Directory UMM :Data Elmu:jurnal:J-a:Journal of Computational And Applied Mathematics:Vol99.Issue1-2.1998:

Journal of Computational and Applied Mathematics 99 (1998) 129–141

Marcinkiewicz inequalities based on Stieltjes zeros
Sven Ehrich a; ∗ , Giuseppe Mastroianni b
a

GSF Research Center for Environmental and Health, Institute of Biomathematics and Biometry, Ingolstadter
Landstr. 1, 85764 Neuherberg, Germany
b
Dipartimento di Matematica, Universita degli Studi della Basilicata, 85100 Potenza, Italy
Received 15 December 1997; received in revised form 30 April 1998

Abstract
The authors nd necessary and sucient conditions for GDT weighted Marcinkiewicz inequalities based at Stieltjes
c 1998 Elsevier Science B.V. All rights reserved.
zeros.
Keywords: Marcinkiewicz inequalities; Stieltjes polynomials

1. Introduction
If P is an arbitrary polynomial of degree m − 1 and u is a weight function in [−1; 1]; then the
following identity is well known:

1

Z

−1

|P(x)u(x)|2 d x =

m
X

m (u2 ; yk )|P(yk )|2 ;

k=1

where m (u2 ; t) is the mth Christoel function with respect to the weight function u2 and yk ;
k = 1; : : : ; m; are the zeros of the mth orthogonal polynomial associated with the weight function
u2 :
The above identity is generally false if we replace 2 by an arbitrary p ∈ (0; +∞): Thus, we can
investigate on the validity of the inequality

Z

1

−1

p

|P(x)u(x)| d x 6 C

m
X

m (up ; yk )|P(yk )|p ;

k=1

for any absolute constant C¿0 and for given −1 6 y1 ¡ · · · ¡ym 6 1:
∗

Corresponding author.

c 1998 Elsevier Science B.V. All rights reserved.
0377-0427/98/$ – see front matter
PII: S 0 3 7 7 - 0 4 2 7 ( 9 8 ) 0 0 1 5 1 - 4

(1)

130

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

This problem, which is very far from having a complete solution, is strongly connected with
the weighted Lp convergence of Lagrange interpolating polynomials. In particular, if P is the
Lagrange polynomial Lm (f) interpolating a continuous function f on the knots yk ; k = 1; : : : ; m;
then (1) becomes
Z

1

−1

|Lm (f; x)u(x)|p d x 6 C

m
X

m (up ; yk )|f(yk )|p :

k=1

By using some imbedding theorems and some results of [8], from the above inequality we can
deduce theorems on the uniform boundedness of the operator Lm in some Besov space.
However, if the points yk ; k = 1; : : : ; m; are zeros of orthogonal polynomials associated with any
weight function w (i.e. yk = xm; k (w); k = 1; : : : ; m) and u and w are GDT weight functions (see
Eq. (2) in Section 1.1), then in [8] it has been proved that for 1¡p¡∞ (1) is true if and only if
√
√
w’
′

1
u
1
p
∈ Lp ;
+ ′ = 1; ’(x) = 1 − x2 :
and
∈
L
√
w’
u
p p
Very little is known in literature when the points yk ; k = 1; : : : ; m; are not zeros of orthogonal
polynomials.
In this paper we will assume that the knots yk ; k = 1; : : : ; m; are the zeros of the mth Stieltjes
polynomial Em (x) associated with an ultraspherical weight function
1

w (x) = (1 − x2 )− 2 ;

 ∈ (0; 1):

We recall that {Em (x)}m is not a standard sequence of orthogonal polynomials (see Section 1.2) and
that the previous results in [8] cannot help us. However, in this paper we will nd
necessary and sucient conditions for the validity of (1) when u is a generalized Ditzian–Totik

weight function and yk ; k = 1; : : : ; m + 1; are zeros of Em+1
(x): An analogous theorem is proved if
 

yk ; k = 1; : : : ; 2m + 1; are zeros of Pm Em+1 ; where Pm is the mth ultraspherical polynomial. In the


proof of these theorems, Lemmas 2 and 3 (new bounds for Em+1
and Pm Em+1
) are crucial and can
be used in several contexts. For the sake of brevity, we cannot establish here new results on the
corresponding Lagrange interpolation. We will consider this topic elsewhere.
1.1. Generalised Ditzian–Totik weights

We consider the so-called generalised Ditzian–Totik (GDT ) weights of the form
u(x) =

M
Y
k=0

|x − tk | k !˜ k (|x − tk |k );

(2)

where k ∈ R, −1 = t0 ¡t1 ¡ · · · ¡tM −1 ¡tM = 1, k = 12 if k ∈ {0; M } and k = 1 otherwise. The function !˜ k is either equal to 1 or is a concave modulus of continuity of the rst order, i.e. !˜ k
is semi-additive, nonnegative, continuous and nondecreasing on [0; 1], !˜ k (0) = 0 and 2 !˜ k ((a +
b)=2)¿!˜ k (a) + !˜ k (b) for all a; b ∈ [0; 1], and for every ”¿0, !˜ k (x)=x” is a nonincreasing function on (0; 1) with limx→0+ !˜ k (x)=x” = ∞. Special cases are the generalised Jacobi weights (!˜ k ≡ 1
for k = 0; : : : ; M ) and the Ditzian–Totik (DT ) weights ( k = 0 and !˜ k ≡ 1 for k = 1; : : : ; M − 1). For

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

131

u ∈ GDT being of the type (2) and m ∈ N, we dene
M
−1
Y
√
√
um (x) = ( 1 + x + m−1 )2 0 !˜ 0 ( 1 + x + m−1 )
(|x − tk | + m−1 )
−1

√

k

k=1

−1 2

×!˜ k (|x − tk | + m )( 1 − x + m )

M

√
!˜ M ( 1 − x + m−1 ):

(3)

We recall some results concerning GDT weights from [8]. For the convenience of the reader, we
also include a short proof of the following Lemma 1 (see Section 3). Let the Hilbert transform be
denoted by
Z

1

H (f; t) = −

−1

f(x)
d x:

x−t

(4)

Lemma 1 (Mastroianni and Russo [8, Lemma 2.5]). Let 1¡p¡∞ and U ∈ GDT . Then for every
function f such that fU ∈ Lp we have
kH (f)U kp 6 CkfU kp ;

C 6= C(f);

if and only if
′

U −1 ∈ Lp ;

U ∈ Lp ;

p′ =

p

:
p−1

(5)

Lemma 2 (Mastroianni and Russo [8, Corollary 2.3]). Let w ∈ GDT; w ∈ Lp ; 1 6 p 6 ∞. Then; for
each xed 0 6 a¡m; there exists a positive constant C; depending on a and w; such that for every
P ∈ Pm and E ⊂[−1; 1] with |arccos E| 6 a=m we have
kPwkp 6 CkPwkLp ([−1;1]\E) :
Lemma 3 (Mastroianni and Russo [8, Lemma 2.1]). Let w ∈ GDT . Then there exists a polynomial
Q ∈ Pm ; m¿1; such that for |x|61
wm (x) 6 Q(x) 6 Cwm (x);
√
1 − x2 ′
Q (x) 6 Cwm (x);
m
where C 6= C(m).
1.2. Stieltjes polynomials
In the sequel, C will denote a generic constant independent of the variables in the context. In
dierent formulas, the same symbol C may have dierent values. Moreover, we will write A ∼ B;
for A; B¿0; i there exist two positive constants M1 ; M2 independent of A and B such that
M1 6

 ±1

A
B

6 M2 :

132

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141
1

Given the ultraspherical polynomial Pm ,  ∈ (0; 1), orthogonal with respect to w (x) = (1 − x2 )−2 on

(−1; 1), the Stieltjes polynomial Em+1
is dened, up to a multiplicative constant, by
Z

1

−1


(x) xk d x = 0;
! (x) Pm (x) Em+1

k = 0; 1; : : : ; m:

While this denition is possible for more general orthogonal polynomials, the zeros of the Stieltjes
polynomials are not necessarily real and in (−1; 1). For w and  ∈ (0; 1) this property, as well as

the interlacing property of the zeros of Pm and Em+1
, was proved by Szego in [14]. See the surveys
[5, 10] for a more detailed overview on the history and on basic properties of Stieltjes polynomials.
Zeros of Stieltjes polynomials are used as nodes for Kronrod extensions of Gaussian quadrature
formulas, based on Pm , for a maximum degree of algebraic exactness. Pairs of Gauss and Gauss–
Kronrod formulas are a standard method for error estimation in automatic integration packages, see
[13]. Carrying over this idea to Lagrange interpolation, extended interpolation formulas, based on
the zeros of Stieltjes polynomials or other choices of nodes, have been considered in [3, 4].

In this paper, the Stieltjes polynomials Em+1
are normalised such that we have (cf. [2])

Em+1
(cos ) sin−1




 = cos (m + ) − ( − 1)
2



+ o(1)

uniformly for  ∈ [”;  − ”], ”¿0 xed. The proofs of the results are based on the sharp bounds
sup
x∈[−1+m−2 ;1−m−2 ]

sup
x∈[−1+m−2 ;1−m−2 ]


(x)| 6 C;
|!  (x)Em+1

(6)


(x)| 6 C;
|! (x)Pm (x)Em+1

(7)

2

proved in [4, Theorem 2.1], where the polynomials Pm are normalised such that we have (cf. [2])



(cos ) sin2−1  = cos (2m + 2) − (2 − 1)
Pm (cos )Em+1


2



+ o(1)

uniformly for  ∈ [”;  − ”], ”¿0 xed. Furthermore, we shall use the asymptotic relations (see [4])
1

′
|Em+1
(k; m+1 )|

∼


1
(1 − (k; m+1 )2 ) 2 ;
m

k = 1; : : : ; m + 1;

1
1
∼ (1 − (yk; 2m+1 )2 ) ;
′
m
|K2m+1
(yk; 2m+1 )|

k = 1; : : : ; 2m + 1;

(8)
(9)

respectively
1
;
m
1
∼
m




k; m+1 − k+1;
m+1 ∼ m+1; m+1 ∼  − 1; m+1 ∼

k; 2m+1

−


k+1; 2m+1

∼


m+1; m+1

∼−


1; m+1

(10)
(11)


, respectively y1; 2m+1 ¡ · · · ¡
for the cos arguments of the zeros 1; m+1 ¡ · · · ¡m+1; m+1 of Em+1


 




y2m+1;
2m+1 of K2m+1 = Pm Em+1 , i.e. k; m+1 = cos k; m+1 , yk; 2m+1 = cos k; 2m+1 . The above asymptotic
statements hold uniformly for all k.

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

133

The following results about Stieltjes polynomials are new and of their own interest. They are,
in addition to the above properties, the key to the proofs of the main results (Theorems 2 and 3
below).
Lemma 4. Let  ∈ (0; 1), r ∈ N; Im := [m−1 ;  − m−1 ]. Then

2
Z 
 

(E (cos ) sin−1 )r − cosr (m + ) − ( − 1)   d = 0;
lim
m+1

m→∞
2 
Im



Z 
 
 2
2−1
r
r
(K
(cos
)
sin
)
−
cos
(2m
+
2)
−
(2
−
1)
d = 0:
 2m+1
m→∞
2 

lim

Im

Lemma 5. Let  ∈ (0; 1); 1¡p¡∞; u ∈ Lp ; r ∈ N. Then there exists C¿0 such that






u
r


¿0;
lim inf k(Em+1 ) ukp ¿ C

r
m→∞
(w  )
2

p



u

r

¿0:
lim inf k(K2m+1 ) ukp ¿ C
m→∞
(w )r


(12)

(13)

p

2. Marcinkiewicz inequalities
For a nonnegative weight w with 0¡kwk1 ¡∞, the Christoel function is dened by
m (w; t) =

m+1
X

pk2 (w; t)

k=1

!−1

;

where pk (w) are the orthonormal polynomials with respect to w. In [1] and [7, Theorem 5] it is
proved that, for w ∈ GDT; we have
!
√
1
1 − x2
+ 2 wm (x);
m (w; x) ∼
(14)
m
m
where wm is like um in (3) and |t| 6 1: The following result is proved in [8, Theorem. 2.6].
Theorem 1. Let u ∈ GDT; u ∈ Lp ; 1 6 p¡∞. Let zk = cos k ; k ∈ [0; ]; −1¡z1 ¡ · · · ¡zm ¡1; and
k+1 − k ∼ m ∼  − 1 ∼ m−1 . Then for every polynomial P ∈ Plm and l xed integer; we have
m
X
k=1

p

p

m (u ; zk )|P(zk )|

!1=p

6 CkPukp ;

where C is independent of m and P.

134

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

Remark. The statement is slightly dierent from [8, Theorem 2.6] (with respect to the endpoints),
but the proof can be carried over completely.
The following theorems are the main results of this paper.
Theorem 2. Let  ∈ (0; 1); 1¡p¡∞; u ∈ GDT; u ∈ Lp . The following assertions are equivalent.
(1) For all p ∈ Pm ;
m+1
X

kPukp 6 C

k=1

m+1 (up ; k; m+1 ) |P(k; m+1 )|p

!1=p

;

where C is independent of m and P:
w
′
2
(2)
∈ Lp ;
u
where (1=p) + (1=p′ ) = 1.

(15)

(16)

Theorem 3. Let  ∈ (0; 1); 1¡p¡∞; u ∈ GDT; u ∈ Lp . The following assertions are equivalent.
(1) For all p ∈ P2m ;
2m+1
X

kPukp 6 C

m (u

p

; yk; 2m+1 )|P(yk; 2m+1 )|p

k=1

where C is independent of m and P:
′
w
u
∈ Lp and
(2)
∈ Lp ;
w
u

!1=p

;

(17)

(18)

where (1=p) + (1=p′ ) = 1.
Remark. In view of Lemma 1, a third equivalent statement is the boundedness of the weighted
Hilbert transform in Lp ,












H (f) u
6 C
f u
;




w

w

2

2

p

p

respectively,



H (f) u

w







6 C
f u

w
p





:

p

3. Proofs
Proof of Lemma 1. We have
kH (f)U kp 6 CkfU kp ;

C 6= C(f);

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

135

if and only if U ∈ Ap , i.e. for each interval I ⊂(−1; 1) there holds
kU kLp (I ) kU −1 kLp′ (I ) 6 C|I |;

p′ =

p
;
p−1

(19)

with C being independent of I , and where |I | is the measure of I (cf. [6, 11]). If (19) holds,
then obviously (5) follows. Suppose (5) holds. Since U and U −1 are bounded functions in each
subinterval of [−1; 1] not including the possible singularities tk , k = 0; 1; : : : ; M , it is sucient to
consider U (x) = |x|a w(|x|),
˜
−1=p¡a¡1=p′ , I = (0; d), 0¡d¡1, where w˜ is either equal to 1 or
is a concave modulus of continuity of the rst order that satises the additional assumptions in
Section 1.1. Hence, w(|x|)
˜
is a nondecreasing function, and we have
A :=

d

Z

p

ap

x w˜ (x) d x

0

!1=p

6

U (d) d1=p
w(d)
˜
da+1=p
=
:
(1 + ap)1=p (1 + ap)1=p

”
is a nonincreasing function. For every xed ”¡(1=p′ ) − a,
Since U ∈ GDT , for all ”¿0 w(|x|)=x
˜

B :=

Z

d

0

dx
′
′
ap
x w˜ p (|x|)

! 1′
p

=

Z

d

′

x−ap −”p

′

0

′



x”
w(|x|)
˜

p′

dx

!1=p′

′

d1=p
1
d”
d1=p
6
=
:
′
w(d)
˜
da+” (1 − ap′ − ”p′ )1=p
U (d) (1 − ap′ − ”p′ )1=p′
Thus it follows that A · B 6 C(a; p) d, i.e., (19).
Proof of Lemma 4. Let ”¿0. For the rst statement, we split the integral,
Z

Im


Z ” Z
= 1 +

−

1
m

−”

m

Now, using (6),


Z
|I1 | 6 C 

”
1
m

+

Z



+

−

−”

1
m

Z

−”

=: I1 + I2 :

”



˜
 d 6 C”;

where C˜ is independent of ” and n. Furthermore,
sup
 ∈ [”;−”]




 

E (cos ) sin−1  − cos (m + ) − ( − 1)   = o(1)
 m+1
2 


for m → ∞ (cf. [2]). For A = Em+1
(cos ) sin−1  and B = cos{(m + ) − ( − 1) 2 } we have

|Ar − Br | 6 |A − B|

r−1
X
i=0

|A|r−1−i |B|i

and hence |I2 | → 0 for m → ∞ and xed ”. Now we can choose ” arbitrarily small, which leads to
the rst statement. The proof of the second statement is analogous.

136

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

Proof of Lemma 5. For orthogonal polynomials and r = 1 the assertion was proved in
[12, Theorem 32]. For the present case, the proof is based on Lemma 4 and follows analogously as
in [12, Proof of Theorem 3.2]. Therefore we omit the details.
Proof of Theorem 2. Assume that condition (16) holds. We consider the Lagrange interpolation
operator
Lm+1 (f; x) =

m+1
X

lk; m+1 (x) f(k; m+1 ):

k=1

For every P ∈ Pm we have
kPukp = kLm+1 (P)ukp

= sup
kgkp′ =1

Z

1

−1

p′ =

Lm+1 (P; x)u(x)g(x) d x;

p
:
p−1

′

Let g be any function in Lp such that kgkp′ = 1. We have
Z

1

−1

Lm+1 (P; x)u(x)g(x) d x =

m+1
X
k=1

=

P(k; m+1 )
′
Em+1
(k; m+1 )

Z

1

−1


Em+1
(x)u(x)g(x)
dx
x − k; m+1

m+1
X
P(k; m+1 )(k; m+1 )
k=1

′
Em+1
(k; m+1 )

;

where
(t) =

Z

1

−1



(t)Q(t) g(x)u(x)
Em+1
(x)Q(x) − Em+1
d x;
x−t
Q(x)

with Q being a yet unspecied positive polynomial of degree 6 m, is a polynomial of degree 6 2m.
Using (8), we estimate
Z

1

−1

Lm+1 (P; x)u(x)g(x) d x 6 C

m+1
X
’(k; m+1 )
k=1

=C

m

(
)1=p
m+1
X
’(k; m+1 )
k=1

m



1

(1 − (k; m+1 )2 ) 2 − 2 |(k; m+1 )P(k; m+1 )|
|um (k; m+1 )| |P(k; m+1 )|



(
)1=p′


 1 
’(k; m+1 )

−


2
 (k; m+1 )
2 2 
(1
−
(
)
)
k; m+1


m


×




u
(
)
m k; m+1







S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

6C

(m+1
X ’(k; m+1 )

m

k=1

|um (k; m+1 )|p |P(k; m+1 )|p

137

)1=p

′

1 1=p
′ 
m+1


p′

2 p (2−2) 
X
’(k; m+1 ) |(k; m+1 )| (1 − (k; m+1 ) )
×
:


m
|um (k; m+1 )|p′

k=1

Using (14), we obtain
q

1 − (k; m+1 )2
m

|um (k; m+1 )|p 6 Cm+1 (up ; k; m+1 );

where the constant C is independent of m and k, and hence
Z

1

−1

Lm+1 (P; x)u(x)g(x) d x 6 C

(m+1
X

m+1 (u

p

; k; m+1 )

k=1

|P(k; m+1 )|p

)1=p

Sp′ m ;

where

Sp′ m

′

′ 1=p

 1 p 

m+1
 X

’(k; m+1 )
(1 − (k; m+1 )2 ) 2 − 2
′

=
|(k; m+1 )|p 


m
um (k; m+1 )



k=1

w




6C
2
;
u
′
p

and the last inequality follows from Theorem 1, (10) and (14). From Lemma 2 we obtain
w



2



u

p′

w (Z
w
)
Z 1






1
Em+1
(x)g(x)u(x)
g(x)u(x)

2


2
d x − Em+1 Q
dx
=C
6C


p′
u
u
p′
x
−
·
−1
−1 (x − ·)Q
L (Am )
L (Am )
w


w





gu






6 C
2 H (Em+1
gu)
QH
+ C
2 Em+1

u
p′
u
Q
p′
L (Am )

L (Am )

=I1 + I2 ;

(20)

where
Am = Am (w) = (−1; 1)\A′m (w);
A′m (w) =



1
−1; −1 + 2
m



1
∪ 1 − 2;1 ∪
m




" M −1 
[
k=1

1
1
tk − ; tk +
m
m

and H denotes the Hilbert transform as in (4). We have






I1 6 C
w  Em+1 g

2

Lp′ (Am )

˜
˜
¡Ckgk
p′ = C;

#

;

138

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

using (6). For x ∈ Am , we have um ∼ u, and from Lemma 3 we obtain that there exists a polynomial
Q∗ of degree 6 m such that
u (x) 6 Q∗ (x)¡Cu (x):
m

m

We use again (6) and choose Q = Q∗ in inequality (20). Since u ∈ Lp we have that u=w  ∈ Lp ,
2

 ∈ [0; 1]. We observe that the assumptions of Lemma 1 are satised with U = w  =u, and hence
2


 

Q
gu


I2 6 C
H
u
Q

Lp′ (A

−1
˜
˜
˜
6 CkQQ
gkLp′ (Am ) 6 Ckgk
p′ = C:
m)

Therefore, the condition (16) is sucient for the inequality (15).
We now show the necessity of (16). We rst estimate the fundamental Lagrange polynomials,
using (8),




′
Em+1
(x)


|lk; m+1 (x)| =  ′

 Em+1 (k; m+1 )(x − k; m+1 ) 

 1
’(k; m+1 )
(1 − (k; m+1 )2 ) 2 − 2
¿C
m

and hence, for u being a GDT weight,


E u


m+1


· − k; m+1

p



 E  (x) 

 m+1
;

 x − k; m+1 

C1
(’(k; m+1 )) 6 klk;m+1 ukp
m
6 C2 um (k; m+1 )

(21)
’(k; m+1 )
m

!1=p

;

where the second inequality comes from an application of (15) and (14). Assume
w
′
p
2
∈= Lp ; p′ =
:
u
p−1

Since u ∈ GDT , i.e. !˜ j in the denition of u are of subalgebraic growth, at least one of the following
assertions must be true:
(i)
(ii)
(iii)

 1
1
− − 0 6 − ′;
2 2
p
 1
1
− − M +1 6 − ′ ;
2 2
p
1
for some i = 1; : : : ; M − 1;
i¿ ′
p

where t0 = 1 and tM = 1. Assume (i) holds, the other cases can be treated analogously. For k = 1 we
obtain from (21), using |x − k; m+1 |¡2,

ukp m
kEm+1

−

2
−+2 0 +1
p′

6 C:

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

Using Lemma 5, we have
m

−

2
−+2 0 +1
p′

−1



u

6C


;
w
2

p

p
and since we already proved that uw−1
 ∈ L , we have
2

 1
− −
2 2

0

=−

1
:
p′

Similarly as in (21) we obtain


E u


m+1
sup

6 C:

m∈N
· − k; m+1
p

Let 1 = 1; m+1 . Using

|x − 1 | 6 |x + 1| +

c
;
m2

and dening
A = [−1 + ; 21 (−1 + t1 )];

¡ 41 (−1 + t1 );

we obtain






E u

Em+1
u


m+1
¿
C




| · +1| + m−2

· − 1

:

Lp (A )

p

Using Lemma 5, we have






lim inf m→∞ fm


;
lim sup kEm+1 fm kp ¿ C

w
m→∞


2

p

where

fm (x) =

u(x) A (x)
;
|x + 1| + m−2

where  A denotes the characteristic function of A . Using this inequality, we have






u


¿ C1

|
·
+1|
w



Lp (A )






Em+1
u


C ¿ lim sup

−2
m→∞
| · +1| + m

Z

¿ C2

−1+t1
2

−1+

1

= log p

t1 + 1
;
2

[(x + 1)

1 
0 + − −1 p
2 2
]

2

1=p

d x

Lp (A )


Z

= C2

−1+t1
2

−1+

1=p

(x + 1)−1 

139

140

S. Ehrich, G. Mastroianni / Journal of Computational and Applied Mathematics 99 (1998) 129–141

i.e., there exists a constant D, independent of , such that
log

t1 + 1
6D
2

for all 0¡¡

t1 + 1
;
4

which is a contradiction. Hence, (16) must be valid.
Proof of Theorem 3. We obtain the suciency of (18) as well as the necessity of the second
condition in (18) in a completely analogous way as in the previous proof. To prove the necessity
of the rst condition in (18), assume that for every polynomial P ∈ P2m we have
kPukpp 6 C

2m+1
X

2m+1 (up ; yk; 2m+1 )|P(yk; 2m+1 )|p :

k=1

Then for every continuous function f we have

kL2m+1
(f)ukp 6 C

2m+1
X

2m+1 (up ; yk; 2m+1 )|f(yk; 2m+1 )|

k=1

6 Ckfk∞

2m+1
X
k=1

2m+1 (up ; yk; 2m+1 ) 6 Ckfk∞ kukp 6 Ckfk∞ ;

using Theorem 1 and since u ∈ Lp . Therefore,


kL2m+1
(f)ukp 6 C:
ukp = sup kL2m+1
kfk∞ =1

Applying Theorem 2.2 of [9, p. 433], we get



kK2m+1
ukp :
w k1 kL2m+1
ukp 6 CkK2m+1

Using (7), we have

kK2m+1
w k1 6 C;

C independent of m. Hence


˜
kK2m+1
ukp ¡C¡∞;
ukp ¡CkL2m+1

where C˜ is independent of m. By Lemma 5 with r = 1, this implies the rst condition in (18).
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