View of On some bivariate spline operators

142

^M

BAL'A'zs

⁶ⁱ

The

condition

III. ^{is the}

^same

with 4" if we lake A:0' ^the

^nota-

tion ã : M0 *

^{Bo)t has been used'}

The conditio,',,of the

^{theorem being satisfied we}

lnay

use

it

^{ancl we}

obtain j)-").

MATHEMAîICA

_

REVUE D'AN,{I,YSE NUMÉRIQUE

ET

^{DD TTTÉONTE} DD I,"{PPROXIMÄTION

L'ANALYSE NUMÉRIOI]E ET LA THEORIE I}E

L'APFROXIMATION Tome B, No

2,

1979,

pp. 143-163

RDFERENCE

^S

n.l B a 1 á z

s, M.,

Conttibution

to

the ^{stud,y of} soluing the Equalions in Banack ^sþaces'

óoctor Thesis, Cluj ^(1969)'

l2lBalázs,M.,Goltln.r,C',DifercnlediaizatelnsþaliiBanachçiuneleøþIica!íiale toz lnomanåö.

'ðñitlí

c"rä"teri-u.t"À'fi"ä, ^{7, 21, 985-se6 (1e6s).}

t3l

Balázs,

^M.,

GoI¿ner,

^{G., on} Aþþroximøte soluingby sequenceslhe Equations in Banach ^SPacøs (in _Press)'

t4lPåväloiu,I.,Surl,AþþroùmatioødesSolutionsdesEquation.sat,øidedesSuites a Eléments dans un Esþaae de nono"oi."f!ái-' À,nat''tL¿orie Apptox' 5' 63-67

(1e76).

t5'ì U 1'm S., ^Oø the Ordinary Steffensen's Melhod' for Soluing Nonlinea'r''Oþeralor Equatioøs' Lur v ¡

"' "inî"'räõ.-î"î;¿i"i''ír;i' i

rnr"t' rizl,

/',6,

10e3-10s7 ^(1e64)'

Received 16' II. 1970'

ON SOME BIVARIATE SPLINE OPERATORS

by

P. BLÂGÀ and GI{. COMAN (Cluj-Napoca)

Introd,uction In

_,some

^{previous papers} [9, 6, 5] it

was studied the spline operator

56

aS

a

genéralization

of

Be¡nstein's opefator which asso-

ciates

to a

functiãn

/,

^dãfined

^on the interval la, bl, the

approximation

(1) (Sofl(") :

n*l¡

Ð N,(x)Í(1n),rn>a,h>0,

where

6:{xj!I}, ^with

(2\ x-t: .,. : xo:

^&

(

.ør

(

\/

and

X¡-p

1 I¡,

^'i

-

{ ^xm-r1b : x*:

^xñ+È

(3)

,(

9i _- x¡-¡|...*xi-t

h

lr

+ l, ...,

^ffi

- ^l,

, i:1,...,rn+h,

(4)

^N¡(x)

:trM,(r), i:1, ^...,

⁷ⁿ

+

^Þ

and

where

M,(x): _l*,-o-r, ..., xi; _þ + l)('-*)u*l isthe (Af l)-th divi- ãed

difference

of the function M(x, t) :

(h

+

^1)(l

-

^x)'*

^with

^{respect to,}

the

variable l.

The

approximation sa

/ ^{is a}

^spline

^{function of the}

^{degree h. having}

the knots at the points of

Ä.

using the operator sa and the

method pfesented

in _lL2)

there _{_are} constructãd

so-"- birr"tiatJapproximation

^schemes

on

a triangle, probiem

also

mentioned.

in [13]. As-an applical.ion of

these schemes are given some cubature fo¡mulas

on a

triangle'

ò

¿ve

obtain lor

^g(x,

!) : l, ^x, y

tespectively

(sg)(r, o):i,N,(r) ^'Ë' N|#,h) ,, _,Ð' ^*i (#,r)

ON SOME BIVARIATE SPLINE OPERATORS

and taking into

account

the liniarity of

G1, Gr, 144 145

Now, appliyng to the

function

/

:S¡; with

L,r:ti ,. - ^{lL:O < ri}

^<

P. BLAGA and GH. COMAN 2

l. Let

^Gr,

^G,

^be

two

polynomials

with the

properties

'.Go@)

>G,.(x) Þ 0 for x el0, kf, k>0, D:{(x,y) eR'zl0 < r <¿},

"Gr(ø)

( y < G,(x))

^anð'

y :"=9, f ^{G, an}

application

which

transform

'the

_dornain

D in the

^square 10,

k) x

10,

kl. If

one applies

to the

func- -tion

/: ^D -*R ^the

^operator

^{56 with to the first variable, we}

^obtain

(s^fl(,,%ryt ¡c,@)):i:,r't,çx¡¡ltt' ^{¿+ c,(6,)l'}

fr,,

^¿

⁺ c,(8,)]theoperator

( flr-, th:t'no: ..':ttn¡+rf

N"(4 Ë' ^*¡ (#thr|'"ryï; +

^G.(q,)J

:

n+þ

(ss)(ø,

y) : _D

i:1 ,ft+þ

_s\

_¿-t

j:1

G,(4ì-GlEù y-G,(E¡)'

-1,

:!Ç

*

^G.(€¡)

l: ^r.

,n+b

(SS)(r,

ù:D

_i:1 ^1,N,

N,@)l

^In

'rse have

.(5)

(sf

)(x, v) :

where

tn¡

<t'¡*r,, j :7, ...,

^%i

- l, i - ^1,

^.

'.,

^ltr

*

^h,

The operator

S. generates

the

approximation formula

{6) _f : _S/* R/

where

R is the

remainder operator.

lfaking in

(6) some

particular functions

Gr,

G,

and some convenient

partitions a, Ài there are obtain

approximalion-

formulas for

various domains. Such,

if

Gr

: 0, Gr: k ald

L¡:!-r:,.. : lo:0 1!t.< ^.,.

:!n¡s, i:1, ..., rn+h,

!¡1!¡+,, j:1,...,tu-1,

then the

operator

(5)

becomes m+h r+s

(s/)(", y) : _{Ð D}

wn@)ñ,U)f

G¡,

n¡)

'-L J-t

stud.ied

in _l2l.

If.

Gt(x)

: l, the triangle T^:

tor

(5) becomes

,(7) (s/)(ø, r) :'f^'Ë'

",tøM (h)t(,,,ff ^ni).

_Respecting

to the

remainder

term of the formula

(6),

in this

case,

we have

h

^{G^(E¿)}

_-

^c.(ErJ

Gr(x)

:

h

- ^{x, then the}

^d.omain

^{D is}

transformed

in {(*, y) e

R2lø Þ 0,

y > 0, x *y ( ^h}

^and,

^the

^opera-

^-i

,,,

- ^ri,;+i ---;t- ^*.'

^'

^{' -l ti-t}

and

- -j.

", -

"-r, Fì-l Li

NiU)

: æLll.n*r-,, ^{" ', tiii}

^(s¿

* ^1)(' _-4'i]

'fot j : 1, ...,

^rL¿

*

^s¡,

ⁱ : 1, ...,

^r.n 1- ^h.

Lemma LIf GL,G2ePytkenSg:B, IeP¡i.e. Sg:g,g:

:1,

%, y.

Proof. Using the

identities

m*h ffi+þ

Ð

¡rn(")

: l,

D._,

l,N,{*) : *,

;$ - L'analyse numêrique et la ühéo¡ie de l'approximatiot

- ^Tome g, No. 2, l9Z9

746 P. BLAGA and GH. COMAN 4 ₅

and respectively

t"-tr: .,,:t"o:Q,

ON SOME BIVARIATE SPLINE OPERATO1TS r47,

L

^e rn m

a 2. If Í ₌

^{8,,,r(To) lB),} then the rent'ainder term of the formcùa'

(6)

kas tke reþresentation

,l ^h vrt_, ^,i|t111,ì

-

tvt uilt- | | |

; L1

'rt ! | L

i-I, ...,

^in,

5 0

m-it7

(R/)(ø,

v) :

Kro(x,

y;

s)/{2,0)(s, 0)ds

f

^Kor(x,

y;

t)fto,z\(0,t)dt

+

_-ll

-'tt-il

' ^t:n:'0,

^ti,

:

n+t

=

^/1,

t1t'-h+Þ:

:

ti,-t,+t

:

h,

þ -

^{1, . .} .,.h"'i

:

m

I

^7,

.

.,

^n't,

-l

^h,

+ I\ ^o-(*, o,

^s, l)/(1,1)(s, t)d'sd't,

Th

It

follorvs

that

uhere

I(ro(x,.!i

^s)

: R6,r¡l(x -

^s)+]

Kor(x,

y; t) :

Rø,nl[(y

- ^t)+]

Krr(x, 1',

^s,

t):

^R(", _vll@

-

^t)t+(Y

- ^¿)il'

Proof. Using

^Taylor',s

formula with

the_integral representation

of the remäinder

t-*erm

[B], for _þ :

q

: 7' and taking into

account ^the-

lemma

^1,

we

obtain

(B) _1¡:

i(i-1) L : f

-tlt

L- t, ..,,, þ 2øoh

b(24-1>

- 1\'

₅

+ r, ...,f**o*'''

--t;;-/t,t.:t , " ^-l

h

-

⁴ⁿ

^+k-i+',

^o

:l-+Ol

m

!

^h,. þ

:

_{min {m,,} k), and respectively

i(i -

^r) lo,

j:1,...,þ

2h(m-i. 11)

þ(2i-þ-,1)

! ^(y -

t)+Í(0,2\(0, t)dt

+ ^2h(m-itl\

(R/)(ø,

_-y)

:

R

(, -

^t)+ ¡(z,o)(s, O)ds

f

_r^^^i:ll

t'- tlilt-t+n-j+2,.1- l l, ...,

^/l'1,

-i'+ h+

^1,

tl

þ:t¡¡'in{*-i+I'h}

1,

n 5lt, - ^s)ï(r'-

r)+/(1'1)(s,

t)dsd'tl:5*,' _,¡l(x -

s)11/{2'0)(s,

O)dsf

Tt, ^o

lor i:1, ..., w

and

\,:,ri: i

_tr -- Ir- i+l

t

_h,

j- 1,...,h-i+1,i:1,...,h.

'

'l'he explicit forrn of the operator S will

be given

in

some practicaF

cases.

1".

The :l,inear-linear case

(/r:1,

^{m, >-}

2). Taking into ^account

^(B),

(9)

there

are obtain

1,

-"-J,,,+n;:jrt, ^{j :1, ...,}

^nt

- ^{i +2;}

'i:1, ..., ru'!l

and

the

operator- (7) becomes

ü+1 m-¿+2 / '.-. \ ^/-.

(10) ^(s/)(r, 9: Ð"'-f,' *,(*)*;(h)r(îr,+r),

i:l i:1.

* I ^{t,,, ùl(t} -

t)+lf(0,2,(0, t)dt

+

!) ",",y¡l(x - ^s)î(y -

l)!l/t1'1)(s, ^t)dsdt.

2. In the

next

it ^will

be studied

the

operatot

(7)for

some given

parti- tions A and

^An.

,For nt,: l, ni: ^{1, s} - ^k -'i + l, ^the operator'('Ì)

becomes the Bernstein

operator on the triangle Tt, l12l'

Let A

^{and. Ä¿}

be

given

uniform partitions,

i.e.

*-þ -

: 1,n: Q, xt: ^{^} ! ^h Xnt-t:"' ^nc-|.- k, %*:

?n ^rn : %*¡p: lL:

P. BLAGA tnd GFT COMAN 6 148

rMhefe

I ON SOME BIVARIÀTE SPLINE O?ERATORS r49

with Nt(r) : Itn-

h

fnx ₊

K"o(x,)r;

^s)

: _(r-r)+ -ifr- x- (h-zx)nl(lø-2s)¡-

- |tr. -

^lø)*(h

-

^{s), Ko,(x,}

^{Y; t)} : (1 -t)+-

N;(x):Lutlt _-2)h - ^m'xf+ -

²¹⁽ⁱ

- ^r)h - ^{mxl+ +}

^(ih

- ^m*)+\'

₍₁₃₎

i:2, ,..,

^rrl'

N,+r(ø) : I ^l** -

^(m

-

^1)h')*

^-(k-

_h.(h

^¡(r*2t-

- ^x)

and

I{',(x,y:s,t):(x-,)i(y-,l,*-W(h_2ùo+(k-2t)o+

.ilh): *,tt' ^{- x} -

^(m

^{- i +}

^r)Y1+

.f (*) : *,{tU -

^2)(L

- ^x) -

^(wù

- ^{i + 1)vl*} -

-2t(j - ^r)(k -

^x)

- ^(* -

ⁱ

l,t

^:.t,.

; ^:,|\r,x) -

^(m

- ^{i + l)vl+)'} Nln-¡+z(h):ì;lþn-í ^+\)v - ^{(m-ù(t'- x)7+'} i:1' ""

^rn'

*7,+r - ( _lh ^ny l:t,

-

^{x )}

THDoREM 1.

If Í ₌

^{Bl',f (T,,), th'en}

ll*ll .fillr"r, _{o)ll. äll} ¡o'zt(o, )ll * i llt'r'"ll'

uhere

ll-ll is

tke un'iform ^no/171.

Proof,

In

order

to ^{apply the}

medium theorem one studies

the

func-

tions Kri, Ko,

^anð.

K'

^sign.

Let

us consider

first the function

^{1(2¡ ;}

A.s<1

2

'Ihe

remaind'er

term of the

approximation

formula

generated'

by

^the

"p"'äiàì iiöi-;;

be obtained

bv

remma 2'

For a

d'etailed stud'y,

let us

constder m

:2' fn ^this

^{case we have} _I{ro(x,

r;

^s)

:

x(h - ^2s\

-::::----!- ïOf I <.s

h

-(h-2x)+s ^for ø>

^s

h

It ^{follows that}

^Kro@,

Y; ^{s) < 0,}

^(ø,

Y) = Tt

2lh-x-

^(h _- ^2x)

-x-Y) r(i, ^{o)+ B.} s¡ ⁴

_I

h(h - ^x)

, ^zth- x-

^(h

_{-2,)tl!- f(L,} Lln?x

^-h)+.f

^(h,0) * ^(Rf)(x,

^v),

_j- '

-r@ '\z zl

^h

^-

^h)n(lt'

^{- s)}

for ø {

^s

o ^tot x>

^s,

such

that l{ro(x, Y; ^s) ( 0,

(x,

Y) =

^Ty

fn an

analogous

way one obtains that

^I{or@'

y ; t) <

^0' (x,

y) = ^T¡, t e _l0'

^k).

It is

^{easy seen}

that ^{the functioî Krt}

^changes

^the

^sign

^{on the}

^do-

mattn Tn.

where

bY

lemma ²

h

(R/)(ø, ,, :

I

K,o(x, Y;

s)/(2r)(s, ^O)ds

f

^Kor(*,

^{y i}

t)l9'2\(0, t)dt

+

f12)

It ^{Or,,, ,} y; s,

l)/{1'1)(s, t)dsdt, T¡

150 ^{P. tsLAGA and} cH. COMAñ ₍₎

19 ON SOME BIVARIATE SPLINE OPERATORS 151

In this way

we have

(n/)(",

_1t)

:

_ftz,0t(|, 0)ç,0@,

y) +

Í(0,2)(0, ùç0,@,

y) I + J!

r<,,{"

, ! ⁾

^s, i)/(1,1) (s, t)d,sctt,

Th

'where 1, 1= [0, h,l

and

¿nd

ß2 lt(Sx - ^h)

Ior

^{^h}

x2-

N,(x):(r:r)(+)'-'('=,,,.)--u*', i= ^1,

^.

.., m+1,

ìy'r1

having the

expressions

from the

previous case.

We remark

that

S,

in this

case,

is a

spline polynomial operator.

In detail will

be studied

the practical

case ,1fl'

:2.

One obtaines the appr

oximation

forrnula

l@, ^y) : *þ - ^x)(k-

^x

^-2y)+f(0,

⁰⁾

+ i4r - x)lh x ^y -

(1s) -(h - ^x -2y)+lÍ(o,*)+ _)W - ^{x)(x +2y} -tù+f(o,

^tn)

+

+frx@- ^x -r)r(+,

⁰⁾

* #r¡(i'+)*#¡tr,0) ^{+ (R/)(x, !),}

2 4 2

g2o@,

1t):

*2 _hx

24

lof '1'<-^"h

2

'9or@,

y) :

y(2y lt) _ ^lt(h - ^2x)(x Í ^2y - ^h)

4 4(h_x)

."+and,x!2y)/t,

y(2y

-

^h)

4'

^otherwise,

where

It

follows

that

'(r4)

max

_lgro(x

_, l,)l :

^t''- ,

t h oL rnax _{lgo, @,}

s,)l:

^zs

^-

^{tz ^ftz}

^,,

- 6a tr<-

^{1 ,"}

l6

(R/) Kro(x,

y;

s)/{2,0)(s, O)ls

f ^Krr(x, y;

t)fto'2t(O, t)dt

+

T.

flence

I

TI

h

+ ^Krr(x, y;

^s,

t)ÍÍ'r't(s,

t)d,sd't,

,l(R.f)(*, ù <Lflfe,ot1.,

0)ll

+rftlfaø{0,")ll+ lfa,1)ll\\lN,,t*,y;

^{s, t)ldsd,t.}

Th

Takinginto accountthatll{rr(x, y; s, t)l ç ^{1 on}

Tutheproof follows.

2".

The Bernste.in-Lineør case. Using

the

partitions

L:x-o, :frs:0Íh:x1:.. :xnir,

A,r:l,i-1

:tL:0, ti:

t

ih . i:1. ,t-i ti .,.- ti . ^_

^t,

rn_i+1, J ^t ..., t¡u vt unt-¿lt-þw_i-l-2 -tt,

i:1, .,.,

^j,r,t,

, A*¡r:

t'{+1

: Q,

tT-F'

:

h,

one

obtains '

^l

.,' ^q,.

--

it- | , h- t; ; _?nhm

lL,:----::.r);:r---_:h,

i -1

,i:1, ..., nt.!1, j:1, .., ffi__i+2

with

K,,(x,];

s)

: ^(r _- r)n -ryltn _- ^zrln - i ^{t, -,).|,}

[Io"(x, y I ^t) : (y - ^t)+ - i l, -

^x)ltr,

- ^x - ^y -

^(h

- ^x - ^2y)+] I

_t *yj(h _ _2t)+ _+

_@

t

^2y

^{_ h)*(h _}

ù,

K,,(x, y ; ^{s, t)} : (x --s)iþ -

^r):+

-T _{É-')i(å -} ^r)i

+HEoRÞM

2. If f _=sl',f çr,,¡,

^tken

' _ll*/ll < t _lfe,o)(.,

0)ll

+ L*llf*r,10, .)ll +T

^ll/(1,1)ll

752 ^P. BLAGÀ and GH, coMAN ₁₀

Proof. As in the

previous case

the

functions K2o

and

Ko2 are nega-

tive

and

the functiot Kn

changes

the

sign

on Tr.

So we have

(RÍ)(tt, !) :

ero@,

y)Í(''"(8,

0)

*

^qor@, ,¡¡ro'zt(o' ^'q)

i

I

^{ON SOMÉ BIVARIÀTE SPLINE OPERATORS 153j}

Taking into

account

that

lf ^lo,,{',

^t)ld'sd't

: ^E

^hn

Th

+ \\ *-t*, !; ^s, t)fl,t\(s,

t)d.sd,t,

Th

we

obtain

i

with

_{E, rJ}

= ^{[0, å],}

^where

lR(/)l * fr t+tyop)(.,

^0)ll

a

s¡¡¡to'a10,

.)ll +

13ll/(',1)lll

In an

^{analogous way, using}

the formula

(15), we obtain

Th

I

f

^{(x, y)dxd'y}

:ftftt1o,

o)

+ ^6/(0, f,)+ ^vro,

^h)

^{+ 4r(+,}

^o)

*

* u(+'+)* ^{4r(h, o)f+ Rr),}

It

follows

that

m¿xlero@,

v)l :t*,

\î

^lsor(x,

v)l:L*, m;;xlKrr(x, y; x, t)l:1

and

the

theorem .is proved.

3. Next, using the approximation formulas

(11)

and

(15)

there

are constructed some cubature formulas

on the triangle

1r.

ff we

integrate on

T,

each member

of the formula (10), one

obtains

\\ ^{¡t", y)dxdy: li} I s/(0,

0)

+ tlf _{(o,i) +} 5/(0,

h)

+ er(+,

^o)

*

Tr

+ nr(+ , i) *

^{4f (h,}

^{o)l+ Rff),}

where

lR(/)l * _ffn

talr,,,o).(.,

0)ll a

ol¡¡to'a10,

.ll+

lì./(',')lll.

REFERENCÞS

whe¡e

R(/) :

CzoÍ(z,ot(t,, 0)

¡

cor¡rz,o)(0,

tl) +

with

I I ^t,,{',

/)/{1'1)(s, t)dsd't,, T,n

[1] ^{A r a rn} ä, O., ^{Oø the} þroþerties of monolon'icity of the sequence oJ interþ-olation þolyno-

nials of ^S. N. Bernsteòn øød, their øþþl'ical'ion to the-_stud'y of aþþtoximatiotr' of funcíions (Russian). Mathematica (Cluj) 2(25), 1'.25-40 ltOe!)'

[2] Com^o,-Gh.andFrenf iu,M., Biuøriøle sþline aþþroxirnation. $tud1a univ.,,Babeç- Bolyai", Ser. Math.,59-64 ^(1974).

t3l ^{G o r cl o} n, W. ¡., D.istributiae lattices and, ^I'he øþþroñmalion oJ multiaariøle,funotions.'

fn ,,Apiroximation with ,special emphasis on spline functions" (ed. ! ^{c h o e n-}

b e r g, -f.

J.). ^Academic Þress, Nerv York-I,ouilon, 1969, 223-277'

[a] ^{M a r s cl e} n, lU. f., 'An _àdentily Jor ^sþt'ine funol'ions uith øþþlicalòons lo uariation ili- minishin{ sþline aþþroiimøtion. J. Approx. 7--49 (l?70)'

l5l - ^{On uni'form} sõliie øþþrôximalion. J. ^Approx. l'he :?5.3 ^(1972)'

¡?j u.tsdeî, v. i. ana-Schoenberg, l. _-¡., ^Ott' yg, 'inishing sþline methods'

Mathematica (Cluj), S(31), / 61-82, ^(1966).

[7] popovic ^í:u,, 1., Sur'le ^r:este d.ani certain.s formules li.néøires d"øþroúmation del"analyse. - - Mathematica (Cluj), l(24r, 95-142 ^(1959).

[8] ^{S a r d, À,,} Lineør aþþiòximation. Math. S-urveys No. 9, American Mathematical ^Society,- P¡ovidence, Rhocle Island, 1963.

[9] Schoenberg, I. 1., ou. sþIine fuflclions. In "Inequalities" (eil. shisha, o.). ^Âca-

clemic Press, New York, 1967, 255-291.

[10] Starcu, n. o., ln" remaittd.ei o¡ ceirain li.near aþþroximation.formulas in tuä uaria- ål¿s. SIAM J. ^Numer. Anal. Ser' ß,'l', 137-163 ^(1964)'

l1ll - Eua.luation of ilíe remainder lerm in aþþroximation form.ulas by Bcrnstein þolyno- miøls. MatL.. Comput., 17' 270-278 ^(1963)'

fnl - A ^method, Jor obtøininþ þoþtnomiat of Bernste'in tyþe of tøo uariables' Amer. Ivlath.

Monthly, 70, 260-264 ^(1963).

l-131 - Aþþroximation of biuøriate Junclion.s by m.eans oJ some,Bernslein-lyþe oþeratovs. I*

,,Multivariatê Approximation" ^(ed. 1Iandscomb, D. ^{G.). Acaclemic Press,.}

Lonclon-New Yãk-San Franscisco, 1978, 189-208'

Receivetl 7, Il, ^1979.

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