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

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DOI: 10.4236/am.2017.89095    388 Downloads   655 Views   Citations
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ABSTRACT

In this paper, we investigate the weighted polynomial approximations with several variables. Our study relates to the approximation for by weighted polynomials. Then we will estimate the degree of approximation.

1. Introduction

Let s : = × × × ( s times, s 1 integer) be the direct product space, and let W ( x 1 , x 2 , , x s ) : = w 1 ( x 1 ) w 2 ( x 2 ) w s ( x s ) , where w i ( x i ) 0 are even weight functions. We suppose that for every nonnegative integer n,

0 x n w i ( x ) d x < , n = 0 , 1 , 2 , , i = 1 , 2 , , s .

In this paper, we will study to approximate the real-valued weighted function ( W f ) ( x 1 , x 2 , , x s ) by weighted polynomials ( W P ) ( x 1 , x 2 , , x s ) , where

P ( x 1 , x 2 , , x s ) P n , n , , n ( s ) . Here, P n , n , , n ( s ) ( = : P n ; s ( s ) ) means a class of

all polynomials with at most n-degree for each variable x i , i = 1 , 2 , , s . We need to define the norms. Let 0 < p , and let f : s be measurable. Then we define

W f L p ( s ) : = { [ | ( W f ) ( x 1 , , x s ) | p d x 1 d x s ] 1 / p , if 0 < p < ; sup ( x 1 , , x s ) s | ( W f ) ( x 1 , , x s ) | , if p = .

We assume that for 0 < p the integral is independent of the order of integration with respect to each x i , i = 1 , 2 , , s . When W f L p ( s ) < , we write W f L p ( s ) . If p = , we require that f is continuous and

l i m | X | ( W f ) ( X ) = 0 , where | X | = | ( x 1 , , x s ) | = max | x i | ; i = 1 , 2 , , s .

Our purpose in this paper is to approximate the weighted function W f L p ( s ) by weighted polynomials W P ; P 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 L p ( 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 1 , C 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 1 f ( x ) g ( x ) C f ( 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 ) C f ( 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) lim x Q ( x ) = .

(d) The function

T ( x ) : = x Q ( x ) Q ( x ) , x 0

is quasi-increasing in ( 0, ) , with

T ( x ) Λ > 1 , x \ { 0 } .

(e) There exists C 1 > 0 such that

Q ( x ) | Q ( x ) | C 1 | Q ( x ) | Q ( x ) , a . e . x .

Then we write w = e x p ( Q ) F ( C 2 ) .

Moreover, if there also exists a compact subinterval J ( 0 ) of , and C 2 > 0 such that

Q ( x ) | Q ( x ) | C 2 | Q ( x ) | Q ( x ) , a . e . x \ J ,

then we write w = exp ( Q ) F ( C 2 + ) . If T ( x ) is bounded, then the weight w = e x p ( Q ) F ( C 2 + ) is called a Freud-type weight, and if T ( x ) is unbounded, then w is called an Erdös-type weight.

For w ( x ) = e x p ( Q ( x ) ) F ( C 2 + ) , Q C 3 ( \ { 0 } ) , 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 , C > 0 such that for 0 < λ < 3 2 ,

| Q ( x ) | Q ( x ) λ C , (2.2)

then we write w F λ ( C 3 + ) . Furthermore, if

| Q ( 4 ) ( x ) Q ( 3 ) ( x ) | C | Q ( x ) Q ( x ) | ~ | Q ( x ) Q ( x ) | (2.3)

and the inequality (2.2) with 0 < λ < 4 3 hold, then we write w F λ ( C 4 + ) .

We have some examples satisfying Definition 2.1.

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

1 < Λ 1 ( x Q ( x ) ) Q ( x ) Λ 2 ,

where Λ i , i = 1 , 2 are constants, then we call w = exp ( Q ( x ) ) the Freud weight. The class F ( C 2 + ) contains the Freud weights.

(2) For α > 1 , l 1 we define

Q ( x ) = Q l ; α ( x ) = exp l ( | x | α ) exp l ( 0 ) ,

where exp l ( x ) = exp ( exp ( exp exp x ) ) ( l times ) . Moreover, we define

Q l ; α , m ( x ) = | x | m { exp l ( | x | α ) α * exp l ( 0 ) } , α + m > 1 , m 0 , α 0 ,

where α * = 0 if α = 0 , and otherwise α * = 1 . We note that Q l ; 0, m gives a Freud-type weight, that is, T ( x ) is bounded..

(3) We define

Q α ( x ) = ( 1 + | x | ) | x | α 1 , α > 1.

(4) Let w = e x p ( Q ) F ( C 2 + ) , and let us define

μ + : = lim sup x Q ( x ) Q ( x ) / Q ( x ) Q ( x ) , μ : = lim inf x Q ( x ) Q ( x ) / Q ( x ) Q ( x ) .

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 w F λ ( C 3 + ) . If the weight w is regular and if Q C 3 ( \ { 0 } ) satisfies (2.1), then for the regular weights we have w F λ ( C 3 + ) (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 = e x p ( Q ) F λ ( C 3 + ) , we can construct a new weight

w α F ( C 2 + ) such that

T w ( x ) α w ( x ) ~ w α ( x ) on ,

and for some C 1 ,

a n / C ( w α ) a n ( w ) a C n ( w α ) and T w α ( x ) ~ T w ( x ) = T ( x ) ,

where a n ( w α ) and a n ( w ) are MRS-numbers for the weight w α and w , respectively, and T w α and T w are correspond for w α or w , respectively.

Let { p n } be orthonormal polynomials with respect to a weight w, that is, p n is the polynomial of degree n such that

p n ( x ) p m ( x ) w 2 ( x ) d x = δ m n ( theKroneckerdelta ) .

For 1 p , we denote by L p ( ) the usual L p space on (here for p = , if w f L ( ) , then we require f to be continuous, and w f to have limit 0 at ± ). For w f L p ( ) , we set

s n ( f , x ) : = k = 0 n 1 b k ( f ) p k ( x ) , where b k ( f ) = f ( t ) p k ( t ) w 2 ( t ) d t (2.4)

for n (the partial sum of Fourier-type series). The de la Vallée Poussin mean of order n is defined by

v n ( f , x ) : = j = n + 1 2 n s j ( f , x ) . (2.5)

Let w F ( C 2 + ) . We need the Mhaskar-Rakhmanov-Saff numbers (MRS-numbers) a x ;

x = 2 π 0 1 a x u Q ( a x u ) ( 1 u 2 ) 1 / 2 d u , x > 0.

We easily see

lim x a x = and lim x + 0 a x = 0

and

lim x a x x = 0 and lim x + 0 a x x = .

For w f L p ( ) ( 1 p ) , the degree of weighted polynomial approximation is defined by

E n , p ( w ; f ) : = i n f P P n w ( f P ) L p ( ) .

3. Approximations for Lp-Functions

In this section, we treat the function such as W f L p ( s ) , where 1 p ), and if p = , then we suppose that W f is continuous and

l i m | X | ( W f ) ( X ) = 0 . For any multivariate point X = ( x 1 , , x s ) s , we consider the weights;

W ( X ) : = j = 1 s w j ( x j ) = j = 1 s exp ( Q j ( x j ) ) .

As shown under, we will also use X ( u ) : = ( u 1 , u 2 , , u s ) . Let

w i = exp ( Q i ) F λ ( C 3 + ) , 0 < λ < 3 / 2 , i = 1 , 2 , , s . From Proposition 2.3 we see T i 1 / 4 w i ~ w i , 1 / 4 F ( C 2 + ) , i = 1 , 2 , , s . Then we admit to write

T i 1 / 4 w i F ( C 2 + ) . For the weight W we construct the modulus of continuity of f . It involves the function

Φ t , i ( x i ) : = 1 | x i | σ i ( t ) + 1 T i ( σ i ( t ) ) , i = 1 , 2 , , s ,

where σ i ( t ) is defined by

σ i ( t ) : = inf { a n ( i ) : a n ( i ) n t } , t > 0 ,

where a n ( i ) is the MRS-number for the weight w i ( x ) . If a n ( i ) / n = t , then we have σ i ( t ) = a n ( i ) . In the sequel, if 1 j s is an integer, then f p ; j will denote the L p norm of f taken with respect to the j-th variable. This is a function of the remaining ( s 1 ) variables. For each fixed

X ^ j : = ( x 1 , , x j 1 , x j + 1 , , x s ) j s 1 , we write

f X ^ j ( x ) : = f ( x 1 , , x j 1 , x , x j + 1 , , x s ) , j = 1 , 2 , , s . (3.1)

Using

Δ h f X ^ j ( x ) : = f X ^ j ( x + h 2 ) f X ^ j ( x h 2 ) ,

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

ω ¯ p , j ( f X ^ j , w j ; t ) : = ( 1 t 0 t w j ( x ) ( Δ h f X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h ) 1 / p + inf c j ( constant ) ( f X ^ j ( x ) c j ) w j ( x ) L p ( | x | σ j ( 4 t ) ) .

If w j is Erdös-type, then we define

ω ¯ p , j ( f X ^ j , w j ; t ) : = ( 1 t 0 t w j ( x ) ( Δ h Φ t , j ( x ) f X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h ) 1 / p + inf c j ( constant ) ( f X ^ j ( x ) c j ) w j ( x ) L p ( | x | σ j ( 4 t ) ) .

We remark that if T j ( x ) is bounded, then we see Φ t , 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 v n , j ( f ) , j = 1 , 2 , , s denote the operation to f with respect to j-th co-ordinate, and v n [ j ] will denote the operator v n applied to f with respect to each of the first j co-ordinates. Clearly,

v n [ 1 ] ( f ) = v n , 1 ( f ) , v n [ j ] ( f ) = v n [ j 1 ] ( v n , j ( f ) ) , j = 2 , 3 , , s . (3.2)

Let a n ( j ) 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 < , C c ( s ) is dence in L p ( s ) , where C c ( s ) 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 f K with a compact support [ K , K ] s such that

W ( X ) ( f ( X ) f K ( X ) ) L p ( | X | K ) < 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

w j ( x ) ( f X ^ j ( x ) ( f K ) X ^ j ( x ) ) L p ( | x | K ) < ε (3.4)

holds.

We define a new class of functions L p * ( s ) , 1 p as follows:

L W p * ( s ) : = { f | W f L p ( s ) holds ( 3.4 ) } , (3.5)

where if p = , then L W p * ( s ) = L W p ( s ) and we suppose that f is continuous and

lim | X | W ( X ) f ( X ) = 0

(we write this fact as W f C 0 ( s ) ). We state the theorem in this section.

Theorem 3.3. (1) We suppose

w j = exp ( Q j ) F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , j = 1 , 2 , , s , and let

T j ( a n ( j ) ) c ( n a n ( j ) ) 2 / 3 , j = 1,2, , s . (3.6)

Let n 1, 1 p . Then we have

W ( f v n [ s ] ( f ) ) L p ( s ) j = 1 s C j ( 1 i s , i j w i ) ( k = 1 j 1 T k 1 / 4 ) ω ¯ p , j ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) L p ( j s 1 ) , (3.7)

where

j s 1 : = { ( x 1 , , x j 1 , x j + 1 , , x s ) } , (3.8)

and k = 1 0 T k 1 / 4 = 1 . Especially f L T s W p * ( s ) , then we have

j = 1 s C j ( 1 i s , i j w i ) ( k = 1 j 1 T k 1 / 4 ) ω ¯ p , j ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) L p ( j s 1 ) 0 as n . (3.9)

(2) We suppose w j = exp ( Q j ) F ( C 2 + ) , j = 1 , 2 , , s , and let (3.6) holds. Let n 1, 1 p . Then we have

W ( f v n [ s ] ( f ) ) k = 1 s T k 1 / 4 L p ( s ) j = 1 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 ) . (3.10)

Especially f L W p * ( s ) , then we have

j = 1 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 ) 0 as n . (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 F ( C 2 + ) satisfies T ( a n ) C ( n / a n ) 2 / 3 . Then there exists a constant C > 0 such that when w g L p ( ) , then

w v n ( g ) T 1 / 4 L p ( ) C w g L p ( ) ,

and so,

w v n ( g ) L p ( ) C T ( a n ) 1 / 4 w g L p ( ) .

(2) We assume that w F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) satisfies T ( a n ) C ( n / a n ) 2 / 3 . Then there exists a constant C > 0 such that if T 1 / 4 w g L p ( ) , then

w v n ( g ) L p ( ) C T 1 / 4 w g L p ( ) .

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

(1) Let w F ( C 2 + ) , and n 1 be an integer. Then

w ( g v n ( g ) ) T 1 / 4 L p ( ) C E n , p ( w ; g ) ,

where C do not depend on g and n.

(2) Let w F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , and n 1 be an integer. Then

w ( g v n ( g ) ) L p ( ) C E n , p ( T 1 / 4 w ; g ) ,

where C do not depend on g and n.

Proposition 3.6. ( [6] ) Let w = e x p ( Q ) F ( C 2 + ) , and let 0 < p . Then there exist n 0 and positive constants C1, C2 such that for every w g L p ( ) (and for p = , we require g to be continuous, and w f to vanish at ± ) and every n n 0 ,

E n , p ( g ; w ) C 1 ω ¯ p ( g , w ; C 2 a n n ) ,

where n 0 and C 1 , C 2 do not depend on g and n , and ω p * ( g , w , t ) will be defined in Section 6.

We set

T j : = i = 1 j T i 1 / 4 , ( j ) : = { x j } ,

j j : = { ( x 1 , , x j ) j } , j s j + 1 : = { ( x j , , x s ) s j + 1 } ,

j s 1 : = { ( x 1 , , x j 1 , x j + 1 , , x s ) s 1 } ,

W : = i = 1 s w i , W j : = i = 1 , i j s w i , j = 1 , 2 , , s .

We need the infinite-finite inequality.

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

P ( X ) P n , , n ( s ) ( = : P n ; s ( s ) ) . Then

W ( X ) P ( X ) L p ( s ) C W ( X ) P ( X ) L p ( | x i | a n ( i ) ( 1 L δ n ( i ) ) , i = 1,2, , s ) . (3.12)

If r > 1 , then there exists ε > 0 such that

W ( X ) P ( X ) L p ( i , r s ) C e x p ( n ε ) W ( X ) P ( X ) L p ( s ) , (3.13)

where i , r s : = { x i ; | x i | a r n ( i ) } × i s 1 .

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

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

w ( x ) P ( x ) L p ( ) C w ( x ) P ( x ) L p ( | x | a n ( 1 L δ n ) ) . (3.14)

If r > 1 , then there exists ε > 0 such that

w ( x ) P ( x ) L p ( a r n | x | ) C e x p ( n ε ) w ( x ) P ( x ) L p ( | x | a n ) . (3.15)

Proof of Theorem 3.7. For the proof of (3.12) we use (3.14). We put A for the left side of the above equation. Let 0 < p < . By repeatedly applying Proposition 3.8 (3.14), we have

A p = | w 2 ( x 2 ) w s ( x s ) | p × { | w 1 ( x 1 ) P ( x 1 , , x s ) | p d x 1 } d x 2 d x s C 1 | w 2 ( x 2 ) w s ( x s ) | p a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( 1 ) ( 1 L δ n ( 1 ) ) | w 1 ( x 1 ) P ( x 1 , , x s ) | p d x 1 d x s = C 1 a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( 1 ) ( 1 L δ n ( 1 ) ) w 1 p ( x 1 ) | w 2 ( x 2 ) w s ( x s ) P ( x 1 , , x s ) | p d x 2 d x s d x 1 C 2 a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( 2 ) ( 1 L δ n ( 2 ) ) a n ( 2 ) ( 1 L δ n ( 2 ) ) w 1 p ( x 1 ) w 2 p ( x 2 ) × | w 3 ( x 2 ) w s ( x s ) P ( x 1 , , x s ) | p d x 3 d x d d x 1 d x 2 C s a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( 1 ) ( 1 L δ n ( 1 ) ) a n ( s ) ( 1 L δ n ( s ) ) a n ( s ) ( 1 L δ n ( s ) ) | w 1 ( x 1 ) w s ( x s ) P ( x 1 , , x s ) | p d x 1 d x s = C s W ( X ) P ( X ) L p ( | x i | a n ( i ) ( 1 L δ n ( i ) ) , i = 1 , 2 , , s ) p .

Next, we show the case of p = .

A = sup x s sup x 2 | w 2 ( x 2 ) w s ( x s ) | sup x 1 | w 1 ( x 1 ) P ( x 1 , , x s ) | C 1 sup x s sup x 2 | w 2 ( x 2 ) w s ( x s ) | sup | x 1 | a n ( 1 ) ( 1 L δ n ( 1 ) ) | w 1 ( x 1 ) P ( x 1 , , x s ) | = C 1 sup x s sup x 3 | w 3 ( x 3 ) w s ( x s ) | sup | x 1 | a n ( 1 ) ( 1 L δ n ( 1 ) ) sup x 2 | w 1 ( x 1 ) w 2 ( x 2 ) P ( x 1 , , x s ) | C 1 C 2 sup x s sup x 3 | w 3 ( x 3 ) w s ( x s ) | × sup | x 1 | a n ( 1 ) ( 1 L δ n ( 1 ) ) sup | x 2 | a n ( 2 ) ( 1 L δ n ( 2 ) ) | w 1 ( x 1 ) w 2 ( x 2 ) P ( x 1 , , x s ) | C 1 C 2 C s sup | x 1 | a n ( 1 ) ( 1 L δ n ( 1 ) ) sup | x 2 | a n ( 2 ) ( 1 L δ n ( 2 ) ) sup | x 2 | a n ( s ) ( 1 L δ n ( s ) ) | w 1 ( x 1 ) w 2 ( x 2 ) × × w s ( x s ) P ( x 1 , , x s ) | = C sup | x i | a n ( i ) ( 1 L δ n ( i ) ) , i = 1 , , s | w 1 ( x 1 ) w 2 ( x 2 ) w s ( x s ) P ( x 1 , , x s ) | .

Similarly, using Proposition 3.8 (3.15), we easily have (3.13). #

Lemma 3.9. Let 1 p .

(1) We assume that w i F ( C 2 + ) , i = 1,2, , s satisfies (3.6). Then there exists a constant C > 0 such that when W h L p ( s ) ,

W v n [ j ] ( h ) T j L p ( s ) C W h L p ( s ) , j = 1 , 2 , , s ,

and

W v n [ s ] ( f ) L p ( s ) C i = 1 s T i 1 / 4 ( a n ( i ) ) W f L p ( s ) .

(2) We assume that w i F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , i = 1 , 2 , , s . Let

T s W h L p ( s ) , then

W v n [ j ] ( h ) L p ( s ) C T j W h L p ( s ) , j = 1,2, , s .

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

W v n [ 1 ] ( h ) T 1 L p ( s ) = W v n , 1 ( h X ^ 1 ) T 1 1 / 4 L p ( 1 1 ) L p ( 2 s 1 ) C W h L p ( s ) .

Inductively,

W v n [ j ] ( h ) T j L p ( s ) = W v n [ j 1 ] ( v n , j ( h ) ) T j L p ( j 1 j 1 ) L p ( j s j + 1 ) C W v n , j ( h X ^ j ) T j 1 / 4 L p ( s ) C W h L p ( s ) .

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

W v n [ j ] ( h ) L p ( s ) C i = 1 j T i 1 / 4 ( a n ( i ) ) ( k = j + 1 s w k ) ( i = 1 j w i ) v n [ j ] ( h ) T j L p ( | x i | a 2 n ( i ) ) , 1 i j L p ( j + 1 s j ) C i = 1 j T i 1 / 4 ( a n ( i ) ) W h L p ( s ) , j = 1 , 2 , , s .

Similarly we have the following:

(2) From Theorem 3.4 (2) for j = 1 ,

W v n [ 1 ] ( h ) L p ( s ) = W v n , 1 ( h ) L p ( s ) C W T 1 1 / 4 h L p ( s ) , j = 1 , 2 , , s .

Inductively,

W v n [ j ] ( h ) L p ( s ) = W v n [ j 1 ] ( v n , j ( h ) ) L p ( s ) C W T j 1 v n , j ( h ) L p ( s ) C W T j h L p ( s ) . #

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

w 1 ( x ) ( f X ^ 1 v n , 1 ( f X ^ 1 ) ) L p ( ) C E n , p ( f X ^ 1 ; T 1 1 / 4 w 1 ) C 1 ω ¯ p , 1 ( f X ^ 1 , T 1 1 / 4 w 1 ; c 1 a n ( 1 ) n ) ,

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

w j ( x ) ( f X ^ j v n , j ( f X ^ j ) ) L p ( ) C j ω ¯ p , j ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) . (3.16)

Using f v n [ s ] = ( f v n [ 1 ] ( f ) ) + ( v n [ 1 ] ( f ) v n [ 2 ] ( f ) ) + + ( v n [ s 1 ] ( f ) v n [ s ] ( f ) ) , we get from Lemma 3.9 (2) and (3.2),

W ( f v n [ s ] ( f ) ) L p ( s ) W ( f v n [ 1 ] ( f ) ) L p ( s ) + j = 2 s W ( v n [ j 1 ] ( f ) v n [ j ] ( f ) ) L p ( s ) C [ W ( f v n , 1 ( f ) ) L p ( s ) + j = 2 s W v n [ j 1 ] ( f v n , j ( f ) ) L p ( s ) ] C [ W ( f v n , 1 ( f ) ) L p ( s ) + j = 2 s W ( k = 1 j 1 T k 1 / 4 ) ( f v n , j ( f ) ) L p ( s ) ] C 1 W 1 ω ¯ p , 1 ( f X ^ 1 , T 1 1 / 4 w 1 ; c 1 a n ( 1 ) n ) L p ( 1 s 1 ) + j = 2 s C j W j ( k = 1 j 1 T k 1 / 4 ) ω ¯ p , j ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) L p ( j s 1 )

by (3.16)

j = 1 s C j W j ( k = 1 j 1 T k 1 / 4 ) ω ¯ p , j ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) L p ( j s 1 ) ,

where k = 1 j 1 T k 1 / 4 = 1 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

w 1 ( x ) ( f X ^ 1 v n , 1 ( f X ^ 1 ) ) T 1 1 / 4 ( x ) L p ( ) C E n , p ( f X ^ 1 ; w 1 ) C 1 ω ¯ p , 1 ( f X ^ 1 , w 1 ; c 1 a n ( 1 ) n ) ,

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

w j ( x ) ( f X ^ j v n , j ( f X ^ j ) ) T j 1 / 4 ( x ) L p ( ) C j ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) . (3.17)

Using f v n [ s ] = ( f v n [ 1 ] ( f ) ) + ( v n [ 1 ] ( f ) v n [ 2 ] ( f ) ) + + ( v n [ s 1 ] ( f ) v n [ s ] ( f ) ) , we get from Lemma 3.9 (1) and (3.2),

W ( f v n [ s ] ( f ) ) T s L p ( s ) W ( f v n [ 1 ] ( f ) ) T 1 L p ( s ) + j = 2 s W ( v n [ j 1 ] ( f ) v n [ j ] ( f ) ) T j L p ( s ) C [ W ( f v n , 1 ( f ) ) T 1 1 / 4 L p ( s ) + j = 2 s W v n [ j 1 ] ( { f v n , j ( f ) } ) / T j 1 / 4 T j 1 L p ( s ) ] C [ W ( f v n , 1 ( f ) ) T 1 1 / 4 L p ( s ) + j = 2 s W ( f v n , j ( f ) ) T j 1 / 4 L p ( s ) ] C 1 ( 2 i s w i ) ω ¯ p , 1 ( f X ^ 1 , w 1 ; c 1 a n ( 1 ) n ) L p ( 1 s 1 ) + j = 2 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 )

by (3.17)

j = 1 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 ) .

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 ( 2 t ) . We have

( w f ) ( x ± h Φ t ( x ) ) L p ( ) ~ ( w f ) ( x ) L p ( ) . (3.18)

Proof. We may show

( w f ) ( x ± h 1 | x | σ ( t ) ) L p ( ) ~ ( w f ) ( x ) L p ( ) . (3.19)

Let x > 0 . If we put

x ± h 1 x σ ( t ) = : y , a 2 u 2 u = t , a v v = 2 t , (3.20)

we will see

1 2 d y d x 3 2 . (3.21)

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

d y d x = 1 h 2 σ ( t ) σ ( t ) x .

Since a 2 u / u = 2 t , we see

a v v = a 2 u u > a u u ,

that is, we have

σ ( 2 t ) = a v < a u < σ ( t ) = a 2 u .

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

h 2 σ ( t ) σ ( t ) x t 2 σ ( t ) σ ( t ) σ ( 2 t ) a 2 u 4 u 1 a 2 u a u a 2 u 4 u C 1 T ( a 2 u ) a 2 u 1 4 u C 1 C 2 u 1 δ = C 1 C 2 4 1 u δ 1 2

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

W j T j 1 { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) f X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p L p ( j s 1 ) .

To do so we may estimate

I 1 : = { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) f X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p .

For ε > 0 we take K > 0 large enough, and then by f L T s w p * ( ) we can select a continuous function f K such that

I 1 { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) ( f f K ) X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p + { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) ( f K ) X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p = A + B .

We note w j ( x ) ~ w ( x + ( h / 2 ) Φ t , j ( x ) ) ~ w ( x ( h / 2 ) Φ t , j ( x ) ) (see [7] , Lemma 7). If σ j ( 2 t ) K from our assumption and Lemma 3.10 we have

A C { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( f f K ) X ^ j ( x ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p C ε { 1 t 0 t d h } 1 / p C ε .

If σ j ( 2 t ) > K , then by Lemma 3.10 we see

A C { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( f f K ) X ^ j ( x ) L p ( | x | K ) p d h } 1 / p + { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) f X ^ j ( x ) L p ( K < | x | σ j ( 2 t ) ) p d h } 1 / p C ε + C 1 ε C 2 ε .

When we take t > 0 small enough, we see

B = { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) ( f K ) X ^ j ( x ) ) L p ( | x | K ) p d h } 1 / p { 1 t 0 t ε d h } 1 / p C ε ,

because of the continuity of f K . Therefore we have I 1 < ε . Consequently, we have

W j T j 1 { 1 t 0 t T j 1 / 4 ( x ) w j ( x ) ( Δ h Φ t , j ( x ) f X ^ j ( x ) ) L p ( | x | σ j ( 2 t ) ) p d h } 1 / p L p ( j s 1 ) C ε W j T j 1 L p ( j s 1 ) C ε . (3.22)

Finally, we see

W j T j 1 i n f c j ( constant ) ( f X ^ j ( x ) c j ) T j 1 / 4 ( x ) w j ( x ) L p ( \ [ σ j ( 4 t ) , σ j ( 4 t ) ] ) L p ( j s 1 ) C W j T j 1 f X ^ j ( x ) T j 1 / 4 ( x ) w j ( x ) L p ( \ [ σ j ( 4 t ) , σ j ( 4 t ) ] ) L p ( j s 1 ) C w j 1 / 4 ( σ j ( 4 t ) ) W f L p ( j s ) .

Here, if we set 4 t = a u / u , then we see

w j 1 / 4 ( σ j ( 4 t ) ) = e x p ( 1 4 Q j ( a u ) ) ~ e x p ( u 2 T ( a u ) ) e u δ

for some 0 < δ < 1 , that is,

w j 1 / 4 ( σ j ( 4 t ) ) C e u δ C a u 4 u = C t .

Therefore

W j T j 1 i n f c j ( constant ) ( f X ^ j ( x ) c j ) T j 1 / 4 ( x ) w j ( x ) L p ( \ [ σ j ( 4 t ) , σ j ( 4 t ) ] ) L p ( j s 1 ) C t . (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

W j T j 1 ω ¯ p , j ( f X ^ j , T j 1 / 4 w j , t ) L p ( j s 1 ) < 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 w j = e x p ( Q j ) F ( C 2 + ) , j = 1,2, , s are the Freud-type weights. Let 1 p . Then we have

W ( f v n [ s ] ( f ) ) L p ( s ) j = 1 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 ) .

Especially f L W p * ( s ) , then we have

j = 1 s C j ( 1 i s , i j w i ) ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) L p ( j s 1 ) 0 as n .

Corollary 3.12. We suppose

w j = e x p ( Q j ) F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , j = 1 , 2 , , s , and let (3.6) hold. Let 1 p . If T s W f C 0 ( s ) L p ( s ) ), then we have

W ( f v n [ s ] ( f ) ) L p ( s ) 0 as n .

Moreover, we suppose w j = e x p ( Q j ) F ( C 2 + ) , j = 1,2, , s , and let (3.6) hold. If W f C 0 ( s ) L p ( s ) , then we have

W ( f v n [ s ] ( f ) ) T s L p ( s ) 0 as n .

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

w i ( x i ) = exp ( Q i ( x i ) ) F ( C 2 + ) , i = 1 , 2 , , s . Let r be a positive integer, and | x i | γ > 0 .

W 0 : = i = 1 s w i , 0 ; Q i ( x i ) r w i ( x i ) ~ w i , 0 ( x i ) = exp ( Q i , 0 ( x i ) ) F ( C 2 + ) ,

W ν : = i = 1 s w i , ν ; Q i ( x i ) r ν w i ( x i ) ~ w i , ν ( x i ) = exp ( Q i , ν ( x i ) ) F ( C 2 + ) , ν = 1 , 2 , , r .

Then we see

Q i ( x i ) r ν + 1 w i ( x i ) ~ Q i , ν ( x i ) w i , ν ( x i ) .

Especially, if ν = r , then

w i , r ( x i ) = w i ( x i ) .

Theorem 4.1. Let w i = exp ( Q i ) F ( C 2 + ) , i = 1 , 2 , , s , and let 1 p . Let a constant γ 0 be fixed. We suppose that g : = g ( x 1 , x 2 , , x s ) is absolutely continuous on s and W g ( 1,1, ,1 ) L p ( s ) . Then we have

i = 1 s ( Q i w i ) g L p ( | x i | γ , i = 1 , , s ) C { | x 1 | γ | x s | γ 0 j 1 1 | w 1 , j 1 ( x 1 ) | p 0 j s 1 | w s , j s ( x s ) | p × | g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p , (4.1)

where we set for each j i = 0 or 1, i = 1 , 2 , , s ,

y i = { γ , j i = 0 ; x i , j i = 1.

Furthermore, let r be a positive integer, and let for each i = 1 , 2 , , s , w i = exp ( Q i ) F λ ( C 3 + ) F ( C 2 + ) ( 0 < λ < 3 / 2 ) . We suppose that g ( r 1, r 1, , r 1 ) is absolutely continuous and W g ( r , r , , r ) L p ( s ) . Then we have

{ s | w 1 , j 1 ( x 1 ) | p | w s , j s ( x s ) | p | g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p < , (4.2)

for each 0 j i r , i = 1,2, , s with

y i = { γ , 0 j i r 1 ; x i , j i = r , (4.3)

and

( i = 1 s Q i ) r W g L p ( | x i | γ , i = 1 , , s ) C { | x 1 | γ | x s | γ 0 j 1 r | w 1 , j 1 ( x 1 ) | p 0 j s r | w s , j s ( x s ) | p × | g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p < . (4.4)

Proposition 4.2 ( [8] , Theorem 9, cf. [9] , Lemma 3.4.4). Let

w = e x p ( Q ) F ( C 2 + ) and a constant γ 0 be fixed.

(a) We have

| Q ( x ) w ( x ) γ x w 1 ( t ) d t | C , | x | γ .

(b) Let 1 p , and let r be a positive integer. If g is absolutely continuous, g ( γ ) = 0 and w g L p ( ) , then

Q w g L p ( | x | γ ) C w g L p ( | x | γ ) .

When w = e x p ( Q ) F λ ( C r + 1 + ) F ( C 2 + ) ( 0 < λ < ( r + 1 ) / r ), and g ( r 1 ) is absolutely continuous, g ( j ) ( γ ) = 0 , j = 0 , 1 , , r 1 with w g ( r ) L p ( ) , we see

( Q ) r w g L p ( | x | γ ) C w g ( r ) L p ( | x | γ ) .

Proposition 4.3 ( [3] , Theorem 4.2). Let w = e x p ( Q ) F λ ( C 3 + ) F ( C 2 + ) . Then for α , we can construct a new weight w α F ( C 2 + ) such that

( 1 + | Q ( x ) | ) α w ( x ) ~ w α ( x ) = e x p ( Q α )

on , ( 1 / c ) a n ( w ) a n ( w α ) c a n ( w ) (c is an absolutely constant) on and T w α ( x ) ~ T w ( x ) hold on . Furthermore, we see

Q α ( j ) ( x ) ~ Q ( j ) ( x ) ( j = 0 , 1 ) for | x | γ > 0.

Proof of Theorem 4.1. For the proof of (4.1) we may put r = 1 with w i = exp ( Q i ) F ( C 2 + ) , i = 1 , 2 , , s in the proof of (4.4) below. So we prove only (4.4). We use Proposition 4.2 and 4.3 repeatedly.

{ x s γ x 1 γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( x 1 , , x s ) | p d x 1 d x s } 1 / p = { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | Q 1 r ( x 1 ) w 1 ( x 1 ) g ( x 1 , , x s ) | p d x 1 d x s } 1 / p C 1 , 0 [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | Q 1 , 1 ( x 1 ) w 1 , 1 ( x 1 ) g ( x 1 , , x s ) | p d x 1 d x s } 1 / p ] ,

where ( Q 1 ) r w 1 = Q 1 ( Q 1 ) r 1 w 1 ~ Q 1 w 1 , 1 ~ Q 1 , 1 w 1 , 1 , w 1 , 1 = e Q 1 , 1 ,

C 1 , 0 [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | Q 1 , 1 ( x 1 ) w 1 , 1 ( x 1 ) ( g ( x 1 , , x s ) g ( γ , x 2 , , x s ) ) | p d x 1 d x s } 1 / p + { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | Q 1 , 1 ( x 1 ) w 1 , 1 ( x 1 ) g ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p ] C 1 , 1 [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | w 1 , 1 ( x 1 ) g ( 1 , 0 , , 0 ) ( x 1 , , x s ) | p d x 1 d x s } 1 / p + { x s γ x 1 γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p ]

by Q 1,1 w 1,1 ~ ( Q 1 ) r w 1 ,

C 1 , 1 [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | Q 1 , 2 ( x 1 ) w 1 , 2 ( x 1 ) g ( 1 , 0 , , 0 ) ( x 1 , , x s ) | p d x 1 d x s } 1 / p where w 1 , 1 ~ ( Q 1 ) r 1 w ~ Q 1 , 2 w 1 , 2 , w 1 , 2 = e Q 1 , 2 , + { x s γ x 1 γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p ]

C 1 , 2 [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | w 1 , 2 ( x 1 ) g ( 2 , 0 , , 0 ) ( x 1 , , x s ) | p d x 1 d x s } 1 / p + { x s γ x 1 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × | Q 1 r 1 ( x 1 ) w 1 ( x 1 ) g ( 1 , 0 , , 0 ) ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p + { x s γ x 1 γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p ]

C 1 , r [ { x s γ x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ | w 1 ( x 1 ) g ( r , 0 , , 0 ) ( x 1 , , x s ) | p d x 1 d x s } 1 / p + { x s γ x 1 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × | Q 1 ( x 1 ) w 1 ( x 1 ) g ( 1 , 0 , , 0 ) ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p

+ + { x s γ ... x 1 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × | Q 1 r 1 ( x 1 ) w 1 ( x 1 ) g ( 1 , 0 , , 0 ) ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p + { x s γ ... x 1 γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( γ , x 2 , , x s ) | p d x 1 d x s } 1 / p ]

= C 1 , r { x s γ ... x 2 γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × x 1 γ 0 j 1 r | w 1 , j 1 ( x 1 ) g ( j 1 , 0 , , 0 ) ( y 1 , x 2 , , x s ) | p d x 1 d x s } 1 / p ,

where

y 1 = { γ , 0 j 1 r 1 ; x 1 , j 1 = r .

We continue this manner with respect to x 2 , x 3 , , x s . Then we can easily obtain as follows:

{ | x s | γ | x 1 | γ | i = 1 s ( Q i r ( x i ) w i ( x i ) ) g ( x 1 , , x s ) | p d x 1 d x s } 1 / p C 1 , r { | x s | γ | x 2 | γ | i = 2 s ( Q i r ( x i ) w i ( x i ) ) | p × | x 1 | γ 0 j 1 r | w 1 , j 1 ( x 1 ) g ( j 1 , 0 , , 0 ) ( y 1 , x 2 , , x s ) | p d x 1 d x s } 1 / p

C 2 { | x 1 | γ 0 j 1 r | w 1 , j 1 ( x 1 ) | p | x s | γ | x 3 | γ | i = 3 s ( Q i r ( x i ) w i ( x i ) ) | p × | x 2 | γ 0 j 2 r | w 2 , j 2 ( x 2 ) g ( j 1 , j 2 , 0 , , 0 ) ( y 1 , y 2 , x 3 , , x s ) | p d x 2 d x s d x 1 } 1 / p

= C 2 { | x 1 | γ | x 2 | γ 0 j 1 r | w 1 , j 1 ( x 1 ) | p 0 j 2 r | w 2 , j 2 ( x 2 ) | p | x s | γ | x 4 | γ | i = 4 s ( Q i r ( x i ) w i ( x i ) ) | p × | x 3 | γ | w 3 , j 3 ( x 3 ) g ( j 1 , j 2 , j 3 , 0 , , 0 ) ( y 1 , y 2 , y 3 , x 4 , , x s ) | p × d x 3 d x s d x 2 d x 1 } 1 / p

C r { | x 1 | γ | x s | γ 0 j 1 r | w 1 , j 1 ( x 1 ) | p 0 j s r | w s , j s ( x s ) | p × g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p ,

where for each 0 j i r , i = 1 , 2 , , s . We set (4.3).

Let W g ( r , , r ) L p ( s ) . Then we need to show

A : = { | x 1 | γ | x s | γ 0 j 1 r | w 1 , j 1 ( x 1 ) | p 0 j s r | w s , j s ( x s ) | p × | g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p < .

We rearrange ( x 1 , x 2 , , x s ) as x k i = y i = x i , t + 1 k i s if j i = r , and as x k i = y i = γ < 1 k i t if 0 j i r 1 , where 0 t s . Then we set

f ( x k 1 , x k 2 , , x k s ) : = g ( x 1 , x 2 , , x s ) . We see

g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) = f ( j k 1 , j k 2 , , j k s ) ( γ , , γ , x t + 1 , x t + 2 , , x s ) .

Then we have

A = i = 1 t w k i i = t + 1 s w k i f ( j k 1 , , j k s ) ( γ , , γ , x t + 1 , , x s ) L p ( t + 1 s t ) L p ( t t ) = : i = 1 t w k i | h ( j k 1 , , j k t ) ( γ , , γ ) | L p ( t t ) C | h ( j k 1 , , j k t ) ( γ , , γ ) | i = 1 t w k i L p ( t t ) < . #

We can generalize Theorem 4.1 easily. We give a class of nonnegative integers ( j 1 , j 2 , , j s ) , and set J s : = ( j 1 , j 2 , , j s ) . For r i 1 , i = 1 , 2 , , s we set R s : = ( r 1 , r 2 , , r s ) . Then we consider the order as follows:

K s : = ( k 1 , k 2 , , k s ) R s : = ( r 1 , r 2 , , r s )

means

k i r i ( i = 1 , 2 , , s ) .

Corollary 4.4. Let K s = ( k 1 , k 2 , , k s ) R s = ( r 1 , r 2 , , r s ) be classes of nonnegative integers, where r i 1 , i = 1 , 2 , , s . For each i = 1 , 2 , , s , we

suppose w i = exp ( Q i ) F λ ( C r + 1 + ) F ( C 2 + ) ( 0 < λ < ( r + 1 ) / r ). If

g ( r 1 1, r 2 1, , r s 1 ) is absolutely continuous, and W g ( r 1 , r 2 , , r s ) L p ( s ) , then we see

( i = 1 s Q i ) r i k i W g ( k 1 , , k s ) L p ( | x i | γ , i = 1 , , s ) C { | x 1 | γ | x s | γ k 1 j 1 r 1 | w 1 , j 1 k 1 ( x 1 ) | p k s j s r s | w s , j s k s ( x s ) | p × | g ( j 1 , j 2 , , j s ) ( y 1 , y 2 , , y s ) | p d x s d x 1 } 1 / p < ,

where for each i = 1 , 2 , , s we set

y i = { γ , k i j i r i 1 ; x i , j i = r i .

We remark that W g ( r 1 , , r s ) L p ( s ) means W g ( k 1 , , k s ) L p ( s ) for

0 K s = ( k 1 , , k s ) R s = ( r 1 , , r s ) .

5. Degree of Approximation

We define the degree of approximation for W f L p ( s ) as follows:

E n , p ; s ( W , f ) : = inf P P n ; s ( s ) W ( f P ) L p ( s ) .

Using this E n , p ; s ( W , f ) , we can estimate the degree of approximation of W f L p ( s ) from P n ; s ( s ) .

Theorem 5.1. (1) Let w i F ( C 2 + ) ( i = 1 , 2 , , s ) and let

1 p , W f L p ( s ) . Furthermore, we suppose (3.3). Then we have

W i = 1 s T i ( x i ) 1 / 4 ( f v n [ s ] ( f ) ) L p ( s ) C E n , p ; s ( W , f ) .

(2) If w i F λ ( C 3 + ) ( i = 1 , 2 , , s ) , 0 < λ < 3 / 2 , and let

( i = 1 s T i 1 / 4 ) W f L p ( s ) < , then we have

W ( f v n [ s ] ( f ) ) L p ( s ) C E n , p ; s ( ( i = 1 s T i 1 / 4 ) W , f ) .

(3) Let w i F ( C 2 + ) ( i = 1,2, , s ) and let 1 p , W f L p ( s ) . Then we have

W ( f v n [ s ] ( f ) ) L p ( s ) C ( i = 1 s T i 1 / 4 ( a n ( i ) ) ) E n , p ; s ( W , f ) .

(4) Furthermore, let w i F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , i = 1 , 2 , , s . If f L W δ p * ( s ) for some 0 < δ < 1 , then we have

E n , p ; s ( W , f ) 0 as n .

Proof. (1) There exists P P n such that W ( f P ) L p ( s ) C E n , p ; s ( W , f ) . Therefore, by Lemma 3.9 (1)

W i = 1 s T i 1 / 4 ( f v n [ s ] ( f ) ) L p ( s ) = W i = 1 s T i 1 / 4 ( f P ) L p ( s ) + W i = 1 s T i 1 / 4 v n [ s ] ( f P ) L p ( s ) C W ( f P ) L p ( s ) C E n , p ; s ( W , f ) .

(2) We see W ( X ) i = 1 s T i 1 / 4 ( x i ) ~ W ˜ ( X ) = i = 1 s w ˜ i ( x i ) ,

T i 1 / 4 w ~ w ˜ i F ( C 2 + ) , i = 1 , 2 , , s . Then, there exists P P n such that

W ˜ ( f P ) L p ( s ) C E n 1, p ; s ( W ˜ , f ) . Therefore, by Lemma 3.9 (2)

W ( f v n [ s ] ( f ) ) L p ( s ) = W ( f P ) L p ( s ) + W v n [ s ] ( f P ) L p ( s ) C E n , p ; s ( W ˜ , f ) C E n , p ; s ( ( i = 1 s T i 1 / 4 ) W , f ) .

(3) Similarly, we have (3).

(4) It follows from Theorem 3.3. #

Theorem 5.2. Let w i F λ ( C 3 + ) ( i = 1 , 2 , , s ) , 0 < λ < 3 / 2 , and let 1 p . Then if W f L p ( s ) , we have

W i = 1 s T i ( 2 j i + 1 ) / 4 v n [ s ] ( f ) ( j 1 , , j s ) L p ( s ) C i = 1 s ( n a n ( i ) ) j i W f L p ( s ) ,

and

W v n [ s ] ( f ) ( j 1 , , j s ) L p ( s ) C i = 1 s ( n a n ( i ) ) j i ( i = 1 s T i ( 2 j i + 1 ) / 4 ( a n ( i ) ) ) W f L p ( s ) .

Proposition 5.3 ( [10] , Lemma 2.5, [2] , Corollary 10.2). Let 1 p and w F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) . Then there exists a constant C 1 = C 1 ( w , p ) > 0 such that, if P P n ( ) ( n ),

w T j / 2 P ( j ) L p ( s ) C 1 ( n a n ) j w P L p ( s ) , j ,

and

w P ( j ) L p ( s ) C 1 ( n T ( a n ) 1 / 2 a n ) j w P L p ( s ) , j .

Proof of Theorem 5.2. We use Proposition 5.3 and Lemma 3.7 (1).

W i = 1 s T i ( 2 j i + 1 ) / 4 v n [ s ] ( f ) ( j 1 , , j s ) L p ( s ) C i = 1 s ( n a n ( i ) ) j i W i = 1 s T i 1 / 4 v n [ s ] ( f ) L p ( s ) C i = 1 s ( n a n ( i ) ) j i W f L p ( s ) ,

and further, using Theorem A1 (the Markov-Bernstein inequality) in Appendix,

W v n [ s ] ( f ) ( j 1 , , j s ) L p ( s ) C i = 1 s ( n T i 1 / 2 ( a n ( i ) ) a n ( i ) ) j i W v n [ s ] ( f ) L p ( s ) C i = 1 s ( n T i 1 / 2 ( a n ( i ) ) a n ( i ) ) j i W v n [ s ] ( f ) L p ( | x i | a n ( i ) , i = 1 , 2 , , s ) C i = 1 s ( n a n ( i ) ) j i i = 1 s T i ( 2 j i + 1 ) / 4 ( a n ( i ) ) W i = 1 s T i 1 / 4 v n [ s ] ( f ) L p ( s ) C i = 1 s ( n a n ( i ) ) j i i = 1 s T i ( 2 j i + 1 ) / 4 ( a n ( i ) ) W f L p ( s ) . #

In the rest of only this section, we suppose

w = exp ( Q ) = w i = exp ( Q i ) , i = 1 , 2 , , s ,

so

a n = a n ( i ) , T = T i , i = 1 , 2 , , s .

Let

W i : = W i ( x 1 , , x i 1 , x i + 1 , , x s ) : = j i , 1 j s w j ( x j ) , i = 1 , 2 , , s .

In ( [7] , Corollary 8) we give the Favard-type inequalities:

Proposition 5.4 [7] . Let w F ( C 2 + ) , and let r 0 be an integer. Let 1 p , and let w f ( r ) L p ( ) . Then we have

E p , n ( f , w ) C ( a n n ) k w f ( k ) L p ( ) , k = 1 , 2 , , r ,

and equivalently,

E p , n ( f , w ) C ( a n n ) k E p , n k ( f ( k ) , w ) .

The following theorem is a generalization of Proposition 5.4.

Theorem 5.5. We suppose

w j = exp ( Q j ) F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , j = 1 , 2 , , s , and let (3.3) satisfy, that is,

T j ( a n ) c ( n a n ) 2 / 3 , j = 1 , 2 , , s .

Let W f ( r , r , , r ) L p ( s ) for some positive integer r. Then we have

E n , p ; s ( W ; f ) C ( a n n ) r T s W f ( r , r , , r ) L p ( s ) .

Equivalently,

E n , p ; s ( W ; f ) C ( a n n ) r E n r , p ; s ( T s W ; f ( r , r , , r ) ) .

Proof. Using

f v n [ s ] = ( f v n [ 1 ] ( f ) ) + ( v n [ 1 ] ( f ) v n [ 2 ] ( f ) ) + + ( v n [ s 1 ] ( f ) v n [ s ] ( f ) ) , we get from Lemma 3.9 (2) and (3.2),

E n , p ; s W ( f v n [ s ] ( f ) ) L p ( s ) W ( f v n [ 1 ] ( f ) ) L p ( s ) + j = 2 s W ( v n [ j 1 ] ( f ) v n [ j ] ( f ) ) L p ( s ) C [ W ( f v n , 1 ( f ) ) L p ( s ) + j = 2 s W v n [ j 1 ] ( f v n , j ( f ) ) L p ( s ) ] C [ W ( f v