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Mean and uniform convergence of Lagrange interpolation with the Erdős-type weights

Journal of Inequalities and Applications20122012:237

https://doi.org/10.1186/1029-242X-2012-237

Received: 10 April 2012

Accepted: 2 October 2012

Published: 17 October 2012

Abstract

Let R = ( , ) , and let Q C 1 ( R ) : R R + : = [ 0 , ) be an even function. We consider the exponential-type weights w ( x ) = e Q ( x ) , x R . In this paper, we obtain a mean and uniform convergence theorem for the Lagrange interpolation polynomials L n ( f ) in L p , 1 < p with the weight w.

MSC:41A05.

Keywords

exponential-type weightLagrange interpolation polynomial

1 Introduction and preliminaries

Let R = ( , ) , and let Q C 1 ( R ) : R R + : = [ 0 , ) be an even function, and w ( x ) = exp ( Q ( x ) ) be the weight such that 0 x n w 2 ( x ) d x < for all n = 0 , 1 , 2 ,  . Then we can construct the orthonormal polynomials p n ( x ) = p n ( w 2 ; x ) of degree n with respect to w 2 ( x ) . That is,
p n ( x ) p m ( x ) w 2 ( x ) d x = δ m n ( Kronecker’s delta )
and
p n ( x ) = γ n x n + , γ n > 0 .
We denote the zeros of p n ( x ) by
< x n , n < x n 1 , n < < x 2 , n < x 1 , n < .
We denote the Lagrange interpolation polynomial L n ( f ; x ) based at the zeros { x k , n } k = 1 n as follows:
L n ( f ; x ) : = k = 1 n f ( x k , n ) l k , n ( x ) , l k , n ( x ) : = p n ( x ) ( x x k , n ) p n ( x k , n ) .

A function f : R + R + is said to be quasi-increasing if there exists C > 0 such that f ( x ) C f ( y ) for 0 < x < y .

We are interested in the following subclass of weights from [1].

Definition 1.1 Let Q : R R + be an even function satisfying the following properties:
  1. (a)

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

     
  2. (b)

    Q ( x ) exists and is positive in R { 0 } .

     
  3. (c)

    lim x Q ( x ) = .

     
  4. (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 } .
  1. (e)
    There exists C 1 > 0 such that
    Q ( x ) | Q ( x ) | C 1 | Q ( x ) | Q ( x ) , a.e.  x R { 0 } .
     
Then we write w ( x ) = exp ( Q ( x ) ) F ( C 2 ) . If there also exist 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 R J ,

then we write w ( x ) = exp ( Q ( x ) ) F ( C 2 + ) .

Example 1.2 (1) If T ( x ) is bounded, then the weight w = exp ( Q ) is called the Freud-type weight. The following example is the Freud-type weight:
Q ( x ) = | x | α , α > 1 .
If T ( x ) is unbounded, then the weight w = exp ( Q ) is called the Erdős-type weight. The following examples give the Erdős-type weights w = exp ( Q ) .
  1. (2)
    [2, Theorem 3.1] For α > 1 , l = 1 , 2 , 3 ,
    Q ( x ) = Q l , α ( x ) = exp l ( | x | α ) exp l ( 0 ) ,
     
where
exp l ( x ) = exp ( exp ( exp exp x ) ) ( l -times ) .
More generally, we define for α + u > 1 , α 0 , u 0 and l 1 ,
Q l , α , u ( x ) : = | x | u ( exp l ( | x | α ) α exp l ( 0 ) ) ,
where α = 0 if α = 0 , otherwise α = 1 . (We note that Q l , 0 , u ( x ) gives a Freud-type weight.)
  1. (3)

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

     

In this paper, we investigate the convergence of the Lagrange interpolation polynomials with respect to the weight w F ( C 2 + ) . When we consider the Erdős-type weights, the following definition follows from Damelin and Lubinsky [3].

Definition 1.3 Let w ( x ) = exp ( Q ( x ) ) , where Q : R R is even and continuous. Q exists in ( 0 , ) , Q ( j ) 0 , in ( 0 , ) , j = 0 , 1 , 2 , and the function
T ( x ) : = 1 + x Q ( x ) Q ( x )
is increasing in ( 0 , ) with
lim x T ( x ) = ; T ( 0 + ) : = lim x 0 + T ( x ) > 1 .
(1.1)
Moreover, we assume that for some constants C 1 , C 2 , C 3 > 0 ,
C 1 T ( x ) / ( x Q ( x ) Q ( x ) ) C 2 , x C 3 ,
and for every ε > 0 ,
T ( x ) = O ( Q ( x ) ε ) , x .
(1.2)

Then we write w E .

Damelin and Lubinsky [3] got the following results with the Erdős-type weights w = exp ( Q ) E .

Theorem A ([3, Theorem 1.3])

Let w = exp ( Q ) E . Let L n ( f , x ) denote the Lagrange interpolation polynomial to f at the zeros of p n ( w 2 , x ) . Let 1 < p < , Δ R , κ > 0 . Then for
lim n ( f L n ( f ) ) w ( 1 + Q ) Δ L p ( R ) = 0
to hold for every continuous function f : R R satisfying
lim | x | | f ( x ) w ( x ) ( log | x | ) 1 + κ | = 0 ,
it is necessary and sufficient that
Δ > max { 0 , 2 3 ( 1 4 1 p ) } .

Our main purpose in this paper is to give mean and uniform convergence theorems with respect to { L n ( f ) } , n = 1 , 2 ,  , in L p -norm, 1 < p . The proof for 1 < p < will be shown by use of the method of Damelin and Lubinsky. In Section 2, we write the main theorems. In Section 3, we prepare some fundamental lemmas; and in Section 4, we will prove the theorem for 1 < p < . Finally, we will prove the theorem for the uniform convergence in Section 5.

For any nonzero real-valued functions f ( x ) and g ( x ) , we write f ( x ) g ( x ) if there exist constants C 1 , C 2 > 0 independent of x such that C 1 g ( x ) f ( x ) C 2 g ( x ) for all x. Similarly, for any two sequences of positive numbers { c n } n = 1 and { d n } = 1 , we define c n d n . We denote the class of polynomials of degree at most n by P n .

Throughout C , C 1 , C 2 , denote positive constants independent of n, x, t, and polynomials of degree at most n. The same symbol does not necessarily denote the same constant in different occurrences.

2 Theorems

In the following, we introduce useful notations. Mhaskar-Rakhmanov-Saff numbers (MRS) a x are defined as the positive roots of the following equations:
x = 2 π 0 1 a x u Q ( a x u ) ( 1 u 2 ) 1 2 d u , x > 0 .
The function φ u ( x ) is defined as follows:
φ u ( x ) = { a u u 1 | x | a 2 u 1 | x | a u + δ u , | x | a u , φ u ( a u ) , a u < | x | ,
where
δ x = ( x T ( a x ) ) 2 3 , x > 0 .
We define
Φ ( x ) : = 1 ( 1 + Q ( x ) ) 2 3 T ( x )
and
Φ n ( x ) : = max { δ n , 1 | x | a n } .
Here we note that for 0 < d | x | ,
Φ ( x ) Q ( x ) 1 3 x Q ( x )
and we see
Φ ( x ) C Φ n ( x ) , n 1
(see Lemma 3.3 below). Moreover, we define
Φ ( 1 4 1 p ) + ( x ) : = { 1 , 0 < p < 4 , Φ 1 4 1 p ( x ) , 4 p .

Let 1 < p < . We give a convergence theorem as an analogy of Theorem A for L n ( f ) in L p -norm. We need to prepare a lemma.

Lemma 2.1 ([4, Theorem 1.6])

Let w = exp ( Q ) F ( C 2 + ) .
  1. (a)
    Let T ( x ) be unbounded. Then for any η > 0 , there exists a constant C ( η ) > 0 such that for t 1 ,
    a t C ( η ) t η .
     
  2. (b)
    Assume
    Q ( x ) Q ( x ) λ ( b ) Q ( x ) Q ( x ) , | x | b > 0 ,
    (2.1)
     
where b > 0 is large enough. Suppose that there exist constants η > 0 and C 1 > 0 such that a t C 1 t η . If λ : = λ ( b ) > 1 , then there exists a constant C ( λ , η ) such that for a t 1 ,
T ( a t ) C ( λ , η ) t 2 ( η + λ 1 ) λ + 1 .
(2.2)
If 0 < λ 1 , then for any μ > 0 , there exists C ( λ , μ ) such that
T ( a t ) C ( λ , μ ) t μ , t 1 .
(2.3)
For a fixed constant β > 0 , we define
ϕ ( x ) : = ( 1 + x 2 ) β / 2 .
(2.4)

Using this function, we have the following theorem. We suppose that the weight w is the Erdős-type weight.

Our theorem is as follows. Let f C 0 ( R ) mean that f C ( R ) and lim | x | f ( x ) = 0 .

Theorem 2.2 Let w = exp ( Q ) F ( C 2 + ) , and let T ( x ) be unbounded. Let 1 < p < and β > 0 , and let us define ϕ as (2.4), and λ = λ ( b ) 1 as (2.1). We suppose that for f C ( R ) ,
ϕ 1 ( x ) w ( x ) f ( x ) C 0 ( R ) ,
and
Δ > 9 4 λ 1 3 λ 1 .
(2.5)
Then we have
lim n ( f L n ( f ) ) w Φ Δ + ( 1 4 1 p ) + L p ( R ) = 0 .
We remark that if w F ( C 2 + ) is the Erdős-type weight, then we have λ = λ ( b ) 1 in (2.1). In fact, if λ < 1 , then by Lemma 3.9 below, we see that for x b > 0 ,
T ( x ) = x Q ( x ) Q ( x ) x Q ( x ) Q ( b ) ( Q ( x ) Q ( b ) ) λ = Q ( b ) Q ( b ) λ x Q ( x ) 1 λ 0 as  x .
This contradicts our assumption for T ( x ) . In Example 1.2, we consider the weight w l , α , m = exp ( Q l , α , m ) . In (2.1), we set Q : = Q l , α , m and λ : = λ ( b ) . If w l , α , m is an Erdős-type weight, that is, T ( x ) : = T l , α , m ( x ) is unbounded, then it is easy to show
lim b λ ( b ) = 1 .
Therefore, when we give any Δ > 0 , there exists a constant b large enough such that
Δ > 9 4 λ ( b ) 1 3 λ ( b ) 1 .

Hence, we have the following corollary.

Corollary 2.3 Let 1 < p < and Δ > 0 . Then for the weight w l , α , m = exp ( Q l , α , m ) ( α > 0 ), we have
lim n ( f L n ( f ) ) w l , α , m Φ Δ + ( 1 4 1 p ) + L p ( R ) = 0 .

We also consider the case of p = .

Theorem 2.4 Let w = exp ( Q ) F ( C 2 + ) , and let T ( x ) be unbounded. For every f C 0 ( R ) and n 1 , we have
( f L n ( f ) ) w Φ 3 / 4 L ( R ) C E n 1 ( w ; f ) log n ,
where
E m ( w ; f ) = inf P m P m max x R | ( f ( x ) P m ( x ) ) w ( x ) | , m = 0 , 1 , 2 , .
Moreover, if f ( r ) , r 1 , is an integer, then for n > r + 1 we have
( f L n ( f ) ) w Φ 3 / 4 L ( R ) C ( a n n ) r E n r 1 ( w ; f ( r ) ) log n .

3 Fundamental lemmas

To prove the theorems we need some lemmas.

Lemma 3.1 Let w = exp ( Q ) F ( C 2 + ) . Then we have the following.
  1. (a)
    [1, Lemma 3.11(a), (b)] Given fixed 0 < α , α 1 , we have uniformly for t > 0 ,
    | 1 a α t a t | 1 T ( a t ) ,
     
and we have for t > 0 ,
| 1 a t a s | 1 T ( a t ) | 1 t s | , 1 2 t s 2 .
  1. (b)
    [1, Lemma 3.7 (3.38)] For some 0 < ε 2 , and for large enough t,
    T ( a t ) t 2 ε .
     
Lemma 3.2 Let w = exp ( Q ) F ( C 2 + ) . Then we have the following.
  1. (a)
    [1, Lemma 3.5(a), (b)] Let L > 0 be a fixed constant. Uniformly for t > 0 ,
    Q ( a L t ) Q ( a t ) and Q ( a L t ) Q ( a t ) .
     
Moreover,
a L t a t and T ( a L t ) T ( a t ) .
  1. (b)
    [1, Lemma 3.4 (3.18), (3.17)] Uniformly for x > 0 with a t : = x , t > 0 , we have
    Q ( x ) t T ( x ) a t and Q ( x ) t T ( x ) .
     
  2. (c)
    [1, Lemma 3.8(a)] For x [ 0 , a t ) ,
    Q ( x ) C t a t 1 1 x a t .
     
Lemma 3.3 Let w = exp ( Q ) F ( C 2 + ) . For x R , we have
Φ ( x ) C Φ n ( x ) , n 1 .
Proof Let x = a u , u 1 . By Lemma 3.2(b), we have
u Q ( a u ) T ( a u ) .
So, we have
δ u 1 Q 2 3 ( a u ) T ( a u ) = a u Q ( a u ) Q 1 3 ( a u ) = x Q ( x ) Q 1 3 ( x ) .
(3.1)
Now, if u n 2 , then we have
1 a u a n 1 a n / 2 a n 1 T ( a n ) ( by Lemma 3.1(a) ) 1 ( n T ( a n ) ) 2 3 = δ n ( by Lemma 3.1(b) ) .
So, we have
Φ n ( x ) = 1 a u a n 1 a u a 2 u 1 T ( a u ) ( by Lemma 3.1(a) ) 1 ( u T ( a u ) ) 2 3 = δ u Φ ( x ) ( by Lemma 3.2(b) and (3.1) ) .
Let n 2 < u < n . Then we have
Φ n ( x ) δ n δ u Φ ( x ) ( by Lemma 3.2(a), (b) and (3.1) ) .

 □

Lemma 3.4 Let w F ( C 2 + ) . Then we have the following.
  1. (a)
    [1, Theorem 1.19(f)] For the minimum positive zero x [ n / 2 ] , n ,
    x [ n / 2 ] , n a n n ,
     
and for the maximum zero x 1 , n ,
1 x 1 , n a n δ n .
  1. (b)
    [1, Theorem 1.19(e)] For n 1 and 1 j n 1 ,
    x j , n x j + 1 , n φ n ( x j , n ) .
     
  2. (c)
    [1, p.329, (12.20)] Uniformly for n 1 , 1 k n 1 ,
    φ n ( x k , n ) φ n ( x k + 1 , n ) .
     
  3. (d)
    Let max { | x k , n | , | x k + 1 , n | } a α n , 0 < α < 1 . Then we have
    w ( x k , n ) w ( x k + 1 , n ) w ( x ) ( x k + 1 , n x x k , n ) .
     
So, for given C > 0 and | x | a β n , 0 < β < α , if | x x k , n | C φ n ( x ) , then we have
w ( x ) w ( x k , n ) .
Proof (d) Let max { | x k , n | , | x k + 1 , n | } = | x k , n | (for the case of max { | x k , n | , | x k + 1 , n | } = | x k + 1 , n | , we also have the result similarly). By (b) there exists a constant C > 0 such that
| x k , n x k + 1 , n | C φ n ( x k , n ) .
Then we see
φ n ( x k , n ) a n n 1 | x k , n | a 2 n 1 | x k , n | a n = a n n 1 | x k , n | a n + | x k , n | { 1 a n 1 a 2 n } 1 | x k , n | a n = a n n 1 | x k , n | a n + | x k , n | a n ( 1 a n a 2 n ) 1 | x k , n | a n a n n 1 | x k , n | a n + C | x k , n | a n 1 T ( a n ) 1 | x k , n | a n a n n 1 | x k , n | a n .
(3.2)
Therefore, from (3.2) and Lemma 3.2(c), we have
| Q ( x k , n ) Q ( x k + 1 , n ) | = | Q ( ξ ) ( x k , n x k + 1 , n ) | C | Q ( ξ ) | φ n ( x ) ( x k + 1 , n ξ x k , n ) C | Q ( x k , n ) | a n n 1 | x k , n | a n C n a n 1 1 | x k , n | a n a n n 1 | x k , n | a n C .
Consequently,
w ( x k , n ) w ( x k + 1 , n ) w ( x ) ( x k + 1 , n x x k , n ) .
Let | x x k , n | C φ n ( x ) and | x | a β n . Then we see that there exists n 0 > 0 such that | x k , n | a α n , n n 0 . In fact, we can show it as follows. We use Lemma 3.1(a) and (b). For | x | a β n , we see
| x k , n | | x | + C φ n ( x ) | x | + C a n n 1 | x | a n ,
and if we take n large enough, then we have
d d t ( t + C a n n 1 t a n ) = 1 C 1 n 1 2 1 t a n 1 C 1 n 1 2 1 a n / 3 a n 1 C T ( a n ) 2 n 1 C 1 2 n ε / 2 > 0 ,
that is, g ( t ) = t + C a n n 1 t a n is increasing. So, we see
| x k , n | a β n + C a n n 1 a β n a n a β n + C a n n 1 T ( a n ) .
Therefore, we have
a α n ( a β n + C a n n 1 T ( a n ) ) a n T ( a n ) C a n n 1 T ( a n ) = a n T ( a n ) ( 1 C T ( a n ) n ) a n T ( a n ) ( 1 C 1 n ε / 2 ) > 0 .
Now, we can show (d). Without loss of generality, we may assume x [ x j + 1 , n , x j , n ] { x k , n | | x x k , n | C φ n ( x ) } . We define
x k 1 , n : = min { x k , n | | x x k , n | C φ n ( x ) } , x k 2 , n : = max { x k , n | | x x k , n | C φ n ( x ) } .
Here we note that k 1 , k 2 are decided depending only on the constant C. Then by former result, we have
w ( x k 1 , n ) w ( x k 2 , n ) w ( x ) ( x k 1 , n x x k 2 , n ) .

 □

Lemma 3.5 Let w = exp ( Q ) F ( C 2 + ) . Then we have the following.
  1. (a)
    [1, Theorem 1.17] Uniformly for n 1 ,
    sup x R | p n ( x ) | w ( x ) | x 2 a n 2 | 1 4 1 .
     
  2. (b)
    [1, Theorem 1.19(a)] Uniformly for n 1 and 1 j n ,
    | ( p n w ) ( x j , n ) | φ n 1 ( x j , n ) a n 1 2 ( 1 | x j , n | a n ) 1 4 .
     
  3. (c)
    [1, Theorem 1.19(d)] For x [ x k + 1 , n , x k , n ] , if k n 1 ,
    | p n ( x ) w ( x ) | min { | x x k , n | , | x x k + 1 , n | } a n 1 / 2 φ n ( x ) 1 ( 1 | x k , n | a n ) 1 / 4 .
     

Lemma 3.6 (cf. [5, Theorem 2.7])

Let w F ( C 2 + ) and 0 < p . Then uniformly n 2 ,
Φ ( 1 4 1 p ) + p n w L p ( R ) C a n 1 p 1 2 { 1 , 0 < p < 4 or p = ; log ( 1 + n ) , 4 p ,

where x + = 0 if x 0 , x + = x if x > 0 .

Proof From Lemma 3.3, we know Φ ( x ) Φ n ( x ) , then in [5, Theorem 2.7] we only exchange Φ n with Φ. □

Let f L p , w ( R ) . The Fourier-type series of f is defined by
f ˜ ( x ) : = k = 0 a k ( w 2 , f ) p k ( w 2 , x ) , a k ( w 2 , f ) : = f ( t ) p k ( w 2 , t ) w 2 ( t ) d t .
We denote the partial sum of f ˜ ( x ) by
s n ( f , x ) : = s n ( w 2 , f , x ) : = k = 0 n 1 a k ( w 2 , f ) p k ( w 2 , x ) .
The partial sum s n ( f ) admits the representation
s n ( f , x ) = j = 0 n 1 a j p j ( x ) = f ( t ) K n ( x , t ) w 2 ( t ) d t ,
where
K n ( x , t ) : = j = 0 n 1 p j ( x ) p j ( t ) .
The Christoffel-Darboux formula
K n ( x , t ) = γ n 1 γ n p n ( x ) p n 1 ( t ) p n 1 ( x ) p n ( t ) x t
(3.3)

is well known (see [6, Theorem 1.1.4]).

Lemma 3.7 ([6, Lemma 9.2.6])

Let 1 < p < and g L p ( R ) . Then for the Hilbert transform
H ( g , x ) : = lim ε 0 + | x t | ε g ( t ) x t d t , x R ,
(3.4)
we have
H ( g ) L p ( R ) C g L p ( R ) ,

where C > 0 is a constant depending upon p only.

Lemma 3.8 (see [7, Theorem 1.4, Theorem 1.6])

Let w = exp ( Q ) F ( C 2 ) , 1 p and γ 0 . Then for any ε > 0 , there exists a polynomial P such that
( f ( x ) P ( x ) ) ( 1 + x 2 ) γ w ( x ) L p ( R ) < ε .
Lemma 3.9 Let w F ( C 2 + ) be an Erdős-type weight, that is, T ( x ) is unbounded. Then for any M > 1 , there exist x M > 0 and C M > 0 such that
Q ( x ) C M x M , x x M .
Proof For every M > 1 , there exists x M > 0 such that T ( x ) M for x x M , so that Q ( x ) / Q ( x ) = T ( x ) / x M / x for x x M . Hence, we see
log Q ( x ) Q ( x M ) log ( x x M ) M , x x M ,
that is,
Q ( x ) Q ( x M ) ( x M ) M x M , x x M .

Let us put C M : = Q ( x M ) / ( x M ) M . □

4 Proof of Theorem 2.2 by Damelin and Lubinsky methods

In this section, we assume w F ( C 2 + ) . To prove the theorem we need some lemmas, and we will use the Damelin and Lubinsky methods of [3].

Lemma 4.1 (cf. [3, Lemma 3.1])

Let w F ( C 2 + ) . Let 0 < α < 1 4 and
n ( x ) : = | x k , n | a α n | l k , n ( x ) | w 1 ( x k , n ) .
Then we have for | x | a α n / 2 and | x | a 2 n ,
n ( x ) w ( x ) C .
Moreover, for a α n / 2 | x | a 2 n ,
n ( x ) w ( x ) C ( log n + a n 1 2 | p n ( x ) w ( x ) | T 1 4 ( a n ) ) .

Proof The proof of [3, Lemma 3.1] holds without the condition (1.2) and the second condition in (1.1) and under the assumption of the quasi-increasingness of T ( x ) . The conditions in Definition 1.1 contain all the conditions in Definition 1.3 except for (1.2) and the second condition in (1.1). We see that in [3, Lemma 3.1] we can replace T ( x ) with T ( x ) . □

Lemma 4.2 ([3, Lemma 3.2])

Let 0 < η < 1 . Let ψ : R ( 0 , ) be a continuous function with the following property: For n 1 , there exist polynomials R n of degree n such that
C 1 ψ ( t ) R n ( t ) C 2 , | t | a 4 n .
Then for n n 0 and P P n ,
| x k , n | a η n λ k , n | P ( x k , n ) | w 1 ( x k , n ) ψ ( x k , n ) C a 4 n a 4 n | P ( t ) w ( t ) | ψ ( t ) d t .
Remark 4.3 To prove Lemma 4.7 below, we apply this lemma with ψ ( t ) = ϕ ( t ) = ( 1 + t 2 ) β / 2 , β > 0 . In fact, when ϕ ( x ) = ϕ ( t ) , t = a 4 n x , we can approximate ϕ by polynomials R n P n on [ 1 , 1 ] , that is, for any ε > 0 there exists R n P n such that
| ϕ ( x ) R n ( x ) | < ε , x [ 1 , 1 ] .
Therefore,
| R n ( x ) ϕ ( x ) 1 | < ε ϕ ( x ) , x [ 1 , 1 ] ,
and so there exist C 1 , C 2 > 0 such that
C 1 1 ε ϕ ( x ) | R n ( x ) ϕ ( x ) | < 1 + ε ϕ ( x ) C 2 , x [ 1 , 1 ] .

Now, if we set R n ( t ) = R n ( x ) , then we have the result.

Lemma 4.4 (cf. [3, Lemma 4.1])

Let { f n } n = 1 be a sequence of measurable functions from R R such that for n 1 ,
f n ( x ) = 0 , | x | < a n 9 ; | f n ( x ) | w ( x ) ϕ ( x ) , x R .
Then for 1 p and Δ > 0 , we have
lim n L n ( f n ) w Φ Δ + ( 1 4 1 p ) + L p ( R ) = 0 .
(4.1)
Proof Let | x | a n 18 or | x | a 2 n . We use the first inequality of Lemma 4.1 with α = 1 9 , then from the assumption with respect to f n , we see that
| L n ( f n ; x ) w ( x ) | ϕ ( a n 9 ) | x k , n | a n 9 | l k , n ( x ) | w 1 ( x k , n ) w ( x ) C 1 ϕ ( a n 9 ) .
So,
L n ( f n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 18 o r | x | a 2 n ) ϕ ( a n 9 ) Φ Δ + ( 1 4 1 p ) + L p ( R ) C 2 ϕ ( a n 9 ) = o ( 1 )
(4.2)
by Lemma 3.9 (note the definition of Φ ( x ) ) and the definition of ϕ in (2.4). Next, we let a n 18 | x | a 2 n . From the second inequality in Lemma 4.1, we see that
| L n ( f n ; x ) w ( x ) | ϕ ( a n 9 ) ( log n + a n 1 2 | p n ( x ) | w ( x ) T 1 4 ( a n ) ) .
Also, for this range of x, we see that
Φ ( x ) = 1 ( 1 + Q ( x ) ) 2 3 T ( x ) 1 ( 1 + Q ( a n ) ) 2 3 T ( a n ) T 1 3 ( a n ) n 2 3 T ( a n ) = δ n
by Lemma 3.2(b). So, for n large enough,
Then since Δ > 0 , using Lemma 3.1(a), Lemma 2.1(a), and Lemma 3.6, we have
log n Φ Δ + ( 1 4 1 p ) + L p ( a n 18 | x | a 2 n ) C δ n Δ ( a 2 n a n 18 ) 1 p log n C
and
Therefore, we have by (2.4)
L n ( f n ) w Φ Δ + ( 1 4 1 p ) + L p ( a n 18 | x | a 2 n ) C 4 ϕ ( a n 9 ) = o ( 1 ) .

Consequently, with (4.2) we have (4.1). □

Lemma 4.5 (cf. [3, Lemma 4.2])

Let 1 p . Let { g n } n = 1 be a sequence of measurable functions from R R such that for n 1 ,
g n ( x ) = 0 , | x | a n 9 ; | g n ( x ) | w ( x ) ϕ ( x ) , x R .
(4.3)
Let us suppose
Δ > 9 4 λ 1 3 λ 1 ,
(4.4)
where λ 1 is defined in Lemma 2.1. Then for 1 p , we have
lim n L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) = 0 .
(4.5)
Proof Using Lemma 3.5(b) and Lemma 3.4(b), we have for x a n 8 ,
| L n ( g n ; x ) | | x k , n | a n 9 | l k , n ( x ) | w 1 ( x k , n ) ϕ ( x k , n ) C 1 a n 1 2 | p n ( x ) | | x k , n | a n 9 ( x k , n x k + 1 , n ) ( 1 | x k , n | a n + δ n ) 1 4 | x x k , n | ϕ ( x k , n ) C 2 a n 1 2 | p n ( x ) | a n 9 a n 9 ( 1 | t | a n + δ n ) 1 4 | x t | ϕ ( t ) d t .
(4.6)
Equation (4.6) is shown as follows: First, we see
| x t | | x x k , n | , t [ x k + 1 , n , x k , n ] .
(4.7)
Let | x | a n 8 and t [ x k + 1 , n , x k , n ] . Then
| x t x x k , n 1 | = | t x k , n x x k , n | x k , n x k + 1 , n | x k ± 2 , n x k , n | c < 1 .
Now, we use the fact that x + C φ ( x ) , x > 0 is increasing for 0 < x a n / 2 , and then
x k , n + C φ n ( x k , n ) a n 9 + C φ n ( a n 9 ) a n 8 x .
Here, the second inequality follows from the definition of φ n ( x ) and Lemma 3.1(a), (b). Hence, we have (4.7). Now, we use the monotonicity of ( 1 | x | a n + δ n ) 1 4 ϕ ( x ) . From (4.7) there exists C > 0 such that for t [ x k + 1 , n , x k , n ] ,
( x k , n x k + 1 , n ) ( 1 | x k , n | a n + δ n ) 1 4 | x x k , n | ϕ ( x k , n ) x k + 1 , n x k , n ( 1 | t | a n + δ n ) 1 4 | x x k , n | ϕ ( t ) d t 1 C x k + 1 , n x k , n ( 1 | t | a n + δ n ) 1 4 | x t | ϕ ( t ) d t .
Hence, (4.6) holds. Next, for t [ 0 , a n 9 ] and x a n 8 , we know by Lemma 3.1(a),
1 a n t x t 1 + a n a n 8 a n 8 t 1 + a n a n 8 a n 8 a n 9 1 + C a n 8 a n 9 T ( a n 9 ) T ( a n 8 ) C 3
and
1 | t | a n C 4 1 T ( a n ) δ n .
So, we have
| L n ( g n ; x ) | C 6 a n 1 4 | p n ( x ) | 0 a n 9 ( x t ) 3 4 ϕ ( t ) d t .
Let t = a s , n 9 s 1 . Then, since we know for x a n 8 ,
x t = x ( 1 t x ) a n 8 ( 1 a s a 9 8 s ) C 7 a n T ( a s ) ,
we obtain
| L n ( g n ; x ) | C 8 a n 1 2 | p n ( x ) | 0 a n 9 T 3 4 ( t ) ϕ ( t ) d t C 8 a n 1 2 T 3 4 ( a n ) | p n ( x ) | .
Hence, if 1 λ , then using Lemma 3.6, (3.1) and (2.2), we have
Here, we may consider that above estimations hold under the condition (4.4), because that η > 0 can be taken small enough. Then we have (4.5), that is, for Δ > 9 4 λ 1 3 λ 1 ,
lim n L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) = 0 .

 □

Lemma 4.6 (cf. [3, Lemma 4.3])

Let 1 < p < . Let σ : R R be a bounded measurable function. Let λ = λ ( b ) 1 be defined in Lemma 2.1, and then we suppose
Δ > { 0 , 1 < p 2 ; 3 2 ( λ 1 ) 3 λ 1 p 2 p , 2 < p 4 ; max { λ 1 3 λ 1 p 1 p 1 4 λ + 1 3 λ 1 p 4 p , 0 } , 4 < p .
(4.8)
Then for 1 < p < and the partial sum s n of the Fourier series, we have
s n [ σ ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) C σ L ( R )
(4.9)

for n 1 . Here C is independent of σ and n.

Proof We may suppose that σ L ( R ) = 1 . By (3.3), (3.4) and Lemma 3.5(a),
| s n [ σ ϕ w 1 ] ( x ) | w ( x ) a n 1 2 ( 1 | x | a n ) 1 4 j = n 1 n | H [ σ ϕ p j w ] ( x ) | .
(4.10)
Let us choose l : = l ( n ) such that 2 l n 8 2 l + 1 . Then we know
2 l + 3 n 2 l + 4 .
(4.11)
Define
I k = [ a 2 k , a 2 k + 1 ] , 1 k l + 2 .
For j = n 1 , n and x I k , we split
H [ σ ϕ p j w ] ( x ) w ( x ) = ( 0 + 0 a 2 k 1 + P . V . a 2 k 1 a 2 k + 2 + a 2 k + 2 ) ( σ ϕ p j w ) ( t ) x t d t : = I 1 ( x ) + I 2 ( x ) + I 3 ( x ) + I 4 ( x ) .
(4.12)
Here P . V . stands for the principal value. First, we give the estimations of I 1 and I 2 for x I k . Let x I k . Then we have by Lemma 3.5(a) and Lemma 3.6 with p = 1 ,
| I 1 ( x ) | 0 | ( p j w ϕ ) ( t ) | t + x C 1 a n 1 2 0 a n 2 ϕ ( t ) t + a 2 d t + C 2 a n 1 a n 2 | p j ( t ) | w ( t ) d t C 2 ( a n 1 2 + a n 1 a n 1 1 2 ) C 3 a n 1 2 .
(4.13)
Here we have used
0 ϕ ( t ) 1 + t d t < .
(4.14)
By Lemma 3.5(a), and noting 1 x / a n 1 t / a n for x I k ,
| I 2 ( x ) | 0 a 2 k 1 | ( p j w ϕ ) ( t ) | x t d t C 4 a n 1 2 0 a 2 k 1 ( 1 t a n ) 1 4 x t d t C 4 a n 1 2 ( 1 x a n ) 1 4 0 a 2 k 1 d t x t = C 4 a n 1 2 ( 1 x a n ) 1 4 log ( 1 a 2 k 1 x ) 1 .
Using
1 a 2 k 1 x 1 a 2 k 1 a 2 k C 1 T ( a 2 k ) C 1 T ( x ) ,
we can see
| I 2 ( x ) | C 6 a n 1 2 ( 1 x a n ) 1 4 log ( T ( x ) C ) .
(4.15)
Next, we give an estimation of I 4 for x I k . Let x I k . From Lemma 3.5(a) again,
| I 4 ( x ) | a 2 k + 2 2 a 2 k + 2 | ( p j w ϕ ) ( t ) | t x d t + C 2 a 2 k + 2 | ( p j w ϕ ) ( t ) | t d t ( by  t 2 ( t x ) ) C 7 ( a n 1 2 a 2 k + 2 2 a 2 k + 2 | 1 t a n | 1 4 d t t x + a n 1 2 2 a 2 k + 2 max { 2 a 2 k + 2 , 1 2 a n } ϕ ( t ) t d t + 1 2 a n | ( p j w ) ( t ) | t d t ) C 7 ( a n 1 2 a 2 k + 2 2 a 2 k + 2 | 1 t a n | 1 4 d t t x + C a n 1 2 + a n 1 a n 1 1 2 ) ( by (4.14) and Lemma 3.6 with  p = 1 ) C 8 a n 1 2 [ J + 1 ] ,
(4.16)
where
J : = a 2 k + 2 2 a 2 k + 2 | 1 t a n | 1 4 d t t x .
Since, if
| 1 t a n | 1 2 ( 1 x a n ) ,
then we see
| t x | = a n | ( 1 x a n ) ( 1 t a n ) | a n 2 ( 1 x a n ) .
Now, we have
J C 9 ( ( 1 x a n ) 1 4 | 1 t a n | 1 2 ( 1 x a n ) , t [ a 2 k + 2 , 2 a 2 k + 2 ] 1 t x d t + a n 1 ( 1 x a n ) 1 | 1 t a n | 1 2 ( 1 x a n ) , t [ a 2 k + 2 , 2 a 2 k + 2 ] | 1 t a n | 1 4 d t ) C 10 ( ( 1 x a n ) 1 4 log ( 1 + a 2 k + 2 a 2 k + 2 a 2 k + 1 ) + ( 1 x a n ) 1 | 1 s | 1 2 ( 1 x a n ) | 1 s | 1 4 d s ) C 10 ( ( 1 x a n ) 1 4 log ( 1 + C T ( a 2 k + 2 ) ) + 4 3 ( 1 2 ( 1 x a n ) ) 3 4 ( 1 x a n ) 1 ) C 11 ( 1 x a n ) 1 4 log ( C T ( x ) ) .
So, from (4.16) we have
| I 4 ( x ) | C 12 a n 1 2 ( 1 x a n ) 1 4 log ( C T ( x ) ) .
(4.17)
Therefore, from (4.13), (4.15) and (4.17), we have
| I 1 + I 2 + I 4 | C 13 a n 1 2 ( 1 x a n ) 1 4 log ( C T ( x ) ) .
Hence, with (4.10), (4.12) we have
(4.18)
We must estimate the L p -norm with respect to I 3 , that is, P . V . a 2 k 1 a 2 k + 2 ( σ ϕ p j w ) ( t ) x t d t L p ( I k ) . We use M. Riesz’s theorem on the boundedness of the Hilbert transform from L p ( R ) to L p ( R ) (Lemma 3.7) to deduce that by Lemma 3.5(a) and the boundedness of | σ ϕ | ,
P . V . a 2 k 1 a 2 k + 2 ( σ ϕ p j w ) ( t ) x t d t L p ( I k ) C 15 ( a 2 k 1 a 2 k + 2 | ( σ ϕ p j w ) ( t ) | p d t ) 1 p C 16 a n 1 2 ( 1 a 2 k + 2 a n ) 1 4 ( a 2 k + 2 a 2 k 1 ) 1 p .
(4.19)
So, by (4.18) and (4.19) we conclude
(4.20)
Noting (4.11), we see n 2 l + 3 for k l , so
1 a 2 k + 1 a n 1 a 2 k + 1 a 2 k + 3 C 19 1 T ( a 2 k ) and a 2 k + 1 a 2 k C 20 a 2 k T ( a 2 k ) .
On the other hand, using Lemma 3.2(b), we see Φ ( a t ) δ t . Hence, we have
Φ Δ + ( 1 4 1 p ) + ( a 2 k ) δ 2 k Δ + ( 1 4 1 p ) + = ( 1 2 k T ( a 2 k ) ) 2 3 ( Δ + ( 1 4 1 p ) + ) = { ( 1 2 k T ( a 2 k ) ) 2 3 Δ , 0 < p 4 ; ( 1 2 k T ( a 2 k ) ) 2 3 ( Δ + ( 1 4 1 p ) ) , 4 < p .
Hence, from (4.20) we have
From Lemma 2.1 (2.2), we know
T 2 3 Δ + 1 2 1 p ( a 2 k ) C 1 C ( λ , η ) ( 2 k ) 2 ( η + λ 1 ) λ + 1 max { 2 3 Δ + 1 2 1 p , 0 } ,
and
T 2 3 Δ + 1 3 ( 1 1 p ) ( a 2 k ) C 2 C ( λ , η ) ( 2 k ) 2 ( η + λ 1 ) λ + 1 max { 2 3 Δ + 1 3 ( 1 1 p ) , 0 } .
Therefore, we continue with Lemma 2.1(a) as
C 20 C ( λ , η ) log ( C T ( a 2 k + 1 ) ) × { ( 1 2 k ) 2 3 Δ η p 2 ( η + λ 1 ) λ + 1 max { 2 3 Δ + 1 2 1 p , 0 } , 1 < p 4 ; ( 1 2 k ) 2 3 ( Δ + ( 1 4 1 p ) ) η p 2 ( η + λ 1 ) λ + 1 max { 2 3 Δ + 1 3 ( 1 1 p ) , 0 } , 4 < p .
(4.21)
First, let 1 < p 4 . Then (4.8), that is,
Δ > { 0 , 1 < p 2 ; 3 2 λ 1 3 λ 1 p 2 p , 2 < p 4
implies
Δ > 3 2 λ 1 3 λ 1 p 2 p and Δ > 0
iff
2 3 Δ 2 ( λ 1 ) λ + 1 ( 2 3 Δ + 1 2 1 p ) > 0 and Δ > 0
iff
2 3 Δ 2 ( λ 1 ) λ + 1 max { 2 3 Δ + 1 2 1 p , 0 } > 0 .
This means that there exists a positive constant η 1 > 0 small enough such that
A ( η 1 ) : = 2 3 Δ η 1 p 2 ( η 1 + λ 1 ) λ + 1 max { 2 3 Δ + 1 2 1 p , 0 } > 0 .
Now, let p > 4 . Then (4.8), that is,
Δ > λ 1 3 λ 1 p 1 p 1 4 λ + 1 3 λ 1 p 4 p
implies
Δ > λ 1 3 λ 1 ( 1 1 p ) λ + 1 3 λ 1 ( 1 4 1 p ) and Δ + 1 4 1 p > 0
iff
2 3 ( Δ + ( 1 4 1 p ) ) 2 ( λ 1 ) λ + 1 ( 2 3 Δ + 1 3 ( 1 1 p ) ) > 0
and
2 3 ( Δ + ( 1 4 1 p ) ) > 0
iff
2 3 ( Δ + ( 1 4 1 p ) ) 2 ( λ 1 ) λ + 1 max { 2 3 Δ + 1 3 ( 1 1 p ) , 0 } > 0 .
Similarly to the previous case, this means that there exists a positive constant η 2 > 0 small enough such that
B ( η 2 ) : = 2 3 ( Δ + ( 1 4 1 p ) ) η 2 p 2 ( η 2 + λ 1 ) λ + 1 max { 2 3 Δ + 1 3 ( 1 1 p ) , 0 } > 0 .
Now, we estimate I p , k . From (4.21), we have
For η > 0 small enough, we can see A ( η ) > A ( η 1 ) > 0 and B ( η ) > B ( η 2 ) > 0 . Let τ : = min { A ( η 1 ) , B ( η 2 ) } / 2 . Then for small enough η > 0 , we have
s n [ σ ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( I k ) C 20 C ( λ , η ) log ( C T ( a 2 k + 1 ) ) ( 1 2 k ) 2 τ C 21 C ( λ , η ) ( 1 2 k ) τ ,
because we see that for all k > 0 ,
log ( C T ( a 2 k + 1 ) ) ( 1 2 k ) τ < C 22 .
Therefore, under the conditions (4.8) we have
s n [ σ ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( a 2 | x | a n 8 ) p k = 1 l s n [ σ ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( I k ) p C 21 C ( λ , η ) k = 1 l ( 1 2 k ) τ C 23 C ( λ , η ) .
(4.22)
The estimation of
s n [ σ ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( | x | a 2 ) p
is similar. In fact, for x [ a 2 , a 2 ] , we split
H [ σ ϕ p j w ] ( x ) = ( 2 a 2 + P . V . 2 a 2 2 a 2 + 2 a 2 ) ( σ ϕ p j w ) ( t ) x t d t .
Here we see that
| 2 a 2 ( σ ϕ p j w ) ( t ) x t d t | = | 2 a 2 ( σ ϕ p j w ) ( t ) x + t d t | | 2 a 2 ( σ ϕ p j w ) ( t ) t a 2 d t | = | 0 ( σ ϕ p j w ) ( s 2 a 2 ) s + a 2 d t |
and
| 2 a 2 ( σ ϕ p j w ) ( t ) x t d t | = | 2 a 2 ( σ ϕ p j w ) ( t ) t x d t | | 2 a 2 ( σ ϕ p j w ) ( t ) t a 2 d t | = | 0 ( σ ϕ p j w ) ( s + 2 a 2 ) s + a 2 d s | .
So, we can estimate 2 a 2 and 2 a 2 as we did I 1 before (see (4.12)). We can estimate the second integral as follows: By M. Riesz’s theorem,
P . V . 2 a 2 2 a 2 ( σ ϕ p j w ) ( t ) x t d t L p ( | t | 2 a 2 ) p C 2 a 2 2 a 2 | ( σ ϕ p j w ) ( t ) | p d t C a n p 2 C .
Now, under the assumption (4.8), we can select η 0 > 0 small enough such that
Δ > { 0 , 1 < p 2 ; 3 2 λ + η 0 1 3 λ + 2 η 0 1 p 2 p , 2 < p 4 ; max { λ + η 0 1 3 λ + 2 η 0 1 p 1 p 1 4 λ + 1 3 λ + 2 η 0 1 p 4 p , 0 } , 4 < p .

Consequently, from (4.22) with η 0 we have the result (4.9). □

Let 0 < α < 1 , then for g n in Lemma 4.5 we estimate L n ( g n ) over [ a α n , a α n ] .

Lemma 4.7 (cf. [3, Lemma 4.4])

Let 1 < p < and 0 < ε < 1 . Let { g n } be as in Lemma 4.4, but we exchange (4.3) with
| g n ( x ) w ( x ) | ε ϕ ( x ) , x R , n 1 .
Then for 1 < p < ,
lim sup n L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) C ε .
Proof Let
χ n : = χ [ a n 8 , a n 8 ] ; h n : = sign ( L n ( g n ) ) | L n ( g n ) | p 1 χ n w p 2 Φ ( Δ + ( 1 4 1 p ) + ) p
and
σ n : = sign s n [ h n ] .
We shall show that
L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) ε s n [ σ n ϕ w 1 ] w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) .
(4.23)
Then from Lemma 4.5 we will conclude (4.22). Using orthogonality of f s n [ f ] to P n 1 , and the Gauss quadrature formula, we see that
Here, if we use Lemma 4.2 with ψ = ϕ , we continue as
C ε R | s n [ h n ] ( x ) | ϕ ( x ) w ( x ) d x = C ε R s n [ h n ] ( x ) σ n ϕ ( x ) w 1 ( x ) w 2 ( x ) d x = C ε R h n ( x ) s n [ σ n ϕ w 1 ] ( x ) w 2 ( x ) d x = C ε a n 8 a n 8 h n ( x ) s n [ σ n ϕ w 1 ] ( x ) w 2 ( x ) d x .
Using Hölder’s inequality with q = p / ( p 1 ) , we continue this as

Cancellation of L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) p 1 gives (4.23). □

Proof of Theorem 2.2 In proving the theorem, we split our functions into pieces that vanish inside or outside [ a n 9 , a n 9 ] . Throughout, we let χ S denote the characteristic function of a set S. Also, we set for some fixed β > 0 ,
ϕ ( x ) = ( 1 + x 2 ) β / 2 ,
and suppose (2.5). We note that (2.5) means (4.8). Let 0 < ε < 1 . We can choose a polynomial P such that
( f P ) w ϕ 1 L ( R ) ε
(see Lemma 3.8). Then we have
(4.24)
Here we used that
ϕ Φ Δ + ( 1 4 1 p ) + L p ( R ) < ,
because Δ > 0 and Φ 1 grows faster than any power of x (see Lemma 3.9). Next, let
χ n : = χ [ a n 9 , a n 9 ] ,
and write
P f = ( P f ) χ n + ( P f ) ( 1 χ n ) = : g n + f n .
By Lemma 4.4 we have
lim n L n ( f n ) w Φ Δ + ( 1 4 1 4 ) + L p ( R ) = 0 .
By Lemma 4.5 we have
lim n L n ( g n ) w Φ Δ + ( 1 4 1 4 ) + L p ( | x | a n 8 ) = 0 ,
and by Lemma 4.7,
lim sup n L n ( g n ) w Φ Δ + ( 1 4 1 p ) + L p ( | x | a n 8 ) C ε .

Here we take ε > 0 as ε 0 , then with (4.24) we have the result. □

5 Proof of Theorem 2.4

Lemma 5.1 (cf. [3, Lemma 3.1])

Let w F ( C 2 + ) . Let 0 < α < 1 4 and
n ( x ) : = | x k , n | a α n | l k , n ( x ) | w 1 ( x k , n ) .
Then we have for x R ,
n ( x ) w ( x ) Φ 1 / 4 ( x ) C log n .

Proof From Lemma 4.1 and Lemma 3.6 with p = , we have the result easily. □

Lemma 5.2 Let w F ( C 2 + ) . Let 0 < α < 1 4 and
n ( x ) : = | x k , n | a α n | l k , n ( x ) | w 1 ( x k , n ) .
Then we have
n ( x ) w ( x ) Φ ( x ) 3 / 4 C log n .
Proof By Lemma 3.5(c), Lemma 3.4(d) and Lemma 3.5(b),
n ( x ) = | x k , n | a α n | l k , n ( x ) | w 1 ( x k , n ) = | p n ( x ) | | x x j x , n | | P n ( x j x , n ) | w ( x j x , n ) + | x k , n | a α n , k j x | p n ( x ) | | x x k , n | | P n ( x k , n ) | w ( x k , n ) C w ( x ) 1 + a n 1 / 2 | p n ( x ) | | x k , n | a α n , k j x φ n ( x k , n ) ( 1 | x k , n | a n ) | x x k , n | C w ( x ) 1 + a n 3 / 2 n | p n ( x ) | | x k , n | a α n , k j x 1 | x k , n | a 2 n 1 | x k , n | a n ( 1 | x k , n | a n ) 1 / 4 1 | x x k , n | C w ( x ) 1 + a n 3 / 2 n | p n ( x ) | | x k , n | a α n , k j x ( 1 | x k , n | a n ) 3 / 4 1 | x x k , n | ,
where we used the fact
1 | x k , n | a 2 n 1 | x k , n | a n , | x k , n | a α n .
So,
n ( x ) C w ( x ) 1 + a n 3 / 2 n | p n ( x ) | | x k , n | a α n , k j x ( 1 | x k , n | a n ) 3 / 4 1 | x j x , n x k , n | C w ( x ) 1 + a n 3 / 2 n | p n ( x ) | | x k , n | a α n , k j x ( 1 | x k , n | a n ) 3 / 4 1 j x i k φ n ( x i , n ) C w ( x ) 1 + a n 1 / 2 | p n ( x ) | | x k , n | a α n , k j x ( 1 | x k , n | a n ) 3 / 4 1 j x i k 1 | x i , n | / a n .
Therefore we have by Lemma 3.6 with p = ,
n ( x ) w ( x ) Φ ( x ) 3 / 4 C + C a n 1 / 2 | p n ( x ) | w ( x ) Φ ( x ) 1 / 4 × | x k , n | a α n , k j x ( 1 | x k , n | a n ) 3 / 4 ( 1 | x j x , n | a n ) 1 / 2 1 j x i k 1 | x i , n | / a n C | x k , n | a α n , k j x 1 | j x k | log n .

 □

Lemma 5.3 ([8, Theorem 1])

Let w F ( C 2 + ) . Then there exists a constant C 0 > 0 such that for every absolutely continuous function f with w f C 0 ( R ) (this means w ( x ) f ( x ) 0 as | x | ) and every n N , we have
E n ( w ; f ) C a n n E n 1 ( w ; f ) .
Proof of Theorem 2.4 There exists P n 1 P n such that
| ( f ( x ) P n 1 ( x ) ) w ( x ) | 2 E n 1 ( w ; f ) .
Therefore, by Lemma 5.1 and Lemma 5.2,
Let w f ( r ) C 0 ( R ) . If we repeatedly use Lemma 5.3, then we have
| ( f ( x ) L n ( f ) ( x ) ) w ( x ) Φ 3 / 4 ( x ) | C r ( a n n ) r E n r 1 ( w ; f ( r ) ) log n .

 □

Declarations

Acknowledgements

The authors thank the referees for many kind suggestions and comments.

Authors’ Affiliations

(1)
Department of Mathematics Education, Sungkyunkwan University, Seoul, Republic of Korea
(2)
Department of Mathematics, Meijo University, Nagoya, Japan

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© Jung and Sakai; licensee Springer 2012

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