ON THE ESTIMATION OF GENERALIZED LEBESGUE CONSTANT AND MODULUS OF GENERALIZED SINGULAR QUADRATURE FORMULAS AND ITS APPLICATION

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Acta Mathematica Scientia 2006,26B(4):610-614 www.wipm.ac.cn/publish/

ON THE ESTIMATION OF GENERALIZED LEBESGUE CONSTANT AND MODULUS O F GENERALIZED SINGULAR QUADRATURE FORMULAS AND ITS APPLICATION* Cai Haotao ( &+7$ ) Department of mathematics, Qufu Normal University, Qufu 2731 65, China Academy of Mathematics and Systems Science, CAS, Beijing 100080, China

D u Jinyuan ( f i L 4 7 i ) Department of Mathematics, Wuhan University, Wuhan 430072, China

Abstract This article, first gives the estimaties of two modulus, namely, generalized Lebesgue constant and modulus of generalized singular integral quadrature formulas, then applies them to obtain the error bounds of the operator B L L to the operator B.

Key words Cauchy kernel, Dini conditions, generalized Jacobi weight 2000 MR Subject Classification

1

45E10

Introduction Singular integral equations with Cauchy kernel of the form:

a ( t ) $ ( t )+

$1

-dx 4(x)

-,a:-t

=f(t)

- 1< t

<1

appear frequently in problems of theory of elasticity. Here a ( t ) ,b ( t ) ,f ( t )are Holder-continuous functions, it is required to find the solution function $(t)in class ho in [l].The classical theory of these equations is rather complete. In the past 20 years a great deal of interest has risen in the numerical ways in [2,3]. Regard of uniform convergence, D.Elliott, Giuliana Criculuo, Giuseppe Mastroiann give some results, but those results are not complete. Du Jinyuan put forward to a supposition in [4]. The supposition depends on the estimaties of two operators, namely, generalized Lebesgue constant and modulus of generalized singular quadrature formulas. In this article, we give the estimaties of the two operators, then give their application. *R%bivedMay 10, 2004;revised June 16, 2005

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Cai & Du: ESTIMATION OF GENERALIZED LEBESGUE CONSTANT

611

Notions

In order to draw our conclusion’s, we give some notims. Through the article, let LipL denote the space of Holder-continuous functions, Let 3T be the set of continuous functions satisfying Dini conditions. Namely:

1

f E C[-l,ll;

y

d

6 < +m},

where w ( f , s ) is the continuous modulus of function f. We say w ( t ) is a generalized Jacobi weight (w(t) E GJ,,o) iff w(t) = (1 - t),(l + t ) ” ( t ) , a , p > -1,0 < +(t)E DT. Let { p m ( t ) } ~ = obe the sequence of orthogonal polynomial associated with the weight function w(t), namely:

+

p m ( t ) = amzm lower degree terms,

om > 0.

We denote the zeros of p m ( t ) by t k = cos 8 k l k = 1,2, . . , m, assume they are ordered increasingly: 0 = 80 < 81 < . . . < 8, < em+l < em+l = 7T. Obviously, t o = l , t m + l = -1. Let X k = X m ( Z k ) , where X ( t ) is the m-th Christoffel function. For t E [-II 11, let t , be the closest knot to t , defined by:

Let N be the set of positive integers and C be some constant taking a different value each time used. If A and B are two expressions depending on some variables, we write

A

-

B iff IA-lBI 5 C and IAB-lI 5 C.

For a given weight function w and function f , the corresponding Lagrange interpolation polynomial is denoted as m

Lm(W, f ) ( t )=

1

f(tk)lk(t),

k=l

where the fundamental polynomial

lk

is defined by :

For the convenience, we give some properties of generalized polynomias which will be applied ek+l

Xk

-

1 -(I m

-

-

8k

-

1 uniformly for m E N and 0 5 k 5 m + 1, m

-

tk)’+*(l+

tk)”’

uniformly for m E N and 1 5 IC 5 uniformly for m E N and

1 5 k 5 m,

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ACTA MATHEMATICA SCIENTIA

Vo1.26 Ser.B

Ipm(t) 5 C ~ m l / ~ ( t )uniformly I for m E N and It1 5 1,

-

IZc(t)l

(4)

1 uniformly for m E N and 1 5‘c 5 m and It1 5 1,

where wm(t) = (-+

(5)

$)2af1(m+ Let $)2f10+1.

then we call

the generalized Lebesgue constant and modulus of singular quadrature formulas respectively, ~ if a = O(p = 0), where b ( t ) is a Holder-continuous function which hase relation with C U , as: b(1) = O(b(-1) = 0).

3 Main Results In this section we first give some lemmas which can help us to give the estimation of two modulus, namely, generalized Lebesgue constant and modulus of generalized singular quadra-

Lemma 1 If w(t) E GJ,,fl, a ,/3 2 0, m 2 2, then we have

IlSLll 5 C l n m . m

Proof We first estimate the bounds of the function

&/It - t k l . Let z = C O S ~ . k=l,k#c When c = 1, applying (l),(2) and Jordan inequality $8 5 sin8 5 8, (0 5 8 5 yields

4)

A2

t - t2

5C

sin O2 5 cos el - cos e2

c.

Then we have sin 8

d 8 + C 5 Clnm.

k=2

Similarly we can prove the case of c = m. When 2 5 c 5 m - 1, we can obtain sin 8 dB+C/= cos e - cos 4

sin 8 d8 cos 4 - cos 8

Applying (2), (3) and the Cauchy inequality yields

So Lemma 1 is proved. Applying (4),( 5 ) yields Iw(t)pm(t)l5

c ( m +-)“-i(diTt+ 1 +, m

m

+C 5 Clnm.

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Cai & Du: ESTIMATION OF GENERALIZED LEBESGUE CONSTANT

No.4

then we can obtain the following estimation of generalized Lebesgue constant Theorem 1 If w ( t ) E GJ,,p,a,P 2 0, m 2 2 , we have

Lemma 2 [5] Let p ( t ) E P,, then for arbitrary p , r

E

R, we have

Let

Lemma 3 Assume w(t)

where

6

GJ,,p,

E

Q,

p > 0 , 4 E Lip&, then for m 2 2 , we have

is an arbitrary constant such that 0 < E

< min{a, p } .

3 , ~

9.

Proof We only prove the case It1 < 1. Let € 1 = = %, where 1 = with X satisfying: if cwp > 0, X = min{a, p, p } ; if a = 0, /3 > 0, X = min{P, p } ; if a > 0, p = 0, X = min{P, p } ; if (Y = /3 = 0, X = p. So we have

Applying (4) yields

p = 0; I1

Ic

lnm

a2

f , p > i;

mz and

13

5C

lnm

else, where 6 is an arbitrary positive constant such that: if max{cr,D}. For 12,we have

Obviously, we have

122

a = 0; a > ;,p>

f;

else,

(YP > 0 , s <

5 CIw(t)pm(t)l.By Lemma 2 , we can obtain w(wp,, t ) I C7n3+a+tJtX

Like similar proof as in Lemma 3, we have the following lemma.

min{a,p}; else 6 <

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ACTA MATHEMATICA SCIENTIA

Lemma 4 Assume w ( t ) E GJa,o,a , p 2 0,4(t)E Lip;,

Vo1.26 Ser.B

then for m 2 2, we have

where E is an arbitrary constant such that 0 < E < min{a, p}. Applying above lemmas can yields the following estimation of modulus of singular quadrature formulas. Theorem 2 Assume w(t) E GJa,p,a ,,8 2 0, + ( t )E Lip'l,, then for m 2 2, we have m i ln2m om

5C

ln2m

ap = 0; a , p > f;

m$-minta,81+c

where E is an arbitrary constant such that 0 < E

4

else,

< min{a, p}.

Application We first introduce an operator B as in [4]:

where a@),b ( t ) ,f ( t ) , w ( t ) are defined as before. In order to prove the operator BLmf uniformly convergent to B on the interval [-1,1], we give the following lemma. Lemma 5 [4] If f ( t ) E LipL, let Jm-l be the best approximation polynomial of degree m - 1, Em-1 = f - Jm-1, then we have

JIBE,-1

11 Im-'l+',

where E is an arbitrary constant such that 2~ < p. Theorem 3 Assume w ( t ) E GJa,o,a , p 2 0, $(t) E Li?;, Liph, then for m 2 2, we have

IIB(Lmf) - Bfll 5 C

i

mi-k-v+e m-k-v-e

mi-min{a,p}-v-k+e

where E is an arbitrary constant such that 0 < E

f ( t ) E C(')[-l, 11,f ( k ) E

ap = 0;

ff,P > ; else,

< min{a, p, 5 ) .

References Lu Jianke. Boundary Value Problems for Analytic Functions. Singapore: World Scientific, 1993 Eilliot E. The classical collocation method for singular integral equations. SIAM J Numer Anal, 1982, 19: 816-832 Giuliana Crisyolo, Giuseppe Mastroianni. On the convergence of a n interpolation product rule for evaluating Cauchy kernel vaule integrals. Math of Computation, 1987, 48: 723-735 Du Jinyuan. Singular integral operators and singular quadrature operators associated with singular integral equations. Acta Math Sci, 1998, 18(2): 227-234 Nevai G P. Mean convergence of Lagrange interpolation (I). J of Approximatiom Theory, 1976, 18: 363-377