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The Complex Spline Approximation of Singular Integral Operators over an Open Arc
Journal of Approximation Theory 119, 86–109 (2002) doi:10.1006/jath.2002.3721
The Complex Spline Approximation of Singular Integral Operators over an Open Arc Yong Jia Xu Department of Statistics, Guangdong Business College, Guangzhou, Guangdong 510320, People’s Republic of China E-mail: [email protected] Communicated by Wolfgang Dahmen Received August 29, 2001; accepted in revised form June 26, 2002
This paper mainly considers the smooth complex spline approximation of Cauchytype integral operators over an open arc. First, the smoothness of the operators is investigated, then some properties of complex splines are discussed, and finally the error estimates of the approximation are given. # 2002 Elsevier Science (USA)
1. INTRODUCTION The approximation for Cauchy-type singular integrals and the numerical solution of the corresponding equations have been studied in many papers, and a series of important results have been obtained [3, 7, 8, 13]. However, the rates of convergence for the approximation are not very satisfactory. In fact, the best rates have not been revealed. The reason may be that, as mappings, the smoothness of the integrals is not investigated thoroughly. For example, paper [7] discusses the following operator: Z bðtÞ 1 w2 ðtÞjðtÞ A# ðjÞðtÞ aðtÞw2 ðtÞjðtÞ dt; t 2 ½1; 1; ð1:1Þ p 1 t t where a 2 H½1; 1; b is a polynomial and w2 is a weight function constructed from a and b (for details, see [7] or [6]), and proves that if w2 2 H g ½1; 1 and j 2 C m;m ½1; 1 for 05g; m51; then A# j 2 H l ½1; 1 where l ¼ minðg; mÞ: This means that the smoothness of A# j depends on w2 : But in fact, as an operator, A# is sufficiently smooth and its smoothness is not affected by w2 : This can be illustrated when aðtÞ 0 and bðtÞ 1 in (1.1). In this case, A# becomes Z pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 t2 jðtÞ dt; t 2 ½1; 1 ð1:2Þ TðjÞðtÞ tt p 1 86 0021-9045/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
COMPLEX SPLINE APPROXIMATION
87
and it is bounded on C m;m ½1; 1 because we can easily convert it to a periodic singular integral or a Cauchy-type integral on a unit circle (cf. [15, 18]). Usually, the Cauchy-type singular integrals are of the form Z 1 jðtÞ dt; t 2 G; ð1:3Þ TG ðjÞðtÞ p G tt where G is a smooth or piecewise smooth oriented curve. If G is closed, as a mapping, TG is bounded on C m;m ðGÞ; and if G is an axis, TG has the similar property [14, 15]. However, if G is an open arc, the property disappears. Instead, we consider its ‘‘weighted type’’. This is also a natural approach to the solution of singular integral equations over an open arc. Generally speaking, by a simple transformation, our problems on an open arc can be converted to those on an interval, and then the problems may be solved more conveniently. But this will bring about at least two problems: the first is that the kernels of those integrals may become more complicated and it will then be more difficult for some treatments, such as numerical evaluations; the second is that a smooth function on the arc, as well as the related smooth operators, may become less or even not smooth on the interval if the arc is not perfectly smooth, and it will then impair the rates of its approximation. On the other hand, it is known that complex splines have a lot of advantages over other functions such as polynomials in approximation, especially in the approximation for those functions defined on arbitrary curves. So it will be more meaningful to discuss the singular integral operators and their complex spline approximation directly on arcs. In this paper, our discussion focuses on the Cauchy singular integral operators of the form Z bðtÞ wðtÞjðtÞ Aw ðjÞðtÞ aðtÞwðtÞjðtÞ þ dt; t 2 G; ð1:4Þ pi G t t which are derived from the theory of singular integral equations. Here we assume a; b 2 HðGÞ satisfying a2 ðtÞ b2 ðtÞa0;
t2G
and G is a smooth or piecewise smooth oriented open arc from a to b: The function w is constructed as follows (cf. [5, 6, 12]). Let GðtÞ ¼ aðtÞþbðtÞ aðtÞbðtÞ; and choose a continuous branch of GðtÞ such that 04yðtÞ51;
ð1:5Þ
where yðtÞ ¼ argGðtÞ 2p : Let ½y denote the integral part of the real number y and k ¼ ½yðbÞ:
ð1:6Þ
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YONG JIA XU
Define the canonical function [12, 15] Z 1 ln GðtÞ dt ; X ðzÞ ¼ ðb zÞk exp 2pi G t z
z2 =G
ð1:7Þ
and now let wðtÞ ¼ ½ðaðtÞ bðtÞÞX þ ðtÞ1 ;
t 2 G:
ð1:8Þ
It is easy to verify that wðtÞ 2 HðGÞ and wðtÞa0; t 2 G =fa; bg; which is similar to a weight function. From the above construction, we know the smoothness of the function w is poor. However, we will show in this paper that the operator Aw is very smooth. Furthermore, if wðaÞ ¼ wðbÞ ¼ 0 and b 2 C m;m ðGÞ; Aw is a bounded operator on C m;m ðGÞ: Besides, we make a further study of complex interpolating splines and obtain some interesting properties. And then, there follow the results of the complex spline approximation for the operator Aw : All these appear to be very attractive. Throughout the paper, we assume that: 05m51; m is a nonnegative _ _ integer, G ¼ ab; c0 is a constant such that the arc length j t1 t2 j4c0 jt1 t2 j for all t1 ; t2 2 G; CðGÞ and C m ðGÞ denote the spaces of continuous and mtimes continuously differentiable complex-valued functions on G; respecP ðkÞ tively, jjcjj maxt2G jcðtÞj; jjcjjcm m k¼0 jjc jj; the modulus of continuity for c 2 CðGÞ is denoted by oðc; xÞ; Mm ðcÞ sup05x4 m;m 1 oðc;xÞ ðGÞ fc 2 C m ðGÞ : Mm ðcðmÞ Þ51g; H m ðGÞ C 0;m ðGÞ; HðGÞ xm ; C S 05m51 H m ðGÞ; jjcjjH m jjcjj þ Mm ðcÞ; jjcjjC m;m jjcjjC m þ Mm ðcðmÞ Þ; and c is an absolute positive constant taking different value in different place. The rest of the paper is organized as follows. In the next section, we show the smoothness of the operator Aw : Some properties of complex splines are discussed in Section 3 and the results of spline approximation for Aw are given in Section 4. In Section 5 we illustrate an application for the approximation and remark on the case m ¼ 1; constant c and some other applications. The proof of Lemma 2.2 is longer and thus is put in the final section.
2. SMOOTHNESS OF SINGULAR INTEGRAL OPERATORS In this section, we start our discussion for the smoothness of operator Aw from the introduction of some operators and the proofs of two lemmas. Let u 2 CðGÞ: Define the operator Su;m as Z f ðtÞ Tm ð f Þðt; tÞ Su;m ð f ÞðtÞ ¼ m! ut dt; f 2 C m;m ðGÞ; ðt tÞmþ1 G
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COMPLEX SPLINE APPROXIMATION
where Tm ð f Þðt; tÞ ¼ f ðtÞ þ f 0 ðtÞðt tÞ þ þ f ¼ f ðtÞ: If m ¼ 0; Su;m ð f Þ is written as Su ð f Þ:
ðmÞ
ðtÞ m! ðt
tÞm and T0 ð f Þðt; tÞ
k ð f Þðt;tÞ Let Ck ðt; tÞ ¼ k!uðtÞf ðtÞT ; f 2 C m;m ðGÞ: It is easy to verify that ðttÞkþ1
@ Ck1 ðt; tÞ ¼ Ck ðt; tÞ @t
ð2:1Þ
and Z
jCk ðt; tÞjjdtj51;
supt2G
ð2:2Þ
G
and using dominant convergence theorem, we obtain dk Su ð f ÞðtÞ ¼ Su;k ð f ÞðtÞ; dtk
t2G
ð2:3Þ
for k ¼ 1; 2; . . . ; m: If k5m; then Z t 1 j f ðtÞ Tk ð f Þðt; tÞj ¼ ðz tÞk f ðkþ1Þ ðzÞ dz k! t 1 4 c jj f ðkþ1Þ jj jt tjkþ1 k!
ð2:4Þ
and if k ¼ m; j f ðtÞ Tm ð f Þðt; tÞj Z t 1 m1 ðmÞ ðmÞ ¼ ðz tÞ ½ f ðzÞ f ðtÞ dz ðm 1Þ! t 4
c oð f ðmÞ ; jt tjÞjt tjm m!
ð2:5Þ
for t; t 2 G; therefore Z jSu;k ð f ÞðtÞj4c juðtÞj jj f ðkþ1Þ jj jdtj4c0 jjujj jj f ðkþ1Þ jj;
t2G
ð2:6Þ
G
for k ¼ 1; 2; . . . ; m 1; and jSu;m ð f ÞðtÞj4cjjujj
Z
1 0
Thus, we have
oð f ðmÞ ; yÞ dy; y
t 2 G:
ð2:7Þ
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Lemma 2.1. For the operator Su ; there hold (2.3) and the following estimates: k d Su ð f Þ4cjjujj jj f ðkþ1Þ jj; k ¼ 0; 1; . . . ; m 1; ð2:8Þ dtk if f 2 C m and
if f ðmÞ
m Z 1 d oð f ðmÞ ; yÞ Su ð f Þ4cjjujj dy; dtm y 0 R1 ðmÞ satisfies Dini condition, i.e., 0 oð f x ;xÞdx51:
ð2:9Þ
We also have Lemma 2.2.
There is a constant c > 0 depending on m; m and G such that m
d o Su ð f Þ; x 4cjjujjMm ð f ðmÞ Þxm ð1 þ j ln xjÞ ð2:10Þ dtm
or
m
Z 1 d oðu; yÞ dy Mm ð f ðmÞ Þxm ; o S ð f Þ; x 4c jjujj þ u y dtm 0
ð2:11Þ
if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; where f 2 C m;m ðGÞ and x > 0: Since the proof is longer, we put it in the last section. From Lemmas 2.1 and 2.2, it follows that jjSu ð f ÞjjC m;n 4cNðuÞjj f jjC m;m ;
ð2:12Þ
R1 where 05n5m and NðuÞ ¼ jjujj or 05n4m and NðuÞ ¼ jjujj þ 0 oðu;xÞ x dx if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0: We rewrite the operator Aw as Z 1 bðtÞ bðtÞ 0 jðtÞ dt; t 2 G; ð2:13Þ Aw ðjÞðtÞ ¼ Aw ðjÞðtÞ wðtÞ pi G tt where A0w ðjÞðtÞ aðtÞwðtÞjðtÞ þ
1 pi
Z
wðtÞbðtÞjðtÞ dt; tt G
t2G
ð2:14Þ
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and let KðjÞðtÞ ¼
1 pi
Z wðtÞ G
bðtÞ bðtÞ jðtÞ dt; tt
t2G
or KðjÞðtÞ ¼
1 Swj ðbÞ: pi
ð2:15Þ
Thus, Aw ¼ A0w K:
ð2:16Þ
Theorem 2.3. There exists a constant c > 0 such that k d 0 A ðjÞ4cjjjjj kþ1 ; k ¼ 0; 1; . . . ; m 1 C dtk w
ð2:17Þ
for j 2 C m and jjA0w ðjÞjjC m;n 4cjjjjjC m;m
ð2:18Þ
for j 2 C m;m ðGÞ; where 05n5m: If bðaÞwðaÞ ¼ bðbÞwðbÞ ¼ 0; then (2.18) is valid for n ¼ m: Proof.
A0w can be written as
A0w ðjÞðtÞ ¼ A0w ð1ÞðtÞjðtÞ þ
1 pi
Z
wðtÞbðtÞ½jðtÞ jðtÞ dt; tt G
t2G
or A0w ðjÞðtÞ ¼ pðtÞjðtÞ þ
1 Swb ðjÞðtÞ; pi
ð2:19Þ
where pðtÞ ¼ A0w ð1ÞðtÞ: But
1 ;t pðtÞ ¼ p:p: X ðzÞ
ð2:20Þ
is a k-degree polynomial ðpðtÞ 0 if k50) (cf. [16]), and it is obvious that jjðpjÞðkÞ jj4cjjjjjC k ;
k ¼ 0; 1; . . . ; m
ð2:21Þ
for j 2 C m ðGÞ and Mn ððpjÞðmÞ Þ4cjjjjjC m;m ;
04n4m
ð2:22Þ
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or jjpjjjC m;n 4cjjjjjC m;m
ð2:23Þ
for j 2 C m;m ðGÞ; where the constant c depends on aðtÞ; bðtÞ and m: On the other hand, wb satisfies Dini condition, so that there hold (2.8)–(2.10) for Swb with u ¼ wb and there also holds (2.11) if wðaÞbðaÞ ¼ wðbÞbðbÞ ¼ 0: Thus we obtain (2.17) and (2.18) from (2.19). The proof is complete. ] Let b 2 C m;m ðGÞ: From Lemmas 2.1 and 2.2, k d KðjÞ4c0 jjwjjj jjbðkþ1Þ jj4cjjjjj; k ¼ 0; 1; . . . ; m 1 dtk
ð2:24Þ
and jjKðjÞjjC m;n 4cjjwjjj jjbjjC m;m 4cjjwjj jjbjjC m;m jjjjj;
05n5m
for j 2 CðGÞ; and if wðaÞ ¼ wðbÞ ¼ 0; then
Z 1 wðwj; yÞ 0 dy jjbjjC m;m jjKðjÞjjC m;n 4c jjwjjj þ y 0
Z 1 oðj; yÞ dy 4c jjjjj þ y 0
ð2:25Þ
ð2:26Þ
for j 2 H m ðGÞ; where we have used the fact that oðwj; yÞ4jjwjjoðj; yÞ þ jjjjjoðw; yÞ: Thus, we obtain Theorem 2.4. Let b 2 C m;m ðGÞ: Then there is a constant c > 0 such that k d Aw ðjÞ4cjjjjj kþ1 ; k ¼ 0; 1; . . . ; m 1 ð2:27Þ C dtk for j 2 C m ðGÞ and jjAw ðjÞjjC m;n 4cjjjjjC m;m
ð2:28Þ
for j 2 C m;m ðGÞ; where 05n5m: If wðaÞ ¼ wðbÞ ¼ 0; then (2.28) is valid for n ¼ m: Example 2.1. Consider the operator A# in (1.1) of Section 1. Since a; b are real functions, we have w2 ð1Þbð1Þ ¼ 0 ([16]). And b is a polynomial, so that A# is bounded on C m;m ðGÞ according to Theorem 2.4. An interesting
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COMPLEX SPLINE APPROXIMATION
result is that the operator 1 0 A# ðjÞðtÞ aðtÞw2 ðtÞjðtÞ p
Z
1 1
w2 ðtÞbðtÞjðtÞ dt; tt
t 2 ½1; 1
ð2:29Þ
is bounded on C m;m ðGÞ where we only require a; b 2 H½1; 1: It appears that its smoothness does not depend on a and b:
3. SOME PROPERTIES OF COMPLEX SPLINES Complex splines have [1, 2, 4, 10, 11]). In order to operators, we need to have only refer to the linear and Let
many good properties (for details, see discuss the approximation of singular integral a further investigation for the splines. Here we cubic interpolating splines.
D : a ¼ t0 t1 tN ¼ b _
ðrÞ
be a partition of G; Gj ¼ tj1 tj ; h ¼ max14j4N jDtj j; and fj ¼ f ðrÞ ðtj Þ; where tj1 tj means that tj1 precedes tj ; Dtj ¼ tj tj1 and f 2 C m ðGÞ: We denote the linear and cubic interpolating splines of f by s1 ð f Þ or s1 and s3 ð f Þ or s3 ; respectively. For linear interpolating spline, we have s1 ðtÞ ¼
tj t t tj1 fj1 þ fj ; Dtj Dtj
t 2 Gj ;
j ¼ 1; 2; . . . ; N
and jjs1 ð f Þ f jj4c0 oð f ; hÞ:
ð3:1Þ
Let t; t0 2 G; t0 t and jt t0 j > 0: Then there are k and j; 04k4j4N; such that t 2 Gj and t0 2 Gk : If k5j; we have js1 ðtÞ s1 ðt0 Þj4js1 ðtÞ fj1 j þ j fj1 fk j þ j fk s1 ðt0 Þj: According to the property of modulus of continuity, j fj1 fk j 4oð f ; jtj1 tk jÞ
jtj1 tk j 4 1þ oð f ; jt t0 jÞ; jt t0 j
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jt tj1 j oð f ; jDtj jÞ jDtj j
jt tj1 j jDtj j 1þ 4 oð f ; jt t0 jÞ jDtj j jt t0 j
jt tj1 j 4 c0 þ oð f ; jt t0 jÞ jt t0 j
js1 ðtÞ fj1 j 4
and similarly,
jtk t0 j j fk s1 ðt0 Þj4 c0 þ oð f ; jt t0 jÞ: jt t0 j So we have
jt tj1 j þ jtj1 tk j þ jtk t0 j js1 ðtÞ s1 ðt Þj 4 1 þ 2c0 þ oð f ; jt t0 jÞ jt t0 j 0
4 ð1 þ 3c0 Þoð f ; jt t0 jÞ: If k ¼ j; then jt t0 j oð f ; jDtj jÞ jDtj j
jDtj j jt t0 j 1þ 4 oð f ; jt t0 jÞ jDtj j jt t0 j
js1 ðtÞ s1 ðt0 Þj 4
4ð1 þ c0 Þoð f ; jt t0 jÞ: Thus, we have proved that oðs1 ; xÞ4ð1 þ 3c0 Þoð f ; xÞ;
x50:
ð3:2Þ
x50;
ð3:3Þ
From (3.1) and (3.2), we conclude that oðs1 f ; xÞ4coð f ; xg h1g Þ;
where c ¼ 2 þ 3c0 ; 04g41: We show the conclusion as follows. If 04x4h; then x4xg h1g and oðs1 f ; xÞ 4oðs1 ; xÞ þ oð f ; xÞ 4ð2 þ 3c0 Þoð f ; xÞ 4coð f ; xg h1g Þ
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COMPLEX SPLINE APPROXIMATION
by (3.2); if x > h; then h5xg h1g and oðs1 f ; xÞ 42jjs1 f jj 42oð f ; hÞ 4coð f ; xg h1g Þ by (3.2). Therefore (3.3) follows. Thus, we have obtained Theorem 3.1. For the linear interpolating spline s1 of f 2 CðGÞ; there hold estimates (3.1) and (3.3). For f 2 H m ðGÞ; we let n ¼ gm; then (3.3) becomes oðs1 f ; xÞ4cMm ð f Þxn hmn :
ð3:30 Þ
Furthermore, we have Corollary 3.2.
If f 2 H m ðGÞ; then jjs1 ð f Þ f jjH n 4cMm ð f Þhmn
ð3:4Þ
for 04n4m: Now we discuss the cubic interpolating splines. Here we require that maxj jDtj j 4c1 51; minj jDtj j
max14j4N1
jDtj j þ jDtjþ1 j 4c2 52; jDtj þ Dtjþ1 j
ð3:5Þ
and the boundary conditions are given by s03 ðt0 Þ ¼ 0;
s0 ðtN Þ ¼ 0;
if f 2 CðGÞ;
s03 ðt0 Þ ¼ f 0 ðt0 Þ;
s0 ðtN Þ ¼ f 0 ðtN Þ;
s003 ðt0 Þ ¼ f 00 ðt0 Þ;
s00 ðtN Þ ¼ f 00 ðtN Þ;
if f 2 C 1 ðGÞ; if f 2 C 2 ðGÞ:
For f 2 C m ; m ¼ 0; 1 or 2; we have ðrÞ
jjs3 f ðrÞ jj4coð f ðmÞ ; hÞhmr ;
r ¼ 0; 1; . . . ; m
ð3:6Þ
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and ðmÞ
oðs3 ; xÞ4coð f ðmÞ ; xÞ;
x50:
ð3:7Þ
Because of the similarity, we only give their proof under m ¼ 2: The cubic spline has the following expression: s3 ðtÞ ¼
Mj1 Mj ðtj tÞ3 þ ðt tj1 Þ3 6Dtj 6Dtj
fj1 Mj1 Dtj fj Mj Dtj þ ðtj tÞ þ ðt tj1 Þ; Dtj 6 Dtj 6 t 2 Gj ;
j ¼ 1; 2; . . . ; N;
ð3:8Þ
where Mj ¼ s003 ðtj Þ; j ¼ 0; 1; . . . ; N which are determined by the equation AM ¼ D with M ¼ ½M0 ; M1 ; . . . ; MN T : 2 1 6m 6 1 6 6 A¼6 6 6 4
ð3:9Þ
Here the matrixes A and D are given by 3 0 7 2 l1 7 7 7 .. .. .. 7 . . . 7 7 mN1 2 lN1 5 0
1
and D ¼ ½ f000 ; 6f ½t0 ; t1 ; t2 ; . . . ; 6f ½tN2 ; tN1 ; tN ; fN00 T ; Dt
respectively, where mj ¼ Dtj þDtj jþ1 ; lj ¼ 1 mj and f ½tj1 ; tj ; tjþ1 is the second divided difference of f at the points tj1 ; tj ; tjþ1 : Let CNþ1 be a N þ 1 dimension complex space with maximum norm jj jj and x ¼ ðx0 ; x1 ; . . . ; xN ÞT 2 CNþ1 : Suppose k such that jjx ¼ jxk j: We have ( jxk j if k ¼ 0; N; jjAxjj 5 jmk xk1 þ 2xk þ lk xkþ1 j if 14k4N 1 5ð2 c2 Þjjxjj: It means jjA1 jj41=ð2 c2 Þ:
ð3:10Þ
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COMPLEX SPLINE APPROXIMATION
Let f 00 ¼ ½ f000 ; f100 ; . . . ; fN00 T : From 00 00 6f ½tj1 ; tj ; tjþ1 ðmj fj1 þ 2fj00 þ lj fjþ1 Þ 00 00 ¼ 6ð f ½tj1 ; tj ; tjþ1 12fj00 Þ þ mj ð fj00 fj1 Þ þ lj ð fj00 fjþ1 Þ
and [1] j f ½tj1 ; tj ; tjþ1 12fj00 j4coð f 00 ; jDtj jÞ;
ð3:11Þ
jjD Af 00 jj4coð f 00 ; hÞ:
ð3:12Þ
we have
Then by (3.9), (3.10) and (3.12), jjM f 00 jj ¼ jjA1 ðD Af 00 Þjj4
c oð f 00 ; hÞ 2 c2
or jMj fj00 j4coð f 00 ; hÞ
ð3:13Þ
and also jMk f 00 ðtÞj4coð f 00 ; hÞ;
k ¼ j 1;
j; t 2 Gj
ð3:14Þ
for j ¼ 1; 2; . . . ; N: From (3.8), the second derivative of the cubic spline is tj t t tj1 Mj1 þ Mj ; t 2 Gj ; j ¼ 1; 2; . . . ; N ð3:15Þ s003 ðtÞ ¼ Dtj Dtj and from (3.14) and (3.15), we have tj t t tj1 js003 ðtÞ f 00 ðtÞj ¼ ðMj1 f 00 ðtÞÞ þ ðMj f 00 ðtÞÞ Dtj Dtj jtj tj þ jt tj1 j maxk¼j1;j fjMk f 00 ðtÞjg 4 jDtj j 4 coð f 00 ; hÞ;
t 2 Gj ;
j ¼ 1; 2; . . . ; N:
ð3:16Þ
Thus, we have proved (3.6) for r ¼ 2: If r ¼ 1 or 0; (3.6) is easily obtained from the relations Z tj 1 s03 ðtj1 Þ f 0 ðtj1 Þ ¼ ðtj tÞ½s003 ðtÞ f 00 ðtÞ dt; Dtj tj1 s03 ðtÞ
0
f ðtÞ ¼
Z
t
tj1
½s003 ðtÞ f 00 ðtÞ dt þ s03 ðtj1 Þ f 0 ðtj1 Þ;
t 2 Gj ;
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and s3 ðtÞ f ðtÞ ¼
Z
t
tj1
½s03 ðtÞ f 0 ðtÞ dt;
t 2 Gj ;
where j ¼ 1; 2; . . . ; N: Let t; t0 2 G: If jt t0 j > h; then, by (3.6), js003 ðtÞ s003 ðt0 Þj 4js003 ðtÞ f 00 ðtÞj þ j f 00 ðtÞ f 00 ðt0 Þj þ j f 00 ðt0 Þ s003 ðt0 Þj 4c0 oð f 00 ; hÞ þ oð f 00 ; jt t0 jÞ 4coð f 00 ; jt t0 jÞ;
ð3:17Þ
and if 05jt t0 j4h; from jMj Mj1 j jDtj j 1 00 00 ðjMj fj00 j þ j fj00 fj1 j þ j fj1 Mj1 jÞ 4 jDtj j oð f 00 ; hÞ 4 c0 jDtj j oð f 00 ; hÞ ; t 2 Gj ; j ¼ 1; 2; . . . ; N; 4c h
js000 3 ðtÞj ¼
ð3:18Þ
where we have used (3.13) and (3.5), it follows Z t 00 00 0 000 js3 ðtÞ s3 ðt Þj 4 s3 ðtÞ dt t0
jt t0 j oð f 00 ; hÞ h
jt t0 j h 1þ 4c0 oð f 00 ; jt t0 jÞ h jt t0 j 4c0
4coð f 00 ; jt t0 jÞ:
ð3:19Þ
Hence (3.7) is valid for x > 0: Of course, (3.7) is valid if x ¼ 0; and so (3.7) is true. Similar to (3.3), we also have ðmÞ
oðs3
f ðmÞ ; xÞ4coð f ðmÞ ; xg h1g Þ;
x50;
ð3:20Þ
where 04g41: Thus we have proved Theorem 3.3. For the cubic interpolating spline s3 of f 2 C m ðGÞ; there hold the estimates (3.6) and (3.20) where m ¼ 0; 1; or 2.
99
COMPLEX SPLINE APPROXIMATION
For f 2 C m;m ðGÞ; (3.20) becomes ðmÞ
oðs3 f ðmÞ ; xÞ4cMm ð f ðmÞ Þxn hmn ;
04n4m
ð3:200 Þ
and thus we have Corollary 3.4.
If f 2 C m;m ðGÞ; then jjs3 ð f Þ f jjC m;n 4cMm ð f ðmÞ Þhmn ;
ð3:21Þ
where 04n4m:
4. COMPLEX SPLINE APPROXIMATION In this section, we discuss the complex spline approximation for singular integral operators. For convenience, we assume that the operator Aw is bounded on C m;m ðGÞ and let sðjÞ denote the linear or cubic interpolating spline of j 2 C m;m ðGÞ: Here the splines are those defined in the previous section and m ¼ 0; 1; or 2: In the case of linear splines, m automatically equals 0: It is effective to use complex splines for the numerical evaluation of Cauchy-type singular integrals. The following theorem is about their error estimates. Theorem 4.1.
If f 2 C m;m ðGÞ; then
jjAw sð f Þ Aw ð f ÞjjC k 4cMm ð f ðmÞ Þhmþmk1 ;
04k4m 1
ð4:1Þ
04n4m:
ð4:2Þ
and jjAw sð f Þ Aw ð f ÞjjC m;n 4cjj f jjC m;m hmn ; Proof.
Using Theorem 2.4 and the results in Section 3, we have
jjAw sð f Þ Aw ð f ÞjjC k 4cjjsð f Þ ð f ÞjjC kþ1 4cMm ð f ðmÞ Þhmþmk1 ;
04k4m 1
ð4:3Þ
04n4m:
ð4:4Þ
and jjAw sð f Þ Aw ð f ÞjjC m;n 4cjjsð f Þ f jjC m;n 4cjj f jjC m;m hmn ; if f 2 C m;m ðGÞ: The proof is complete.
]
100
YONG JIA XU
Now we consider the operator approximation by complex splines. Let Aw;D ¼ s*Aw s:
ð4:5Þ
The spline operators s* and s may be different. For convenience, we assume s* ¼ s here. Theorem 4.2.
If j 2 C m;m ðGÞ; then
jjAw;D ðjÞ Aw ðjÞjjC m;n 4cjjjjjC m;m hmn ; Proof.
04n4m:
ð4:6Þ
By Corollary 3.2 or 3.4 and Theorem 2.4, we have jjAw;D ðjÞ Aw sðjÞjjC m;n 4cMm ððAw sðjÞÞðmÞ Þhmn 4cjjsðjÞjjC m;m hmn 4cjjjjjC m;m hmn
ð4:7Þ
and from Theorem 4.1, it follows that jjAw;D ðjÞ Aw ðjÞjjC m;n 4jjAw;D ðjÞ Aw sðjÞjjC m;n þ jjAw sðjÞ Aw ðjÞjjC m;n 4cjjjjjC m;m hmn ; The theorem has been proved.
04n4m:
ð4:8Þ
]
5. APPLICATIONS AND REMARKS As a direct application of the results obtained in the previous section, we give a scheme for the numerical solution of singular integral equations over open arcs. We know that the Cauchy-type singular integral equation Z Z bðtÞ jðtÞ aðtÞjðtÞ dt þ l kðt; tÞjðtÞ dt ¼ f ðtÞ; t2G ð5:1Þ pi G t t G can be regularized as a Fredholm integral equation ðI þ lAw KÞy ¼ f n ; R
ð5:2Þ
where KyðtÞ G w1 ðtÞkðt; tÞyðtÞ dt; y ¼ j=w1 ; w1 ¼ ½ða2 b2 Þw1 ; f n ¼ Aw f þ bNk1 and Nk1 is a given polynomial with degree k 1 ðNk1 ¼ 0 if k40Þ (for details, see [12]). Here all given functions are Ho. lder continuous,
101
COMPLEX SPLINE APPROXIMATION
l is a constant, and undetermined function j is restricted in h0 ; which is a function class whose functions are integrable on G and are Ho. lder continuous on G except at the endpoints. Using Aw;D to replace Aw in (5.2), we have ðI þ lAw;D KÞyD ¼ f n ; and then letting uD ¼ yD f n ; gD ¼ lAw;D Kf n ; we get ðI þ lAw;D KÞuD ¼ gD :
ð5:3Þ
If l is not an eigenvalue of (5.2) and h is small enough, (5.3) has unique solution and the solution is a spline (cf. [9]). Thus, we can obtain the approximate solution of (5.1) by collocation method and the error estimate is easily obtained from Theorem 4.2. The problem will be discussed in detail in another paper. For m ¼ 1; the constants c and some other applications, we present some remarks. Remark 1. The assumption m51 is only for convenience. Examining the proof of Lemma 2.2, we see that if m ¼ 1 (2.10) is still true but (2.11) is not. Remark 2. The constants c in theorems of Section 2 depend on a; b; m; m and G: In Section 3, constants c depend on G and also depend on c1 if the splines are cubic. Thus, in Theorem 4.1 and 4.2 the constants are related to a; b; m; m; G and c1 ; but do not depend on n: Remark 3.
Consider the following Cauchy-type integral: Z f ðtÞ Tf ðtÞ ¼ dt; Gtt
ð5:4Þ
which is approximated by TD f ðtÞ ¼
Z
sð f ÞðtÞ dt: G tt
ð5:5Þ
If gðaÞ ¼ gðbÞ ¼ 0; it is easy to prove jTgðtÞj4c
Z
1 0
oðg; yÞ dy; y
t 2 G:
ð5:6Þ
Thus, by (3.3) or (3.20) we have jðT TD Þf ðtÞj 4c
Z 0
t2G
1
oð f sð f Þ; yÞ c dy4 y 1g
Z
hg 0
oð f ; yÞ dy; y ð5:7Þ
102
YONG JIA XU
where 04g51: This means that if f satisfies Dini condition then TD f is uniformly convergent to Tf on G: We can also obtain c oð f ðmÞ ; hÞhm1þg ; jðT TD Þf ðtÞj4 t2G ð5:8Þ 1g if f 2 C m ; m ¼ 1 or 2. Remark 4. If function w has integrable singularities at the endpoints of G; such as w1 mentioned above, we can make a transformation with w multiplied by a simple polynomial sðtÞ ¼ b t; t a; or ðb tÞðt aÞ such that the operator becomes a smooth operator divided by sðtÞ (cf. [16]). For example, the operator Z 1 1 1 f ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dt; t 2 ð1; 1Þ ð5:9Þ T1 f ðtÞ ¼ 2 p 1 1 t t t can be transformed into 1 1 T1 f ðtÞ ¼ sðtÞ p
Z
! pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 t2 f ðtÞ 1 1 tþt pffiffiffiffiffiffiffiffiffiffiffiffiffi f ðtÞ dt ; dt þ tt p 1 1 t2 1 1
ð5:10Þ
where sðtÞ ¼ 1 t2 : 6. PROOF OF LEMMA 2.2 Proof.
Equations (2.10) and (2.11) are equivalent to jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4cMm ð f ðmÞ Þdm ð1 þ j ln djÞ
or
Z jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4c jjujj þ 0
1
oðu; yÞ dy Mm ð f ðmÞ Þdm ; y
ð6:1Þ
ð6:2Þ
if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; where t1 ; t2 2 G and jt2 t1 j ¼ d: Now we prove (6.1) and (6.2). For convenience, suppose 05d51 and a t1 t2 b: (i) If jt1 aj > d and jb t2 j4d; let jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj 0 1 " # Z Z f ðtÞ Tm ð f Þðt; t2 Þ f ðtÞ Tm ð f Þðt; t1 Þ ¼ m!@ _ þ _ AuðtÞ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ t1 b at1 ¼ jI1 þ I2 j:
103
COMPLEX SPLINE APPROXIMATION _
_
_
From (2.5) and j t1 b j ¼ j t1 t2 j þ j t2 b j42c0 d; it follows that " # Z f ðtÞ T ð f Þðt; t Þ f ðtÞ T ð f Þðt; t Þ m 2 m 1 jI2 j ¼ m! _ uðtÞ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ t1 b Z 4 cjjujjMm ð f ðmÞ Þ _ ðjt t2 jm1 þ jt t1 jm1 jÞj dtj: t1 b
4 cjjujjMm ð f ðmÞ Þdm :
ð6:3Þ
Define the function hðtÞ ¼ Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ;
t 2 G:
ð6:4Þ
Then hðtÞ ¼ hðt2 Þ þ h0 ðt2 Þðt t2 Þ þ þ
hðmÞ ðt2 Þ ðt t2 Þm m!
ð6:5Þ
because hðtÞ is an m-degree polynomial of t: Noting that hðkÞ ðt2 Þ ¼ Tmk ð f ðkÞ Þðt2 ; t1 Þ f ðkÞ ðt2 Þ
ð6:6Þ
and (2.5), we have ! m m! ðkÞ jh ðt2 Þj4c Mm ð f ðmÞ Þdmþmk k! k
ð6:7Þ
for k ¼ 1; 2; . . . ; m; therefore Z T ð f Þðt; t Þ T ð f Þðt; t Þ m 2 m 1 i1 m! _ uðtÞ dt mþ1 ðt t2 Þ t1 b Z m X jhðkÞ ðt2 Þj juðtÞj jdtj _ mkþ1 k! at1 jt t2 j k¼0 ! Z m X m mkþm juðtÞj ðmÞ 4cMm ð f Þ jdtj d _ mkþ1 k at1 jt t2 j k¼0
1 m ðmÞ 4cMm ð f Þjjujjd 1 þ ln : d 4m!
ð6:8Þ
104
YONG JIA XU
If u satisfies Dini condition and uðbÞ ¼ 0; Z Z Z juðtÞj juðtÞ uðbÞj oðu; jt bjÞ jdtj ¼ _ jdtj4 _ jdtj: _ jt t2 j jt t2 j at1 jt t2 j at1 at1 _
For t 2 at1 ; _
_
_
c0 jt t2 j5j tt2 j ¼ j tt1 j þ j t1 t2 j5jt2 t1 j ¼ d;
ð6:9Þ
but jb t2 j4d; thus jb t2 j4c0 jt t2 j and jt bj4jt t2 j þ jt2 bj4ð1 þ c0 Þjt t2 j: So we have Z Z Z 1 juðtÞj oðu; ð1 þ c0 Þjt t2 jÞ oðu; yÞ dy jdtj4 jdtj4c _ _ jt t2 j y 0 at1 jt t2 j at1
ð6:10Þ
ð6:11Þ
and Z i1 4cMm ð f ðmÞ Þ jjujj þ
1
0
oðu; yÞ dy dm : y
By (2.5), " # Z 1 1 i2 ¼ m! _ uðtÞ½ f ðtÞ Tm ð f Þðt; t1 Þ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ at1 Z mþm mþ1 mþ1 jt t1 j jðt t1 Þ ðt t2 Þ j 4 cjjujjMm ð f ðmÞ Þ _ jdtj mþ1 jt t1 j jt t2 jmþ1 at1 m Z X jt t1 jkþm1 4 cjjujjMm ð f ðmÞ Þjt1 t2 j jdtj: _ kþ1 at1 jt t2 j k¼0 _
ð6:12Þ
ð6:13Þ
_
Let l1 ¼ j at1 j; l2 ¼ j at2 j and s be the parameter of arc length. Using integration by parts and l1 5l2 ; Z
jt t1 jkþm1 _
at1
jt t2 jkþ1
jdtj 4c
_
Z
l1
j tt1 jkþm1 _
ds ¼ c
Z
l1
ðl1 sÞkþm1
ðl2 sÞkþ1 " # Z k þ 1 l1 ðl1 sÞkþm 1 l1kþm ds 4c k þ m 0 ðl2 sÞkþ2 k þ ml2kþ1 0
j tt2 jkþ1
0
ds
105
COMPLEX SPLINE APPROXIMATION
kþ1 4c kþm
Z
l1
ðl2 sÞkþm
ds ðl2 sÞkþ2 " # kþ1 1 1 1 4c ; k þ m 1 m ðl2 l1 Þ1m l21m 0
_
but l2 l1 ¼ j t1 t2 j > d; then Z
jt t1 jkþm1 _ at1
jt t2 j
kþ1
jdtj4c0
kþ1 1 1 4cdm1 k þ m 1 m ðl2 l1 Þ1m
ð6:14Þ
for k ¼ 0; 1; . . . ; m; so we obtain i2 4cMm ð f ðmÞ Þdm :
ð6:15Þ
Thus from I1 ¼ m!
"
Z _
uðtÞ
f ðtÞ Tm ð f Þðt; t2 Þ
at1
¼ m!
Z _ at1
( uðtÞ
ðt t2 Þmþ1
f ðtÞ Tm ð f Þðt; t1 Þ
#
ðt t1 Þmþ1
dt
Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ ðt t2 Þmþ1
þ ½ f ðtÞ Tm ð f Þðt; t1 Þ
1 ðt t2 Þmþ1
!)
1 ðt t1 Þmþ1
jI1 j4i1 þ i2 8 m ðmÞ > < cMm ð f Þjjujjd ð1 þ jln djÞ
4 R 1 oðu; yÞ ðmÞ > dy dm : cMm ð f Þ jjujj þ 0 y
dt;
ð6:16Þ
if uðbÞa0; if uðbÞ ¼ 0
ð6:17Þ
and together with (6.3), we obtain jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4jI1 j þ jI2 j 8 m ðmÞ > < cMm ð f Þjjujjd ð1 þ jln djÞ
4 R 1 oðu; yÞ > dy dm : cMm ð f ðmÞ Þ jjujj þ 0 y and there follows (6.1) and (6.2).
if uðbÞa0; if uðbÞ ¼ 0
ð6:18Þ
106
YONG JIA XU
(ii) If jt1 aj4d and jb t2 j > d; similarly, we also have (6.18) but a in place of b: Hence, (6.1) and (6.2) are valid. (iii) If jt1 aj > d and jb t2 j > d; let jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj 0 1 " Z Z Z f ðtÞ Tm ð f Þðt; t2 Þ ¼ m!@ _ þ _ þ _ AuðtÞ ðt t2 Þmþ1 t1 t2 t2 b at1 # f ðtÞ Tm ð f Þðt; t1 Þ dt ðt t1 Þmþ1 ¼ jI1 þ I2 þ I3 j:
ð6:19Þ
Similar to the proof of (6.3), jI2 j4cMm ð f ðmÞ Þdm :
ð6:20Þ
We rewrite I1 þ I3 as i1 þ i2 where Z Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ dt i1 ¼ m! _ uðtÞ ðt t2 Þmþ1 at1 Z Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ þ m! _ uðtÞ dt; ðt t1 Þmþ1 t2 b
i2 ¼ m!
Z _
uðtÞ½ f ðtÞ Tm ð f Þðt; t1 Þ
at1
þ m!
Z _
1 ðt t2 Þmþ1
uðtÞ½ f ðtÞ Tm ð f Þðt; t2 Þ
t2 b
!
1 ðt t1 Þmþ1
1 ðt t2 Þmþ1
ð6:21Þ
dt
1
!
ðt t1 Þmþ1
dt: ð6:22Þ
Using (6.4) and (6.5), we also rewrite i1 as 2 3 Z Z m1 ðkÞ ðkÞ X h ðt Þ uðtÞ h ðt Þ uðtÞ 2 1 4 m! dt þ dt5 _ _ mkþ1 mkþ1 k! k! at1 ðt t2 Þ t2 b ðt t1 Þ k¼0 0 1 Z Z uðtÞ uðtÞ dt þ _ dtA½ f ðmÞ ðt1 Þ f ðmÞ ðt2 Þ þ @ _ t t t t 2 1 at1 t2 b ¼ i11 þ i12 :
ð6:23Þ
COMPLEX SPLINE APPROXIMATION
107
Similar to the proof of (6.8) and (6.15), we have ji11 j4cjjujjMm ð f ðmÞ Þdm ;
ð6:24Þ
1 ji12 j4cjjujjMm ð f ðmÞ Þdm 1 þ ln d
ð6:25Þ
ji2 j4cjjujjMm ð f ðmÞ Þdm :
ð6:26Þ
and
If u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; we have Z Z uðtÞ uðtÞ _ t t dt þ _ t t dt 2 1 t2 b at1 Z Z uðtÞ uðt2 Þ uðtÞ uðt1 Þ ¼ _ dt þ _ dt t t t t1 2 t2 b at1 Z Z 1 1 þ uðt2 Þ _ dt þ uðt1 Þ _ dt at1 t t2 t2 b t t1 Z Z oðu; jt t2 jÞ oðu; jt t1 jÞ jdtj þ _ 4 _ jdtj jt t t t1 j 2 at1 t2 b
t1 t2 b t1 þ iy1 þ uðt1 Þ ln þ iy2 ; þ uðt2 Þ ln at t t 2
1
ð6:27Þ
2
where y1 is the angle between t1 t2 and a t2 and y2 is the angle between b t1 and t2 t1 ; and uðt2 Þ lnt1 t2 þ iy1 þ uðt1 Þ ln b t1 þ iy2 a t t t 2 1 2 ¼ j½uðt2 Þ uðt1 Þ lnjt2 t1 j ½uðt2 Þ uðaÞ lnja t2 j þ ½uðt1 Þ uðbÞ lnjb t1 j þ i½y1 uðt2 Þ þ y2 uðt1 Þ
1 4c jjujj þ sup oðu; xÞ ln ; x 05x41
ð6:28Þ
108 therefore
YONG JIA XU
Z
Z 1 Z uðtÞ uðtÞ oðu; yÞ dy ; _ t t dt þ _ t t dt4c jjujj þ y 2 1 0 t2 b at1
so that ji12 j4cMm ð f
ðmÞ
Z Þ jjujj þ
1
0
oðu; yÞ dy dm : y
Now we obtain ji1 j 4ji11 j þ ji12 j 8
R 1 oðu; yÞ > ðmÞ > > dy dm cM ð f Þ jjujj þ m < 0 y
4 > 1 > m ðmÞ > ð f Þjjujjd 1 þ ln cM : m d
ð6:29Þ
ð6:30Þ
if uðaÞ ¼ uðbÞ ¼ 0; ð6:31Þ otherwise
and together with (6.20) and (6.26), (6.1) and (6.2) are valid in this _ case. _ _ (iv) If jt1 aj4d and jb t2 j4d; then jGj ¼ j at1 j þ j t1 t2 j þ j t2 b j43c0 d: From (2.5), jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj Z 4cjjujjMm ð f ðmÞ Þ ðjt t1 j1þm þ jt t2 j1þm Þjdtj G
4cjjujjMm ð f
ðmÞ
ÞjGjm
43c0 cjjujjMm ð f ðmÞ Þdm
ð6:32Þ
and then (6.1) and (6.2) are valid. Now we have proved that (6.1) and (6.2) are true in each case. The proof is complete. ] ACKNOWLEDGMENTS The author is grateful to the editor and the reviewers for their valuable suggestions. The author also thanks Professor Wei Lin, Dr. John H Easton and Dr. Xiao Bin Liu for their generous support and assistance.
REFERENCES 1. J. H. Ahlberg, E. N. Nilson, and J. L. Wash, Complex cubic splines, Trans. Amer. Math. Soc. 129 (1967), 391–418.
COMPLEX SPLINE APPROXIMATION
109
2. J. H. Ahlberg, E. N. Nilson, and J. L. Wash, Properties of analytic splines. I. Complex polynomial splines, J. Math. Anal. Appl. 27 (1969), 262–278. 3. D. Berthold and D. Junghanns, New error bounds for the quadrature method for the solution of Cauchy singular integral equations, SIAM J. Numer. Anal. 30 (1993), 1351–1372. 4. H. L., Cheng, On complex splines, Sci. Sinica 24 (1981), 160–169. 5. J. Y. Du, Some systems of orthogonal polynomials associated with singular integral equations, Acta Math. Sci. 7 (1987), 85–96. 6. D. Elliott, Orthogonal polynomials associated with singular integral equations having a Cauchy kernel, SIAM J. Math. Anal. 13 (1982), 1041–1052. 7. D. Elliott, Rates of convergence for the method of classical collocation method for solving singular integral equations, SIAM J. Numer. Anal. 21 (1984), 136–148. 8. X. L. Huang, Approximate solutions to singular integral equations, Acta Math. Sci. 12 (1992), 1, 75–85. 9. R. Kress, ‘‘Linear Integral Equation,’’ Springer-Verlag, Berlin, 1989. 10. C. K. Lu, Error analysis for interpolating complex cubic splines with deficiency 2, J. Approx. Theory 36 (1982), 183–196. 11. C. K. Lu, The approximation of Cauchy-type integrals by some kinds of interpolatory splines, J. Approx. Theory 36 (1982), 197–212. 12. C. K. Lu, ‘‘Boundary Value Problems for Analytic Functions,’’ World Scientific, Singapore, 1993. 13. J. K. Lu and J. Y. Du, Numerical methods of singular integral equations, Advanced Math. 23 (1994), 424–431. 14. S. G. Mikhlin and S. Pro. ssdorf, ‘‘Singular Integral Operators,’’ Springer-Verlag, Berlin, 1986. 15. N. I. Mushelishvili, ‘‘Singular Integral Equations,’’ Noordhoff, Gro. ningen, 1962. 16. Y. J. Xu, On stability of negative index singular integral equations over an interval, Integral Equations Operator Theory, to appear. 17. Y. J. Xu, A kind of stability of solution of singular integral equations, Acta. Math. Sci. 11 (1991), 448–456. 18. A. Zygmund, ‘‘Trigonometric Series,’’ Cambridge Univ. Press, Cambridge, UK 1968.
The Complex Spline Approximation of Singular Integral Operators over an Open Arc Yong Jia Xu Department of Statistics, Guangdong Business College, Guangzhou, Guangdong 510320, People’s Republic of China E-mail: [email protected] Communicated by Wolfgang Dahmen Received August 29, 2001; accepted in revised form June 26, 2002
This paper mainly considers the smooth complex spline approximation of Cauchytype integral operators over an open arc. First, the smoothness of the operators is investigated, then some properties of complex splines are discussed, and finally the error estimates of the approximation are given. # 2002 Elsevier Science (USA)
1. INTRODUCTION The approximation for Cauchy-type singular integrals and the numerical solution of the corresponding equations have been studied in many papers, and a series of important results have been obtained [3, 7, 8, 13]. However, the rates of convergence for the approximation are not very satisfactory. In fact, the best rates have not been revealed. The reason may be that, as mappings, the smoothness of the integrals is not investigated thoroughly. For example, paper [7] discusses the following operator: Z bðtÞ 1 w2 ðtÞjðtÞ A# ðjÞðtÞ aðtÞw2 ðtÞjðtÞ dt; t 2 ½1; 1; ð1:1Þ p 1 t t where a 2 H½1; 1; b is a polynomial and w2 is a weight function constructed from a and b (for details, see [7] or [6]), and proves that if w2 2 H g ½1; 1 and j 2 C m;m ½1; 1 for 05g; m51; then A# j 2 H l ½1; 1 where l ¼ minðg; mÞ: This means that the smoothness of A# j depends on w2 : But in fact, as an operator, A# is sufficiently smooth and its smoothness is not affected by w2 : This can be illustrated when aðtÞ 0 and bðtÞ 1 in (1.1). In this case, A# becomes Z pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 t2 jðtÞ dt; t 2 ½1; 1 ð1:2Þ TðjÞðtÞ tt p 1 86 0021-9045/02 $35.00 # 2002 Elsevier Science (USA) All rights reserved.
COMPLEX SPLINE APPROXIMATION
87
and it is bounded on C m;m ½1; 1 because we can easily convert it to a periodic singular integral or a Cauchy-type integral on a unit circle (cf. [15, 18]). Usually, the Cauchy-type singular integrals are of the form Z 1 jðtÞ dt; t 2 G; ð1:3Þ TG ðjÞðtÞ p G tt where G is a smooth or piecewise smooth oriented curve. If G is closed, as a mapping, TG is bounded on C m;m ðGÞ; and if G is an axis, TG has the similar property [14, 15]. However, if G is an open arc, the property disappears. Instead, we consider its ‘‘weighted type’’. This is also a natural approach to the solution of singular integral equations over an open arc. Generally speaking, by a simple transformation, our problems on an open arc can be converted to those on an interval, and then the problems may be solved more conveniently. But this will bring about at least two problems: the first is that the kernels of those integrals may become more complicated and it will then be more difficult for some treatments, such as numerical evaluations; the second is that a smooth function on the arc, as well as the related smooth operators, may become less or even not smooth on the interval if the arc is not perfectly smooth, and it will then impair the rates of its approximation. On the other hand, it is known that complex splines have a lot of advantages over other functions such as polynomials in approximation, especially in the approximation for those functions defined on arbitrary curves. So it will be more meaningful to discuss the singular integral operators and their complex spline approximation directly on arcs. In this paper, our discussion focuses on the Cauchy singular integral operators of the form Z bðtÞ wðtÞjðtÞ Aw ðjÞðtÞ aðtÞwðtÞjðtÞ þ dt; t 2 G; ð1:4Þ pi G t t which are derived from the theory of singular integral equations. Here we assume a; b 2 HðGÞ satisfying a2 ðtÞ b2 ðtÞa0;
t2G
and G is a smooth or piecewise smooth oriented open arc from a to b: The function w is constructed as follows (cf. [5, 6, 12]). Let GðtÞ ¼ aðtÞþbðtÞ aðtÞbðtÞ; and choose a continuous branch of GðtÞ such that 04yðtÞ51;
ð1:5Þ
where yðtÞ ¼ argGðtÞ 2p : Let ½y denote the integral part of the real number y and k ¼ ½yðbÞ:
ð1:6Þ
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YONG JIA XU
Define the canonical function [12, 15] Z 1 ln GðtÞ dt ; X ðzÞ ¼ ðb zÞk exp 2pi G t z
z2 =G
ð1:7Þ
and now let wðtÞ ¼ ½ðaðtÞ bðtÞÞX þ ðtÞ1 ;
t 2 G:
ð1:8Þ
It is easy to verify that wðtÞ 2 HðGÞ and wðtÞa0; t 2 G =fa; bg; which is similar to a weight function. From the above construction, we know the smoothness of the function w is poor. However, we will show in this paper that the operator Aw is very smooth. Furthermore, if wðaÞ ¼ wðbÞ ¼ 0 and b 2 C m;m ðGÞ; Aw is a bounded operator on C m;m ðGÞ: Besides, we make a further study of complex interpolating splines and obtain some interesting properties. And then, there follow the results of the complex spline approximation for the operator Aw : All these appear to be very attractive. Throughout the paper, we assume that: 05m51; m is a nonnegative _ _ integer, G ¼ ab; c0 is a constant such that the arc length j t1 t2 j4c0 jt1 t2 j for all t1 ; t2 2 G; CðGÞ and C m ðGÞ denote the spaces of continuous and mtimes continuously differentiable complex-valued functions on G; respecP ðkÞ tively, jjcjj maxt2G jcðtÞj; jjcjjcm m k¼0 jjc jj; the modulus of continuity for c 2 CðGÞ is denoted by oðc; xÞ; Mm ðcÞ sup05x4 m;m 1 oðc;xÞ ðGÞ fc 2 C m ðGÞ : Mm ðcðmÞ Þ51g; H m ðGÞ C 0;m ðGÞ; HðGÞ xm ; C S 05m51 H m ðGÞ; jjcjjH m jjcjj þ Mm ðcÞ; jjcjjC m;m jjcjjC m þ Mm ðcðmÞ Þ; and c is an absolute positive constant taking different value in different place. The rest of the paper is organized as follows. In the next section, we show the smoothness of the operator Aw : Some properties of complex splines are discussed in Section 3 and the results of spline approximation for Aw are given in Section 4. In Section 5 we illustrate an application for the approximation and remark on the case m ¼ 1; constant c and some other applications. The proof of Lemma 2.2 is longer and thus is put in the final section.
2. SMOOTHNESS OF SINGULAR INTEGRAL OPERATORS In this section, we start our discussion for the smoothness of operator Aw from the introduction of some operators and the proofs of two lemmas. Let u 2 CðGÞ: Define the operator Su;m as Z f ðtÞ Tm ð f Þðt; tÞ Su;m ð f ÞðtÞ ¼ m! ut dt; f 2 C m;m ðGÞ; ðt tÞmþ1 G
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COMPLEX SPLINE APPROXIMATION
where Tm ð f Þðt; tÞ ¼ f ðtÞ þ f 0 ðtÞðt tÞ þ þ f ¼ f ðtÞ: If m ¼ 0; Su;m ð f Þ is written as Su ð f Þ:
ðmÞ
ðtÞ m! ðt
tÞm and T0 ð f Þðt; tÞ
k ð f Þðt;tÞ Let Ck ðt; tÞ ¼ k!uðtÞf ðtÞT ; f 2 C m;m ðGÞ: It is easy to verify that ðttÞkþ1
@ Ck1 ðt; tÞ ¼ Ck ðt; tÞ @t
ð2:1Þ
and Z
jCk ðt; tÞjjdtj51;
supt2G
ð2:2Þ
G
and using dominant convergence theorem, we obtain dk Su ð f ÞðtÞ ¼ Su;k ð f ÞðtÞ; dtk
t2G
ð2:3Þ
for k ¼ 1; 2; . . . ; m: If k5m; then Z t 1 j f ðtÞ Tk ð f Þðt; tÞj ¼ ðz tÞk f ðkþ1Þ ðzÞ dz k! t 1 4 c jj f ðkþ1Þ jj jt tjkþ1 k!
ð2:4Þ
and if k ¼ m; j f ðtÞ Tm ð f Þðt; tÞj Z t 1 m1 ðmÞ ðmÞ ¼ ðz tÞ ½ f ðzÞ f ðtÞ dz ðm 1Þ! t 4
c oð f ðmÞ ; jt tjÞjt tjm m!
ð2:5Þ
for t; t 2 G; therefore Z jSu;k ð f ÞðtÞj4c juðtÞj jj f ðkþ1Þ jj jdtj4c0 jjujj jj f ðkþ1Þ jj;
t2G
ð2:6Þ
G
for k ¼ 1; 2; . . . ; m 1; and jSu;m ð f ÞðtÞj4cjjujj
Z
1 0
Thus, we have
oð f ðmÞ ; yÞ dy; y
t 2 G:
ð2:7Þ
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YONG JIA XU
Lemma 2.1. For the operator Su ; there hold (2.3) and the following estimates: k d Su ð f Þ4cjjujj jj f ðkþ1Þ jj; k ¼ 0; 1; . . . ; m 1; ð2:8Þ dtk if f 2 C m and
if f ðmÞ
m Z 1 d oð f ðmÞ ; yÞ Su ð f Þ4cjjujj dy; dtm y 0 R1 ðmÞ satisfies Dini condition, i.e., 0 oð f x ;xÞdx51:
ð2:9Þ
We also have Lemma 2.2.
There is a constant c > 0 depending on m; m and G such that m
d o Su ð f Þ; x 4cjjujjMm ð f ðmÞ Þxm ð1 þ j ln xjÞ ð2:10Þ dtm
or
m
Z 1 d oðu; yÞ dy Mm ð f ðmÞ Þxm ; o S ð f Þ; x 4c jjujj þ u y dtm 0
ð2:11Þ
if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; where f 2 C m;m ðGÞ and x > 0: Since the proof is longer, we put it in the last section. From Lemmas 2.1 and 2.2, it follows that jjSu ð f ÞjjC m;n 4cNðuÞjj f jjC m;m ;
ð2:12Þ
R1 where 05n5m and NðuÞ ¼ jjujj or 05n4m and NðuÞ ¼ jjujj þ 0 oðu;xÞ x dx if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0: We rewrite the operator Aw as Z 1 bðtÞ bðtÞ 0 jðtÞ dt; t 2 G; ð2:13Þ Aw ðjÞðtÞ ¼ Aw ðjÞðtÞ wðtÞ pi G tt where A0w ðjÞðtÞ aðtÞwðtÞjðtÞ þ
1 pi
Z
wðtÞbðtÞjðtÞ dt; tt G
t2G
ð2:14Þ
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COMPLEX SPLINE APPROXIMATION
and let KðjÞðtÞ ¼
1 pi
Z wðtÞ G
bðtÞ bðtÞ jðtÞ dt; tt
t2G
or KðjÞðtÞ ¼
1 Swj ðbÞ: pi
ð2:15Þ
Thus, Aw ¼ A0w K:
ð2:16Þ
Theorem 2.3. There exists a constant c > 0 such that k d 0 A ðjÞ4cjjjjj kþ1 ; k ¼ 0; 1; . . . ; m 1 C dtk w
ð2:17Þ
for j 2 C m and jjA0w ðjÞjjC m;n 4cjjjjjC m;m
ð2:18Þ
for j 2 C m;m ðGÞ; where 05n5m: If bðaÞwðaÞ ¼ bðbÞwðbÞ ¼ 0; then (2.18) is valid for n ¼ m: Proof.
A0w can be written as
A0w ðjÞðtÞ ¼ A0w ð1ÞðtÞjðtÞ þ
1 pi
Z
wðtÞbðtÞ½jðtÞ jðtÞ dt; tt G
t2G
or A0w ðjÞðtÞ ¼ pðtÞjðtÞ þ
1 Swb ðjÞðtÞ; pi
ð2:19Þ
where pðtÞ ¼ A0w ð1ÞðtÞ: But
1 ;t pðtÞ ¼ p:p: X ðzÞ
ð2:20Þ
is a k-degree polynomial ðpðtÞ 0 if k50) (cf. [16]), and it is obvious that jjðpjÞðkÞ jj4cjjjjjC k ;
k ¼ 0; 1; . . . ; m
ð2:21Þ
for j 2 C m ðGÞ and Mn ððpjÞðmÞ Þ4cjjjjjC m;m ;
04n4m
ð2:22Þ
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or jjpjjjC m;n 4cjjjjjC m;m
ð2:23Þ
for j 2 C m;m ðGÞ; where the constant c depends on aðtÞ; bðtÞ and m: On the other hand, wb satisfies Dini condition, so that there hold (2.8)–(2.10) for Swb with u ¼ wb and there also holds (2.11) if wðaÞbðaÞ ¼ wðbÞbðbÞ ¼ 0: Thus we obtain (2.17) and (2.18) from (2.19). The proof is complete. ] Let b 2 C m;m ðGÞ: From Lemmas 2.1 and 2.2, k d KðjÞ4c0 jjwjjj jjbðkþ1Þ jj4cjjjjj; k ¼ 0; 1; . . . ; m 1 dtk
ð2:24Þ
and jjKðjÞjjC m;n 4cjjwjjj jjbjjC m;m 4cjjwjj jjbjjC m;m jjjjj;
05n5m
for j 2 CðGÞ; and if wðaÞ ¼ wðbÞ ¼ 0; then
Z 1 wðwj; yÞ 0 dy jjbjjC m;m jjKðjÞjjC m;n 4c jjwjjj þ y 0
Z 1 oðj; yÞ dy 4c jjjjj þ y 0
ð2:25Þ
ð2:26Þ
for j 2 H m ðGÞ; where we have used the fact that oðwj; yÞ4jjwjjoðj; yÞ þ jjjjjoðw; yÞ: Thus, we obtain Theorem 2.4. Let b 2 C m;m ðGÞ: Then there is a constant c > 0 such that k d Aw ðjÞ4cjjjjj kþ1 ; k ¼ 0; 1; . . . ; m 1 ð2:27Þ C dtk for j 2 C m ðGÞ and jjAw ðjÞjjC m;n 4cjjjjjC m;m
ð2:28Þ
for j 2 C m;m ðGÞ; where 05n5m: If wðaÞ ¼ wðbÞ ¼ 0; then (2.28) is valid for n ¼ m: Example 2.1. Consider the operator A# in (1.1) of Section 1. Since a; b are real functions, we have w2 ð1Þbð1Þ ¼ 0 ([16]). And b is a polynomial, so that A# is bounded on C m;m ðGÞ according to Theorem 2.4. An interesting
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COMPLEX SPLINE APPROXIMATION
result is that the operator 1 0 A# ðjÞðtÞ aðtÞw2 ðtÞjðtÞ p
Z
1 1
w2 ðtÞbðtÞjðtÞ dt; tt
t 2 ½1; 1
ð2:29Þ
is bounded on C m;m ðGÞ where we only require a; b 2 H½1; 1: It appears that its smoothness does not depend on a and b:
3. SOME PROPERTIES OF COMPLEX SPLINES Complex splines have [1, 2, 4, 10, 11]). In order to operators, we need to have only refer to the linear and Let
many good properties (for details, see discuss the approximation of singular integral a further investigation for the splines. Here we cubic interpolating splines.
D : a ¼ t0 t1 tN ¼ b _
ðrÞ
be a partition of G; Gj ¼ tj1 tj ; h ¼ max14j4N jDtj j; and fj ¼ f ðrÞ ðtj Þ; where tj1 tj means that tj1 precedes tj ; Dtj ¼ tj tj1 and f 2 C m ðGÞ: We denote the linear and cubic interpolating splines of f by s1 ð f Þ or s1 and s3 ð f Þ or s3 ; respectively. For linear interpolating spline, we have s1 ðtÞ ¼
tj t t tj1 fj1 þ fj ; Dtj Dtj
t 2 Gj ;
j ¼ 1; 2; . . . ; N
and jjs1 ð f Þ f jj4c0 oð f ; hÞ:
ð3:1Þ
Let t; t0 2 G; t0 t and jt t0 j > 0: Then there are k and j; 04k4j4N; such that t 2 Gj and t0 2 Gk : If k5j; we have js1 ðtÞ s1 ðt0 Þj4js1 ðtÞ fj1 j þ j fj1 fk j þ j fk s1 ðt0 Þj: According to the property of modulus of continuity, j fj1 fk j 4oð f ; jtj1 tk jÞ
jtj1 tk j 4 1þ oð f ; jt t0 jÞ; jt t0 j
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YONG JIA XU
jt tj1 j oð f ; jDtj jÞ jDtj j
jt tj1 j jDtj j 1þ 4 oð f ; jt t0 jÞ jDtj j jt t0 j
jt tj1 j 4 c0 þ oð f ; jt t0 jÞ jt t0 j
js1 ðtÞ fj1 j 4
and similarly,
jtk t0 j j fk s1 ðt0 Þj4 c0 þ oð f ; jt t0 jÞ: jt t0 j So we have
jt tj1 j þ jtj1 tk j þ jtk t0 j js1 ðtÞ s1 ðt Þj 4 1 þ 2c0 þ oð f ; jt t0 jÞ jt t0 j 0
4 ð1 þ 3c0 Þoð f ; jt t0 jÞ: If k ¼ j; then jt t0 j oð f ; jDtj jÞ jDtj j
jDtj j jt t0 j 1þ 4 oð f ; jt t0 jÞ jDtj j jt t0 j
js1 ðtÞ s1 ðt0 Þj 4
4ð1 þ c0 Þoð f ; jt t0 jÞ: Thus, we have proved that oðs1 ; xÞ4ð1 þ 3c0 Þoð f ; xÞ;
x50:
ð3:2Þ
x50;
ð3:3Þ
From (3.1) and (3.2), we conclude that oðs1 f ; xÞ4coð f ; xg h1g Þ;
where c ¼ 2 þ 3c0 ; 04g41: We show the conclusion as follows. If 04x4h; then x4xg h1g and oðs1 f ; xÞ 4oðs1 ; xÞ þ oð f ; xÞ 4ð2 þ 3c0 Þoð f ; xÞ 4coð f ; xg h1g Þ
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COMPLEX SPLINE APPROXIMATION
by (3.2); if x > h; then h5xg h1g and oðs1 f ; xÞ 42jjs1 f jj 42oð f ; hÞ 4coð f ; xg h1g Þ by (3.2). Therefore (3.3) follows. Thus, we have obtained Theorem 3.1. For the linear interpolating spline s1 of f 2 CðGÞ; there hold estimates (3.1) and (3.3). For f 2 H m ðGÞ; we let n ¼ gm; then (3.3) becomes oðs1 f ; xÞ4cMm ð f Þxn hmn :
ð3:30 Þ
Furthermore, we have Corollary 3.2.
If f 2 H m ðGÞ; then jjs1 ð f Þ f jjH n 4cMm ð f Þhmn
ð3:4Þ
for 04n4m: Now we discuss the cubic interpolating splines. Here we require that maxj jDtj j 4c1 51; minj jDtj j
max14j4N1
jDtj j þ jDtjþ1 j 4c2 52; jDtj þ Dtjþ1 j
ð3:5Þ
and the boundary conditions are given by s03 ðt0 Þ ¼ 0;
s0 ðtN Þ ¼ 0;
if f 2 CðGÞ;
s03 ðt0 Þ ¼ f 0 ðt0 Þ;
s0 ðtN Þ ¼ f 0 ðtN Þ;
s003 ðt0 Þ ¼ f 00 ðt0 Þ;
s00 ðtN Þ ¼ f 00 ðtN Þ;
if f 2 C 1 ðGÞ; if f 2 C 2 ðGÞ:
For f 2 C m ; m ¼ 0; 1 or 2; we have ðrÞ
jjs3 f ðrÞ jj4coð f ðmÞ ; hÞhmr ;
r ¼ 0; 1; . . . ; m
ð3:6Þ
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YONG JIA XU
and ðmÞ
oðs3 ; xÞ4coð f ðmÞ ; xÞ;
x50:
ð3:7Þ
Because of the similarity, we only give their proof under m ¼ 2: The cubic spline has the following expression: s3 ðtÞ ¼
Mj1 Mj ðtj tÞ3 þ ðt tj1 Þ3 6Dtj 6Dtj
fj1 Mj1 Dtj fj Mj Dtj þ ðtj tÞ þ ðt tj1 Þ; Dtj 6 Dtj 6 t 2 Gj ;
j ¼ 1; 2; . . . ; N;
ð3:8Þ
where Mj ¼ s003 ðtj Þ; j ¼ 0; 1; . . . ; N which are determined by the equation AM ¼ D with M ¼ ½M0 ; M1 ; . . . ; MN T : 2 1 6m 6 1 6 6 A¼6 6 6 4
ð3:9Þ
Here the matrixes A and D are given by 3 0 7 2 l1 7 7 7 .. .. .. 7 . . . 7 7 mN1 2 lN1 5 0
1
and D ¼ ½ f000 ; 6f ½t0 ; t1 ; t2 ; . . . ; 6f ½tN2 ; tN1 ; tN ; fN00 T ; Dt
respectively, where mj ¼ Dtj þDtj jþ1 ; lj ¼ 1 mj and f ½tj1 ; tj ; tjþ1 is the second divided difference of f at the points tj1 ; tj ; tjþ1 : Let CNþ1 be a N þ 1 dimension complex space with maximum norm jj jj and x ¼ ðx0 ; x1 ; . . . ; xN ÞT 2 CNþ1 : Suppose k such that jjx ¼ jxk j: We have ( jxk j if k ¼ 0; N; jjAxjj 5 jmk xk1 þ 2xk þ lk xkþ1 j if 14k4N 1 5ð2 c2 Þjjxjj: It means jjA1 jj41=ð2 c2 Þ:
ð3:10Þ
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COMPLEX SPLINE APPROXIMATION
Let f 00 ¼ ½ f000 ; f100 ; . . . ; fN00 T : From 00 00 6f ½tj1 ; tj ; tjþ1 ðmj fj1 þ 2fj00 þ lj fjþ1 Þ 00 00 ¼ 6ð f ½tj1 ; tj ; tjþ1 12fj00 Þ þ mj ð fj00 fj1 Þ þ lj ð fj00 fjþ1 Þ
and [1] j f ½tj1 ; tj ; tjþ1 12fj00 j4coð f 00 ; jDtj jÞ;
ð3:11Þ
jjD Af 00 jj4coð f 00 ; hÞ:
ð3:12Þ
we have
Then by (3.9), (3.10) and (3.12), jjM f 00 jj ¼ jjA1 ðD Af 00 Þjj4
c oð f 00 ; hÞ 2 c2
or jMj fj00 j4coð f 00 ; hÞ
ð3:13Þ
and also jMk f 00 ðtÞj4coð f 00 ; hÞ;
k ¼ j 1;
j; t 2 Gj
ð3:14Þ
for j ¼ 1; 2; . . . ; N: From (3.8), the second derivative of the cubic spline is tj t t tj1 Mj1 þ Mj ; t 2 Gj ; j ¼ 1; 2; . . . ; N ð3:15Þ s003 ðtÞ ¼ Dtj Dtj and from (3.14) and (3.15), we have tj t t tj1 js003 ðtÞ f 00 ðtÞj ¼ ðMj1 f 00 ðtÞÞ þ ðMj f 00 ðtÞÞ Dtj Dtj jtj tj þ jt tj1 j maxk¼j1;j fjMk f 00 ðtÞjg 4 jDtj j 4 coð f 00 ; hÞ;
t 2 Gj ;
j ¼ 1; 2; . . . ; N:
ð3:16Þ
Thus, we have proved (3.6) for r ¼ 2: If r ¼ 1 or 0; (3.6) is easily obtained from the relations Z tj 1 s03 ðtj1 Þ f 0 ðtj1 Þ ¼ ðtj tÞ½s003 ðtÞ f 00 ðtÞ dt; Dtj tj1 s03 ðtÞ
0
f ðtÞ ¼
Z
t
tj1
½s003 ðtÞ f 00 ðtÞ dt þ s03 ðtj1 Þ f 0 ðtj1 Þ;
t 2 Gj ;
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and s3 ðtÞ f ðtÞ ¼
Z
t
tj1
½s03 ðtÞ f 0 ðtÞ dt;
t 2 Gj ;
where j ¼ 1; 2; . . . ; N: Let t; t0 2 G: If jt t0 j > h; then, by (3.6), js003 ðtÞ s003 ðt0 Þj 4js003 ðtÞ f 00 ðtÞj þ j f 00 ðtÞ f 00 ðt0 Þj þ j f 00 ðt0 Þ s003 ðt0 Þj 4c0 oð f 00 ; hÞ þ oð f 00 ; jt t0 jÞ 4coð f 00 ; jt t0 jÞ;
ð3:17Þ
and if 05jt t0 j4h; from jMj Mj1 j jDtj j 1 00 00 ðjMj fj00 j þ j fj00 fj1 j þ j fj1 Mj1 jÞ 4 jDtj j oð f 00 ; hÞ 4 c0 jDtj j oð f 00 ; hÞ ; t 2 Gj ; j ¼ 1; 2; . . . ; N; 4c h
js000 3 ðtÞj ¼
ð3:18Þ
where we have used (3.13) and (3.5), it follows Z t 00 00 0 000 js3 ðtÞ s3 ðt Þj 4 s3 ðtÞ dt t0
jt t0 j oð f 00 ; hÞ h
jt t0 j h 1þ 4c0 oð f 00 ; jt t0 jÞ h jt t0 j 4c0
4coð f 00 ; jt t0 jÞ:
ð3:19Þ
Hence (3.7) is valid for x > 0: Of course, (3.7) is valid if x ¼ 0; and so (3.7) is true. Similar to (3.3), we also have ðmÞ
oðs3
f ðmÞ ; xÞ4coð f ðmÞ ; xg h1g Þ;
x50;
ð3:20Þ
where 04g41: Thus we have proved Theorem 3.3. For the cubic interpolating spline s3 of f 2 C m ðGÞ; there hold the estimates (3.6) and (3.20) where m ¼ 0; 1; or 2.
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COMPLEX SPLINE APPROXIMATION
For f 2 C m;m ðGÞ; (3.20) becomes ðmÞ
oðs3 f ðmÞ ; xÞ4cMm ð f ðmÞ Þxn hmn ;
04n4m
ð3:200 Þ
and thus we have Corollary 3.4.
If f 2 C m;m ðGÞ; then jjs3 ð f Þ f jjC m;n 4cMm ð f ðmÞ Þhmn ;
ð3:21Þ
where 04n4m:
4. COMPLEX SPLINE APPROXIMATION In this section, we discuss the complex spline approximation for singular integral operators. For convenience, we assume that the operator Aw is bounded on C m;m ðGÞ and let sðjÞ denote the linear or cubic interpolating spline of j 2 C m;m ðGÞ: Here the splines are those defined in the previous section and m ¼ 0; 1; or 2: In the case of linear splines, m automatically equals 0: It is effective to use complex splines for the numerical evaluation of Cauchy-type singular integrals. The following theorem is about their error estimates. Theorem 4.1.
If f 2 C m;m ðGÞ; then
jjAw sð f Þ Aw ð f ÞjjC k 4cMm ð f ðmÞ Þhmþmk1 ;
04k4m 1
ð4:1Þ
04n4m:
ð4:2Þ
and jjAw sð f Þ Aw ð f ÞjjC m;n 4cjj f jjC m;m hmn ; Proof.
Using Theorem 2.4 and the results in Section 3, we have
jjAw sð f Þ Aw ð f ÞjjC k 4cjjsð f Þ ð f ÞjjC kþ1 4cMm ð f ðmÞ Þhmþmk1 ;
04k4m 1
ð4:3Þ
04n4m:
ð4:4Þ
and jjAw sð f Þ Aw ð f ÞjjC m;n 4cjjsð f Þ f jjC m;n 4cjj f jjC m;m hmn ; if f 2 C m;m ðGÞ: The proof is complete.
]
100
YONG JIA XU
Now we consider the operator approximation by complex splines. Let Aw;D ¼ s*Aw s:
ð4:5Þ
The spline operators s* and s may be different. For convenience, we assume s* ¼ s here. Theorem 4.2.
If j 2 C m;m ðGÞ; then
jjAw;D ðjÞ Aw ðjÞjjC m;n 4cjjjjjC m;m hmn ; Proof.
04n4m:
ð4:6Þ
By Corollary 3.2 or 3.4 and Theorem 2.4, we have jjAw;D ðjÞ Aw sðjÞjjC m;n 4cMm ððAw sðjÞÞðmÞ Þhmn 4cjjsðjÞjjC m;m hmn 4cjjjjjC m;m hmn
ð4:7Þ
and from Theorem 4.1, it follows that jjAw;D ðjÞ Aw ðjÞjjC m;n 4jjAw;D ðjÞ Aw sðjÞjjC m;n þ jjAw sðjÞ Aw ðjÞjjC m;n 4cjjjjjC m;m hmn ; The theorem has been proved.
04n4m:
ð4:8Þ
]
5. APPLICATIONS AND REMARKS As a direct application of the results obtained in the previous section, we give a scheme for the numerical solution of singular integral equations over open arcs. We know that the Cauchy-type singular integral equation Z Z bðtÞ jðtÞ aðtÞjðtÞ dt þ l kðt; tÞjðtÞ dt ¼ f ðtÞ; t2G ð5:1Þ pi G t t G can be regularized as a Fredholm integral equation ðI þ lAw KÞy ¼ f n ; R
ð5:2Þ
where KyðtÞ G w1 ðtÞkðt; tÞyðtÞ dt; y ¼ j=w1 ; w1 ¼ ½ða2 b2 Þw1 ; f n ¼ Aw f þ bNk1 and Nk1 is a given polynomial with degree k 1 ðNk1 ¼ 0 if k40Þ (for details, see [12]). Here all given functions are Ho. lder continuous,
101
COMPLEX SPLINE APPROXIMATION
l is a constant, and undetermined function j is restricted in h0 ; which is a function class whose functions are integrable on G and are Ho. lder continuous on G except at the endpoints. Using Aw;D to replace Aw in (5.2), we have ðI þ lAw;D KÞyD ¼ f n ; and then letting uD ¼ yD f n ; gD ¼ lAw;D Kf n ; we get ðI þ lAw;D KÞuD ¼ gD :
ð5:3Þ
If l is not an eigenvalue of (5.2) and h is small enough, (5.3) has unique solution and the solution is a spline (cf. [9]). Thus, we can obtain the approximate solution of (5.1) by collocation method and the error estimate is easily obtained from Theorem 4.2. The problem will be discussed in detail in another paper. For m ¼ 1; the constants c and some other applications, we present some remarks. Remark 1. The assumption m51 is only for convenience. Examining the proof of Lemma 2.2, we see that if m ¼ 1 (2.10) is still true but (2.11) is not. Remark 2. The constants c in theorems of Section 2 depend on a; b; m; m and G: In Section 3, constants c depend on G and also depend on c1 if the splines are cubic. Thus, in Theorem 4.1 and 4.2 the constants are related to a; b; m; m; G and c1 ; but do not depend on n: Remark 3.
Consider the following Cauchy-type integral: Z f ðtÞ Tf ðtÞ ¼ dt; Gtt
ð5:4Þ
which is approximated by TD f ðtÞ ¼
Z
sð f ÞðtÞ dt: G tt
ð5:5Þ
If gðaÞ ¼ gðbÞ ¼ 0; it is easy to prove jTgðtÞj4c
Z
1 0
oðg; yÞ dy; y
t 2 G:
ð5:6Þ
Thus, by (3.3) or (3.20) we have jðT TD Þf ðtÞj 4c
Z 0
t2G
1
oð f sð f Þ; yÞ c dy4 y 1g
Z
hg 0
oð f ; yÞ dy; y ð5:7Þ
102
YONG JIA XU
where 04g51: This means that if f satisfies Dini condition then TD f is uniformly convergent to Tf on G: We can also obtain c oð f ðmÞ ; hÞhm1þg ; jðT TD Þf ðtÞj4 t2G ð5:8Þ 1g if f 2 C m ; m ¼ 1 or 2. Remark 4. If function w has integrable singularities at the endpoints of G; such as w1 mentioned above, we can make a transformation with w multiplied by a simple polynomial sðtÞ ¼ b t; t a; or ðb tÞðt aÞ such that the operator becomes a smooth operator divided by sðtÞ (cf. [16]). For example, the operator Z 1 1 1 f ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi dt; t 2 ð1; 1Þ ð5:9Þ T1 f ðtÞ ¼ 2 p 1 1 t t t can be transformed into 1 1 T1 f ðtÞ ¼ sðtÞ p
Z
! pffiffiffiffiffiffiffiffiffiffiffiffiffi Z 1 t2 f ðtÞ 1 1 tþt pffiffiffiffiffiffiffiffiffiffiffiffiffi f ðtÞ dt ; dt þ tt p 1 1 t2 1 1
ð5:10Þ
where sðtÞ ¼ 1 t2 : 6. PROOF OF LEMMA 2.2 Proof.
Equations (2.10) and (2.11) are equivalent to jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4cMm ð f ðmÞ Þdm ð1 þ j ln djÞ
or
Z jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4c jjujj þ 0
1
oðu; yÞ dy Mm ð f ðmÞ Þdm ; y
ð6:1Þ
ð6:2Þ
if u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; where t1 ; t2 2 G and jt2 t1 j ¼ d: Now we prove (6.1) and (6.2). For convenience, suppose 05d51 and a t1 t2 b: (i) If jt1 aj > d and jb t2 j4d; let jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj 0 1 " # Z Z f ðtÞ Tm ð f Þðt; t2 Þ f ðtÞ Tm ð f Þðt; t1 Þ ¼ m!@ _ þ _ AuðtÞ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ t1 b at1 ¼ jI1 þ I2 j:
103
COMPLEX SPLINE APPROXIMATION _
_
_
From (2.5) and j t1 b j ¼ j t1 t2 j þ j t2 b j42c0 d; it follows that " # Z f ðtÞ T ð f Þðt; t Þ f ðtÞ T ð f Þðt; t Þ m 2 m 1 jI2 j ¼ m! _ uðtÞ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ t1 b Z 4 cjjujjMm ð f ðmÞ Þ _ ðjt t2 jm1 þ jt t1 jm1 jÞj dtj: t1 b
4 cjjujjMm ð f ðmÞ Þdm :
ð6:3Þ
Define the function hðtÞ ¼ Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ;
t 2 G:
ð6:4Þ
Then hðtÞ ¼ hðt2 Þ þ h0 ðt2 Þðt t2 Þ þ þ
hðmÞ ðt2 Þ ðt t2 Þm m!
ð6:5Þ
because hðtÞ is an m-degree polynomial of t: Noting that hðkÞ ðt2 Þ ¼ Tmk ð f ðkÞ Þðt2 ; t1 Þ f ðkÞ ðt2 Þ
ð6:6Þ
and (2.5), we have ! m m! ðkÞ jh ðt2 Þj4c Mm ð f ðmÞ Þdmþmk k! k
ð6:7Þ
for k ¼ 1; 2; . . . ; m; therefore Z T ð f Þðt; t Þ T ð f Þðt; t Þ m 2 m 1 i1 m! _ uðtÞ dt mþ1 ðt t2 Þ t1 b Z m X jhðkÞ ðt2 Þj juðtÞj jdtj _ mkþ1 k! at1 jt t2 j k¼0 ! Z m X m mkþm juðtÞj ðmÞ 4cMm ð f Þ jdtj d _ mkþ1 k at1 jt t2 j k¼0
1 m ðmÞ 4cMm ð f Þjjujjd 1 þ ln : d 4m!
ð6:8Þ
104
YONG JIA XU
If u satisfies Dini condition and uðbÞ ¼ 0; Z Z Z juðtÞj juðtÞ uðbÞj oðu; jt bjÞ jdtj ¼ _ jdtj4 _ jdtj: _ jt t2 j jt t2 j at1 jt t2 j at1 at1 _
For t 2 at1 ; _
_
_
c0 jt t2 j5j tt2 j ¼ j tt1 j þ j t1 t2 j5jt2 t1 j ¼ d;
ð6:9Þ
but jb t2 j4d; thus jb t2 j4c0 jt t2 j and jt bj4jt t2 j þ jt2 bj4ð1 þ c0 Þjt t2 j: So we have Z Z Z 1 juðtÞj oðu; ð1 þ c0 Þjt t2 jÞ oðu; yÞ dy jdtj4 jdtj4c _ _ jt t2 j y 0 at1 jt t2 j at1
ð6:10Þ
ð6:11Þ
and Z i1 4cMm ð f ðmÞ Þ jjujj þ
1
0
oðu; yÞ dy dm : y
By (2.5), " # Z 1 1 i2 ¼ m! _ uðtÞ½ f ðtÞ Tm ð f Þðt; t1 Þ dt mþ1 mþ1 ðt t2 Þ ðt t1 Þ at1 Z mþm mþ1 mþ1 jt t1 j jðt t1 Þ ðt t2 Þ j 4 cjjujjMm ð f ðmÞ Þ _ jdtj mþ1 jt t1 j jt t2 jmþ1 at1 m Z X jt t1 jkþm1 4 cjjujjMm ð f ðmÞ Þjt1 t2 j jdtj: _ kþ1 at1 jt t2 j k¼0 _
ð6:12Þ
ð6:13Þ
_
Let l1 ¼ j at1 j; l2 ¼ j at2 j and s be the parameter of arc length. Using integration by parts and l1 5l2 ; Z
jt t1 jkþm1 _
at1
jt t2 jkþ1
jdtj 4c
_
Z
l1
j tt1 jkþm1 _
ds ¼ c
Z
l1
ðl1 sÞkþm1
ðl2 sÞkþ1 " # Z k þ 1 l1 ðl1 sÞkþm 1 l1kþm ds 4c k þ m 0 ðl2 sÞkþ2 k þ ml2kþ1 0
j tt2 jkþ1
0
ds
105
COMPLEX SPLINE APPROXIMATION
kþ1 4c kþm
Z
l1
ðl2 sÞkþm
ds ðl2 sÞkþ2 " # kþ1 1 1 1 4c ; k þ m 1 m ðl2 l1 Þ1m l21m 0
_
but l2 l1 ¼ j t1 t2 j > d; then Z
jt t1 jkþm1 _ at1
jt t2 j
kþ1
jdtj4c0
kþ1 1 1 4cdm1 k þ m 1 m ðl2 l1 Þ1m
ð6:14Þ
for k ¼ 0; 1; . . . ; m; so we obtain i2 4cMm ð f ðmÞ Þdm :
ð6:15Þ
Thus from I1 ¼ m!
"
Z _
uðtÞ
f ðtÞ Tm ð f Þðt; t2 Þ
at1
¼ m!
Z _ at1
( uðtÞ
ðt t2 Þmþ1
f ðtÞ Tm ð f Þðt; t1 Þ
#
ðt t1 Þmþ1
dt
Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ ðt t2 Þmþ1
þ ½ f ðtÞ Tm ð f Þðt; t1 Þ
1 ðt t2 Þmþ1
!)
1 ðt t1 Þmþ1
jI1 j4i1 þ i2 8 m ðmÞ > < cMm ð f Þjjujjd ð1 þ jln djÞ
4 R 1 oðu; yÞ ðmÞ > dy dm : cMm ð f Þ jjujj þ 0 y
dt;
ð6:16Þ
if uðbÞa0; if uðbÞ ¼ 0
ð6:17Þ
and together with (6.3), we obtain jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj4jI1 j þ jI2 j 8 m ðmÞ > < cMm ð f Þjjujjd ð1 þ jln djÞ
4 R 1 oðu; yÞ > dy dm : cMm ð f ðmÞ Þ jjujj þ 0 y and there follows (6.1) and (6.2).
if uðbÞa0; if uðbÞ ¼ 0
ð6:18Þ
106
YONG JIA XU
(ii) If jt1 aj4d and jb t2 j > d; similarly, we also have (6.18) but a in place of b: Hence, (6.1) and (6.2) are valid. (iii) If jt1 aj > d and jb t2 j > d; let jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj 0 1 " Z Z Z f ðtÞ Tm ð f Þðt; t2 Þ ¼ m!@ _ þ _ þ _ AuðtÞ ðt t2 Þmþ1 t1 t2 t2 b at1 # f ðtÞ Tm ð f Þðt; t1 Þ dt ðt t1 Þmþ1 ¼ jI1 þ I2 þ I3 j:
ð6:19Þ
Similar to the proof of (6.3), jI2 j4cMm ð f ðmÞ Þdm :
ð6:20Þ
We rewrite I1 þ I3 as i1 þ i2 where Z Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ dt i1 ¼ m! _ uðtÞ ðt t2 Þmþ1 at1 Z Tm ð f Þðt; t1 Þ Tm ð f Þðt; t2 Þ þ m! _ uðtÞ dt; ðt t1 Þmþ1 t2 b
i2 ¼ m!
Z _
uðtÞ½ f ðtÞ Tm ð f Þðt; t1 Þ
at1
þ m!
Z _
1 ðt t2 Þmþ1
uðtÞ½ f ðtÞ Tm ð f Þðt; t2 Þ
t2 b
!
1 ðt t1 Þmþ1
1 ðt t2 Þmþ1
ð6:21Þ
dt
1
!
ðt t1 Þmþ1
dt: ð6:22Þ
Using (6.4) and (6.5), we also rewrite i1 as 2 3 Z Z m1 ðkÞ ðkÞ X h ðt Þ uðtÞ h ðt Þ uðtÞ 2 1 4 m! dt þ dt5 _ _ mkþ1 mkþ1 k! k! at1 ðt t2 Þ t2 b ðt t1 Þ k¼0 0 1 Z Z uðtÞ uðtÞ dt þ _ dtA½ f ðmÞ ðt1 Þ f ðmÞ ðt2 Þ þ @ _ t t t t 2 1 at1 t2 b ¼ i11 þ i12 :
ð6:23Þ
COMPLEX SPLINE APPROXIMATION
107
Similar to the proof of (6.8) and (6.15), we have ji11 j4cjjujjMm ð f ðmÞ Þdm ;
ð6:24Þ
1 ji12 j4cjjujjMm ð f ðmÞ Þdm 1 þ ln d
ð6:25Þ
ji2 j4cjjujjMm ð f ðmÞ Þdm :
ð6:26Þ
and
If u satisfies Dini condition and uðaÞ ¼ uðbÞ ¼ 0; we have Z Z uðtÞ uðtÞ _ t t dt þ _ t t dt 2 1 t2 b at1 Z Z uðtÞ uðt2 Þ uðtÞ uðt1 Þ ¼ _ dt þ _ dt t t t t1 2 t2 b at1 Z Z 1 1 þ uðt2 Þ _ dt þ uðt1 Þ _ dt at1 t t2 t2 b t t1 Z Z oðu; jt t2 jÞ oðu; jt t1 jÞ jdtj þ _ 4 _ jdtj jt t t t1 j 2 at1 t2 b
t1 t2 b t1 þ iy1 þ uðt1 Þ ln þ iy2 ; þ uðt2 Þ ln at t t 2
1
ð6:27Þ
2
where y1 is the angle between t1 t2 and a t2 and y2 is the angle between b t1 and t2 t1 ; and uðt2 Þ lnt1 t2 þ iy1 þ uðt1 Þ ln b t1 þ iy2 a t t t 2 1 2 ¼ j½uðt2 Þ uðt1 Þ lnjt2 t1 j ½uðt2 Þ uðaÞ lnja t2 j þ ½uðt1 Þ uðbÞ lnjb t1 j þ i½y1 uðt2 Þ þ y2 uðt1 Þ
1 4c jjujj þ sup oðu; xÞ ln ; x 05x41
ð6:28Þ
108 therefore
YONG JIA XU
Z
Z 1 Z uðtÞ uðtÞ oðu; yÞ dy ; _ t t dt þ _ t t dt4c jjujj þ y 2 1 0 t2 b at1
so that ji12 j4cMm ð f
ðmÞ
Z Þ jjujj þ
1
0
oðu; yÞ dy dm : y
Now we obtain ji1 j 4ji11 j þ ji12 j 8
R 1 oðu; yÞ > ðmÞ > > dy dm cM ð f Þ jjujj þ m < 0 y
4 > 1 > m ðmÞ > ð f Þjjujjd 1 þ ln cM : m d
ð6:29Þ
ð6:30Þ
if uðaÞ ¼ uðbÞ ¼ 0; ð6:31Þ otherwise
and together with (6.20) and (6.26), (6.1) and (6.2) are valid in this _ case. _ _ (iv) If jt1 aj4d and jb t2 j4d; then jGj ¼ j at1 j þ j t1 t2 j þ j t2 b j43c0 d: From (2.5), jSu;m ð f Þðt1 Þ Su;m ð f Þðt2 Þj Z 4cjjujjMm ð f ðmÞ Þ ðjt t1 j1þm þ jt t2 j1þm Þjdtj G
4cjjujjMm ð f
ðmÞ
ÞjGjm
43c0 cjjujjMm ð f ðmÞ Þdm
ð6:32Þ
and then (6.1) and (6.2) are valid. Now we have proved that (6.1) and (6.2) are true in each case. The proof is complete. ] ACKNOWLEDGMENTS The author is grateful to the editor and the reviewers for their valuable suggestions. The author also thanks Professor Wei Lin, Dr. John H Easton and Dr. Xiao Bin Liu for their generous support and assistance.
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COMPLEX SPLINE APPROXIMATION
109
2. J. H. Ahlberg, E. N. Nilson, and J. L. Wash, Properties of analytic splines. I. Complex polynomial splines, J. Math. Anal. Appl. 27 (1969), 262–278. 3. D. Berthold and D. Junghanns, New error bounds for the quadrature method for the solution of Cauchy singular integral equations, SIAM J. Numer. Anal. 30 (1993), 1351–1372. 4. H. L., Cheng, On complex splines, Sci. Sinica 24 (1981), 160–169. 5. J. Y. Du, Some systems of orthogonal polynomials associated with singular integral equations, Acta Math. Sci. 7 (1987), 85–96. 6. D. Elliott, Orthogonal polynomials associated with singular integral equations having a Cauchy kernel, SIAM J. Math. Anal. 13 (1982), 1041–1052. 7. D. Elliott, Rates of convergence for the method of classical collocation method for solving singular integral equations, SIAM J. Numer. Anal. 21 (1984), 136–148. 8. X. L. Huang, Approximate solutions to singular integral equations, Acta Math. Sci. 12 (1992), 1, 75–85. 9. R. Kress, ‘‘Linear Integral Equation,’’ Springer-Verlag, Berlin, 1989. 10. C. K. Lu, Error analysis for interpolating complex cubic splines with deficiency 2, J. Approx. Theory 36 (1982), 183–196. 11. C. K. Lu, The approximation of Cauchy-type integrals by some kinds of interpolatory splines, J. Approx. Theory 36 (1982), 197–212. 12. C. K. Lu, ‘‘Boundary Value Problems for Analytic Functions,’’ World Scientific, Singapore, 1993. 13. J. K. Lu and J. Y. Du, Numerical methods of singular integral equations, Advanced Math. 23 (1994), 424–431. 14. S. G. Mikhlin and S. Pro. ssdorf, ‘‘Singular Integral Operators,’’ Springer-Verlag, Berlin, 1986. 15. N. I. Mushelishvili, ‘‘Singular Integral Equations,’’ Noordhoff, Gro. ningen, 1962. 16. Y. J. Xu, On stability of negative index singular integral equations over an interval, Integral Equations Operator Theory, to appear. 17. Y. J. Xu, A kind of stability of solution of singular integral equations, Acta. Math. Sci. 11 (1991), 448–456. 18. A. Zygmund, ‘‘Trigonometric Series,’’ Cambridge Univ. Press, Cambridge, UK 1968.