View of Approximation of complex variable functions and applications for solving singular integral equations

Roger'Webstor 10 REVUE D'ÄNALYSE ^NUMÉRIQUE ET DE THÉORIE DE L'APPROXIMATION

258 Tome XXVI, Nd ^{1-2, 1997,} PP' ^259-268

RE FERENCES

1.E.Artin,EínfihrungindieTheoríe¿¿¡'Qammanktio-n,.Teubnar'-Laipng'1931'

2. H. Bohr ^{and J.} Moll-erup,

iirr"øog

^í Mathematik Analyse

III,

Koponhagen, 1922, pp. 149-164.

¡. N. drlurtaf<i, Élëments ^de Mathématique,BooklV, ^Chaptor

YIl

Lafonctíon gamma,Pafis, 1951.

¿. p. ¡. bavis , ^Leonhar¿ Euler's íntegral. A hístorícal profile of the gamma functíon, ^Amar' Math. MonthlY 66 ^(1959), 849-869'

5.

F. \---'"

_{7r (1939)' 175-189'}

6.

R.

^theorem' Amor' Math'

a ^

_(1938), ^{x f(x)' Acta Math' 70}

5',7-62.

s.A.v/.RobertsandD.E.VarbergConvøtFunctions,Acadgrlichess,NowYork'1973.

9. R. W"brtur, Convøcíty, Oxford Ùniversity ^Pross, Oxford' 7994'

APPROXIMATIONoFCOMPLEXVARIABLEFI-INCTIONS

AND APPLICATIONS FOR SOLVING ^SINGULAR

INTEGRAL ^EQUATIONS

V.

ZOLOTAREVSKI,'V. ^SEICHIUK

Recoivod MaY 15, 1996 Pure Mathematics Section School of Mathematiu and ^Statístics

UniversitY of Sheffield Sheffield 53 7RH

England

E-maíl: R.J.WEBSTER@SHEFFIELD'AC'UK

The direct methods of solving singular integral ^equations (SIE) use results concerning ^the approximation of functions of complex ^variables by polynomials' These functions ^are defined on closed ^smooth contours' some results ^{have been} giveninourpapefs[1_3]withoutploof.InthisNotewewillprovesomeassertions about the approximationuy polynomials of functions defined ^on arbitrary ^closed smooth contours. ^These resrilts havebeenused in ^numerical anaþsis (collocation'

-q*O*t

.,

reduction ^and spline methods) for solving SIE'

l.Furtheru'eneedthefollowingdefinitionsandnotations:Byf'wewill

denote an arbitrary closed smooth contour bounding ^a simply-connected region ^F+

containing ^the point

i=

^{o. By F_ we}

will

denote the complement

of 4 uf

^{in the}

entire complex ^plane. Let

c(i¡

^{be the space} of continuous functions ^on

f

^{. Assume}

,* ;;(ai

^{is a set of} tuncìions defined ^on

f

^and satisffing ^a Hölder condition with the exponent B, 0 < _B <

l.

The nolm on

¡¡u(r)

is defined [4,

p'

173] by

llqllp

Tgl'( ⁾⁺

q)

-

^.p(t'

= llell" +

ã(q;Þ)

t sup

lÉ12

lþt2el

l', -'rl'

(1)

From [4,

p' ll3lwe

^know

^that Í/u(f) with

the norm

(l) is

a Banach ^space' According to [5,

p.

109],

if

^{0 <}

^p

^<

^ø (

_{1' then}

_H*(f) c Au(f)'

^{In this case}

it

also frol<ls _llellu

. _r',llql[,

^{where tp €}

^/r-(f)'

^This

^is

a consequence

of

the relation

1991 AMS Subject Classification: 30E10' 65R20'

260

(s)

V. Zolotarevski, V. Soichiuk 2

Proof. Letus first

assume

that lrr-rrlr-!Q,,,r, ef).

^Then,

^fron

^(4),

Solving Singular Integral Equations

t1)

-,pl(tt) -

^{q(t2) + q*n(tz}

4) -

^q(tz

26t

lq(t,)-

^q(,,)l

l',

-

',1' sup

lrxt2 tr,lTel

< max{ I sup

,

ll,'-øl=t

3

we have

Let

xn ₌ X,,(f)

^{be the set of} all

poþomials

of the type

Ø ^{q'(r) =} f'o'o'/e r'

k=-n where roare arbitrary real or complex numbers,

. For the function

q

the value 8,,(q)

=

infenexnllq

- ^q,ll"

is called the best uniform approximation by

poþomials

of the form (2), The polynomiat

9l

^for

which

E,(q) _{= ll*} - _{*; ll.} ^*

^{called the best} polynomial uniform approximation of g. Recall _16, p. 4321that for each flrnction

q

c

C(f) in X,,

there always exists such aunique

poþomial

rpi,

The module of continuity _[5, p. 107] of function

g

is

r(ô, e) =

^sup _^le(r,)

-

^{.p(¿r)l (a}

t

^{o; ty,t2 e}

^t)

^,

lr,-trl<ô'

If

^<p e

¡/"(f) (o . o <

1), then

(3)

c,)(¡,q)

<

ä(<o;c,)ô".

Itisknown l7,p.3lll ^thatif e

^e

c(l),

ttren

n,(q)sþ2,'[;,-),

rvhere

p,

are constants not depending on

r

^and on the function

q.

Fròm tÉese two last inequalities we deduce that

if

<p e

.H.(I'),

^then

(4) ^{E,(q) < p,#H(q;o).}

TrTEoREM

L Let

<p u

ã"(f)

^and

q)

be the polynomial of the best uniform approxímation of p. Then

ltt

1

,"-ø

lq(q)

- ^qi(,') - ^{q(,,) * qi(r,)l}

< 2*qðlq(,) - el.)l,p

₌

#H(q;o).

l', -',1'

^'er

If

_lr,

- ^,rl .I,

according to [3,

p.

42],wereceive

(6) ^{r(ô,q;) 3} þt'

^o(õ,

^q),

since

s . H"(r),

^trren _le(4)

-

^,pþr)l

.

_lr'

-

^hl"

^H(qta).

^{From (3) and (6)}

we deduce

lei(,') -

^,p;(/r)l

^<

^p¿lr'

-

^trl"

^H(rp;a)'

^Therefore,

sup

o<14-trl<t

),

^suP

I rtl2 lt,l2el

lq(t')- ^q(t')l l\ -

"1"

1,,

- ^,,

1,,

- ^,,

lle

-

^eillu

. H(q;") (0. ^p

^<

o <

¹⁾

=F$H(q;o) ^. +#H(q;o) ^.

^t

^s#H(q;o)

Lelr¡tøe

L

For each polynomial rpn

from X,

^{we have}

llqill" <

^pu

',llq,ll".

Proof

qrþ) -- i*, ^{= {'(r-n} r r-n¡{*'..+,nt2') = t-nP2,,(t), '

k=-n

q', (t) = -nt-n-rP2,(r) * t-n(r2,Q))'= -u!rqr(r)

⁺

t-'(r2,Q))'

'

Assume that the

function , = \f(w) =

^{cw + cs + ctw-t} _-+ '.

'

^{performs a}

conformal mapping of the exterior of the unit circle _{l

,

_I

t l}

onto F.- so that

V(*) ⁼

^{co and}

V'(*) ⁼

^c

^>

^0.

According to the Cauchy theorem about derivatives ^{o f} the analytic functions, we have

(7) P',zn(t)=*!,#0"=*I,ffi*,

262 V. Zolotarovski, V. Soichiuk 4 ₅ Solving Singular Integral Equations 263

where

fo ₌

_{/

;t = V(pw),lr,l = t,p t l}

is the level line for some p

>

1, We may construct a

circle lr - ^tl =

^¿(p

- ^t)

^b

^>

^0,

^with

the center on an arbitrary

point r

e

f

^{such that}

^it

^{does not} intersect the level

line Ç.

^{Let us choose}

t ₌ y(rv),r ₌ V(pr),Thenweobtain

l' -

^rl

⁼

l.r,(or)

- v(,)l = lv'(6)llo", - *l= _{lv'(E)(p -}

^r),

From[7,p,

1s1]

0<m <lv'(E)l <M <-,lEl>1.

^Then

*(p-t) <l'-4<u(p-t).

In this case, the circle

lr -

4

= î(p ^{- f)}

^does not intersect

I,

^{. gV the Cauchy}

theorem and from (7) it follows that

P'r,(t)=* J -.1'(') .,o,

It-ll=ó(P-l)

r-n(r -

^t)"

Further,

(x + å(p

-

^l)cos

v)' +(t *

b(p -l)sin<p)2

x'+ y'

f

dq=

:u+m5i

^{x2 + y2}

^{* b'(p - t)'*}

^zb(p

^-

¹

.xcosg+ysin

^t2

22

dg=

x +y

ó(p

-

^t)

2

l2

+ (.r cos <p + _-1, sin q)

1+2b(P

I

_{) 22}

x +y

^d<p

Leto=t+a _'nb

^Then

2Í

J t*

0

ålo

-

¹⁾

T+(.rcos9+Ysinq)

tC +1t22

2

2b(P Ð

dq=

l,-'

r',,(,)l _{= l,-'l} * n_uJ^,_,,ffi ø

₌

',ft_-rj,l*, ol+*{,, lËl# ⁼

^B

+¡*ir,[' lil'ø,

where

' =:flËhlq'(")l

Let t -

^t = b(p-t)eio (o <,p

<2n). Then r =t +b(p-r)e¡v.

^{dr =}

å(p-t)iejetfq;

lo"l =

(p-t)aa9,Let t ₌

x

+ry.

Then

l,-'r,,,,(,)l =, +dry'Ï)

ⁿ oro-,)<re ₌

= sL, I, ^{2flx+¿y+ å(p-}

rXcos<p+¡sin<p)l'

IrrT tl ;,

-l

o'=

=T'-,1

I

2n**cosq+/smq

1

x

22

^+y ur={['.1 ^o

_--an1, t2^

I

^¿.p<

I

exp

2tt

I

=J

0

^ i

^x

^cos9+/sm9

¿n

x z2 ^+y

^d9

rcosg+ysin<p

,'*y' ^dg,

where

^ I

)r+*cosq+/smq

*'*y'

However,

rcosq+ysing

cosq+ -l ,sing=cos(ï-q) <l

x'+ y'

*'*y'

6 ,| Solving Singular Intograt Equations 265

,t el

264

Hence

V. Zolotarevski, V. Soicbiuk

l,

-.

r,,.(,

_)l

.

^B

+, *r(^¡r)r- ^=*n

['

. ffi)"t ^þ,

^{I tc}

^ft

^{-, | = ¿ ( p - r) )}

Let us denote

by rp

the inverse

of

the frrnction ry. Then we perfonn a conformal mapping of ^the exterior

of f

_{onto {lwl} ^>

^t}

^{so that}

g-(æ) =

^{oo. Recall}

that

q-

^{has the form}

_e-(r) = I, *? . þ .

Let,l?

>

1 and

f"

^{be a} level line of mapping

V for

_lwl=

l?.

Consider the Q,

function

(_ I

^. This function ^is analytic onF_,

inc

ling thepoint ^{co. Therefore,} by the maiinium principle, we have

*lffil .***ffi=#xl#

Since D,*

c F*,

^then

23+le,(r)le.t,)]'l ⁼ -sl*,(')[e.(')]"1 ⁼ T.irle,(')[*.(')]'l ^{= 1e*le'(r)l}

In this relation we consider that

for r

e

F, lq.(t)l ⁼ I

. Therefore,

(e) llq"ll"r,;1"

^<

llq,ll.t.l'

The function ^{<p, is} analytic in ^the ring Fbounded

by f,

and

fo

_' ^{Hence, by}

the maximum principie ^and by inequalities (8) ^and (9), it follows that

i.Ele,

^(r)l

^{= -u*{ll*,}

ll"1ç¡ ^; lle, 11",,^,1

='.*{il*,

11"1.¡

},

^ilr, 11",.,

^' }'

Let r =t-!

^{and R}

⁼ l+!.tn.n

^Rn

^<

^eand

^{rn > 1} nn+

so

llq,ll.rot

= allç"ll.rrl

and, rinally,

llq,ll"r.l <

^t urllq,,ll.rr¡: ^rüh€re

znltl'z

[,.**,[,.+)),n

⁼

_

maxr.r;l ,p"(t)l

.R,

We note

that ¡t; is

^{apart of F-, which is the} exterior for the level

line fo

We also note that

for I

^e

l^

^{we ftuut} _l,n-(r)l

=

^{.R, Since}

Dl c f

^{, then}

-,{ffil _{= -r,lffil} ⁼ **]# ⁼

^mqxl

^e,(')

^I

Now we use that

fo¡ r

e

r,

_le-(r)l

-

^{1. Hence}

(8) 1i3¡le,(r)l = +g.l,p,(,)li?"

Further, let

w =

^{rp* (r) be the mapping which is the} interior of

f

^on

_{lrl . t}

suchthat,p.(0)

= 0.Itisknownthat

^q.(¿)

= t +yrtz rTzt3 +...

.Then

[*.]'*,

is an analytic function onF'n.

Letr

^< 1,

f,be

^the level line of themapping

9*(r) for

_lwl

= r.

Note tnat

[e.(r)]=t i,

un inverse function

of

<p* and

D"*

is a part of

F*,

which is in the interior o1

f,

^{, Then, by the} maximum principle,

ry+1e,,(')[e.(,)]'l ⁼ *f{|q,,(')llq.(')1"} ⁼ llq,ll.r,,r' /'.

Tmoneu 2. ^Let g,

^€

X,.

^Then

(10) llq,llp<p,rnpllq,llc, o(p<1'

Proof.

^{First assum ethatlt,}

- ^{bl> j,{,r,t, el)'}

^Then

1

lÌ.

< znpllq,,ll.

Now consider that _lr,

- ^{rrl.L .mthis}

^{case, from}

le,(r,)

-,p,,(,r)l

=

|

I *',,,"ru"

= llq',11" 'leng'thtrt,

3

constllc',ll"l,t

- ^trl

and Lemma 1, we have

266

(q) - ^q,(t, _{< cotst}

¡rø

.rllqrll

,lrt - ^rrl'-u ₌

^const

'þs.nß llqrll, l', -'rlu

From these cases we deduce (10), where p7

=

^{max(l + const' po;3),}

Further, we establish the estimation relation of the interpolating Lagrange polynomial for the function

q

defined on

f

^.

LEtvfl\l,t 2.

Let

,p €

ã"(f)

^and

^q,,

e X,, be a poþnomial such that

llq- ^q,ll"

^{< (p',}

*¡rnhn)ø,(rp):

Then

(11) llq-q"llu .@j#'¡aã(q;o) ^{(o.p < o <}

^r)

Proof.

Let

<pi be the polynomial of the best approximation for the func- tion <p. From (5) and (10) it follows that

lle

-

^e,llu

_{= ll*} -

^eillu

^* _llol -

e,llu

< p,#ã(q;o) * p,,ullel -

^r,,11,.

Using (a) for the second term of the last inequality, we deduce that

llel -e,ll. _{= ll*i} -*ll. ^*llq" -qll. , *,#n(q;o)+(p,,

+pnrnn)^E',,(rp) <

< (t * pr *

ps

lnn)¡:"r# r@to)

₌ (p,,, + _¡r,,,ln

lþ ^u@;").

Then

for

_Fro

=

F¡

* _þtl\z

ifrrd ^prrr

=

pzlqs we obtain inequality (11), Let _{t r}'"-_rbe ^{a consequ ence} of

2n+

1 distinct points from

f

^and

2n

Solving Singular Integral Equations ^26',7

u,(s, t) = iî,1,¡gç,,¡

^.

t=0

It is evident that

U,X, ₌

Xn. As in _[5, p. 539], we establish that

lr,(t,¿)-

^g(¿)l

^<

^(t

* )',)n^(e), g(r)

e

c(r),

where

2t

L, = TS,.IlLt'l¡. _-

_j=o

Let us

consider the

function

_V,

_V(w) =

^{cw +} "orr-' ^{+. ..}

, ^which

^maps

conformally ^{the exterior of the} unit circle with the center 0 onto

Fl

^{so that}

V(*) ⁼ -

^,

v(-) ^{= c>}

^o'

Let tu, =

^eXp

#htt ^{-n) (i'=:1,i =}

^0,1,

^{...,2n) be a system of}

equidistance points on

lo

=

{lrl : l}

_, and

(12) ^t,=v(w,), i=0,1,...,2n.

TuBonRv 3.

If

tj,

i ^{:0, l,}

^...

^,2n

^are defined by (12), then

I, I[rr++¡r*lnn,

The proof of this theorem is very long, We

will

omit it.

2. Now, using our previous results, we propose a substantiation

of

the collocation method.

We consider ^SIE with the Cauchy kernel

(13)

^c(r)e(r)

.+{9 a,+};[*r',t)<p(t}1t = f(t),t ^et,

in the Banach space

Hp(f)

, 0 < p < 1. Here c, d,

K

andf are known functions in

øu(f) '

_According ^and

^q

^{is unknown,}_to the collocation method, we seek an approximate solution of (13) in the form of

apoþomial

n ,,

.p"(r)

= luf){, ^ter

k=-n

{"f'}-=_, = {c,r}'r=-,

^are found from the following

V. Zolotarovski, V. Seichiuk 8 9

I¡(')

₌

fl('-'o)

k=0; k+

j

fl ^{(,, -,0)}

k=0;k+

j

tj

2n t

n

= >^?* ^{(re r, i=o2n)}

k=-n

By

Unwe ^denote the operator which maps any function g continuous on

I

onto its interpolatin

g

Lagrange poþnomial defined by using the nodes

{t,}',',=o

This is apolynomial of the form

(14)

The unlarown ^c oeffrcienfs system of linear eqttations

268 V. Zolotarevski, V. Soichiuk 10

(1s) "(',)ä.'otj

⁺

r(t,)/,ootl *

þ,*r*[*r,,,,),e.rr ^{= Í'(,,),}

where a(t)

= c(t)+d(t), b(t)= {t)-a(t)andt,i:0,1,...,2nform

^{aset of}

distinctpoints

onf.

. Tmoneu

^4. Supposethefunclions ^{a, b and} K(uniformwithrespectto both variables) belong

to

the spdce

¡1"(I.),

0

<

_P

a a Í ^l,

^and

^let

thefollowing condítions hold:

t)

a(t)b(t) É 0,

/

e F,

2)

intd a(t)U-'(t)

=

^0,

,

€ I-,

3) the lrernel of ^the operator cowesponding to ^the left side of

(I3)

is empty.

In addition, let t¡,

i ⁼

^{0,2n, be} calcalated according ^to (I2).

Then,

þr

stfficiently large n, the system

(I5)

has a uníque

solutø"

{op]J'o=

,,

The ap¡troximatesolutions ⁽¹⁴⁾ convergein thenorm of

Hr(f)

as n

-+

æ ^{to the} exact solution ^<p

of (I3),

whatever ^the function

f

^e

H"(f). For

the rrtte oJ convergence, the following ^{estimate holds}

llq

-

^q"llu

³

^{(P,u + P,r ln}

n)nþ-"n(e;ø)'

For the proofsee [1-3].

REFERENCES

RDVTIE D'ANALYSE NT]MÉRIQUE ET DE THÉONTN DE L'APPROXIMATION Tome XXVI, N* 1-2, 1997, pp. 269-275

BOOK REVIEV/S

Partial Differential Equatíons and Functional Anølysis,lnl. Memory of Píerre Grisvard, edíleó,

by J. CEA, D. CHENAIS, G. GEYMONAT and J.-L. LIONS, Progress ín Nonlînear

Dffirential Equations and Their Applícatíons, Yol. 22, Birkhäuser, Boston-Basel- Berlin, 1996, 264 pp., ISBN 0-8176-3839-3.

Piorro Grisvard, one of tho most distinguished contomporary French mathomaticians, died on April 22, 1994. A conforonco was hold in November 1994 out of which grew tho invited articlos contained in this votumo. All ^papers are rolated to functional analysis applied to partial differential equations, which was Grisvard's speciality. Indood, his knowledge of this ar€a was extremely broad. Functional analysis applied to partial differontial equations is an a¡oa enjoying rebirth and reexamination in many parts of tho world, particularþ with rospect to its recent thrusts into mathomatical physics. Griwa¡d also becamo a specialist in tho study of opti- mal rogularity for partial difforontial equations with bounda¡y conditions. Ho oxaminod singularitios coming ftom coefhcients, boundary conditions, and mainly non-smooth domains.

Griward left ^a logacy of preciso results vory usoful in applied mathematics, which ^wero published in journals and books. This volume contains a bibliography of his works ^as woll as his last paper, which is published horo posthumously.

Tho contents aro as follows: P. Griwmd, Problèmes aux limitets dans des domainec avec points de rebroussemenf; T. Apel and S. Nicaisa, Ellíptic problems in domains with edges:

anisotropic regularity and anisotropicfníte elernent meshes; M. S. Baouendi and L. Preiss Rothschild, (Inique continuation of harmoníc functíons at boundary poinß and applications to problems in complex analysis; C. Ba¡dos and M. Bolyshov, The wave shapíng problem; P. G.

Ciarlet, Modélisation mathématique ^des coques linéøirement élastiques; G. Da Prato, Fully non- Iinear equ,ations by linearization and maximal regularity and applicatiorls; M. Dauge, Strongly ellíptic problems near cuspidal poinß and edges; L. Boutot do Monvol, Star produît associë à un crochet de Poisson de.rang constant; P. Destuynder and F. Santi, La méthode de,s lâchéø _Qe' tourbíllons pour lecalcul ^des efforß aérodynamíques; A. Favini, Sum of operators' method in abstract equations; G. Goymonat ancl O. Tcha-Kondor, Constructive ^methods

þr

abstract dif- þrential ^{equations and} applícatiozs,' V. A. Kondratiov and O. A. Oleinik, On asymptotics oJ

solutíotts of second order elþtic equations in cylindrical domains; V. A. Kozlov ancl V. G.

M*'yq

Singularities to solutions in mathetnatical physics problems in non-smooth domaíns; J.-L.

Lions and E. Sanchoz-Palancia, Problèmes sensítifs et coques ëlastiques minces; M. T. Niano, Contrôlabílité eractefrontière de l'ëquøtion ^dec ondes en présence de singularités; E. Sinestrari, Interpolation and extrapolation spaces in evolution equations; M. Zamar, Localísatíon des

singularit* sur la frontière ^et partitions de I'unitë.

Ths rosoarch or oxpository articles included in this volumo, many of them with diroct roforences to Grisvard's contributions, will ^bo of ^groat intorost for all spocialists in nonlinear partial difforontial oquations and thoir applications.

R Precup 1991 AMS Subject Classification: 65899

l.Y.

Zolotarevski and V. ^{Seichiuk, The} collocation methodfor ^solving singular integral equations gíven on a Lyapunov contour of íntegration, Soviot. Math. Dokl.23 (1981),

468470.

2. V. Seichiuk, On the rate of convergence of collocation and mechanícal quadratures method

ral

equ

apunov contout', Chislonnoe roshonie zadach ,

$tiints

^{108-118 (in Russian).}

3.

V. riimens

^{lving singulør} integral equations on closed contours of íntegration, Çtiintsa, Chiçinãu (1991) (in Russian).

4. N. Mushelisvili, Siigular Integral Equations, Nauka, Moscow, 1968 (in Russian); English transl., Noordhoff, 1972.

5. I. Natanson, Constructívè Functíon Theory, GITTL, Moscow, 1949 (in Russian); English transl., Vol. III, Ungar, Now York, 1965.

6. J. Walsh, Interpolating and Approximation by Rational Functions ín the Complex Domain, Moscow, 1961 (in Russian).

7. P. Suetin, The series on Faber's Polynomíals, Nauka, Moscow, 1984 (in Russian).

Received May 15, 1996 State Uníversíly of Moldova ChiSínãu

Moldova