SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS ASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS

1998,18(2):227-240

SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS ASSOCIATED WITH SIN·GULAR INTEGRAL EQUATIONS 1 Du Jinyuan

({.l~.it.)

Department of Mathematics, Wuhan University, Wuhan 430072, China

Abstract In this paper, the author discusses some singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, and obtain some useful properties for them. These results improve both the classical theory of singular integral equation and the classical theory of singular quadrature.

Key words Singnlar integral operators, singular quadrature operators, discretization matrices.

1 Introduction Singular integral equations (SIEs) with Cauchy type kernels of the form

a(t)ip(t) +. -b(t) 71'

1 1

),1

(r) ~dr +-

-1 T -

t

1

71'_1

k(r, t)ip(r)dr

= f(t), -1 < t < 1,

(1.1)

appear frequently in problems of the theories of elasticity. Here the input functions a, b, f, k are the Holder-continuous functions for their variables, ), is a given constant, and it is required to find the solution ip in the class ho [1,2]. The classical theory of these equations is rather complete [1,2]. In the past twenty years a great deal of interest has arisen in their numerical solution. Various collocation methods for SIEs have appeared, for which some references can be found in the survey papers [3,4]. But the analytic theory for these collocation methods seems complex and incomplete. The early works in the field is to study the numerical solutions for the SIEs of the first kind (i.e., a 0, b 1) and the slightly more general case when a and b are arbitrary constants, early in the eighties D. Elliott more successfully studied first ones for the variable coefficients SIE (1.1) in a series of important papers [5-12]. Our purpose is to give a unified framework for various collocation methods of SIE (1.1). To do this, we must discuss some singular integral operators, singular quadrature operators and discretization matrices associated with SIE (1.1). In this paper, we shall establish some such operators and see that these operators possess very interesting and similar properties. These results improve

=

=

1 Received Oct.l0,1996. Supported by the National Natural Science Foundation of China and the Foundation of the State Education Commission of China.

228

ACTA MATHEMATICA SCIENTIA

VaLlS

both the classical theory of singular integral equations and the theory of singular quadrature. Using them we may give a unified framework for various collocation methods of SIE(1.1). Under the framework, the analytic theory of collocation methods is very simple and obvious. For the SIEs of the first kind, the applications have been given in [13]. In order to keep the present paper within reasonable bounds, the applications for the collocation methods of the SIEs of the second kind are postponed to another paper [14]. From [15], we know that the normalized equation of SIE (1.1) is the following

a(t)w1(t)y(t)

bet) + --;-

1 1

-1

Al

w (r)y(r) \ _ t dr +;:

l

-1

w1(r)k(r,t)y(r)dr

= J(t), -1 < t < 1,

(1.2)

where W1(t) = z(t)/r(t) is theweight function of the first kind associated with SIE (1.1), z(t) is the fundamental function of (1.1), ret) = JaZ(t) + bZ(t) f. 0 {normal type). With the relation

cp(t) = W1(t)y(t),

(1.3)

the solutions of SIE (1.1) in the class ho are equivalent to the solutions of SIE (1.2) in the class H [15]. Therefore we shall discuss the latter.

2 Singular Integral Operators Introduce the singular integral operators (SIOs)

b~)

(Ay)(t) = a(t)w1(t)y(t) +

ill

y w1;r2 dr, t(r)

-1

wz(r)y(r) dr, T - t

»:

(By)(t) = a(t)wz(t)y(t) _

1l"

and their adjoint operators

(A *y)(t)

= a(t)w1(t)y(t) + ~

(B*y)(t) = a(t)wz(t)y(t) -

ill w1(r~b~rly(r)

.!..1 1l"

1

-1

dr,

wz(r)b(r)y(r) dr, r - t

where wz(t) = l/r(t)z(t) is the weight function of the second kind associated with SIE (Ll) . Remark 2.1 For convenience thereinafter, here we denote the adjoint operator of A as B* , of B as A ", but as A' and B ' in general text. Again introduce the integral operators

.,

(D y)(t ) = 1

.!..1 1l"

1

w1(r)[P.P(X-

-1

( )( )_ 1 Dzy t -

~

1

1l"

-1

1)(r) T

- P,P(X- t

1)(t)]y(r)

dr,

wz(r)[P.P(X)(r) - P.P(X)(t)]y(r)d r, T' - t

where X is the canonical function (in the class ho) of SIE (1.1), P,P(~)(z) denotes the principal part of the sectionally holomorphic function ~ with the jump line [-1,1].

No.2

Du: SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS 229

Remark 2.2 If (1.2)[1,2].

K,

2

0 then D 2

= 0, if :s 0 then D 1 = 0, K,

where

K,

is the index of SIE

Let II n denote the family of all polynomials of degree not greater than n and regard < O. In [15] we had proved the following theorems. Theorem 2.1 AB = 1+ bD 2 , BA = 1- bDl, where I is the identity operator. Theorem 2.2 If 7I"n E II n, then A*7I"n = P.P(X7I"n), B*7I"n = P.p(X-171"n).

II n = {O} if n

when

Remark 2.3 From Theorem 2.1 and Remark 2.2 we know that, A is a left inverse of B K, 2 0 and A is a right inverse of B when K, :s O. This result is first obtained by Elliott

[6-9].

Corollary 2.1 ker(A) = bII"_1, ker(B) = bII_"_1 and Ima(D 1) = II"-ll Ima(D 2) = II-,,-1, more precisely, if 71",,-1 E II"-1 then D 1b7l""_1 = 71",,-1 and if 11"-,,-1 E II-"-1 then D 2b7l"_"_1 = -71"-,,-1. Proof Obviously, by Theorem 2.2, bII"_1 ~ ker(A), and from the definition of Dl, we get Ima(D 1) ~ II"-1' On the other hand, by Theorem 2.1, if Y E ker(A) then Y E bII,,_1, i.e., ker(A) ~ bII,,_1' Again if y E II"-ll then by = bD 1(by), treating b =P 0 (the case b = 0 is trivial because K, = 0 when b(t) == 0) and noting that D1(by) is a polynomial, then get y = D (by ), i.e., II"-l ~ Imal Dj ). The proof of the other equalities is similar. Corollary 2.2 bD 1 and -bD 2 are idempotent, D 1 and D2 are the left zero divisors of B and A respectively, bD 1 and bD z are the right zero divisors of A and B respectively, i.e.,

1

D 1B

= D 2A = AbD 1 = BbD 2 = O. Corollary 2.3 If = 0, then Ay = f K,

possesses a unique solution y

=

= Bf; if = =

K,

> 0 and

N"-l E II"-l is a given polynomial, then under the condition DIY N"-l, Ay f possesses a unique solution y Bf +bN"_l; if K, < 0, the condition of solvability for Ay f is Dzf 0,

=

=

and it possesses a unique solution y = Bf when the condition of solvability is fulfilled. So D 1 is called the unisolving operator, D z is called the restricting operator. Proof When K, = 0, by Remark 2.3 A and B are inverse to each other, so this case is

trivial. When K, > 0, obviously y = Bf + bN"_l is the solution of Ay = f and D 1y = N"-l by Remark 2.3, Corollary 2.1 and Corollary 2.2. Conversely if Ay = f and D 1y = N"-1 then by Theorem 2.1 y = Bf + bD 1y = Bf + bN,,_l' When K, < 0, if Ay = f then Dz! = 0 by Corollary 2.2 and y = Bf by Remark 2.3, conversely, the latter two equations implicate Ay = f by Theorem 2.1. Remark 2.4 Obviously, for the SIE By its condition of solvability is Dd = O. Remark 2.5 If K, < 0, Dzf

=f

= 0 is equivalent to

[11 wz(r)r f(r)dr = 0 j

if

K,

we also have the similar results. For example,

(j

that in the classical theory [1,2]

= 0,···,

-K, -

1);

(2.1)

> 0, Dd = 0 is equivalent to (2.2)

In fact, let

_ P.P(X- 1)(r ) - P.P(X- 1)(t) X 1 ( r, t ) . r-t

(2.3)

230

ACTA MATHEMATICA SCIENTIA

VaLlS

Since P.P(X- 1) is a polynomial of degree K" we know that X 1(r, t) = 2:;~: Aj(r)t j with the A j is the polynomial of degree i. thus {Aj, j = 0, ... , K, - I} are linearly independent. Therefore

Dd = 0 {:=:::> J~1 W1( r)Aj(r)f(r)dr = 0 (j = 0", ., K, - 1) {:=:::> J~1 W1(r)r j f( r)dr 0, ... , K,

-

= 0 (j =

1). The rest of the proof is similar.:

Thereinafter we assume always that bet) in (1.2) is a polynomial of degree 1'. Theorem 3.3 Ifr ~ K,+J.L-l then A(II r ) ~ II r - l< , ifr ~ -K,+J.L-l then B(II r ) ~ II r + I< ' Proof By Theorem. 2.2, if 1f r E lIn then

(A1fr)(t) = P.P(X1fr)(t) so A(1fr ) E II r -

I< '

+.: -1

1f

w1(r)[b(r) - b(t)]1fr(r) dr, r-t

The rest of the proof is similar.

3 Singular Quadrature Operators In this section, we shall construct some singular quadrature operators (SQOs) possessing properties similar to those of SIOs A and B in the last section.

Definition 3.1 If Pn is a polynomial with all simple zeros a.n,jU = 1"", n) E [-1,1], qm is a polynomial with all simple zeros f3m,j(j = 1,···, m) E [-1,1], n = m+K, and Apn = qm, Bqm = Pn, then we call (Pn, qm) a pair of associated polynomials of SIOs (A, B). For the remainder of this paper we will always assume that min{n, m} ~ 1'. Under this hypothesis there are many of such pairs of polynomials, for example, Elliott first successfully found one in [6] based on the Gauss quadrature rule and the present author further gave more ones in [15] based on the Markov quadrature rule. Introduce the discretization operator and the Lagrange interpolation polynomial operator at the set of zeros tj U = 1,2,· " , n) of a polynomial x; as follows: r~f

= (f(t 1),···,f(tn))'!,

t

(3.1)

(L~J)(r) = j=1 1f1 (t~)(~~t./(tj), n

J

(3.2)

J

If (Pn, qm) is a pair of associated polynomials, we know that the interpolation type singular quadrature operators Q~A for A and Q~ for B, are as follows [16]:

(Q~A J)(t) = ~:g? f(t) + b(t)E~Dd~ (. ~ t) ~f, where

E!:D

= (U n,1, ... , un,n)

(3.3)

(3.4)

with

(3.5) and the diagonal matrix

df. (_1) .- t

= diag

(

1 , ... , 1 ) a n,1 - t an,n - t

j

(3.6)

Du: SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS 231

No.2

and

(Q~ J)(t) = ::~~~ f(t) - b(t)E~Rd'!n (. ~ t )r'!nf,

(3.7)

E~R = (V m,l,"',Vm,m),

(3.8)

where with

(3.9) and the diagonal matrix q ( 1)_. (1 1) dm . _ t - diag 13m,l - t"", 13m,m - t .

(3.10)

If (Pn,qm) is a pair of associated polynomials of SIOs (A,B), the (Q~A, Q~) given by (3.3) and (3.7) is called a pair of associated SQOs relative to (A, B).

Definition 3.2

Remark 3.1 diag(u n , l " ' " un,n)r~b=r~(qm/p~), diag( Vm,ll" " vm,m)r'!nb=-r'!n(Pn/q:"). For example, noting that WI only possibly possesses the weak singularity at -1 and 1, we have

Let C' (7rn) denote the family of all continuous functions on [-1, 1] which are differentiable at the zeros of 7rn • Remark 3.2 From (3.3) and (3.7) we know Q~A : C'(Pn) -+ C'(qm), Q~ : C'(qm)-+

C'(Pn)' We recall that the definition of the algebraic precision of a quadrature formula [16,17], for example, the algebraic precision of Q~A, denoted by pr(Q~A), is defined to be the number max{j, IIj ~ ker(Q~A -An. We know that a quadrature formula is of interpolation type iff its algebraic precision P 2:: n - 1, n being the number of its nodes [16,17]. Remark 3.3 pr(Q~A) 2:: n, pr(Q~) 2:: m. We point out a fact used thereinafter: the matrix E~D is just the matrix of the interpolation type quadrature formula for the integral operator

111

Df = -

7r

-1

w1(r)f(r)dr,

(3.11)

and the matrix E7nR is just the matrix of the interpolation type quadrature formula for the integral operator

111

Rf = -

7r

• w2(r)f(r)dr.

-1

(3.12)

In other words, setting

(3.13) we have Remark 3.4 pr(Q~D) 2:: n - 1 and pr(Q'!nR ) 2:: m - 1. Using the EK D we set the quadrature formula of D 1

(3.14)

232

ACTA MATHEMATICA SCIENTIA

Val.IS

where

(3.15) which originates from the kernel Xl in D}, and Xl is as in (2.3). Similarly, using the E~I( we set the quadrature formula of D z

(3.16) with d~ (X z(" t))

= diag(Xz(,6m,}, t),' .. ,Xi(,Bm,m,i)),

(3.17)

which originates from the kernel X z in D z , and

_ P.P(X)(r) - P.P(X)(t) X 2 ( r,t ) . r-t

(3.18)

The two quadrature formulae are also the interpolation type quadrature formulae. This fact is not obvious. To do this, we must improve the results in Remark 3.4. First we point out a obvious fact:

2: 0 then Q;,f2 = 0, if", :s; 0 then Qf.D, = O. We introduce two spaces L~,[-l, 1] and L;2[-1,1] equipped with the inner products respectively: Remark 3.5 If '"

(3.19) Definition 3.3 Let

11""

be a polynomial of degree n. If for any

f

E II v -

1

(3.20) and there exists a polynomial g of degree v such that

(3.21) then we say 11"" to be a (l/)-quasi-orthogonal polynomial in L;, [-1, 1] (L~2[-1, 1]) and v to be its degree of orthogonality, which is denoted by

(3.22) Obviously, any polynomial of degree n does not possess the orthogonality of degree greater than n.

Lemma 3.1 Odel(P,,) = v and pr(Qf.D) = n -1 + v are equivalent, Odez(qm) = v and pr( QaI() = m - 1 + v are equivalent. Proof Noting that for any h(r) = r j there are 1I"j_" E II j _ " and r,,-l E II"-l such

that Ii = 1I"j_"p" + r,,_l, from Remark 3.4 we get Dfj - Qf.D Ii = D1I"j_"p", hence the first conclusion is proved. The proof of the second conclusion is similar.

Such Qf.D and Qa,R in Lemma 3.1 are called (v)-quasi-Gaussian quadrature formula and (v)-quasi-Gaussian quadrature formula respectively. When v = n and v = m they are just the classical Gaussian quadrature formulae. Lemma 3.2 Odel(p,,) ~ max{O, ",}, Odez(qm) ~ max{O, -",}.

No.2

Du: SINGULAR INTEGRAL OPERATORS AND 'SINGULAR QUADRATURE OPERATORS 233

This results from the definitions of Pn and qm, Remark 2.4-2.5 and Corollary 2.3. Lemma 3.3 pr(Q~D) 2:: max{n -1,n+ 1f,-1}, pr(Q~R) 2:: max{m-1,m- 1f,....:.1}. The following two lemmas are its consequences. Lemma 3.4 pr(Q~A) 2:: max{n,n+If,}, pr(Q~) 2:: max{m,m- If,}. Lemma 3.5 Q~Dl I =D1L~/, Qr.f2 I = D2L~/, more precisely, pr(Q~Dl) 2::n, pr(Qr.f2) 2:: m.

This results from X 1(',t)EII"-1l X2(·,t)ETI-I<-1l Lemma 3.3, Remark 2.2 and Remark 3.5. The following theorem is parallel to Theorem 2.1. Theorem 3.1 If (Q~A, Q~) is associated relative to (A, B), then Q~A Q~ = 1 + bQr.f2, Q~ Q~A = 1 - bQ~Dl. In particular, Q~A Q~ = 1 if I\, 2:: 0 and Q~ Q~A = 1 if I\, ::; O. Proof Noting Remark 3.2, we know that the compositions of Q~A and Q~ are reasonable. First suppose I E ker(~), we get Q~Q~A I = I, for a general function I, we have

Qr!Q~A I

=I

- L~I + BAL~I

=I

- L~I + (1 - bD1)L~1

=I

- bQ~Dl f.

By Remark 3.5, obviously Q~Q~A = 1 if If, ::; O. The rest of the proof is similar. Introduce the adjoint SQOs of SQOs Q~A and Q~, which are the SQOs for SIOs A * and B* respectively, as follows

(Q~A' J)(t) = ~:g; I(t) + E~D~ (. ~ t) ~bf,

(3.23)

~ t) r;"bf.

(3.24)

(Q~B' J)(t) = ::~~~ I(t) -

E!,.Rd;" (.

Theorem 3.2

(QpA' J)(t) _ qm(t)/(t) - (L~qmJ)(t) (QqB' J)(t) 'm n Pn(t)

= Pn(t)f(t) -

(L'!np,.!)(t). qm(t)

Proof Only prove the first equality.

Using Theorem 3.1 and Theorem 3.2 we may get the following corollaries, which are parallel to Corollary 2.1-2.3, and their prooffollows trivially in a completely analogous way to that used in Corollary 2.1-2.3. Corollary 3.1ker(Q~A) = bII"-ll ker(Q~) = bII_,,_1, Ima(QtD1) = TI"-1, Ima(Qaf 2) = II_"-1, more precisely, if 11',,-1 E II"_1 then Q~Dl(b1l'''_1) = 11',,-1 and if 11'_,,_1 E TI- K - 1 then Qr.f2(b1l'-,,_d = -11'-,,-1' Corollary 3.2 bQf,D 1 and -bQr.f2 are idempotent and the right zero divisors of Qf,A and Q~ respectively, Qf,D 1 and Qr.f2 are the left zero divisors of Q~ and Q~A respectively. Corollary 3.3 If I\, = 0, then Q~Ay = I possesses a unique solution y = Q~ f; if If, > 0 and N"-1 E II_ K - 1 is a given polynomial, then under the condition Q~Dly = N K - 1, Qf.Ay = f

234

ACTA MATHEMATICA SCIENTIA

VaLlS

possesses a unique solution y = Q7:f + bN"_l; if K, < 0, the condition of solvability for Q~Ay = f is Qr,f2 f = 0, and it possesses a unique solution y = Q7: f when the condition of solvability is fulfilled. Remark 3.6 If K, < 0 Qr,f2 f = 0 is equivalent to that q (f3j)r q f - 0 (Jo - 0 ... EqRd mm m-"

where

din(f3j) if

K,

>0

Q~Dl f

-K, -

1) ,

= diag(f3~,l' ... ,f3!n,m);

(3.25)

(3.26 )

= 0 is equivalent to that (3.27)

where

d~(ai)

= diag(a!"l""

,a~,n)'

(3.28)

Theorem 3.3 If r ~ K, + f..l- 1, Q~A(IIr) ~ II r-,,; if r ~ -K, + f..l- 1, Qa~(IIr) ~ II r +" . It follows from Theorem 3.2 using a proof similar to that used in Theorem 2.3.

4 Discretization Matrices In this section, we discuss the compositions of the associated SQOs and the discretization operators. Applying r'/n to Q~A we get

(4.1) where (m,n) matrix Am,n is just

Am,n with

~,;

= {

= (ai,j),

(4.2)

b(f3m,i)Un,j an,j - f3m,i '

if an,j

q~(f3m,i) _ b'({3 0)

0

) m,a un,J' P,n( an,j In (4.3), the case an,j "# f3m,i is obvious, if an.j b(f3m,i)Un,j = 0, hence

a',J o

•

_

li

im

t-+f3m,i

(qm(t) ( ) Pnt

+ b(t)Un,j) an,j-t

if an,j

"# f3m,i,

= f3m,i'

= f3m,i,

(4.3)

from Remark 3.1 we know

_ q~(f3m,i) _ b'(f3') . m,' un,J' p,,(an,j)

-

The matrix Am,n arises from discretizing Q~A by r'/n, we call it the discretization matrix of Q~A. By analogy, discretizing Q~ by ~, we also have qf r PQqBf-B n,mrm' n m

where the (n,m) matrix Bn,m is

Bn,m

= (bj,r),

(4.4)

(4.5)

Du: SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS 235

No.2

with

b( O'.n,j )Vm,r b. _ i-r -

{3m,r -

{

I (

if O'.n,j

O'.n,j' )

Pn O'.n,j I ({3 ) qm nl,T

f. {3m,r, (4.6)

+ b (O'.n,j )Vm,Tl I

if O'.n,j = {3m,r'

which is called the discretization matrix of Q~ .

Definition 4.1 The pair (Am,n, Bn,m) given by (4.2) and (4.5) is called a pair of associated matrices of (Q~A , Q~), or simply, is associated. Remar k 4.1 From the source of (Am,n, Bn,m), we have ra, Q~A

= Am,n r~,

Bn,mr'!n. Lemma 4.1 If (Am,n, Bn,m) is associated, then when", 2: O,Am,nBn,m Bn,mAm,n = In where In is the unit square matrix of order n.

r~ Qaf1

=

= 1m; when ",:SO,

Proof We only prove the first equality. From Theorem 3.1 and Remark 3.5, we have Q~AQ~ = I, Again applying r'!n to both sides of this equality and noting Remark 4.1, get

= Am,nr~Q'!nB! = Am,nBn,mr'!n! = raJ. Noting that raJ is arbitrary, finally, Am,nBn,m = 1m . Remark 4.2 When K, = 0, denote Am,n and Bn,m, which are square matrices of order n, as An and B,,, then they are inverse to each other. Now we come to find the similitude of An and B n when K, f. O.

r'!nQ~AQ~!

First we treat [-1. 1], let

K,

>

O. Arbitrarily choosing

s,,(t)

=

K,

different points {3m,i (i

= m + 1, ... , n)

III

n

II

(t - (3m,i)'

(4.7)

i=m+l

Discretizing Qf..D 1 by r~, we get

(4.8) where (K" n) matrix A",n is (4.9) with (4.10) Let the partitioned matrix (square matrix of order n)

).

(4.11)

Again let (4.12) where (4.13) with

(4.14)

236

ACTA MATHEMATICA SCIENTIA .

S",r(t)

s,,(t)

= s", (,Bm,m+r )(t -

,Bm,m+r

Vol.IS

)"

(4.15)

Remark 4.3 ~ (bL~J) = Bn,,,r~f. We may prove AnBn = In later, so An and B n are called the supplemented matrices of Am,n and Bn,m respectively. By analogy, when If, < 0 we arbitrarily choose -If, different points an,n+i(i = 1"", -If,) in [-1,1] and set m

II

s : .. (t) =

(t - an,i)'

(4.16)

q! = B -.. ,mrm'

(4.17)

i=n+l

From

• .. QqD,! r_ m we get the (-If" m) matrix

B- ..,m = (bn+i,r),

(4.18)

where (4.19) Let

s.; = (Bn,m

B- ..,m

),

(4.20)

and (4.21) with

A m,-..

= (ai,n+i)'

(4.22) (4.23)

ai,n+i = -b(,Bm,i)S-",i(,Bm,i), .( ) _ s., ..,} t -

s- .. (t) , ( )( ) an,n+i t - an,n+i

(4.24)

s_ ..

Remark 4.4 r'!n(-bL"-..J) = Am,_ .. r"-.. !. Theorem 4.1 If If, 0 then AnBn In, where An and B n are given by (4.2) and (4.5); if If, > 0 then AnBn = In, where An and B n are given by (4.11) and (4.12); if If, < 0 then BmA m = 1m, where B m and Am are given by (4.20) and (4.21). So we also call Am and B m the supplemented matrices of Am,n and Bn,m respectively. To prove this theorem, we first give some lemmas.

=

=

=

=

Lemma 4.2 When If, > 0, Am,nBn,.. Om,..; when If, < 0, Bn,mAm,-" On,- .., where Om,.. is the (m, If,) zero matrix. Proof Only prove the first equality. By Corollary 3.1, we have QtA(bL~f) = 0, afterwards, by Remark 4.1 and Remark 4.3, r'!nQ~A(bL~J) Am,nr~(bL~J) Am,nBn,..r~! Om,l, noting that r~! is arbitrary, we get Am,nBn,.. = Om,n»

=

=

=

=

Lemma 4.3 When If, > O,A..,nBn,m O..,m; when If, < O,B- ..,mAm,n O- ..,n' Proof From Corollary 3.2, we have Q~l Q~! = 0, hence, by (4.8) and Remark 4.1, r~QtDl Q~! = A ..,nrt Q~! = A ..,nBn,mr'!n! = 0 ..,1> finally, A ..,nBn,m = O..,m' Lemma 4.4 When If, > 0, A ..,nBn,.. = I..; when If, < 0, B_ ..,mAm,_.. = 1_...

No.2

On: SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS

Proof By Corollary 3.1, Q~Dl(bL~f)

= LU. Applying r~

237

to both sides of this equality

and noting (4.8), Remark 4.3, we have A",nBn,,,r~1 = r~/, i.e., A",nBn,,, = I". Proof of Theorem 4.1 The case of /'i, = 0 is just Remark 4.2. The proofs of the other two cases are similar, we only prove the case of r: > O. Quoting Lemma 4.1- 4.4, we have

o.; ) -

I"

I n'

For some special cases (the SlEs of the first kind and the case when a and bare arbitrary constants), N. 1. loakimidis, P. S. Theocaris, G. Tsamasphyros and A. Gerasoulis proved the result of Theorem 4.1 via very technical calculations respectively [18-20]. For the SlEs of the second kind, when Pn and qm are respectively the orthogonal polynomials associated with 1111 and 1112 Elliott first proved the result of Lemma 4.1 based on the properties of the Gauss quadrature formulae [5,9]. The author obtained some results for more general cases still using specific computation [21-24]. Here Theorem 4.1 is obtained in a different way by the abstract properties of the aforementioned operators, which is very simple and obvious. Corollary 4.1 If

K,

~

0 then the linear algebraic equation AnOn = In' where

(4.25) possesses a unique solution On = Enln, If K, < 0 then the condition of solvability for Am,non = 1m is B-li,m!m = 0 and it possesses a unique solution On = Bn,m!m when the condition of solvability is fulfilled. Proof The case of K, ~ 0 is obvious. By Lemma 4.1 and Lemma 4.3 we know that if Am,non = i-; then On = Bn,m/m and B-li,m/m = 0, conversely, if the last two equalities hold then !m = AmBm!m = Am,nBn,m/m, so On = Bn,m/m is the solution of Am,non = L«.

5 The Convergence of SQOs Suppose that there are a sequence of the pair (Pn' qm) of associated polynomials of SlOs (A,B), say {Pn' qm}, we discuss 'now the convergence of SQOs established before. We again suppose that the spaces considered are equipped with the Chebyshev's norm, namely, the maximum of the absolute values of a function. Obviously, n

IIDII = D1, IIRII = R1, IIQ~DII =

L j=l

m

IUn,jl, IIQ~RII

=

L

Iv""jl·

(5.1)

j=l

Theorem 5.1 If the sequence {Q~D} ({Q?r~}) is uniformly bounded, then the sequence {Qt } ({ Q~R}) pointwisely converges to the operator D (R), more precisely, IIR~D III ::; 12(IIDII+IIQtD ll)w(f, (n-l)-l) (IIR~~/II ::;12(1!RII + IIQaRII)0.,(/, (m-1)-l» for f E C[-l, 1], D

where RtD =D-Qf,D (R~=R-Q~) and ""J.') is the modulus of continuity of I. Proof Denoting the best approximate polynomial of degree n-l of I as In-l, by Jackson theorem and Lemma 3.3, IIRf,D III = IIR~D(f - I n - 1)1 1::; 12(IIDII + IIQ~D II)w (f, (n - 1)-1). Observing that {QtD } is uniformly bounded, the assertion follows.

238

ACTA MATHEMATICA SCIENTIA

VaLl8

Remark 5.1 If {Q~D} ({ Qa"R}) is a sequence of nonegative operators, i.e.,

Un,j

20 (j =

1"," n) (vm,j 20 (j = 1"", m)), then IIQ~DII = 2: =1 Un,j = Dl = IIDII (1IQi,[i11 = IIRII), of course, it is uniformly bounded, in particular, the sequence of the Gauss and Markov quadrature formulae is uniformly bounded since they are the positive operators [17,25]. Theorem 5.2 If {Qi,[i} is uniformly bounded, then the sequence {Q?,f} pointwisely con-

1

verges to the operator Q~ in G'[-I, 1]' more precisely, IIRi,~ III::; 121Ibll(IIRI~IIQi,[ill)w(f', (m1)-1) for f E G'[-l, 1], where R~B = B- Q;'~. Proof Fixing t, let 6(7) = 1(7) - I(t), if

7

f::. t,

if

7

= t.,

G'[-I, 1] and w(f*, 0 ::; w(f', O. Obviously, from Lemma 3.4 and Theorem 5.1, IIRi,~ III = IIRi,~ 611 :S Ilbllll(Ra,R)f*II::;1211bll(IIRII + II Qi,[iIl)w(f', (m - 1)-1). From [25] we know that

f*

E

From Remark 3.2 we know that, in general, Qi,~ is not defined if I E H~ (the class of all Holder continuous functions with the index 1/ (0 < 1/ ::; 1)). This is not convenient in applications, therefore we must improve Qa"B. We introduce the interpolation operator La" (f I--> Li,,f), which possesses the following properties: rinL';,,f

= rinl,

(5.2)

namely, the interpolation property at the zeros of qm, and (5.3) There are many such operators, for example, the aforementioned Lagrange interpolation polynomial operator given by (3.2) is just one. For the discussion of the convergence of {Q?,f L?n}, we must show that B is well defined on [-1,1]. To do this, we give the following remark. Remark 5.2 B: H -+ H. In fact, if I E H then BI is well defined at -1 and 1, for example, if 1 is nonspecial end of (1.2) then 1112 (1) = 0, if 1 is special end of (1.2) then b( 1) = 0 [15], hence BI E H [26]. Theorem 5.3 If IE H V and IIQ~L?nll = o(mV ) , then limn~
E m-1 = 1- J m- 1, then /lE m- 111 ::; Gd(m - l )" (G1 is some constant), quoting Kalandiya's lemma [27] we have I(BEm-d(t)1

/Em_1(t)(Bl)(t) -

<

/lB1111IEm-1/1

b~) 1

11 1112(7)

[E

m- 1(7;

=~m-1(t)] d71

+ GZ~b111l111 1112(7)17 - W-1d711

en ~

1) v-2£

,

where Cz is the constant which is independent of m, 0 < 2f < 1/, again IIQ~Li,tEm-111 ::; IIQi,~Li,tIIlIEm-111, and (B - Q?,fLa,,)Em - 1 = (B - Q~La,,)1 by Lemma 3.4 mid (5.3), hence

lim'H
No.2

Du: SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS

If we take

L¥,.

as the Lagrange interpolating polynomial operator

L¥,.,

239

then

m

~)Bqm,j )(t)f({3m,j ) j=1

Q~L~J)(t)

t[

;n(t)~Pn({3m,j). + b(t)-b({3~,j)vm,j]f({3m,j),

j=1 qm({3m,J)(t-{3m,J)

where

t-f3m,J

qm(r) (.J = 1, 2, ... , m ) qm,l'(r ) =, qm({3m,j )( -r - {3m,j )

(5.4)

(5.5)

are the fundamental polynomials of the Lagrange interpolation corresponding to qm' The Q~L¥,. is called the Paget-Elliott type SQO, which only has the nodes {{3m,j,j = 1,···, m}. In this instance, we have (5.6) IIQ~L¥"II :::; IlallA m + am and

IIQ~L~II :::; 1Ib'III1Q~RII :::; where

(5.7)

m

= L II w 2qm,j ll

Am is the Lebesgue number of

Ilb'lI Am + Pm,

L¥,.

j=1

associated with weight

(5.8)

W2,

(5.9)

Pm

=

til

;n(t) - Pn({3m,j) II· j=1 qm({3m,j)(t - {3m,j)

(5.10)

For some {qm} used often, we easily estimate IIQ~L¥"II by estimating Am, am and Pm [5,8,20,21,28]. Acknowledgment research ..

I would like to thank Professor Lu Jianke for his assistance in this

References 1 Lu Jianke. Boundary Value Problems for Analytic Functions. World Scientific,1993 2 Muskhelishvili N I. Singular Integral Equations. 2nd ed. Noordhoff, 1968 3 Venturino E.Recent developments in the numerical solution of singular integral equations. J. of Math. Anal. & Appl., 1986, 115:239-277 4 Lu Jianke, Du Jinyuan. Numerical methods of solution for singular integral equations. Advances in Mathematics, 1991,20(3): 278-293 5 'Elliott D. The classical collocation method for singular integral equations. SIAM J. Numer. Anal., 1982,19:816-832 6 Elliott D.Orthogonal polynomials associated with singular integral equations having a Cauchy kernel. SIAM J. Math. Anal., 1982,13:1041-1052 7 Elliott D.A Galerkin-Petrov method for singular integral equations. J. Austr. Math. Soc. Ser. B,1983,25: 261-275

240

ACTA MATHEMATICA SCIENTIA

VoLl8

8 Elliott D.Rates of convergence for the method of classical collocation for solving singular integral equations. SIAM J. Numer. Anal., 1984,21:136-148 9 Elliott D.A convergence theorem for singular integral equations. J, Austral.Math. Soc. Ser. B,1981,22 :539552 10 Elliott D.The numerical treatment of singular integral equations-a review. In: Baker C T H, Miller C F (Editors). Treatment of Integral Equations by Numerical Methods. New York: Acad. Press., 1982 11 Elliott D.Singular integral equations on arc (-1,1): theory and approximate solution, Part 1. Theory, Technical Report No. 218,Dept. of Math., University of Tasmania, Australia,1987 12 Elliott D.Projection methods for singular integral equations, J. of Integral Equations and Applications, 1989, 2:95-106 13 Du Jinyuan. Singular integral operators and singular quadrature operators associated with singular integral equations of the first kind and their applications. Acta Math.Sci., 1995,15(2): 219-234 14 Du Jinyuan. The classical collocation methods for singular integral equations with Cauchy kernels(has been submitted to J. of Math. Anal. & Appl.) 15 Du Jinyuan. Some systems of orthogonal polynomials associated with singular integral equations. Acta Math. Sci.,1987,7(1): 85-96 16 Du Jinyuan. On the numerical evaluation of singular integrals. J. of Central China Teachers College, 1985,2: 15-28 17 Du Jinyuan. The quadrature formulas for singular integral of higher order. Chinese Math. AIm., 1985,6A(5): 625-636 18 Ioakimidis N I, Theocaris P S. A comparison between the direct and the classical numerical methods for the solution of Cauchy type singular integral equations. SIAM. J. Numer. Anal.,1980, 17: 115-118 19 Gerasoulis A.On the existence of approximate solutions for singular integral equations of Cauchy type discretized by Gauss-Chebyshev quadrature formulae. BIT, 1981,21: 377-380 20 Tsarnasohyros G, Theocaris P S. Equivalence and convergence of direct and indirect methods for the numerical solution of singular integral equations. Computing,1981,27: 17-80 21 Du Jinyuan. On methods for numerical solutions for singular integral equations by interpolatory quadrature formulae (I). Acta Math. Sci., 1985,5(2): 205-223 22 Du Jinyuan. On methods for numerical solutions for singular integral equations by interpolatory quadrature formulae (II). Acta Math. Sci., 1985,5(4): 433-443 23 Du Jinyuan. On methods for numerical solutions for singular integral equations (I). Acta Math. Sci., 1987,7(2):169-189 24 Du Jinyuan. On methods for numerical solutions for singular integral equations (II). Acta Math. Sci., 1988,8(1):33-45 25 Du Jinyuan. The quadrature formulas of the closed type and the transformed weight type for singular integrals of higher order.•J. of Math. (PRC), 1986,6(4): 439-454 26 Du Jinyuan. A theorem on boundary values of integrals of Cauchy type and its applications. J. of Math. (PRC),1982,2 (2) :115-126 27 Kalandiya A I. On a direct method for solution of equation of wing theory and its application in theory of elasticity. Math. Sbornik,1957,42(2): 249-272 28 Tsa.masphyros G, Theocaris P S. On the convergence of a Gauss quadrature rule for evaluation of Cauchy type singular integrals. BIT, 1977,17: 458-464