 , 1,2, ,

, , d d d d

, , d d

.

n n

s s

n n n n

s s

n n n n

i i

p

i n n

a L

p p

a L

p

s s s d

a L a L p

s _a _L _a _L s s s s

p

s _L _x _a _L _i _s

w x w x

w x w x P x x x x x x

C w x w x P x x x x

C W X P X

δ δ

δ

−

− −

∞ ∞

−∞ −∞

− −

− − − −

≤ − =

×

≤

=

∫



   



   

Next, we show the case of p= ∞.

( )

( ) (

)

( )

(

( )

)

( ) (

)

( )

(

( )

)

( ) ( ) (

)

( )

2 1

1 1

2 ₁

1 1

3 ₁ 2

3

2 2 1 1 1

1 2 2 1 1 1

1

1 3 3 1 1 2 2 1

1

1 2 3 3

sup sup sup , ,

sup sup sup sup , ,

sup sup

sup s

s _n _n

s

s s s

x x x

s s s

x x _x _a _L

s s s

x x _x _a _L x

s s

x x

A w x w x w x P x x

C w x w x w x P x x

C w x w x w x w x P x x

C C w x w x

δ

∈ ∈ ∈

∈ ∈ _≤ ₋

∈ ∈ _≤ ₋ ∈

∈ ∈

=

≤

=

≤

×

  

 

  

 

  

 

( )

(

( )

)

( )

(

( )

)

( ) ( ) (

)

( )

(

( )

)

( )

(

( )

)

( )

(

( )

)

( ) ( )

( ) (

)

( )

(

( )

)

( ) ( )

( ) (

)

1 1 2 2 1 2

1 1 2 2

1 2 2

1 1 2 2 1

1 1

1 2 1 1 2 2

1 1 1

1

1 1 2 2 1

1 , 1, ,

sup , ,

sup sup sup

, ,

sup , , .

n n n n

s s

n n n n n n

i i

i n n

s

x a L x a L

s

x a L x a L x a L

s s s

x a L i s

w x w x P x x

C C C w x w x

w x P x x

C w x w x w x P x x

δ δ

δ δ δ

δ

≤ − ≤ −

≤ − ≤ − ≤ −

≤ − =

≤

× ×

=



 

Similarly, using Proposition 3.8 (3.15), we easily have (3.13). # Lemma 3.9. Let 1≤ ≤ ∞p .

(1) We assume that

(

2

)

, 1, 2, ,

i

w∈ C + i=  s satisfies (3.6). Then there

exists a constant C>0 such that when

( )

s

p

(10)

[ ]

( )

, 1, 2, , , p s

p s

j n

L j

L Wv h

C Wh j s

T

≤ =

 



and

[ ]

( )

1 1 4

( )

( ) p

( )

s.

p s

s

s i

n _L i i n L

Wv f ≤C

∏

₌T a Wf _



(2) We assume that

(

3

)

₀

(

_{3 2 , 1, 2,}

)

_,

i

w∈_λ C + < <λ i=  s. Let

( )

s p s

T Wh∈L  , then

[ ]

( )

p

( )

s , 1, 2, , .

p s

j j

n

L L

Wv h ≤C T Wh j=  s

 

Proof. (1) From Theorem 3.4 (1), for j=1

[ ]

( )

( ) ( )

( )

1 1

1 1 2

1

ˆ ,1

1 4 1

1

. p s p s

p

p s

n X n

L L

L L Wv h

Wv h

C Wh T

T

≤ −

≤

= ≤ _

 _



Inductively,

[ ]

( )

[ ]

(

( )

)

(

) ( )

( )

1

1 1

1 ,

ˆ ,

1 4 .

p s p j

j p s j

j

p s

j j

n n j

n

j j

L L

L

n j X

L j

L

Wv v h

Wv h

T T

Wv h

C C Wh

T

−

≤ − − + ≤

−

=

≤ ≤



 

 

For the second formula, using Theorem 3.7 and the above inequality, we have

[ ]

( )

(

) (

)

[ ]

( )

(

)

( )

2

1

1 1 4

1 1

,1

1 4

1 ,

1, 2, , .

p s

i p

i n p s j

j

p s

j n

L

j j

i n

i

j i s

i n k

i k j j

L x a i j

L

j i

i n

i L

Wv h

w v h

C T a w

T

C T a Wh

j s

− + ≤

=

= = +

 _≤  _{≤ ≤}

 

 

=

≤

=

∏



 



Similarly we have the following: (2) From Theorem 3.4 (2) for j=1,

[ ]

( )

1 1 4

,1 1 ,

1, 2, , .

p s p s

p s

n n _L _L

L

Wv h Wv h C WT h

j s

= ≤

=

 





(11)

[ ]

( )

[ ]

(

( )

)

( )

1 1

, ,

.

p s

p s p s

p s

j j j

n n n j n j

L

L L

j L

Wv h Wv v h C WT v h

C WT h

− −

= ≤

≤



 



#

Proof of (3.7) in Theorem 3.3. By Proposition 3.5 (2) and Proposition 3.6, we get

( )

(

( )

)

( )

(

)

( )

1 1 1 1

1

1 4 1 4

ˆ ˆ ˆ ˆ

1 ,1 , ; 1 1 1 ,1 , 1 1; 1 ,

p

n

n n p p

X X X X

L

a

w x f v f CE f T w C f T w c

n

ω  

− ≤ ≤ _ _

 



where the constant C1 and c1 is independent of Xˆ1. Similarly, for j=1, 2,,s,

( )

(

( )

)

( )

( ) 1 4

ˆ_j , ˆ_j , ˆ_j, ; .

p

j n

j X n j X j p j X j j j

L

a

w x f v f C f T w c

n

ω  

− ≤ _ _

 

 (3.16)

Using [ ]s

(

[ ]1

( )

)

(

[ ]1

( )

[ ]2

( )

)

(

[ ]s1

( )

[ ]s

( )

)

n n n n n n

f −v = f −v f + v f −v f + + v − f −v f ,

we get from Lemma 3.9 (2) and (3.2),

[ ]

( )

(

)

( )

[ ]

( )

(

)

( )

(

[ ]

( )

[ ]

( )

)

( )

(

)

( )

[ ]

(

( )

)

( )

(

)

( )

(

)

(

( )

)

( )

1

1 1

2

1

,1 ,

2

1 1 4

,1 1 ,

2 1 1 4 ˆ

1 1 ,1 1 1 1

1 4 2

, ;

p s

p s p s

p s _p _s

p s

s n

L

s

j j

n n n

L _j L

s j

n n n j

L _j L

s

j

n _L k k n j

L j

n

p X

L s

j j k k

j

W f v f

W f v f W v f v f

C W f v f Wv f v f

C W f v f W T f v f

a C W f T w c

n

C W T

ω

−

=

−

=

− = =

= =

−

≤ − + −

 

≤ _ − + − _

 

 

≤  − + − 

 

 

 

≤ _ _

 

+

∑

∏

∑



 

 _



(

)

( )

1

1 1 4

ˆ ,

1 j, ;

p s

j

j _n

p j X j j j L a f T w c

n

ω

−

−  

 

 

∏



by (3.16)

(

)

( )

1

1 _{1 4} _{1 4}

ˆ , 1 1

, ; ,

j

p s j

j s

j _n

j j k k p j X j j j

j

L a

C W T f T w c

n

ω

−

− = =

 

≤ _ _

 

∑

∏



where 1 1 4

1 1

j k

k T

−

= =

∏

for j=1. Hence we obtain (3.7).

Proof of (3.10) in Theorem 3.3. By Proposition 3.5 (1) and Proposition 3.6, we get

( )

(

( )

)

( )

(

)

( )

1 1

1

ˆ ˆ

1 ,1

ˆ ˆ

, 1 1 ,1 1 1

1 4 1

; , ; ,

p

n

X X _n

n p X p X

L

w x f v f _a

CE f w C f w c

n

T x ω

− _ _

≤ ≤ _ _

 



(12)

( )

(

( )

)

( )

ˆ , ˆ

ˆ ,

1 4 , ; .

j j

j

p

j

j X n j X

n j p j X j j j

L w x f v f

a

C f w c

n

T x

ω

− _ _

≤ _ _

 



(3.17)

Using [ ]s

(

[ ]1

( )

)

(

[ ]1

( )

[ ]2

( )

)

(

[ ]s1

( )

[ ]s

( )

)

n n n n n n

f −v = f −v f + v f −v f + + v − f −v f ,

we get from Lemma 3.9 (1) and (3.2),

[ ]

( )

(

)

( )

[ ]

( )

(

)

( )

[ ]

( )

[ ]

( )

(

)

( )

(

)

( )

[ ]

(

{

( )

}

)

( )

(

)

( )

(

)

( )

(

)

1 ( )

1 1

1

2

1 1 4

, ,1

1 4 1

2 1

, ,1

1 4 1 4

2 1

1 ˆ

1 2 ,1 , 1; 1

p s

p s p s

p s _p _s

p s p s

p

s n s

L

j j

s

n n n

j j

L L

j s

n n j j

n

j j

L _L

s

n j n

j j

L L

n

i p X

i s

L W f v f

T

W f v f W v f v f

T T

Wv f v f T

W f v f C

T T

W f v f W f v f

C

T T

a

C w f w c

n

ω

−

=

−

− =

=

≤ ≤

−

− −

≤ +

 

− −

 

≤ _ + _

 

 

 

− −

 

≤ _ + _

 

 

 

≤ _ _

 

∑

∏



 

 _

 



( )

(

)

( )

1 1

1

ˆ , 1 ,

2

, ; s

j

p s

j

j s

n j i s i j i p j X j j j

L a

C w f w c

n

ω

−

≤ ≤ ≠ =

 

+ _ _

 

∑

∏



by (3.17)

(

)

( )

1

ˆ , 1 ,

1

, ; .

j

p s j

j s

L a

C w f w c

n

ω

−

≤ ≤ ≠ =

 

≤ _ _

 

∑

∏



Hence we obtain (3.10).

To prove (3.9) and (3.11) we need a lemma:

Lemma 3.10. Let

0

< ≤

h

1

,

σ

( )

t ≥1 and x ≤

σ

( )

2t . We have

( )

wf

(

x± Φh t

( )

x

)

_Lp_{( )}_ ~

( )( )

wf x _Lp_{( )}_ . (3.18)

Proof. We may show

( )

( )( )

_{( )}

1 ~ p .

p

L L

x

wf x h wf x

t

σ

 

 ± − 

 

  _  (3.19)

Let x>0. If we put

( )

2

1 : , , 2 ,

2

u v

a a

x

x h y t t

t u v

σ

(13)

we will see

1 d 3

.

2 d 2

y x

≤ ≤ _(3.21)

Then we conclude (3.19). Now, from (3.20)

( ) ( )

d

1 .

d 2

y h

x σ t σ t x

=

− 

Since a2u u=2t, we see

2

,

v u u

a a a

v = u > u

that is, we have

( )

2t av au

( )

t a2u.

σ

= < <

σ

=

Then, using ([2], Lemmas 3.6, 3.7), we see

( ) ( )

( )

2 2

2

2 1 1 2

1 1 2

2

1 4

2 2 2

1 1 1

4 4 4 2

u

u u

u

a

h t

u a a

t t x t t t

T a

a C C

C C C u

u a u u

δ

σ σ σ σ σ

−

≤ ≤

−

− −

≤ ≤ = ≤

for some 0< ≤δ 1 and

u

large enough. We have (3.21). So we conclude

(3.19). #

Proof of (3.9). We will estimate

( ) ( )

(

( )

)

( )

,

1

1 1 4

ˆ 0

2

1

d .

t j j p

j _p _s

j

p p

t j

j j j h x X

L x t

L

W T T x w x f x h

t σ

−

Φ

≤

 

 _∆ 

 

 



∫





To do so we may estimate

( ) ( )

(

_{( )}

( )

)

( )

,

1 1 4

ˆ

1 ₀

2

1

: d .

t j j p

j

p p

t

j j h x X

L x t

I T x w x f x h

t Φ ≤σ

 

 

=_ ∆ _

 



∫



For

ε

>0 we take K>0 large enough, and then by *

( )

s

p

T w

f ∈L  we can

select a continuous function fK such that

( ) ( )

(

( )

(

) ( )

)

( )

( ) ( )

(

( )

( ) ( )

)

( )

,

1 1 4

ˆ

1 ₀

2 1 1 4

ˆ 0

2

1

d

1

d

.

t j j p

j

t j j p

j

p p

t

j j h x K X

L x t p p

t

j j h x K X

L x t

I T x w x f f x h

t

T x w x f x h

t

A B

σ

σ Φ

≤

Φ

≤

 

 

≤ ∆ − 

 

 

 

 

+_ ∆ _

 

 

= +

∫

(14)

( ) ( )(

) ( )

( )

1 1 4

ˆ

0 ₂

1 0

1

d

1

d .

p j

j

p p

t

j j K X

L x t p

t

A C T x w x f f x h

t

C h C

t

σ

ε ε

≤

 

 

≤ _ − _

 

 

 

≤ _ _ ≤

 

∫

If

σ

j

( )

2t >K, then by Lemma 3.10 we see

( ) ( )(

) ( )

( )

( ) ( )

( )

1 1 4

ˆ 0

1 1 4

ˆ

0 ₂

1 2

1

d

1

d

.

p j

j

p p

t

j j K X

L x K p p

t

j j X

L K x t

A C T x w x f f x h

t

T x w x f x h

t

C C C

σ

ε ε ε

≤

< ≤

 

≤ _ − _

 

 

 

+  

 

 

≤ + ≤

∫

When we take t>0 small enough, we see

( ) ( )

(

( )

( ) ( )

)

( )

,

1 1 4

ˆ 0

1 0

1

d

1

d ,

t j j p

p p

t

j j h x K X

L x K p

t

B T x w x f x h

t

h C

t ε ε

Φ

≤

 

 

=_ ∆ _

 

 

 

≤_ _ ≤

 

∫

because of the continuity of fK. Therefore we have I1<ε. Consequently, we

have

( ) ( )

(

( )

)

( )

,

1 1

1

1 1 4

ˆ 0

2

1

d

.

t j j p

j

p s j

p p

t j

j j j h x X

L x t

L j

j L

W T T x w x f x h

t

C W T C

σ

ε ε

−

Φ

≤

−

 

 _∆ 

 

 

 

≤ ≤

∫



(3.22)

Finally, we see

( )

(

( )

)

( ) ( )

(

_{( ) ( )}

)

( )

( ) ( )

(

)

( )

(

)

( )

1

1 1 4

ˆ

constant _\ _{4 ,} ₄

1 1 4

ˆ

\ 4 , 4

1 4

inf

4 .

j p

j _j _j _p _s

j

p j

j j p s

j

p s j

j

j _c X j j j

L t t

L

j

j X j j

L t t

L

j j L

W T f x c T x w x

C W T f x T x w x

Cw t Wf

σ σ

σ

−

− 

 

−

− 

 

−

≤



 

Here, if we set 4t=a uu , then we see

( )

(

)

( )

1 4 1

4 exp ~ exp e

4 2

u

j j j u

u u

w t Q a

T a

δ

σ ₌ ₋  _₋ __≤ −

  _ _

  _ _

for some 0< <

δ

1, that is,

( )

(

)

1 4

4 e .

4

u u

j j

a

w t C C Ct

u

δ

σ

_≤ − _≤ ₌

(15)

( )

(

( )

)

( ) ( )

(

_{( ) ( )}

)

( )

1

1 1 4

ˆ

constantinf _\ _{4 ,} ₄

.

j p

j _j _j

p s

j

j _c X j j j

L t t

L

W T f x c T x w x

Ct

σ σ −

−

− 

 

−

≤



 (3.23)

For given

ε

>0 if we take K>0 large enough and then t>0 small

enough, then by (3.22) and (3.23) we have

(

)

( )

1

1 1 4

ˆ

, _j, , .

p s

j

j p j X j j

L

W T ω f T w t ε

−

− _<



Consequently, we have (3.9).

Proof of (3.11). If in the proof of (3.9) we set as T=1 (constant), then we

obtain (3.11). #

Corollary 3.11. We suppose that _exp

( )

(

2

)

_{, 1, 2,} _,

j j

w = −Q ∈ C + j=  s are

the Freud-type weights. Let 1≤ ≤ ∞p . Then we have

[ ]

( )

(

)

( )

(

)

( )

1

ˆ , 1 ,

1

, ; .

p s

j

p s

j

s n

L

j s

n

j i s i j i p j X j j

j

L

W f v f

a

C w f w c

n

ω

−

≤ ≤ ≠ =

−

 

≤ _ _

 

∑

∏



Especially p*

( )

s

W

f ∈L  , then we have

(

)

( )

1

ˆ , 1 ,

1

, ; 0 as .

j

p s j

j s

L a

C w f w c n

n

ω

−

≤ ≤ ≠ =

 

→ → ∞

 

 

∑

∏



Corollary 3.12. We suppose

( )

(

3

)

(

)

exp 0 3 2 , 1, 2, ,

j j

w = −Q ∈_λ C + < <λ j=  s, and let (3.6) hold. Let

1≤ ≤ ∞p . If s ₀

( )

s p

( )

s

T Wf∈C  L  ), then we have

[ ]

( )

(

)

( )

0 as .

p s

s n

L

W f −v f → n→ ∞



Moreover, we suppose

( )

(

2

)

exp , 1, 2, ,

j j

w = −Q ∈ C + j=  s, and let (3.6)

hold. If 0

( )

s p s

Wf∈C  L  , then we have

[ ]

( )

(

)

( )

0 as .

p s

s n s

L W f v f

n T

−

→ → ∞



4. A Property of Higher Order Derivatives

In this section we show an important theorem which is useful in approximation theory. We use the following notations for

( )

(

( )

)

(

2

)

exp , 1, 2, ,

i i i i

w x = −Q x ∈ C + i=  s. Let r be a positive integer, and

0

i

x ≥ >

γ

.

( ) ( )

( )

(

( )

)

(

2

)

0: 1 ,0; ~ ,0 exp ,0 ,

r s

i i i i i i i i i

i

(16)

( )

(

( )

)

(

2

)

, , ,

1

: ; ~ exp ,

1, 2, , .

r s

i i i i i i i i i

i

W w Q x w x w x Q x C

r

ν

ν ν ν ν

ν

−

= ′

= = − ∈ +

=

∏





Then we see

( )

1

( )

, ,

~ .

r

i i i