Let ℝ s : = ℝ × ℝ × ⋯ × ℝ ( 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 ( ℝ s ) . Here, P n , n , ⋯ , n ( ℝ s ) ( = : P n ; s ( ℝ 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 : ℝ s → ℝ be measurable. Then we define

‖ W f ‖ L p ( ℝ s ) : = { [ ∫ − ∞ ∞ ⋯ ∫ − ∞ ∞ | ( W f ) ( x 1 , ⋯ , x s ) | p d x 1 ⋯ d x s ] 1 / p ,     if   0 < p < ∞ ; sup ( x 1 , ⋯ , x s ) ∈ ℝ 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 ( ℝ s ) < ∞ , we write W f ∈ L p ( ℝ 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 ( ℝ s ) by weighted polynomials W P ; P ∈ P n ; s ( ℝ 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 ( ℝ 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.

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 ⊂ ℝ is a subset.

We say that f : ℝ → [ 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 : ℝ → [ 0, ∞ ) be a continuous and even function, and satisfy the following properties:

(a) Q ′ ( x ) is continuous in ℝ , with Q ( 0 ) = 0 .

(b) Q ″ ( x ) exists and is positive in ℝ \ { 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 ∈ ℝ \ { 0 } .

(e) There exists C 1 > 0 such that

Q ″ ( x ) | Q ′ ( x ) | ≤ C 1 | Q ′ ( x ) | Q ( x ) , a . e . x ∈ ℝ .

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

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

Q ″ ( x ) | Q ′ ( x ) | ≥ C 2 | Q ′ ( x ) | Q ( x ) , a . e . x ∈ ℝ \ 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 ( ℝ \ { 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 ( ℝ \ { 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 ∈ ℝ . 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 ℝ ,

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

In this section, we treat the function such as W f ∈ L p ( ℝ 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 ) ∈ ℝ 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 ) ∈ ℝ 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 v_n 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 ( ℝ s ) is dence in L p ( ℝ s ) , where C c ( ℝ s ) is a set of all continuous functions with a compact support on ℝ 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 * ( ℝ s ) , 1 ≤ p ≤ ∞ as follows:

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

where if p = ∞ , then L W p * ( ℝ s ) = L W p ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ j s − 1 ) , (3.7)

where

ℝ 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 * ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ j s − 1 ) . (3.10)

Especially f ∈ L W p * ( ℝ 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 ( ℝ 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 ( ℝ ) , then

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

and so,

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

(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 ( ℝ ) , then

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

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 ( ℝ ) ≤ 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 ( ℝ ) ≤ 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 ∈ ℕ and positive constants C₁, C₂ such that for every w g ∈ L p ( ℝ ) (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 , ℝ ( j ) : = { x j ∈ ℝ } ,

ℝ ≤ j j : = { ( x 1 , ⋯ , x j ) ∈ ℝ j } , ℝ j ≤ s − j + 1 : = { ( x j , ⋯ , x s ) ∈ ℝ s − j + 1 } ,

ℝ j s − 1 : = { ( x 1 , ⋯ , x j − 1 , x j + 1 , ⋯ , x s ) ∈ ℝ 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 ( ℝ s ) ( = : P n ; s ( ℝ s ) ) . Then

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

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

‖ w ( x ) P ( x ) ‖ L p ( ℝ ) ≤ 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 ∈ ℝ ⋯ sup x 2 ∈ ℝ | w 2 ( x 2 ) ⋯ w s ( x s ) | sup x 1 ∈ ℝ | w 1 ( x 1 ) P ( x 1 , ⋯ , x s ) | ≤ C 1 sup x s ∈ ℝ ⋯ sup x 2 ∈ ℝ | 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 ∈ ℝ ⋯ sup x 3 ∈ ℝ | w 3 ( x 3 ) ⋯ w s ( x s ) | sup | x 1 | ≤ a n ( 1 ) ( 1 − L δ n ( 1 ) ) sup x 2 ∈ ℝ | w 1 ( x 1 ) w 2 ( x 2 ) P ( x 1 , ⋯ , x s ) | ≤ C 1 C 2 sup x s ∈ ℝ ⋯ sup x 3 ∈ ℝ | 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 ( ℝ s ) ,

‖ W v n [ j ] ( h ) T 〈 j 〉 ‖ L p ( ℝ s ) ≤ C ‖ W h ‖ L p ( ℝ s ) , j = 1 , 2 , ⋯ , s ,

and

‖ W v n [ s ] ( f ) ‖ L p ( ℝ s ) ≤ C ∏ i = 1 s T i 1 / 4 ( a n ( i ) ) ‖ W f ‖ L p ( ℝ 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 ( ℝ s ) , then

‖ W v n [ j ] ( h ) ‖ L p ( ℝ s ) ≤ C ‖ T 〈 j 〉 W h ‖ L p ( ℝ s ) , j = 1,2, ⋯ , s .

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

‖ W v n [ 1 ] ( h ) T 〈 1 〉 ‖ L p ( ℝ s ) = ‖ ‖ W v n , 1 ( h X ^ 1 ) T 1 1 / 4 ‖ L p ( ℝ ≤ 1 1 ) ‖ L p ( ℝ 2 ≤ s − 1 ) ≤ C ‖ W h ‖ L p ( ℝ s ) .

Inductively,

‖ W v n [ j ] ( h ) T 〈 j 〉 ‖ L p ( ℝ s ) = ‖ ‖ W v n [ j − 1 ] ( v n , j ( h ) ) T 〈 j 〉 ‖ L p ( ℝ ≤ j − 1 j − 1 ) ‖ L p ( ℝ j ≤ s − j + 1 ) ≤ C ‖ W v n , j ( h X ^ j ) T j 1 / 4 ‖ L p ( ℝ s ) ≤ C ‖ W h ‖ L p ( ℝ s ) .

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

‖ W v n [ j ] ( h ) ‖ L p ( ℝ 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 ( ℝ j + 1 ≤ s − j ) ≤ C ∏ i = 1 j T i 1 / 4 ( a n ( i ) ) ‖ W h ‖ L p ( ℝ 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 ( ℝ s ) = ‖ W v n , 1 ( h ) ‖ L p ( ℝ s ) ≤ C ‖ W T 1 1 / 4 h ‖ L p ( ℝ s ) , j = 1 , 2 , ⋯ , s .

Inductively,

‖ W v n [ j ] ( h ) ‖ L p ( ℝ s ) = ‖ W v n [ j − 1 ] ( v n , j ( h ) ) ‖ L p ( ℝ s ) ≤ C ‖ W T 〈 j − 1 〉 v n , j ( h ) ‖ L p ( ℝ s ) ≤ C ‖ W T 〈 j 〉 h ‖ L p ( ℝ 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 ( ℝ ) ≤ 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 ( ℝ ) ≤ 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 ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( ∏ k = 1 j − 1 T k 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ ) ≤ 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 ( ℝ ) ≤ 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 ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) T 〈 1 〉 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) T 〈 j 〉 ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) T 1 1 / 4 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( { f − v n , j ( f ) } ) / T j 1 / 4 T 〈 j − 1 〉 ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) T 1 1 / 4 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( f − v n , j ( f ) ) T j 1 / 4 ‖ L p ( ℝ s ) ] ≤ C 1 ‖ ( ∏ 2 ≤ i ≤ s w i ) ω ¯ p , 1 ( f X ^ 1 , w 1 ; c 1 a n ( 1 ) n ) ‖ L p ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ ) ~ ‖ ( w f ) ( x ) ‖ L p ( ℝ ) . (3.18)

Proof. We may show

‖ ( w f ) ( x ± h 1 − | x | σ ( t ) ) ‖ L p ( ℝ ) ~ ‖ ( w f ) ( x ) ‖ L p ( ℝ ) . (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 ( ℝ 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 * ( ℝ ) 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 ( ℝ j s − 1 ) ≤ C ε ‖ W j T 〈 j − 1 〉 ‖ L p ( ℝ 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 ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ j s − 1 ) ≤ C ‖ W j T 〈 j − 1 〉 ‖ f X ^ j ( x ) T j 1 / 4 ( x ) w j ( x ) ‖ L p ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ j s − 1 ) ≤ C w j 1 / 4 ( σ j ( 4 t ) ) ‖ W f ‖ L p ( ℝ 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 ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ j s − 1 ) .

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

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ 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 ( ℝ s ) ∩ L p ( ℝ s ) , then we have

‖ W ( f − v n [ s ] ( f ) ) T 〈 s 〉 ‖ L p ( ℝ s ) → 0 as   n → ∞ .

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 ℝ s and W g ( 1,1, ⋯ ,1 ) ∈ L p ( ℝ 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 ( ℝ s ) . Then we have

{ ∫ ℝ 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 ( ℝ ) , 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 ( ℝ ) , 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 ∈ ℝ , 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 ℝ , ( 1 / c ) a n ( w ) ≤ a n ( w α ) ≤ c a n ( w ) (c is an absolutely constant) on ℕ and T w α ( x ) ~ T w ( x ) hold on ℝ . 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 ( ℝ 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 ( ℝ t + 1 ≤ s − t ) ‖ L p ( ℝ ≤ t t ) = : ‖ ∏ i = 1 t w k i | h ( j k 1 , ⋯ , j k t ) ( γ , ⋯ , γ ) | ‖ L p ( ℝ ≤ t t ) ≤ C | h ( j k 1 , ⋯ , j k t ) ( γ , ⋯ , γ ) | ‖ ∏ i = 1 t w k i ‖ L p ( ℝ ≤ 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 ( ℝ 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 ( ℝ s ) means W g ( k 1 , ⋯ , k s ) ∈ L p ( ℝ s ) for

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

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

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

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

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

1 ≤ p ≤ ∞ , W f ∈ L p ( ℝ 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 ( ℝ 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 ( ℝ s ) < ∞ , then we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ 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 ( ℝ s ) . Then we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ 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 * ( ℝ 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 ( ℝ 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 ( ℝ s ) = ‖ W ∏ i = 1 s T i 1 / 4 ( f − P ) ‖ L p ( ℝ s ) + ‖ W ∏ i = 1 s T i 1 / 4 v n [ s ] ( f − P ) ‖ L p ( ℝ s ) ≤ C ‖ W ( f − P ) ‖ L p ( ℝ 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 ( ℝ s ) ≤ C E n − 1, p ; s ( W ˜ , f ) . Therefore, by Lemma 3.9 (2)

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) = ‖ W ( f − P ) ‖ L p ( ℝ s ) + ‖ W v n [ s ] ( f − P ) ‖ L p ( ℝ 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 ( ℝ s ) , we have

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

and

‖ W v n [ s ] ( f ) ( j 1 , ⋯ , j s ) ‖ L p ( ℝ 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 ( ℝ 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 ( ℝ ) ( n ∈ ℕ ),

‖ w T j / 2 P ( j ) ‖ L p ( ℝ s ) ≤ C 1 ( n a n ) j ‖ w P ‖ L p ( ℝ s ) , j ∈ ℕ ,

and

‖ w P ( j ) ‖ L p ( ℝ s ) ≤ C 1 ( n T ( a n ) 1 / 2 a n ) j ‖ w P ‖ L p ( ℝ s ) , j ∈ ℕ .

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 ( ℝ 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 ( ℝ s ) ≤ C ∏ i = 1 s ( n a n ( i ) ) j i ‖ W f ‖ L p ( ℝ s ) ,

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

‖ W v n [ s ] ( f ) ( j 1 , ⋯ , j s ) ‖ L p ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ 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 ( ℝ ) . Then we have

E p , n ( f , w ) ≤ C ( a n n ) k ‖ w f ( k ) ‖ L p ( ℝ ) , 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 ( ℝ 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 ( ℝ 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 ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( ∏ k = 1 j − 1 T i 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ]

We estimate each term. From Proposition 5.4 with the weight T j 1 / 4 w j ,

‖ W ( ∏ k = 1 j − 1 T k 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) = ‖ ‖ W i ( ∏ k = 1 j − 1 T k 1 / 4 ) w j ( f X ^ j − v n , j ( f X ^ j ) ) ‖ L p ( ℝ ( j ) ) ‖ L p ( ℝ j s − 1 ) ≤ C ‖ W i ( ∏ k = 1 j − 1 T k 1 / 4 ) E n , p ( T j 1 / 4 w j , f X ^ j ) ‖ L p ( ℝ j s − 1 ) ≤ C ( a n n ) r ‖ ∏ i ≠ j w i ( ∏ k = 1 j − 1 T k 1 / 4 ) ‖ T j 1 / 4 w j f X ^ j ( r , 0 , ⋯ , 0 ) ‖ L p ( ℝ ( j ) ) ‖ L p ( ℝ j s − 1 ) .

Now, we use Theorem 4.1 and the fact

T i ( x i ) ≤ Q ′ i ( x i ) r , i = 1,2, ⋯ , s ,

then we have

‖ W ( ∏ k = 1 j − 1 T k 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) r ‖ ∏ i ≠ j Q ′ i ( x i ) r w i ‖ T j 1 / 4 w j f X ^ j ( r , 0 , ⋯ , 0 ) ‖ L p ( ℝ ( j ) ) ‖ L p ( ℝ j s − 1 ) ≤ C ( a n n ) r ‖ T 〈 s 〉 W f ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) .

Consequently, we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) r ‖ T 〈 s 〉 W f ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) .

Corollary 5.6. Under the conditions of Theorem 5.5, if w is a Freud-type weight, then

E n , p ; s ( W ; f ) ≤ C ( a n n ) r ‖ W f ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) .

Equivalently,

E n , p ; s ( W ; f ) ≤ C ( a n n ) r E n − r , p ; s ( W ; f ( r , r , ⋯ , r ) ) .

Let 1 ≤ p ≤ ∞ . For ‖ W f ‖ L p ( ℝ s ) < ∞ we define the K-functional K r , p ( W ; f , δ ) by

K r , p ; s ( W ; f , δ ) : = i n f g { ‖ W ( f − g ) ‖ L p ( ℝ s ) + δ r ‖ W g ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) } ,

where the infimum is over all functions g ( r − 1, r − 1, ⋯ , r − 1 ) which are absolutely

continuous and ‖ W g ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) < ∞ . We have the following.

Theorem 5.7. 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 ) ≤ c ( n a n ) 2 / 3 , j = 1 , 2 , ⋯ , s .

Let 1 ≤ p ≤ ∞ , and let ‖ T 〈 s 〉 W f ‖ L p ( ℝ s ) < ∞ . Then we have

E n , p ; s ( W , f ) ≤ K r , p ; s ( T 〈 s 〉 W , f , a n n ) .

Proof. We take g as

‖ T 〈 s 〉 W ( f − g ) ‖ L p ( ℝ ) + δ r ‖ T 〈 s 〉 W g ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) ≤ C K r , p ; s ( T 〈 s 〉 W ; f , δ ) ,

and for this g we select P ∈ P n ( ℝ s ) such that

‖ W ( g − P ) ‖ L p ( ℝ s ) ≤ E n , p ; s ( W ; g ) .

Then, from Theorem 5.5 we see

E n , p ; s ( W , f ) ≤ ‖ W ( f − P ) ‖ L p ( ℝ s ) ≤ ‖ W ( f − g ) ‖ L p ( ℝ s ) + ‖ W ( g − P ) ‖ L p ( ℝ s ) ≤ ‖ W ( f − g ) ‖ L p ( ℝ s ) + C E n , p ; s ( W , g ) ≤ ‖ T 〈 s 〉 W ( f − g ) ‖ L p ( ℝ s ) + C ( a n n ) r ‖ T 〈 s 〉 W g ( r , r , ⋯ , r ) ‖ L p ( ℝ s ) ≤ C K r , p ( T 〈 s 〉 W , f , a n n ) . #

Corollary 5.8. Let 1 ≤ p ≤ ∞ , and let w ∈ F ( C 2 + ) be a Freud-type weight. If

‖ W f ‖ L p ( ℝ s ) < ∞ , then we have

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

Let 0 < p ≤ ∞ . Damelin [11] gives a K-functional as follows:

K ¯ r , p ( f , w , t r ) : = i n f P ∈ P n { ‖ w ( f − P ) ‖ L p ( ℝ ) + t r ‖ P ( r ) Φ t r w ‖ L p ( ℝ ) } ,

where t > 0 and r ≥ 1 are chosen in advance and

n = n ( t ) : = inf { k ; a k k ≤ t } .

We recall the r-th order of the modulus of smoothness ω r , p ( w ; f ; t ) , which is defined as follows (cf. [6] and [11] ). Let r be a positive integer, and let 0 < p ≤ ∞ . We set

Δ h r ( f , x ) : = ∑ i = 0 r ( r i ) ( − 1 ) i f ( x + r h 2 − i h ) , x ∈ ℝ .

For the Freud-type weight,

ω ¯ r , p ( f , w , t ) : = ( 1 t ∫ 0 t ‖ w Δ h r ( f , x ) ‖ L p ( | x | ≤ σ ( 2 t ) ) d h ) 1 / p + i n f P ∈ P r − 1 ‖ ( f − P ) w ‖ L p ( | x | ≤ σ ( 4 t ) ) .

For the Erdös-type weight,

ω ¯ r , p ( f , w , t ) : = ( 1 t ∫ 0 t ‖ w Δ h Φ t ( x ) r ( f , x ) ‖ L p ( | x | ≤ σ ( 2 t ) ) d h ) 1 / p + i n f P ∈ P r − 1 ‖ ( f − P ) w ‖ L p ( | x | ≤ σ ( 4 t ) ) ,

where

Φ t ( x ) : = | 1 − | x | σ ( t ) | + 1 T ( σ ( t ) ) .

We remark that if T ( x ) is bounded then we see Φ t ( x ) ~ 1 . So, we may consider for only the Erdös-type weight. Then the following proposition holds.

Proposition 5.9 ( [11] , Theorem 1.2, 1.3). Let 0 < p ≤ ∞ , r ≥ 1 , and let w ∈ E 1 (contains F ( C 2 + ) ). Let f : ℝ → ℝ for which w f ∈ L p ( ℝ ) (for p = ∞ , we require f to be continuous, and f w to vanish at ± ∞ ). Then we have

E n , p ( w , f ) ≤ C 1 ω ¯ r , p ( f , w , C 2 a n n ) ~ K ¯ r , p ( f , w , t r ) .

On ℝ s we define

Ω r , p ( f , W , t ) : = max i = 1 , ⋯ , s ‖ W i ω ¯ r , p ( f X ^ i , w i , t ) ‖ L p ( ℝ s − 1 ) ,

K r , p ( f , W , t r ) : = max i = 1 , ⋯ , s ‖ W i K ¯ r , p ( f X ^ i , w i , t r ) ‖ L p ( ℝ s − 1 ) .

We see Ω r , p ( f , W , t ) ~ K r , p ( f , W , t r ) . Then we have the following:

Theorem 5.10. We suppose

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

T i ( a n ) ≤ c ( n a n ) 2 / 3 , i = 1 , 2 , ⋯ , s .

Then

E n , p ; s ( W , f ) ≤ C Ω r , p ( f , T 〈 s 〉 W , t ) ~ K r , p ( f , T 〈 s 〉 W , t 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 Proposition 3.3 (2) and (3.2),

E n , p ; s ( W , f ) ≤ ‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( ∏ k = 1 j − 1 T k 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ] ≤ C ∑ j = 1 s ‖ W j ( ∏ k = 1 j − 1 T k 1 / 4 ) E n , p ( T j 1 / 4 w j , f X ^ i ) ‖ L p ( ℝ j s − 1 ) ≤ C ∑ j = 1 s ‖ W j ( ∏ 1 ≤ k ≤ s , k ≠ j T k 1 / 4 ) ω ¯ r , p ( f X ^ j , T j 1 / 4 w j , a n n ) ‖ L p ( ℝ j s − 1 ) ~ ∑ j = 1 s ‖ W j ( ∏ 1 ≤ k ≤ s , k ≠ j T k 1 / 4 ) K ¯ r , p ( f X ^ j , T J 1 / j w j , ( a n n ) r ) ‖ L p ( ℝ j s − 1 ) ≤ C Ω r , p ( f , T 〈 s 〉 W , t ) ~ K r , p ( f , T 〈 s 〉 W , t r ) . #

We define the modulus of continuity. For the Freud-type weight W (all of weights w i are Freud-type), we define

ω p , j * ( f X ^ j , w j , t ) : = sup 0 < h ≤ t ‖ w j ( x ) ( Δ h f X ^ j ( x ) ) ‖ L p ( | x | ≤ σ j ( 2 t ) )   + ‖ ( f X ^ j ( x ) − f X ^ j ( 0 ) ) w j ( x ) ‖ L p ( | x | ≤ σ j ( 4 t ) ) .

If W is Erdös-type (some weights w i are Erdös-type), then we define

ω p , j * ( f X ^ j , w j , t ) : = sup 0 < h ≤ t ‖ w j ( x ) ( Δ h Φ t , j ( x ) f X ^ j ( x ) ) ‖ L p ( | x | ≤ σ j ( 2 t ) )   + ‖ ( f X ^ j ( x ) − f X ^ j ( 0 ) ) w j ( x ) ‖ L p ( | x | ≤ σ j ( 4 t ) ) .

It is sufficient to consider only the modulus for the Erdös-type.

Assumption 6.1. Let w i ∈ F ( C 2 + ) ( i = 1 , 2 , ⋯ , s ) , and let a n ( i ) ~ a n , i = 1 , 2 , ⋯ , s . Suppose that f is continuous on ℝ s , and f X ^ i ( x ) has a bounded variation on any compact interval in ℝ s with

∫ ℝ i s − 1 W i ∫ ℝ w i ( x ) | d f X ^ i ( x ) | d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s < ∞ , i = 1 , 2 , ⋯ , s , (6.1)

and if p = ∞ , then we further suppose

| W ( X ) f X ^ i ( x i ) | → 0 , as | x i | → ∞ , i = 1 , 2 , ⋯ , s . (6.2)

In (6.7) and (6.8) below, we put t = a n / n , where a n ~ a n ( i ) , i = 1 , 2 , ⋯ , s . Especially, in (6.8) we set

‖ W i ω ∞ , i * ( f X ^ i , w i ; a n ( i ) n ) ‖ L ∞ ( ℝ i s − 1 ) = o n ( 1 ) . (6.3)

Theorem 6.2. We suppose w j = exp ( − Q j ) ∈ F ( C 2 + ) , j = 1 , 2 , ⋯ , s , and let (3.5) hold. Let n ≥ 1 . Then we have

‖ W ( f − v n [ s ] ( f ) ) ∏ k = 1 s T k 1 / 4 ‖ L 1 ( ℝ s ) ≤ ∑ j = 1 s C j ‖ W j ω p , j * ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) . (6.4)

Now Assumption 6.1 holds. Then we have

∑ j = 1 s C j ‖ W j ω p , j * ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) ≤ C o n ( 1 ) 1 − 1 / p ( a n n ) 1 p . (6.5)

Proof of (6.4). By Proposition 3.3 (1) and Proposition 3.4, we get

‖ w 1 ( x ) ( f X ^ 1 − v n , 1 ( f X ^ 1 ) ) T 1 1 / 4 ( x ) ‖ L p ( ℝ ) ≤ C E 1 , n ( f X ^ 1 , w 1 ) ≤ C 1 ω p , 1 * ( f X ^ 1 , w 1 ; c 1 a n ( 1 ) n ) ,

where the constant C 1 and c 1 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 ( ℝ ) ≤ C j ω p , j * ( f X ^ j , w j ; c j a n ( j ) n ) . (6.6)

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 3.7 (1) and (6.6),

‖ W ( f − v n [ s ] ( f ) ) T 〈 s 〉 ‖ L p ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) T 〈 1 〉 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) T 〈 j 〉 ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) T 1 1 / 4 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( { f − v n , j ( f ) } ) / T j 1 / 4 T 〈 j − 1 〉 ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) T 1 1 / 4 ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( f − v n , j ( f ) ) T j 1 / 4 ‖ L p ( ℝ s ) ] ≤ C 1 ‖ W 1 ω p , 1 * ( f X ^ 1 , w 1 ; c 1 a n ( 1 ) n ) ‖ L p ( ℝ 1 s − 1 ) + ∑ j = 2 s C j ‖ W j ω p , j * ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) ≤ ∑ 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 ( ℝ j s − 1 ) .

Hence, we have (6.4). #

To prove Theorem 6.2 (6.5) we need the following two theorems.

Theorem 6.3 (cf. [12] , Proposition 3.2). Let W = ∏ i = 1 s w i , w i ∈ F ( C 2 + ) , i = 1 , 2 , ⋯ , s . Let f hold Assumption 6.1, especially (6.1) and (6.2) hold. Then there exists a constant C > 0 such that for every t > 0 and i = 1 , 2 , ⋯ , s ,

‖ W i ω 1 , i * ( f X ^ i , w i ; t ) ‖ L 1 ( ℝ i s − 1 ) ≤ C t ∫ ℝ i s − 1 W i ∫ ℝ w i ( x ) | d f X ^ i ( x ) | d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s (6.7)

and

lim t → 0 ‖ W i ω ∞ , i * ( f X ^ i , w i ; t ) ‖ L ∞ ( ℝ i s − 1 ) = 0. (6.8)

Proof. Let g X ^ i ( x ) : = f X ^ i ( x ) − f X ^ i ( 0 ) . For t > 0, we write σ i ( t ) = a u ( i ) , and let

0 < h ≤ t and | x = x i | ≤ σ i ( 2 t ) < a u ( i ) . Since Φ t ( x ) ≤ 2 for | x | ≤ σ i ( 2 t ) , if we take t small enough, then

∫ | x | ≤ σ i ( 2 t ) | { g X ^ i ( x + h 2 Φ t ( x ) ) − g X ^ i ( x − h 2 Φ t ( x ) ) } w ( x ) | d x ≤ ∫ | x | ≤ σ i ( 2 t ) ∫ x − h 2 Φ t ( x ) x + h 2 Φ t ( x ) w i ( y ) | d g X ^ i ( y ) | d x ≤ ∫ ℝ ∫ x − h x + h w i ( y ) | d g X ^ i ( y ) | d x ≤ ∫ ℝ w i ( y ) ∫ y − h y + h d x | d g X ^ i ( y ) | ≤ h ∫ ℝ w i ( y ) | d f X ^ i ( y ) | . (6.9)

On the other hand, by Proposition 4.2 with p = 1 ,

∫ σ i ( 4 t ) ∞ | { f X ^ i ( x ) − f X ^ i ( 0 ) } w i ( x ) | d x ≤ 1 Q ′ i ( σ i ( 4 t ) ) ∫ σ i ( 4 t ) ∞ Q ′ i ( x ) | g X ^ i ( x ) | w i ( x ) d x ≤ C t ∫ 0 ∞ Q ′ i ( x i ) w i ( x i ) ∫ 0 x | d f X ^ i ( y ) | d x i ≤ C t ∫ 0 ∞ ( ∫ y ∞ Q ′ i ( x i ) w i ( x i ) d x ) | d f X ^ i ( y ) | = C t ∫ 0 ∞ w i ( y ) | d f X ^ i ( y ) | .

Similarly,

∫ − ∞ − σ i ( 4 t ) | { f X ^ i ( x ) − f X ^ i ( 0 ) } w i ( x ) | d x ≤ C t ∫ − ∞ 0 w i ( y ) | d f X ^ i ( y ) | .

Therefore, we see

∫ − ∞ ∞ | { f X ^ i ( x ) − f X ^ i ( 0 ) } w i ( x ) | d x ≤ C t ∫ − ∞ ∞ w i ( y ) | d f X ^ i ( y ) | . (6.10)

Consequently, from (6.9) and (6.10) we have

‖ W i ω 1 , i * ( f X ^ i , w i ; t ) ‖ L 1 ( ℝ i s − 1 ) ≤ C t ∫ ℝ i s − 1 W i ∫ ℝ w i ( x ) | d f X ^ i ( x ) | d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s ,

that is, we have (6.7). We show (6.8). Let ε > 0 , | x | ≥ L for large L > 0 . Then

| W i w i ( x ) ( f X ^ i ( x + h 2 Φ t ( x ) ) − f X ^ i ( x − h 2 Φ t ( x ) ) ) | ≤ 2 C | W i w i ( x ) f X ^ i ( x ) | ≤ 2 C ε .

Next, for | x | ≤ L we see

| W i w i ( x ) ( f X ^ i ( x + h 2 Φ t ( x ) ) − f X ^ i ( x − h 2 Φ t ( x ) ) ) | → 0 as t → 0,

hence for 0 < t ≤ t 0 small enough,

| W i w i ( x ) ( f X ^ i ( x + h 2 Φ t ( x ) ) − f X ^ i ( x − h 2 Φ t ( x ) ) ) | ≤ ε .

On the other hand,

‖ W i w i ( x ) ( f X ^ i ( x ) − f X ^ i ( 0 ) ) ‖ L ∞ ( | x | ≥ σ i ( 4 t ) ) ≤ ‖ W i w i ( x ) f X ^ i ( x ) ‖ L ∞ ( | x | ≥ σ i ( 4 t ) ) + | f X ^ i ( 0 ) | ‖ W i w i ( x ) ‖ L ∞ ( | x | ≥ σ i ( 4 t ) ) → 0 as t → 0.

Hence for 0 < t ≤ t 0 small enough,

‖ W ( X ) ( f X ^ i ( x ) − f X ^ i ( 0 ) ) ‖ L ∞ ( | x | ≥ σ i ( 4 t ) ) ≤ ε .

Consequently, we have

l i m t → 0 ‖ W i ω ∞ , i * ( f X ^ i , w i ; t ) ‖ L ∞ ( ℝ i s − 1 ) = 0,

that is, (6.8). #

Theorem 6.4. Under Assumption 6.1, we have

∫ ℝ i s − 1 | W i ω p , i * ( f X ^ i , w i : t ) | p D X i ≤ C o t ( 1 ) p − 1 t ,

where

D X i : = d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s , i = 1 , 2 , ⋯ , s

and

o t ( 1 ) : = ‖ W i ω ∞ , i * ( f X ^ i , w i ; t ) ‖ L ∞ ( ℝ i s − 1 ) .

Proof. Let g X ^ i ( x ) : = f X ^ i ( x ) − f X ^ i ( 0 ) .

∫ | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | p d x i ≤ [ sup | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | p − 1 + sup | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | p − 1 ] × [ ∫ | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | d x i + ∫ | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | d x i ]

≤ [ sup | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | + sup | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | ] p − 1 × [ ∫ | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | d x i + ∫ | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | d x i ] = ω ∞ , i * ( f X ^ i , w i ; t ) p − 1 × ω 1 , i * ( f X ^ i , w i ; t ) .

Similarly,

∫ | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | p d x i ≤ [ sup | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | p − 1 + sup | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | p − 1 ] × [ sup | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | + ∫ | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | d x i ]

≤ [ sup | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | + sup | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | ] p − 1 × [ ∫ | x i | ≥ σ i ( 4 t ) | { f X ^ i ( x i ) − f X ^ i ( 0 ) } w i ( x i ) | d x i + ∫ | x i | ≤ σ i ( 2 t ) | { g X ^ i ( x i + h 2 Φ t ( x i ) ) − g X ^ i ( x i − h 2 Φ t ( x i ) ) } w i ( x i ) | d x i ] = ω ∞ , i * ( f X ^ i , w i ; t ) p − 1 × ω 1 , i * ( f X ^ i , w i ; t ) .

Hence,

ω p , i * ( f X ^ i , w i , t ) ≤ C ( ω ∞ , i * ( f X ^ i , w i , t ) p − 1 × ω 1 , i * ( f X ^ i , w i , t ) ) ,

Consequently, we have

∫ ℝ i s − 1 | W i ω p , i * ( f X ^ i , w i ; t ) | p D X i ≤ C ∫ ℝ i s − 1 { W i ω ∞ , i * ( f X ^ i , w i ; t ) } p − 1 { W i ω 1 , i * ( f X ^ i , w i ; t ) } D X i ≤ C ‖ W i ω ∞ , i * ( f X ^ i , w i ; t ) ‖ L ∞ ( ℝ i s − 1 ) p − 1 ∫ ℝ i s − 1 W i ω 1 , i * ( f X ^ i , w i ; t ) D X i ≤ C o t ( 1 ) p − 1 t ,

where

o t ( 1 ) : = ‖ W i ω ∞ , i * ( f X ^ i , w i ; t ) ‖ L ∞ ( ℝ i s − 1 ) . #

Proof of Theorem 6.2 (6.5). Using Theorems 6.3 and 6.4 with t = a n / n , we easily obtain (6.5). #

We will give an analogy of Theorem 6.2. To do so we use the following weights . They are guaranteed by Proposition 2.3.

w i ( x i ) T i 1 / 4 ( x i ) ~ w i $ ∈ F ( C 2 + ) , i = 1 , 2 , ⋯ , s , W $ : = ∏ i = 1 s w i $ , W i $ : = ∏ 1 ≤ j ≤ s , j ≠ i w j $ . (6.11)

Assumption 6.5. Let w i ∈ F ( C 3 + ) , 0 < λ < 3 / 2 ( i = 1 , ⋯ , s ) . Suppose that f is continuous on ℝ s , and f X ^ i ( x ) has a bounded variation on any compact interval in ℝ s with

∫ ℝ i s − 1 W i $ ∫ ℝ w i $ ( x ) | d f X ^ i ( x ) | d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s < ∞ , i = 1 , 2 , ⋯ , s , (6.12)

and if p = ∞ , then we further suppose

| W $ ( X ) f X ^ i ( x i ) | → 0 , as | x i | → ∞ , i = 1 , 2 , ⋯ , s , (6.13)

where the weights w i $ , W $ and W i $ are defined by (6.11).

Theorem 6.6. 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 . (6.14)

Let n ≥ 1,1 ≤ p ≤ ∞ . Then we have

‖ W ( f − v n ( s ) ( f ) ) ‖ L p ( ℝ s ) ≤ ∑ 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 ( ℝ j s − 1 ) . (6.15)

Now Assumption 6.5 holds. Then we have

∑ 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 ( ℝ j s − 1 ) ≤ C o n ( 1 ) 1 − 1 / p ( a n n ) 1 p . (6.16)

Proof of (6.15). By Proposition 3.3 (2) and Proposition 3.4, we get

‖ w 1 ( x ) ( f X ^ 1 − v n , 1 ( f X ^ 1 ) ) ‖ L p ( ℝ ) ≤ C E p , n ( 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 are independent of X ^ 1 . Similarly, for j = 1 , 2 , ⋯ , s ,

‖ w j ( x ) ( f X ^ j − v n , j ( f X ^ j ) ) ‖ L p ( ℝ ) ≤ C j ω p , j * ( f X ^ j , T j 1 / 4 w j ; c j a n ( j ) n ) . (6.17)

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 3.7 (2) and (6.17),

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ ‖ W ( f − v n [ 1 ] ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W v n [ j − 1 ] ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ] ≤ C [ ‖ W ( f − v n , 1 ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( ∏ k = 1 j − 1 T k 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ 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 ( ℝ j 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 ( ℝ j s − 1 )

by (3.11)

≤ ∑ 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 ( ℝ j s − 1 ) ,

where ∏ k = 1 j − 1 T k 1 / 4 = 1 for j = 1 , that is, we have (6.15).

(6.16) follows from the following theorems. #

In Theorems 6.3 and 6.4 we replace w i , W with w i $ , W $ of (6.11), where w i ∈ F λ ( C 3 + ) , 0 < λ < 3 / 2 ( i = 1 , 2 , ⋯ , s ) , then we easily have the following Theorem 6.7.

Theorem 6.7 (cf. [Theorem 6.3 in this paper]). Let W $ = ∏ i = 1 s w i $ , w i $ ∈ F ( C 2 + ) , i = 1 , 2 , ⋯ , s . Let f hold Assumption 6.5, especially (6.18) and (6.19) hold. Then there exists a constant C > 0 such that for every t > 0 and i = 1 , 2 , ⋯ , s ,

‖ W i $ ω 1 , i * ( f X ^ i , w i $ ; t ) ‖ L 1 ( ℝ i s − 1 ) ≤ C t ∫ ℝ i s − 1 W i $ ∫ ℝ w i $ ( x ) | d f X ^ i ( x ) | d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s , (6.18)

and

lim t → 0 ‖ W i $ ω ∞ , i * ( f X ^ i , w i $ ; t ) ‖ L ∞ ( ℝ i s − 1 ) = 0. (6.19)

Theorem 6.8. Under Assumption 6.5, we have

∫ ℝ i s − 1 | W i $ ω p , i * ( f X ^ i , w i $ ; t ) | p D X i ≤ C o t ( 1 ) p − 1 t ,

where

D X i : = d x 1 ⋯ d x i − 1 d x i + 1 ⋯ d x s , i = 1,2, ⋯ , s

and

o t ( 1 ) : = ‖ W i $ ω ∞ , i * ( f X ^ i , w i $ ; t ) ‖ L ∞ ( ℝ i s − 1 ) .

Through this section we consider the weight w = e x p ( − Q ) ∈ F λ ( C 3 + ) .

Theorem 7.1. (1) We suppose w j = exp ( − Q j ) ∈ F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , j = 1 , 2 , ⋯ , s , and let (3.6) hold. Let n ≥ 1, 1 ≤ p ≤ ∞ . Then we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ ∑ 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 ( ℝ j s − 1 ) , (7.1)

where ℝ j s − 1 is defined in (3.8). Now, we suppose that f is continuous on ℝ s . Let ‖ W δ f ‖ L ∞ ( ℝ ) < ∞ for some 0 < δ < 1 , and let W δ f ∈ L i p ( α ) for some 0 < α ≤ 1 , that is,

| W δ f ( X 1 ) − W δ f ( X 2 ) | ≤ | X 1 − X 2 | α . (7.2)

Then we have

∑ 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 ( ℝ j s − 1 ) ≤ C ∑ i = 1 s ( a n ( i ) n ) α . (7.3)

(2) We suppose w j = e x p ( − Q j ) ∈ F ( C 2 + ) , j = 1 , 2 , ⋯ , s , and hold (3.6). Let n ≥ 1, 1 ≤ p ≤ ∞ . Then we have

‖ W ( f − v n [ s ] ( f ) ) ∏ k = 1 s T k 1 / 4 ‖ L p ( ℝ s ) ≤ ∑ j = 1 s C j ‖ W j ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) . (7.4)

Now, we suppose that f has the conditions as (1). Then we have

∑ j = 1 s C j ‖ W j ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) ≤ C ∑ i = 1 s ( a n ( i ) n ) α . (7.5)

Proof. We suppose 0 < δ < 1 .

(1) (7.2) follows from (3.7). We will show (7.3). Now, we use T 〈 j − 1 〉 : = ∏ i = 1 j − 1 T i 1 / 4 , where T 〈 0 〉 = 1 , and W j : = ∏ 1 ≤ i ≤ s , i ≠ j w i . 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 ( ℝ j s − 1 ) .

From W δ f ∈ L i p ( α ) and

w j ( x ) ~ w ( x + ( h / 2 ) Φ t , j ( x ) ) ~ w ( x − ( h / 2 ) Φ t , j ( x ) ) (see [3] , Lemma 7) we have

( 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 = ( 1 t ∫ 0 t ‖ T j 1 / 4 ( x ) w j ( x ) { f X ^ j ( x + h 2 Φ t , j ( x ) ) − f X ^ j ( x − h 2 Φ t , 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 ) w j δ ( x + h 2 Φ t , j ( x ) ) [ | w j δ ( x + h 2 Φ t , j ( x ) ) f X ^ ( x + h 2 Φ t , j ( x ) ) − w j δ ( x − h 2 Φ t , j ( x ) ) f X ^ ( x − h 2 Φ t , j ( x ) ) | + | { w j δ ( x + h 2 Φ t , j ( x ) ) − w j δ ( x − h 2 Φ t , j ( x ) ) } f X ^ j ( x − h 2 Φ t , j ( x ) ) | ] ‖ L p ( | x | ≤ σ j ( 2 t ) ) p d h ) 1 / p ≤ C ( 1 t ∫ 0 t ‖ T j 1 / 4 ( x ) w j ( 1 − δ ) ( x ) [ | w j δ ( x + h 2 Φ t , j ( x ) ) f X ^ ( x + h 2 Φ t , j ( x ) ) − w j δ ( x − h 2 Φ t , j ( x ) ) f X ^ ( x − h 2 Φ t , j ( x ) ) | + | { w j δ ( x + h 2 Φ t , j ( x ) ) − w j δ ( x − h 2 Φ t , j ( x ) ) } f X ^ j ( x − h 2 Φ t , j ( x ) ) | ] ‖ L p ( | x | ≤ σ j ( 2 t ) ) p d h ) 1 / p = : I 1 + I 2 .

By (7.2) we see

‖ W j T 〈 j − 1 〉 I 1 ‖ L p ( ℝ s − 1 ) = ‖ W j 1 − δ T 〈 j − 1 〉 ( 1 t ∫ 0 t ‖ ( W X ^ j δ f X ^ j ) ( x + h 2 Φ t , j ( x ) ) − ( W X ^ j δ f X ^ j ) ( x − h 2 Φ t , j ( x ) ) ‖ L p ( | x | ≤ σ j ( 2 t ) ) p d h ) 1 / p ‖ L p ( ℝ s − 1 ) ≤ C ‖ W j 1 − δ T 〈 j − 1 〉 ( 1 t ∫ 0 t h p α d h ) 1 / p ‖ L p ( ℝ s − 1 ) ≤ t α ‖ W j 1 − δ T 〈 j − 1 〉 ‖ L p ( ℝ s − 1 ) ≤ C t α .

On the other hand, for I 2 we estimate

= C ‖ W j T 〈 j − 1 〉 ( 1 t ∫ 0 t ‖ T j 1 / 4 ( x ) w j ( 1 − δ ) / 2 ( x ) h Φ t , j ( x ) w j δ ( ξ ) × δ Q ′ j ( ξ ) f X ^ j ( x − h 2 Φ t , j ( x ) ) ‖ L p ( | x | ≤ σ j ( 2 t ) ) p d h ) 1 / p ‖ L p ( ℝ s − 1 ) ,       x − h 2 Φ t , j ( x ) < ξ < x + h 2 Φ t , j ( x ) ,

≤ C ‖ W j 1 − δ T 〈 j − 1 〉 ( 1 t ∫ 0 t h p | w j ( 1 − δ ) / 2 ( x ) Q ′ j ( ξ ) ( W X ^ j δ f X ^ j ) ( x − h 2 Φ t , j ( x ) ) | d h ) 1 / p ‖ L p ( ℝ s ) ,

by the boundedness of T j 1 / 4 ( x ) w j ( 1 − δ ) / 2 ( x )

≤ C t ‖ W δ f ‖ L p ( ℝ s ) ,

by the boundedness of w j ( 1 − δ ) / 2 ( x ) | Q ′ j ( ξ ) | (note | Q ′ | ≤ Q C ) and Lemma 3.10

≤ C t α ( by   w δ f ∈ L p ( ℝ ) ) .

Hence we conclude

‖ 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 ( ℝ j s − 1 ) ≤ C t α . (7.6)

Now, we see

‖ W j T 〈 j − 1 〉 inf c j ( constant ) ‖ ( f X ^ j ( x ) − c j ) w j ( x ) ‖ L p ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ j s − 1 ) ≤ ‖ W j T 〈 j − 1 〉 ‖ f X ^ j ( x ) w j ( x ) ‖ L p ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ j s − 1 ) ≤ w j 1 − δ ( σ j ( 4 t ) ) ‖ W δ f ‖ L p ( ℝ s ) ≤ C w j 1 − δ ( σ j ( 4 t ) ) .

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

w j 1 − δ ( σ j ( 4 t ) ) = exp ( − ( 1 − δ ) Q j ( a u ) ) ~ exp ( − C u T ( a u ) ) ≤ e − u η

for some 0 < η < 1 , that is,

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

Therefore, we have

‖ W j T 〈 j − 1 〉 inf c j ( constant ) ‖ ( f X ^ j ( x ) − c j ) w j ( x ) ‖ L p ( ℝ \ [ − σ j ( 4 t ) , σ j ( 4 t ) ] ) ‖ L p ( ℝ j s − 1 ) ≤ C t . (7.7)

Consequently, by (7.6) and (7.7) we have

‖ W j T 〈 j − 1 〉 ω ¯ p , j ( f X ^ j , w j ; t ) ‖ L p ( ℝ j s − 1 ) ≤ C t α .

So we have (7.3), that is,

∑ j = 1 s C j ‖ W j T 〈 j − 1 〉 ω ¯ p , j ( f X ^ j , w j ; c j a n ( j ) n ) ‖ L p ( ℝ j s − 1 ) ≤ C ∑ i = 1 s ( a n ( i ) n ) α .

(2) (7.4) follows from (3.7). The estimate (7.5) follows as (1). We omit the proof. #

In this section, we treat the differentiable functions.

Let s > 1 , r ≥ 1 be fixed integers, and let w j ∈ F λ ( C 3 + ) , j = 1 , ⋯ , s . We suppose that the multivariate function f ( x 1 , ⋯ , x s ) is r-times partial differentiable, and then with norm:

‖ W f ‖ W r , p : = ∑ i = 1 s ‖ w i T i 1 / 4 D i r f ‖ L p ( ℝ s ) < ∞ ,

where

D i r f : = ∂ r f ∂ r x i , i = 1 , 2 , ⋯ , s .

The class of all functions f ( x 1 , ⋯ , x s ) with ‖ W f ‖ W r , p < ∞ will be denoted by W r , p . In the sequel, if 1 ≤ i ≤ s is an integer, then ‖ f ‖ L p , i ( ℝ ) will denote the L p -norm of f taken with respect to the i-th variable.

Theorem 8.1. We suppose w j = e x p ( − Q j ) ∈ F λ ( C 3 + ) and r ≥ 1 is an integer. Let n ≥ 1, 1 ≤ p ≤ ∞ , and let W f ∈ W r , p . Then we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) r ‖ W f ‖ W r , p , (8.1)

where a n = max { a n ( i ) , i = 1 , 2 , ⋯ , s } .

Remark 8.2. Especially, for r = 1 we have

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) ∑ i = 1 s ‖ w i T i 1 / 4 D i 1 f ‖ L p ( ℝ s ) .

Theorem 8.3 ( [7] , Cororally 8). We suppose w ∈ F ( C 2 + ) . Let 1 ≤ p ≤ ∞ and r ( ≥ 1 ) is an integer. If g ( r − 1 ) be absolutely continuous and w g ( r ) ∈ L p ( ℝ ) , then we have

E p , n ( g ) ≤ C ( a n n ) r ‖ w g ( r ) ‖ L p ( ℝ ) .

Equivalently,

E p , n ( g ) ≤ C ( a n n ) r E p , n − 1 ( g ( r ) ) .

Proof of Theorem 8.1. We use Proposition 2.3, that is, T α w ~ w α ∈ F ( C 2 + ) . In view of Theorem 3.3 (1) and a repeated application of Theorem 8.3, we get

‖ w 1 ( x ) ( f X ^ 1 − v n [ s ] ( f X ^ 1 ) ) ‖ L p ( ℝ ) ≤ E p , n ( T 1 1 / 4 w 1 , f X ^ 1 ) ≤ C ( a n ( 1 ) n ) r ‖ T 1 1 / 4 ( x ) w 1 ( x ) f X ^ 1 ( r ) ‖ L p ( ℝ ) = C 1 ( a n ( 1 ) n ) r ‖ w 1 ( x ) T 1 1 / 4 ( x ) D 1 r f ‖ L p ( ℝ ) ,

where the constant C 1 is independent of X ^ 1 , and D 1 denotes differentiation with respect to the first variable. Similarly, for j = 1 , 2 , ⋯ , s ,

‖ w j ( f X ^ j − v n , j ( f X ^ j ) ) ‖ L p ( ℝ ) ≤ C j ( a n ( j ) n ) r ‖ w j T j 1 / 4 D j r f ‖ L p ( ℝ ) , (8.2)

where D j r denotes the derivative with respect to the j-th variable.

Using Lemma 3.7 and (8.2), we obtain for integer j ≥ 2 ,

‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) = ‖ W v n [ j − 1 ] ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ≤ C ‖ W ( ∏ i = 1 j − 1 T i 1 / 4 ) ( f − v n , j ( f ) ) ‖ L p ( ℝ s ) ≤ ( a n ( j ) n ) r ‖ w j T j 1 / 4 D j r f ‖ L p ( ℝ s ) . (8.3)

From (8.2) and (8.3), we get

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) = ‖ W ( f − v n [ 1 ] ( f ) ) ‖ L p ( ℝ s ) + ∑ j = 2 s ‖ W ( v n [ j − 1 ] ( f ) − v n [ j ] ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) r ∑ j = 1 s ‖ W ( ∏ i = 1 j T i 1 / 4 ( x i ) ) D j r f ‖ L p ( ℝ s ) .

Therefore, we conclude

‖ W ( f − v n [ s ] ( f ) ) ‖ L p ( ℝ s ) ≤ C ( a n n ) r ∑ j = 1 s ‖ W T 〈 s 〉 D j ( r ) f ‖ L p ( ℝ s ) = C ( a n n ) r ‖ W f ‖ W r , p . #

Sakai, R. (2017) A Study of Weighted Polynomial Approximations with Several Variables (I). Applied Mathematics, 8, 1267-1306. https://doi.org/10.4236/am.2017.89095

In this Appendix we state two inequalities which play important roles in the study of approximation theory. In fact, we use Theorem A 1 in the proof of Theorem 5.2. Let a n ( i ) be the MRS number of w i = exp ( − Q i ) .

Theorem A 1 (Markov-Bernstein inequalities). Let 0 < p ≤ ∞ , and let r 1 , ⋯ , r s ≥ 0 be integers. There exists C ≠ C ( n , P ) > 0 such that for n ≥ 1 and P ∈ P n ; s ( ℝ s ) .

(1) if w i ∈ F ( C 2 + ) , then we have

‖ P ( X ) ( r 1 , ⋯ , r j ) W ( X ) ‖ L p ( ℝ s ) ≤ C ∏ i = 1 s ( n ( T i ( a n ( i ) ) ) 1 / 2 a n ( i ) ) r i ‖ P ( X ) W ( X ) ‖ L p ( ℝ s ) .

(2) if w i ∈ F λ ( C 3 + ) , then we have

‖ P ( X ) ( r 1 , ⋯ , r j ) W ( X ) ∏ i = 1 s T i r i / 2 ‖ L p ( ℝ s ) ≤ C ∏ i = 1 s ( n a n ( i ) ) r i ‖ P ( X ) W ( X ) ‖ L p ( ℝ s ) .

The following, so called, Nikolskee-type inequality is useful.

Theorem A 2 (Nikolskii-type inequality). Let

w i = exp ( − Q ) ∈ F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , and let P ∈ P n ; s . For 0 < p ≤ q ≤ ∞ , we have

‖ W P ‖ L p ( ℝ s ) ≤ C ∏ i = 1 s ( a n ( i ) ) 1 p − 1 q ‖ W P ‖ L q ( ℝ s ) ,

and for 1 ≤ q < p ≤ ∞ , we have

‖ ∏ i = 1 s T i 1 2 ( 1 p − 1 q ) W P ‖ L p ( ℝ s ) ≤ C ∏ i = 1 s ( n a n ( i ) ) 1 q − 1 p ‖ W P ‖ L q ( ℝ ) .

To prove Theorem A 1 we need the Proposition 5.3.

Proof of Theorem A 1 . To prove (1), we use the second inequality in Proposition 5.3, repeatedly. Then we easily obtain the result.

(2) Using the first inequality in Proposition 5.3, repeatedly, we have the result. #

The proof of Theorem A 2 is obtained by repeatedly using the following proposition.

Proposition A₃ ( [8] , Theorem 18). Let w = exp ( − Q ) ∈ F λ ( C 3 + ) ( 0 < λ < 3 / 2 ) , and let P ∈ P n . For 0 < p ≤ q ≤ ∞ , we have

‖ w P ‖ L p ( ℝ ) ≤ C a n 1 p − 1 q ‖ w P ‖ L q ( ℝ ) ,

and for 1 ≤ q < p ≤ ∞ , we have

‖ T 1 2 ( 1 p − 1 q ) w P ‖ L p ( ℝ ) ≤ C ( n a n ) 1 q − 1 p ‖ w P ‖ L q ( ℝ ) .