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

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

ISSN Print: 2152-7385

DOI: 10.4236/am.2017.89093 Sep. 6, 2017 1239 Applied Mathematics

A Study of Weighted Polynomial

Approximations with Several Variables (II)

Ryozi Sakai

Department of Mathematics, Meijo University, Tenpaku-ku, Nagoya, Japan

Abstract

In this paper we investigate weighted polynomial approximations with several variables. Our study relates to the approximation for p

( )

s

Wf ∈L  by weighted

polynomial. Then we will give some results relating to the Lagrange interpola-tion, the best approximainterpola-tion, the Markov-Bernstein inequality and the Ni-kolskii-type inequality.

Keywords

Weighted Polynomial Approximations, the Lagrange Interpolation, the Best of Approximation, Inequalities

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 be 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, n n_{, , ,}n

( )

s

(

=:n s_;

( )

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 (II). Applied Mathematics, 8, 1239-1256.

https://doi.org/10.4236/am.2017.89093

Received: August 13, 2017 Accepted: September 3, 2017 Published: September 6, 2017

http://creativecommons.org/licenses/by/4.0/

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DOI: 10.4236/am.2017.89093 1240 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_→∞W X f X =0, where X =

(

x₁,,x_s

)

=max x i_i ; =1, 2,,s. Then

we write 0

( )

s

Wf∈C  .

Our purpose in this paper is to approximate the weighted function

( )

p s

Wf ∈L  by weighted polynomials ; ;

( )

s n s

WP P∈  . In Section 2, we give

a class of the weights which are treated in this paper. In Section 3, we state our main theorems. First, we consider the Lagrange interpolation polynomials. Next, we give the necessary and sufficient conditions for the best approximation. In Sections 4 and 5, we will prove theorems.

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 1

( )

C− f x ≤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

′ ′′

≥ ∈

′ 

then we write

( )

2

exp

(3)

DOI: 10.4236/am.2017.89093 1241 Applied Mathematics

( )

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. Let

( )

(

( )

)

( )

2

exp

w x = −Q x ∈ C + , 0< <

λ

(

m+2

) (

m+1

)

and m≥1 be

an integer. Then we write

(

m 2

)

w∈_λ C + + if Q is Cm+2-class and there exist 1

C≥ and K≥1 such that for all x ≥K,

( )

Q x C Q x λ

′

≤ _(2.1)

and

( )

_{( )}

( )

_{( )}

1

~

k

Q x Q x

Q x Q x

+

′′ ′

for every k=2,,m and also

( )

_{( )}

( )

_{( )}

( )

_{( )}

( )

_{( )}

2 1

1 .

m m

Q x Q x

+ +

+ ≤

Specific examples are shown in the following:

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, 1l≥ we define

( )

l;

( )

expl

( )

exp 0 ,l

( )

Q x =Qα x = xα −

where expl

( )

x =exp exp exp

(

expx

)



)

times

(

l

)

. Moreover, we define

( )

{

( )

*

( )

}

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

m

l m l l

Qα x = x xα −α α+ >m m≥ α≥

where *

0

α = _if α =0, and otherwise α =* 1. We note that Ql;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}

(1), (2) and (3) are regular.

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

(

m 2

)

(4)

DOI: 10.4236/am.2017.89093 1242 Applied Mathematics the weight w is regular and if 2

(

{ }

)

\ 0 m

Q∈C +  satisfies definition (2.1), then

for the regular weights we have

(

m 2

)

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

Proposition 2.3 ([3], Theorem 4.2 and (4.11)). Let m be a positive integer,

(

) (

)

0< <

λ

m+2 m+1 and let

( )

(

2

)

exp m

w= − ∈Q _λ C + + . Then for , , ,

µ ν α β∈__{, we can construct a new weight}

(

1

)

( )

2 , , ,

m

w_{µ ν α β}∈_λ C + + ⊂ C +

such that

( )

(

2

)

(

( )

)

(

( )

)

( )

, , ,

1 1 1 ~ on ,

w

T x µ +x ν +Q x α + Q x′ βw x w_{µ ν α β} x 

and for some C≥1,

(

)

( )

(

)

, , ,

( )

/ , , , , , , and ~ ,

n C n Cn w w

a wµ ν α β ≤a w ≤a wµ ν α β T _{µ ν α β} x T x =T x

where an

(

wµ ν α β, , ,

)

and a wn

( )

are MRS-numbers for the weight wµ ν α β, , , or

w, respectively, and Twµ ν α β_{, , ,} , Tw are correspond for wµ ν α β, , , 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 _Lp space on  (here for

p= ∞, if wf∈L∞

( )

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

limit 0 at ±∞). Let

( )

2

w∈ C + . 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 see easily

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

→∞ = →+ = ∞

For wf∈Lp

( ) (

 1≤ ≤ ∞p

)

the degree of weighted polynomial approximation

is defined by

(

)

(

)

_{( )}

, ; : inf p .

n

n p _L

P

E w f w f P

∈

= − _



3. Main Results

Let

( )

2

, 1, 2, , i

w∈ C + i=  s, and let W X

( )

=

∏

is=₁w xi

( )

i , where

(

1, 2, ,

)

s s

X = x x  x ∈ . Then we have the following theorem.

Theorem 3.1 ([4], Theorem 3.3). We suppose

( )

3

(

)

exp 0 3 2

j j

w = −Q ∈λ C + < <

λ

, j=1, 2,,s and let

( )

2 3

, 1, 2, , .

j

j n j

n

T a c j s

a

  ≤ _ _ =

(5)

DOI: 10.4236/am.2017.89093 1243 Applied Mathematics If

{

1 4

}

( )

0 1

s s

i i

i= T w f∈C

∏

 , then there exist ;

( )

, 1, 2, 3,

s n n s

P∈  n=  such

that we have

(

n

)

_L

_{( )}

s 0 as .

W f −P ∞_ → n→ ∞

First, we consider the Lagrange interpolation operators. We construct the orthonormal polynomials pn i, with respect to the weight wi for each

1, 2, ,

i=  s . Let xn n i, , <xn−1, ,n i<<x1, ,n i, 1, 2,i= ,s are zeros of the

orthonormal polynomial pn i, , that is, pn i,

(

xk n i_i, ,

)

=0, 1, 2,ki = ,n and put

(

)

{

1, ,1, , s, , ; 1 , 1, 2, ,

}

n k n k n s i

S = x  x ≤ ≤k n i=  s . Then for Wf∈C0

( )

 we define the Lagrange interpolation polynomial on Sn as

(

)

(

₁

)

( )

1

1 , ,1 , , 1 , ,

1 1

; , , , , ,

s i

s

n n

s

n s k n k n s i k n i i

k k

L f x x f x x ₌l x

= =

=

∑ ∑

∏

   (3.1)

where

( )

(

,

) (

)

, ,

, , , , ,

.

i

i i

n i i k n i i

i k n i n i k n i

p x

l x

x x p x

=

′

− (3.2)

In the rest of this paper, if w ii, 1, 2,= ,s are the Freud-type weights then

we suppose ( )

( )

2 3

1 i n

a =o n .

Theorem 3.2. Let

( )

2

, 1, 2, , i

w∈ C + i=  s , and let f :s → be

continuous. If

(

)

( )

{

2

}

(

)

( )

2

1 0

=1 1 , ,

s s

i i i s

i x w x f x x C

β

+ ∈

∏

  (3.3)

holds, then there exists n0>0 such that for n≥n0

(

)

(

)

( )

{

}

(

)

( )

1

, , , ,1 , , 1

1 1

2 2

1 1

, ,

1 , , ,

i s

s

n n

s

k n i k n k n s i

k k

s

i i i s

i

L

f x x

C x βw x f x x

λ

∞

= = =

=

≤ +

∑ ∑∏

∏



 



(3.4)

where for each i=1, 2,,s, xk n i_i_{, ,} 1, 2,

(

ki = ,n

)

are zeros of pn i,

( )

xi . In

particular,

(

)

( ) (

)

1 1

, , , ,1 , , 1

1 1 2

1 1

1

lim , ,

, , d d .

i s

s

n n

s

k n i k n k n s

i n

k k

s

i i s s

i

f x x

w x f x x x x

λ

= →∞ = =

∞ ∞ = −∞ −∞

=

∑ ∑∏

∏

∫

 

  

(3.5)

Theorem 3.3. Let

( )

3

(

)

0 3 2 , 1, 2, 3, ,

i

w∈_λ C + < <

λ

i=  s. Let β >1 2,

and let : s

f  → satisfy (3.3), then we have for n=1, 2,,

( )

_{( )}

{

(

)

( )

}

(

)

( )

2

1

1 s 1 1 , , _s ,

s s

i n i i i s

i _L i

L

w L f _∞ C x β w x f x x

∞

= ≤ = +

∏



 _ (3.6)

where for each i=1, 2,,s, xk n i_i, ,, 1, 2,ki= ,n are zeros of pn i,

( )

xi . In

particular, if

{

(

2

)

( 2) 1 4

( ) ( )

}

(

)

( )

1 0

1 1 , ,

s s

i i i i i s

i x T x w x f x x C

β

= + ∈

(6)

DOI: 10.4236/am.2017.89093 1244 Applied Mathematics have

( )

(

)

2

_{( )}

1

lim si i n _s 0.

n→∞ =w f L f L

− =

∏

_

For p≠2 we also obtain the similar results. We need a function as follows:

( )

(

)

2 3

( )

1

: .

1 x

Q x T x

Ψ =

+ (3.7)

Theorem 3.4. Let

( )

3

(

)

, 0 3 2 1, 2, ,

i

w∈_λ C + < <

λ

i=  s . Let 1< ≤p 2 and 1 p

β > , and let Ψ

( )

x be defined by (3.7) for each w ii, 1, 2,= ,s. If :

f → is continuous, and satisfies

(

)

( )

{

2

}

( ) ( )

2 1 2 1 4

1 1 ,

, 1, 2, , , s

i i i i i

i

s

x T x x W X f X

X i s

β ₋

= + Ψ < ∞

∈ =

∏



 (3.8)

then we have

( )

_{( )}

{

(

)

( )

}

( )

2

2 1 2 1 4

1 1 ,

1, 2, .

p s

s s

n L i i i i i i

L

WL f C x T x x Wf

n

β

∞

− =

≤ + Ψ

=

∏



 _ _(3.9)

Especially, if f satisfies

(

)

( )

{

₂ 2 _{3 4} _{1 4}

}

( ) ( )

( )

1 1 ,

s s

i i i i i o

i x T x x W X f X C

β ₋

= + Ψ ∈

∏

 (3.10)

then we have

( )

(

)

_{( )}

lim n _Lp s 0.

n→∞ W f −L f  = (3.11)

For 2< ≤ ∞p we have the following:

Theorem 3.5. Let

( )

3

(

)

, 0 3 2 1, 2, ,

i

w∈λ C + < <

λ

i=  s , and let satisfy

( )

i 1 2

i n

T a ≤Cn . Let 2< ≤ ∞p and β >1 p. Furthermore we assume

( )

(

)

(

)

4

log , 1, 2, , .

4

i

i n

i i n

a

Q_ _≥C Q a i= s

   (3.12)

If f :s→ is continuous, and satisfies

(

)

( )

{

2

}

( ) ( )

2 1 2 1

1 1 , ,

s _s

i i i i i

i x T x x W X f X X

β ₋

= + Ψ < ∞ ∈

∏

 (3.13)

then we have

( )

{

}

( )

(

)

( )

{

}

_{( )}

3 4 1

2

2 1 2 1 4

1 1 .

p s

s

i i n

i

L

s

i i i i i

i

L

x WL f

C x β T x x Wf

∞

=

− =

Ψ

≤ + Ψ

∏



(7)

( )

{

3 4

}

(

( )

)

_{( )}

1

lim 0.

p s s

i i n

i

n→∞

∏

=Ψ x W f −L f _L _ =

Remark 3.6. (1) We note that (3.13) means

(

1 4

)

*

( )

0

1 ,

s s

i

i=T W f ∈C

∏



where

{

(

2

)

/2 1 2

( )

1 4

( )

}

*

(

2

)

1 1 ~

s

i i i i i

i x T x x W W C

β ₋

= + Ψ ∈ +

∏

 (see Theorem

2.3).

(2) All examples in Example 2.2 hold (3.12).

(3) To prove Theorem 3.5 we use Proposition 4.5. Then Assumption (3.12) plays an important role.

Next, we characterize the best approximation polynomial (cf. [5]).

Theorem 3.7. Let 0< ≤ ∞p . There is a best approximation polynomial

;

f n s

P ∈ such that

(

)

(

)

_{( )}

(

)

_{( )}

;

, ; , : inf p s ; p s .

n s

p n s _L n f

P L

E W f W f P W f P

∈

= − _ = −





Theorem 3.8 (Kolmogorov-type theorem). P∈n s; is a best of approximation for a continuous function f with ₍ ₎

(

) (

)

1, , 1 1

lim , , , , 0

s s s

xx →∞W x  x f x  x = , if

and only if for each polynomial Q∈n s; ,

(1, , )

(

1

) (

(

1

)

(

1

)

(

1

)

max , , , , , , , , 0,

s s s s s

x x ∈A W x x f x x P x x Q x x

 −  ≥

 

     (3.14)

where A denotes the set (which depends on f and P ) of all points

(

x1,,xs

)

∈A for which

(

1, , s

) (

(

1, , s

)

(

1, , s

)

(

)

_L

_{( )}

s

W x  x f x  x −P x  x = W f −P ∞_ .

Theorem 3.9. Let

( )

2

1 , , 1, ,

s i i i

W =

∏

₌w w∈ C + i=  s and 1≤ < ∞p . Let K

φ _{, where}K=

(

k1,,ks

)

(

0≤kj≤n

)

(

j=1,,s

)

be a linearly independent

system satisfying

( )

s

K p

W

φ

∈L  , and we consider polynomials

( )

₍₁ _{) (}₁ ₎

( )

1

, , , , 1 1

.

s s

s

n n

k k k k

k k

Q X c φ X

= =

=

∑ ∑

_ _ _ _(3.15)

Let

( )

s p

Wf∈L  . The polynomial

( )

₍[ ] ₎

( )

1 1 1

0

, , , ,

1 1 s s

s

n n

k k k k

k k

P X c φ X

= =

=

∑ ∑

 _ _ (3.16) is a polynomial of the best approximation for f if and only if for every

polynomial (3.15) the following equality (3.17) holds.

( ) ( )

( )

(

( )

)

( )

(

) (

)

(

)

(

)

(

)

(

)

(

)

1

1 1 1

1 1 1 1

sign

: , , , , , ,

sign , , , , , , d d 0

s

p _p

p

s s s

p

s s s s

Q X f X P X f X P X W X DX

Q x x f x x P x x

f x x P x x W x x x x

−

 

− _ − _

= −

 

×_ − _ =

∫

∫ ∫

   



(8)

DOI: 10.4236/am.2017.89093 1246 Applied Mathematics holds. If p=1, we also assume that f X

( )

−P X

( )

vanish only on a set of

measure zero.

4. Proofs of Theorems 3.2, 3.3, 3.4 and 3.5

Lemma 4.1. Let xn n i, , <xn−1, ,n i<<x1, ,n i be zeros of the orthonormal

polynomial pn i, , and let

(

)

1

1 1 2 , , 1

0 1, 1,2, ,

, , , s,

s i

j j

n s j j s

j n i s

P− x x x a x x

≤ ≤ − =

=

∑

_



 

where aj1, ,js are coefficients. If for every

(

xk n1, ,1,,xk n ss, ,

)

,

(

1

)

1 , ,1, , _s, , 0, 1 , 1, , ,

n k n k n s i

P− x  x = ≤ ≤k n i=  s (4.1)

then we have Pn−1=0. Therefore, for 1

(

1, 2, ,

)

1

( )

s

n s n

P− x x  x ∈−  we have

( )

1 1 .

n n n

P− =L P− (4.2) Proof. Now we fix any

(

₂

)

1

, ,2, , _s, ,

s

k n k n s

x  x ∈− , and then we consider the

polynomial Qn−1

( )

x1 in n−1

( )

 such that

( )

2 1

1 2 1

1

1 1 , , , ,2 , , 1

0 0 1, 2,3, ,

.

s

s s

i

n

j

j j

n j j k n k n s

j j n i s

Q x a x x x

− −

= ≤ ≤ − =

 

=  

 

∑



 

Since

(

1

)

1 , ,1 0, 1 1, 2, ,

n k n

Q− x = k =  n

(see (4.1)), all coefficients of Qn−1

( )

x1 equal to zero, that is,

2

1, , 2, ,2 , , 1

0 1, 2,3, ,

0, for each 0,1, , 1.

s

s s

i

j j

j j k n k n s

j n i s

a x x j n

≤ ≤ − =

= = −

∑



   (4.3)

Next, we fix any

(

)

(

₃

)

2

1 0 1 1 and , ,3, , s, , ,

s

k n k n s

j ≤ ≤ −j n x  x ∈−

and we consider Rn−1∈n−1

( )

 such that

( )

3 2

1 3 2

1

1 2 , , , ,3 , , 2

0 0 1, 3,4, ,

.

s

s s

i

n

j j j

n j j k n k n s

j j n i s

R x a x x x

− −

= ≤ ≤ − =

 

=  

 

∑



 

Then by (4.3), we see

(

2

)

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

n k n

R− x = k =  n

Hence,

3

1, , 3,3 , 1 2

0 1, 3,4, ,

0, for each , 0,1, , 1.

s

s s

i

j j

j j k s k

j n i s

a x x j j n

≤ ≤ − =

= = −

∑



  

If we continue this method inductively, then we have

1

, , , , 1 2 1

0

0, for each , , , 0,1, , 1.

s s s s

n

j

j j k n s s

j

a x j j j n

−

− =

= = −

(9)

DOI: 10.4236/am.2017.89093 1247 Applied Mathematics We put Hn−1∈n−1

( )

 as

( )

1 ₁

1 , ,

0

,

s s s

n

j

n s j j s

j

H x a x

− −

=

∑

_

then from (4.4) we have Hn−1

(

xk n ss, ,

)

=0, 1, 2,ks = ,n. Therefore, we conclude

(

)

1, ,s 0, 0,1, , 1 1, 2, , ,

j j i

a _ = j =  n− i=  s

that is, Pn−1=0. #

In the rest of this paper, we use the following notations:

(

1

)

( ) (

1

)

1 , , , , , ,

s

i s s

i

W =

∏

₌w X = x  x X u = u  u

( )

d 1 d , s

(

( )

)

d 1 d .s

D X = x x D X u = u  u

We also use

( )

{

(

)

}

; : , , : 1, , are polynomials with degree

for each , 1, , .

s s

n s n n n n s

i

P P x x n

x i s

= = ≤

=

 



 

 

Proposition 4.2 (cf. [6], Theorem 1.2.2). Let

( )

2

, 1, 2, , i

w∈ C + i=  s, and

let n≥1 be an integer. Then for all 2 1;

( )

s n s

P∈ −  , we have

( )

(

)

(

( )

)

(

( )

)

(

)

1 1

1

2

, ,1 , , , ,1 , ,

=1 =1

, , ,

s

s s

s

n n

k n k n s k n k n s

k k

P X u W X u D X u

P x x

λ λ

=

∫

∑ ∑

   

(4.5)

where

( )

2

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

i i

k n i lk n iwi xi xi i s

λ =

_∫

_−∞∞ = _ _(4.6)

Proof. (see [6], pp.12-13). Let 1;

( )

s n s

P∈−  . From (3.1) and (4.2) we see

( )

(

)

( )

(

)

1 1

1

, ,1 1 , , , ,1 , ,

1 1

;

, , .

s s

s n

n n

k n k n s s k n k n s

k k

P X L P X

l x l x P x x

= =

=

∑ ∑

  

Hence we have

( )

(

)

(

( )

)

(

( )

)

(

)

1 1

1

2

, ,1 , , , ,1 , ,

1 1

, , ,

s

s s

s

n n

k n k n s k n k n s

k k

P X u W X u D X u

P x x

λ λ

= =

=

∫

∑ ∑

   

that is, (4.5) holds. Now, we see that (4.5) holds for any 2 1;

( )

s n s

P∈ −  . In fact, for 2 1;

( )

s n s

P∈ −  we set

(

)

(

)

( )

(

)

( )

1 1 1 , 1

1, , 1

, , , , , , ,

, .

s

s s i n i i s

s

n n

P x x Q x x p x R x x

Q R

= − −

= +

∈

∏



  



(10)

DOI: 10.4236/am.2017.89093 1248 Applied Mathematics Then

( )

(

)

(

( )

)

(

( )

)

( )

(

)

( )

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

( )

(

)

(

( )

)

(

( )

)

(

)

(

)

(

) (

)

(

)

1 1 1

1

, 1

, ,1 , , , ,1 , ,

1 1

, ,1 , , , ,1 , , 1 , , , , ,1 , ,

1 1

1

, ,

, , , ,

s

s _i _i

s

s s

s

s s i s

s

n j j i

n n

k n k n s k n k n s

k k

n n

s

k n k n s k n k n s i n i k n i k n k n s

k k

n

k

P X u W X u D X u

Q X u p u W X u D X u

R X u W X u D X u

R x x

Q x x p x R x x

λ λ λ λ = = = = = = = = + = = = + =

∫

∏

∫

∑ ∑

∏

∑

          

(

)

1 1 1

, ,1 , , , ,1 , ,

1

, , ,

s s

n

k n k n s k n k n s k

P x x

λ λ

=

∑

  

that is, (4.5) holds for any 2 1;

( )

s n s

P∈ −  . #

Lemma 4.3 ([7], Theorem 2.1). Let

( )

2

exp ,

w= − ∈Q  C + b∈. If w is a

Freud-type weight, then we assume

( )

2 3

1 n

a =o n . Then there exist constants

1, 2 0

C C > such that for every integer n≥1,

(

2

)

2

( )

(

2

)

(

2

)

1 , , , 2

1

1 d 1 1 d .

n n

n

b b b

a a

k n k n k n

a a

j

C ₋ x x λ w− x x C ₋ x x

=

+ ≤

∑

+ ≤ +

∫

Proof of Theorem 3.2.

(

)

(

)

( )

{

}

(

)

( )

(

)(

)

{

}

(

)

( )

{

}

(

)

( )

[ ] [ ] [ ] [ ]

(

)

1 1 1 1

, , , ,1 , , 1 1 1 2 2 1 1 2 2 , , , , , , 1 1 1 2 2 1 1 2 1 1 , ,

1 , ,

1

1 , ,

1 d d

i s

s

i i i

s s n n s n n n n s

k n i k n k n s

i

k k

s

i i i s

i

L

n n

s

k n i i k n i k n i i

s i

s

i i i s

i

L

a a s

i s

i

a a

f x x

x w x f x x

w x x

x w x f x x

x x x

β β β β λ λ ∞ ∞ = = = = − − = = = = − = − − ≤ + ⋅ + ≤ + ⋅ +

∑ ∑∏

∏

∑ ∑∏

∏

∫

        

by Lemma 4.3

(

)

( )

{

}

(

)

( )

2 2

1

1 1 , , _s .

s

i i i s

i

L

C x βw x f x x

∞

=

≤

∏

+ _



Therefore we have (3.4). To prove (3.5) we use (3.4) and Proposition 4.2. For

( )

;

s n s

(11)

{

}

(

)

(

) ( )

( )

{

}

(

)

(

)

(

)(

) ( )

( )

{

}

(

)

1 1 1 1 1 1

, , , ,1 , , 1

1 1

2 2

1 1 1 1

, , , ,1 , , 1

1 1

2 2

1 1 1 1

, , , ,1 1

1 1

, , d d

i s s i s s i s n n s

k n i k n k n s

i

k k

s s s s

n n

s

k n i k n k n s

i

k k

s s s s

n n

s

k n i k n

i

k k

f x x

f x x w x w x x x

f P x x

f P x x w x w x x x

f P x x

λ λ λ = = = ∞ ∞ −∞ −∞ = = = ∞ ∞ −∞ −∞ = = = − ≤ − + − ≤ −

∑ ∑ ∏

∫

∑ ∑ ∏

∫

∑ ∑ ∏

            

(



)

(

)(

) ( )

( )

(

)

( )

{

}

(

)(

)

( )

(

)

( )

{

}

(

)(

)

( )

(

)

(

)

( )

{

}

(

)(

)

( )

, , 2 2

1 1 1 1

2 2 1 1 2 2 1 1 2 1 1 2 2 1 1

, , d d

1 , ,

1 d d

1 , , .

s

s k n s

s s s s

s

i i i s

i

L

s

i i i s

i L s i s i s

i i i s

i

L

f P x x w x w x x x

C x w x f P x x

x w x f P x x

x x x

C x w x f P x x

β β β β ∞ ∞ ∞ ∞ ∞ −∞ −∞ = = − ∞ ∞ = −∞ −∞ = + − ≤ + − + + − × + ≤ + −

∫

∏

∫

∏

           

Now, we can take P=Pn as

(

)

( )

{

}

(

)(

)

( )

2 2 1 1

lim 1 , , 0.

s s

i i i n s

i n

L

x βw x f P x x

∞

=

→∞

∏

+ −  _ = (4.7)

In fact, when we put

(

2

)

2

( )

(

2

)

11 ~

s

i i i

i x w x W C

β

= + ∈ +

∏

 (see Proposition

2.3), from (3.3) we see

( )

(

)

{

1 4 2 2

}

{

(

2

)

( )

}

( )

0

1 1 1 1 .

s s s

i i i i i i i

i T x x w f C i x w x f C

β β

= + ≤ = + ∈

∏



Hence from Theorem 3.1 with W_β we have (4.7). Therefore we conclude

(3.5). #

Proof of Theorem 3.3. By Proposition 4.2 with 2

( )

n

P=L f and Theorem 3.2

with (3.4),

( )

_{( )}

{

}

{

(

)

}

{

}

{

(

)

}

(

)

( )

{

}

{

(

)

}

( )

(

)

( )

{

}

(

)

( )

2 1 1 1 1 1 2

, , , ,1 , ,

1

1 1

2

, , , ,1 , ,

1 1 1 2 2 2 1 1 2 2 2 1 1 ;

1 , ,

1 , , ,

s i s s i s s s s s i n i _L n n s

k n i n k n k n s i

k k

n n

s

k n i k n k n s i

k k

s

i i i s

i

L

s

i i i s

i

L

w L f

L f x x

f x x

C x w x f x x

C x w x f x x

(12)

DOI: 10.4236/am.2017.89093 1250 Applied Mathematics that is, we have (3.6). From Theorem 4.1 with

(

)

2

( )

(

)

2 2 2

2

11 ~

s

i i i

i x w x W C

β

= + ∈ +

∏

 (note Proposition 2.3) and our

assumption, there exists Pn−1∈n−1 such that

(

)

( )

{

}

(

)(

)

( )

2 2

2

1 1

1

lim 1 , , 0.

s s

i i i n s

i n

L

x β w x f P x x

∞

− =

→∞

∏

+ −  _ =

Then

( )

(

)

_{( )}

(

)

_{( )}

(

)

_{( )}

(

)

( )

{

}

(

)(

)

( )

(

)

( )

(

)

( )

{

}

(

)(

)

( )

(

)

( )

{

}

(

)(

)

( )

2 2

2

1

2

2 2

1 1

2 2

2

1 1 1

2 2

2

1 1 1

1 , , 1

1 , ,

1 , , 0 as .

s s

s

s s

s

n _L n _L n _L

s s

i i i n s i

i i

L L

s

i i i n s

i

L

s

i i i n s

i

L

W f L f W f P WL f P

x w x f P x x x

x w x f P x x

C x w x f P x x n

β β

∞

−

− −

= =

− =

− ≤ − + −

≤ + − +

+ + −

≤ + − → → ∞

∏



 



 



#

We know the following propositions with respect to one variable. Proposition 4.4 ([8], Theorem 2.7). Let

( )

3

, 0 3 2

w∈_λ C + < <

λ

. Let 1< ≤p 2 and β >1 p. If : s

f  → is continuous, and satisfies

(

)

2

( )

( ) ( ) ( )

2 1 2 1 4

1+x β T x Ψ− x w x f x < ∞, x∈,

then we have

( )

_{( )}

(

)

( )

2

2 1 2 1 4

1 , 1, 2, .

p

n _L

L

wL f C x β T x x wf n

∞

−

≤ + Ψ = _

 _ (4.8)

Especially, if

(

)

2

( )

( ) ( ) ( )

( )

2 1 2 1 4

0

1+x β T x Ψ− x w x f x ∈C  ,

then we have

( )

(

)

_{( )}

lim n _Lp 0.

n→∞ w f −L f  =

Proposition 4.5 ([8], Theorem 2.8). Let

( )

3

, 0 3 2

w∈λ C + < <

λ

, and let

satisfy

( )

( )i 1 2

i n

T a ≤Cn . Let 2< ≤ ∞p and β >1 p. Furthermore we assume

( )

(

)

(

)

4

log .

4 n

n a

Q_  ≥_ C Q a

 

If f :→ satisfies

(

₂

)

2 _{1 2}

( )

_{1 4}

( ) ( ) ( )

1+x β T x Ψ− x w x f x ≤C x, ∈,

then we have

( )

_{( )}

(

)

( )

2

3 4 2 1 2 1 4

1 ,

1, 2, .

p

n _L

L

x wL f C x T x x wf

n

β

∞

−

Ψ ≤ + Ψ

= 

(13)

DOI: 10.4236/am.2017.89093 1251 Applied Mathematics Especially, if

(

)

2

( )

( ) ( ) ( )

( )

2 1 2 1 4

0

1+x β T x Ψ− x w x f x ∈C  ,

we have

( )

(

)

_{( )}

lim n p 0.

L

n→∞ w f −L f  =

Proof of Theorem 3.4. We use Proposition 4.4 (4.8).

( )

_{( )}

( )

{

}

_{( )}

( )

(

)

( )

{

}

(

)

( )

( ) ( ) (

)

( )

1 ₁ ₁

2 1 2 1 1 2 2 1

1 1 , ,1 , , , ,1 1 1

2

1 1 1

, , 2

2

1 2

1 1

2

2 1 2 1 4

1 1 1 1 1 1 1 1 2

, ,

, , d

d d

1 , , ,

p s s _s s i s s i p n _L p

n n _s p n

i i k n k n s k n

i

k k k

p s

k n i i s

i

n n p

s i i i k k p s L

k n i i

WL f

w x w x f x x l x x

l x x x

C w x

x T x x w x f x x x

l x β − ≤ − ≤ ∞ = = = = = = = = − = × ≤

× + Ψ

×

∑ ∑

∏

∑

∫

∏

∑ ∑ ∏

∫

         

{

}

(

)

( )

( ) ( )

{

}

( )

{

}

(

)

( )

{

}

( )

(

)

( )

( ) ( )

(

)

( )

( ) ( )

{

}

( )

{

}

1 2 2 1 1 3 3 2 2 2

2 1 2 1 4

1 1 1 1 1 1 1 1

1 2 2

1 1

, , 2

2

2 2

2 1 2 1 4 2 1 2 1 4

2 1 1 1 1 1 1 1 2 2 1 2 2 2 2

3 1 1 d d 1 , , , d d 1 1 s s i s s s s i p

n n _s p _p

i i s

i

k k

s

k n i i s

i L p n n s i i i k k x x

C x T x x w x

w x f x x x

l x x x

C x T x x w x x T x x w x

w x β β β − ≤ ∞ − ≤ = − = = = = − − = = =

= + Ψ

×

≤ + Ψ + Ψ

×

∏

∑ ∑ ∏

∫

∏

∑ ∑ ∏

∫

       

(

)

( )

{

}

₍ ₎

(

)

( )

( ) ( )

{

}

(

)

( )

2 1 2

, , 3

3

2

2 1 2 1 4

1 1 2

1

, , ,

d d

1 , , , .

i s p _p s p s

k n i i s

i

L

p s

s i i i i i i i s

L

f x x x

l x x x

C x β T x x w x f x x x

∞ ≤ ∞ = − = × ≤

≤ + Ψ

∏

     

Hence we have (3.9).

Next we show (3.10). There exists ;

( )

s n n s

P ∈  such that

(

)

( )

{

}

(

)

( )

(

)

( )

{

}

2

2 1 2 1 4

1

2

2 1 2 1 4

, ; 1

1

1 ,

s s

i i i i i n

i

L

s

n s i i i i i i

x T x x W f P

CE x T x x W f

β β ∞ − = − ∞ =

+ Ψ −

 

≤ _ + Ψ _

 

∏



(

)

( )

{

}

(

)

( )

(

)

( )

{

}

2

2 1 2 1 4

1

2

2 1 2 1 4

, ; 1

1

1 , ,

s s

i i i i i n

i

L

s

n s i i i i i i

x T x x W f P

CE x T x x W f

β β ∞ − = − ∞ =

+ Ψ −

 

≤ _ + Ψ _

 

∏

(14)

DOI: 10.4236/am.2017.89093 1252 Applied Mathematics where Pn∈n s; . Then

( )

(

)

_{( )}

(

)

_{( )}

(

)

_{( )}

(

)

_{( )}

{

(

)

( )

}

(

)

( )

(

)

( )

{

}

(

)

( )

(

)

( )

{

}

_{( )}

(

)

( )

{

}

(

)

( )

(

)

2

2 1 2 1 4

1

2

2 1 2 1 4

1

2

2 1 2 1 4

1

2

2 1 2 1 4

1

2 , ;

1

p s p s

p s

s

p s

s

n _L n _L n n _L

s

n L i i i i i i n

L

s

i i i i i n

i

L

s

i i i i i

i

L

s

i i i i i n

i

L

n s i

W f L f W f P WL f P

W f P C x T x x W f P

x T x x W f P

x T x x

C x T x x W f P

CE x

β β

β

∞

− =

− ₋

=

− =

∞

− ≤ − + −

≤ − + + Ψ −

≤ + Ψ −

× + Ψ

+ + Ψ −

≤ +

∏

 



 _



( )

{

2 _{1 2} _{1 4}

}

1 , 0 as 0.

s

i i i i

i T x x W f n

− =

 _Ψ _→ _→

 



∏



The last convergence follows from (3.9) and Theorem 3.1 with

(

)

( )

{

2

}

(

)

2 1 2 1 4 * 2

1 1 ~

s

i i i i i

i x T x x W C

β ₋

= + Ψ ∈ +

∏

 (note Proposition 2.3).

Consequently, we have (3.11). #

Proof of Theorem 3.5. As the proof of Theorem 3.4 we can show Theorem 3.5 by Proposition 4.5 (4.9). Then we also note Remark 3.6 (1) and (3). #

5. Proofs of Theorems 3.7, 3.8 and 3.9

In this section, we characterize the best approximation polynomial (cf. [5]). Proof of Theorem 3.7. We consider the polynomial class

(

)

1

(

)

_{( )}

1

1 , , 1

0,1

, , s; .

p s p s

s i

n

k k

s k k s L L

k i s

P x x a x x f P W fW

= ≤ ≤

 

 

=_ = − ≤ _

 

 

∑

    



Since

(

f −0

)

W _Lp

_{( )}

_s = fW _Lp

_{( )}

_s ,

the set  is not empty. Now we select the sequence

(

)

{

1

}

1

, 1, , 0,1 , , ; 1 ₀

s s

i

n k k

m n s k i s k k m s _m

P x x a x x

∞

= ≤ ≤ ₌

=

∑

_

  such that

(

)

_{( )}

(

)

, , ; 1

, , ;

inf p s ; .

k k ms

m n _L p n s

a _ f −P W  =E W f

Here we see that am : max= kj=0,1, , ;1n ≤ ≤j s ak1, ,k ms; is bounded. In fact, if it is

unbounded, then for

(

)

(

)

1

, 1 , 1 , , ; 1

0,1

, , , , s,

s i

n

k k

m n s m n s m k k m s

k i s

Q x x P x x a b x x

= ≤ ≤

= =

∑

_

  

we see bk1, ,k ms; ≤1. Then we can take a subsequence

{ }

ml l1

∞

= and a fixed term

1;0 ;0

1

s

k k

s

x x such that

(

)

1,0 ,0 1

1 ,0

, 1 1 , , ; 1

0, ,1

, , s s.

l s l

i i i

n

k k k k

m n s s k k m s

k k k i s

Q x x x x b x x

= ≠ ≤ ≤

= +

∑

_