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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:

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 weight
  • Lagrange 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, 110-745, Republic of Korea
(2)
Department of Mathematics, Meijo University, Nagoya 468-8502, Japan

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© Jung and Sakai; licensee Springer 2012

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