# View of A boolean method in bivariate interpolation

## Full text

(1)

MATI]trMATICA

### _

REVUE D'ANALYSE NUMÉRIQUE ET DE TFIÉORIE Dtr L'APPROXIMATION

NUMÉRIQUE

### ET I,A TIIEORIE I}E

L'AI'PROXIN'IATION Tome

No

1980, PP.

by

F.J. DDÍ,VOS and

### II.

POSDORF

(Siegen, West Germany) (Bochum, West Gennany)

Introduction

rruefe introdu-

fund.amental

shown

commutative

a

### distributive lattice

each of 'whose

arametric extensions

univariate

J"","i

"

iog an va-

have

### the

same Boolean

(2)

36 F. J, DELVOS and H. POSDORF

2 3 A BOOLEAN METHOD IN B1VARIÄTE INTERPOLATION 37

1.

projector

S

La,

be

square and.

### let B(S)

denote a sutspace

space

### C(S) of

continuous functions

### on S,

F'urthermore, let

operators

operate on

as a

such

let

a

operators

operate on

as a

such

Nor'v consider

sequence

numbers

<

and let

<

k,

cardinal

### functions,

r,vith respect to

1. e.

### (1) A"(L;,"u):8;,r, l<i, j3no, l<h<I<.

Furthermote, we suppose

let

<

u.nivariate

respect to

{A,i

<l?,

i.e.

h,

and suppose

Thus

can construct

set

associated

sets

h'

P',"u(f)

we construct

set

associated

Âí

anð'

absorPtive, i.e.

<

we assume

sets

commute

B(S) :

Thus

follows

P'n,,

. .

P'|,r, commute

generate

connoN

### [4j).

Special" elements

PirP'1,¡,

P'-,P'; j(f

### ä É,

A':,( f ) Lí,,,(x) Lü,,, ¡(v) .

### obviously, the product projectors enjoy the following

interpolation properties :

AíA'|,(PL P'Jj(.f))

(e)

%i,

### | 3l/rl 1 M¡, 7 < i, I

SK).

(3)

5 A BOOLEAN METHOD IN BIVARIATE INTERPOLATION 39

.)Õ F, J. DELVOS and H, POSDORF 4

DEFTNTTToN

### L

The generøIized..B'iermønn

d.efined. as

tke Booleøtø sum

tke þrojectors

Px

### :

Pi,nPi,K @ P:,^P';.o_, @

pL,r_rp,i," @

(,Note

Roolean

### súÍr of two

commuting projectors

and

is

defined

@

### . -The

generalized, Bierø,t,ønn þrojector

møxírnø|, sittce

### is the

rnøximal elentent of tlte sublølt,ice generøted,by

r

enjoys

rHÞoREM 7. þ-ov

B(S), the

g

interþol,øtes

the sense that

(1

=

=,/Tt<+t-,.

Proof

The

### lattice

theoretical construction

Po yields

Thus

### the

interpolation properties

Boolean sum

### from the

corresponding properties

prôjectors

(see (e)).

Our

objective is

d.erive an

### explicit

expression of the projector

sumsof

R.

T,EMMA

generalized,

following

reþresentøtion

aøl,id. :

p,l,x-,

Proof

representation

The general case

### can be proved by induction ("f.

DÐr,vos-posDoRF [3]).

### Pu:

P'*^P'i,"@ P:,,P'i,@

P',,,Pi,,) @

Pi,,Pi,,

P;,,

P:," P'r:,,

P',,,Lt'/,,

P",,P'1,,

P:,"P',i" 1-

P;,P'i',

THEoREM

### 2. Let.f =

B(S), then the general'ized' Biermønn 'ínterþolatnt

of

### f

has the following reþresentat'ion

.Il-s s krll

"K-t

### (11) P*(fl(x,y) :ÐD D Ð AiAiU)Lri(x,y),

r0 r=0 i:ttr1_l j =nyç_1_s*1

uhere L¿,¡(x,

d'enote

cørd:ina.l,

,4j¡

giaenby

L;,¡(x,

(6)

### the

representation formula (10) yields

R-l nt+l "R-t

K-l fr, nK-,

/:1 i:1 j:l

### Thus

rve obtain

K-l nr-Fl K-l n(-s (" \

t:0 i:t\ll

s:/ i:,tff-l-s+l

\r:/ )

### - iì ,:þ,*,Ð ,:;Ð; ,*'AiA';(Í)(þ''""o' r;"'*-lv)):

K- l nt*l K-l nr-s / s \

### : Ð

t:0 i:n +l s:t j:l¡<-l -s+1

\¿:t

### Li,'*-,{r)J -

K-2 nttt K-l nr-s / s \

### -D D D D AíAí(f)1.D.ri,,,1*)L'i,,*-,g)l:

7O ¿-or+t t:-r-¡1 ¡:rrrj-"+t \':/+l I

### - /-r ¿-r I

t:O i-n¡+,l 5:7 ¡:r¡ç_1-"{1

L't,,,(x)

s-0 /:0

Ai

### {Ð rr,,,,., çx)L,i,,*_,u) -

i-nrll i-nç-t-s*l

, -''n t(x)

ì

### l'

I(-l s

(4)

40 F, J. DELVOS and H. POSDORF

o 7 A BOOLEÄN METIIOD IN BIVARIATE INTERPOLATION 41

case we have

application

Theo-

rern

yields:

.and

gre

now merely have

prove

given

functions.

Theorem 1

for

B(S)

=

r

nu.

Then

I nus

Ln,t

rvhich implies

A:

@k,ù

A,;(,i)

8¿,¿ 8,;¿.

proves

theorem.

considet

### r.oy in

greater detail the_two cases. nt

â"ttd, %,

casõ we have

### N:1(. ei, ãïpii"rtion

ot Theorem 2 yields :

### Þ^ È or*,

Aío-, (f)L,+t,w-,(x, y),

s:0 /:0

s

Lr+t,N-s (*,

### y) :Ðtl*r,;+t

(x)L'io-,,*-, (y)

i:t1-l

Note

### that

the well known bivariate

and

### the

Biermann form.ula.

^

ortainea

### -ny ;p;"irli;'i'ö tîe

operators and cardinal

ãs-follows :

!

h',

### (ii)

Biermann formula:

N.

L|u (x)

O

x¡) I

x¡)

i+i

L(,n (y)

ro

J+¿

\ _/1V

I(-1 s 2/+2

:2(K-s) -

I

A',j (Í)L;,¡

y),

2K -2s (13)

tj

s

L,,¡ (x,

@)Li,z¡r-ub)

### -,Ð,

Li,2¡ (x)Li,21v-n U),

(13)

folloi,ving

easily

derived:

(x,

Ltv+r1,2(K-s)-r

AL,+,

Y)

AL,+t

reduced

### Hermite

interpolation schemes

þ?oposed.

F'or

purpose

1]) :

Y)

### the

corresponding cardinal

are

LL¡-,,"u @)

I-î¡,rn (x)

-',"0

x)'

functions

and

### L'í¡,*(y)

are d.efined analogously.

### F;; th" á;ii"otiä"'äf ihc

cardiäaf iunctions see pHrrrrrrps [7.]).

F'rom

' '

### .,

(17) we now obtain the following

1HE9REM

### 3.

The reþresentation forntul,a

### for

the red'uced' Herm'ite inter-

of

gíuen bY :

K-l s

P

Ð, ^ {D

### ;Di

-t -""/ ( 1, 1 ) (

t ¡

t

### -' !'

L z, + t,,to -o - r ( 1

y)

Dr,Drc-r-y(1,

L",+r21n-"¡-r(1

0)

### l< j<h, 1<h.sI{)

(5)

42 F. J. DELVOS ônd t{. POSDORF

B

### I

A BOOLEAN METIIOD IN BIVARIATE INTERPOLATION 43

witlt

application

Lemma

reduced

yields THEoREM

Let

B(S):C2K+2(S),

Then tloe

remøind,ev

### P*(Í)

o"f tlte reduced' Herm'ite 'interþolant

g'iuen by

"

o?," "f

vo)

### r

s

Lr,) r,"(,, ,¡-r(ø,

LL.t.t,zQ+tt(x)

r,rto_nU)

### - -,Ð,

LL, +r,u(x) Lís, _,t _ r,rv, _rt(r)

### 2.

Rcrnaindel r,cpresentation,

sectiolr we

derive a

represe'tation

### formula

for gerreralized Bicrmann projector

anð.

this

,"ao"ed Her-

interpolation.

Lernma 1

obtain the

### follolvirg

reprcsentation formula of

(PL,*,

pi,,) p1,,,-,

and

wher.e

denotes

### the

zero_projector and

denotes

### the

identity-projector.

## +l ffi

DTD"rrc-'tr(x,,

,,*1(*

1)/+t yK-t(y

:

Pz(t+tl D1,""'

Q?

t))t(z(K

r))l

Proof

Using

estimates

interpolation

theorem

proved' bv

K-l K-1

t:l r:O

### which is

an immediate conseqllence

Lemma 2.

TiÐI,rr,rA

th,e retnø,inder

reþresentøt,ion

### fonnulø

'is uþ_rojector o.f

d.enoted.

ttte fotlo-

ctl,id, :

LI< 1)

-

K I(_l

### Ðo1,o,,,_, - Ð p;,*rr,;o_, 3.

Concluding remarhs

a recent

### papef

ryATKINS and LaNCaSrnn [9]. extend-ed the method

red.uced

gaps

### in the finite

element construction of recfangular elements'

is the

forthco-

he e-íements of

extended Melkes

can

etalized

particular

construction

### of the

corresponding cardinal functions.

by

account

(18)

lemma

proved

P;,) Pü,,-,

K K

PL,)

APPENDIX

{Pi,,

P',,,) P'1, r,

K { e i,,

P'n,) Pi, *

### -, : As an

instance we present

inter

### multiple

node consisting o

and including

W

PL,)

### : r

(6)

44 F, J. DELVOS and lJ. POSDORF

10 1t A BOOLEAN METT{OD ]N BIVARTATE INTERPOLATION 45

### K-2

tll Biermann, O., Über näherungsaeise Kwbøluren. REFERENCES i:[/.orats1n. math. Phys. lLt 271-225

( 1e03).

l2lDavis, Pl., J., Interþolation and Aþþroxitnøtioø.Blaisdell Publishing Company, [on- don (1963).

[3] Delvos, F.J., Posdo¡f, Ir,, N-th ord'er blending. In: ,,Constructive Theory of Func- tions of Several Va¡iables". Ecls. : W. Schempp, K. Zeller. Irecture Notes in Mathematics, 671, 53-64 (1577).

[4] G o r d o n, W.J., Distributiae Lattices and the Aþþroxirnation of Mul'tiuøriate Functions.

Proc. Symp. Approximation with Special Emphasis on Spline Functions, (Madison, Wisc. 1969). Ed. : I.J. Schoenberg. Acaclemic Press, New-York, 1969,

pp.223-277.

tsl Gorclon, W. J., Blend,ing-Function Methods of Bi.uariate ønd Multiuariate Inlerþolation a.nd' Aþþrotrimøtion. SIAI|/,, J. Num. Anal., Vol. B, No' 1, 158-177 (1971)' [6] Melkes, F., Red,uced, Pieceuise Biuaviate Hermite Interþolation. Num. Math. l9' 326-

-s40 (1s72).

t7l P h i 11i p s, G. M., Exþli'cit Forms for Cerlain Hermi'le Aþþroximations. BIT 13, 177 -180

(1s73).

tB]

### Stancu,

D. D., Theyemøind,er of certøinlinear øþþroximationformuløsin twoua,r¿o,bles.

SIA1\[, J. Num. Anal., Ser. B 1, 137-163 (1964).

[9]

D. S.,

### I,anc aster,

P', Sovne Farnil'ies

### of

Finite Elemenls. J. Inst.

lVlaths Applics 19, 385-397 (1977).

Dr.

J. DETTVOS

Lehvslukl,

Matherrxa,t'ih

### I

Uniuersity of Siegen

Höld.erlirstr. 3

D-5900 Siegen 21

West Germany

Dr.

### II.

POSDORF

Reohenzentrum Uniuersity of Bochum Universitätsstr. 150

D-4630 Bochum West Getmany

Fig. 1

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