TWO THEOREMS IN THE APPROXIMATION OF FUNCTIONS OF TWO VARIABLES BY POLYNOMIALS OF THE BERNSTEIN TYPE EUR 505 e

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This reprint is intended for restricted distribution only. It reproduces, by kind permission of the publisher, an article from "SIMON STEVIN, WIS­ EN NATUURKUNDIG TIJDSCHRIFT", 36* Jaarg., Afl. III ­ 1963. For further copies please apply to Prof. J. Bilo ­ 164, Oude Brusselse Weg — Gentbrugge (België).

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Questo estratto è destinato esclusivamente ad una diffusione limitata. Esso è stato riprodotto, per gentile concessione dell'Editore, da SIMON STEVIN, WIS­ EN NATUURKUNDIG TIJD­ SCHRIFT», 36e Jaarg., Afl. Ill ­ 1963. Ulteriori copie dell'articolo

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Deze overdruk is slechts voor beperkte verspreiding bestemd. Het artikel is met welwillende toestemming van de uitgever overgenomen uit „SIMON STEVIN, WIS­ EN NATUURKUNDIG TIJD­ SCHRIFT", 36e Jaarg., Afl. III ­ 1963. Meer exemplaren kunnen

E U R 5 0 5 . e

R E P R I N T

T W O T H E O R I E S I N T H E A P P R O X I M A T I O N O F FUNCTIONS O F TWO V A R I A B L E S BY POLYNOMIALS O F T H E B E R N S T E I N ­ T Y P E by P.V. LAMBERT, (university of Lonvain)

European Atomic Energy Community ­ EURATOM. Joint Nuclear Research Centre.

Ispra Establishment.

Scientific Information Processing Centre (CETIS).

Work performed under E u r a t o m contract No. 011­61­1 CETB with the University of Louvain.

Reprinted from "Simon Stevin ­ Wis­ en Natuurkundig Tijdschrift", Vol. 36, February 1963.

This paper studies the approximation of functions of two variables in the triangle

S = [{χ,y); o < χ, o ^ y, χ + y < 1]

E U R 5 0 5 . e

R E P R I N T

TWO T H E O R I E S IN T H E A P P R O X I M A T I O N O F FUNCTIONS O F TWO VARIABLES BY POLYNOMIALS O F T H E B E R N S T E I N ­ T Æ P E by P.V. LAMBERT, (university of Louvain)

European Atomic Energy Community ­ EURATOM. Joint Nuclear Research Centre.

Ispra Establishment.

Scientific Information Processing Centre (CETIS).

Work performed under Euratom contract No. 011­61­1 CETB with the University of Louvain.

Reprinted from "Simon Stevin ­ Wis­ en Natuurkundig Tijdschrift", Vol. 36, February 1963.

This paper studies the approximation of functions of two variables in the triangle

S = [(x,y) ; o < x, o ^ y, χ ­f­ y ^ 1]

E U R 5 0 5 . e

R E P R I N T

TWO T H E O R I E S IN T H E A P P R O X I M A T I O N O F FUNCTIONS O F TWO V A R I A B L E S BY POLYNOMIALS O F T H E B E R N S T E I N ­ T Y P E by P.V. LAMBERT, (university of Louvain)

European Atomic Energy Community ­ EURATOM. Joint Nuclear Research Centre.

Ispra Establishment.

Scientific Information Processing Centre (CETIS).

Work performed under Euratom contract No. 011­61­1 CETB with the University of Louvain.

Reprinted from "Simon Stevin ­ Wis­ en Natuurkundig Tijdschrift", Vol. 36, February 1963.

This paper studies the approximation of functions of two variables in the triangle

by polynomials constructed by assigning t h e trinomial weights

Ml

Pn,k,m (x,y) = %kym (i—x-^y)n-k~m

kl ml (n—h—m)\

to the functionvalues a t the points

(

this being done for each fixed n = 1, 2

Using the C­norm, wc first establish an estimation formula of the approxima­ tion degree in function of n and afterwards prove a n asymptolic relation for functions with 2nd order differential.

by polynomials constructed by assigning the trinomial weights

n\

Pn.k.m (x,y) = Xkym (i—x—y)n-k-m

kl ml (n—h—m)\

t o the functionvalues a t the points

Ik m\ ( k = o, l,..., n; m = o, 1,..., n;

\n n j { k + m < n ;

this being done for each fixed n = 1 , 2 ,

Using the C­norm, we first establish an estimation formula of the approxima­ tion degree in function of n and afterwards prove an asymptolic relation for functions with 2nd order differential.

by polynomials constructed by assigning t h e trinomial weights

ni

Pn.k.m [x,y) = xkym (i—x—y)n-k-m

M ml (n—h—m)\

to the functionvalues a t the points

(k m\ Í k = o, l,..., n; m = ο, I,..., n;

\n n ] [ k ­\­ m < n;

this being done for each fixed n = 1, '2,

SIMON STEVIN, Wis­ en Natuurkundig Tijdschrift

36e Jaargang, Aflevering III (Februari 1963)

TWO THEOREMS IN THE APPROXIMATION OF FUNCTIONS OF TWO VARIABLES BY POLYNOMIALS

OF THE BERNSTEIN­TYPE (*)

Pol V. LAMBERT

(University of Louvain, Seminarium Prof. Florin)

I. INTRODUCTION

This paper studies the approximation of functions of two varia­ bles in the triangle

S = {{χ, y); 0 ^ χ, 0 «s ν, χ + y ^ 1 } by two­dimensional polynomials of the Bernstein­type.

For every positive integer n, we consider the trinomial weights defined for every (x, y) in S by

( " » ­ " ■ ^ t o K , " ­ ] , ­ , ) ! ' ­ " ^ · " ­ 1 ­ ' ' · " ­

Í1·1·1) j /c = o, 1, ...,n; m = 0, 1, . . . , « ; k + m^ n.

It is known that for every point (x, v) in S

(1.1.2) 0 < pn,k.m(x, V) < 1 , 2 Pn,k,m(x,y) = 1·

This probability distribution over the pointsi — , — J of S has similar convergence propeities (for /? ­> x ) as the one­dimensional Bernoulli distribution given for every χ in [0, 1] by

(1.1.3) Ρ « ^ ) = ^ ( π % ! · * * ­( 1­ *) η~ *

cfr [1] pp. 3­4.

C1) This research was sponsored by the European Commission for Atomic

Energy under contract 011/61/1 DOB.

Over the set of functions ƒ defined and finite in S we define

for every positive integer n the linear operator Bn such that

(1.1.4) BnUix, y)] = 2 f(l7> τ) · />«.*.»(*. y) ■

A convergence theorem (cfr. theorem 3.1) is given in [1] p. 51. Our aim is to prove first an estimation formula of the approximation degree and afterwards an asymptotic relation for functions with 2n d derivatives which generalizes Woronowskaja's result to the

two­dimensional case.(*)

II. LEMMAS

We need first some estimation formulae which are similar to the one­dimensional case because of equality­relations existing between the moments P„,s(x, y) and Qn,s(x, v) defined by

ÍPn,s(x,y) = 2 (k—nx)s.pn,k,m(x,y)

1 OsSfc+TOsSM

(2.0.1)

[QnAx,y)= 2 (m­ny)s.pn,k,m{x,y) 5 = 0,1,2,....

and the one­dimensional moments Tn,s(x) defined by

n

(2.0.2) Tn,s(x) = 2 ik­nxy . pn,k{x)\ 0 ^ χ ^ 1 .

k=0

We have indeed :

LEMMA 2.1

For every s = 0, 1, ..., P„.,s(x, y) is independent of v, Qn,s{x, y)

is independent of χ and :

(2.1.1) Pn,s(x, v) = Tn,s(x) ; Qnjx, y) = TnJy).

(*) This study was part of the author's dissertation requirements to acquire his academic degree at the University of Louvain, Belgium, and he expresses his deepest feelings of gratitude to Prof. H. FLORIN, Dr. M. NEUTS

and Dr. THE TJOE TIE for valuable comments.

Proof :

We only need to examine Pn,s(x, y). We have :

Pn,s(x,y) = 2 (k­nx)spn,k,m(x,y) =

Onk+mun

= > {k­nx)s . , , ­t ' jc­j . xk .

ΖΛ k\(n­k)\

■ ¿0m\(n­k­m)\ } K y)

But

n­k

Í "

m

■ (" k\.y™.(\­x­y)n­k­m = [y + ( l ­ x ­ v ) ] " ­f c =

This yields the result.

COROLLARIES :

We now recall (6) p. 15 in [1], i.e.

(2.2.1) 0 s: Tn,2s(x) < A(s) . ns ; 0 ^ χ s= 1 ; s = 0, 1, 2, ...,

Λ being a constant depending only on 5. So by (2.2.1) and lemma 2.1 follows

(2.2.2) 0< Pn,2s(x,y) < ¿(¿) · «s ; (x.jO e S; s = 0, 1, 2, ...,

and similar results for <2ra,2s(x, v)·

We now have the following estimation formula for every

δ > 0 and (x, v) in S :

(2.2.3) 2 Pn,k,m{x,}') < ^ ^ ; · (*) 0ssfc+m«M

\k/n—x\»õ

k

In order to prove this, we first notice that χ > δ implies

(k­ηχ) φ 0. Then by (1.1.2) and (2.2.2) we get :

2 Pn,k,m(x,y) = 2 (Ιτ — „χ\2ι ■ Pn,k,m(x,y)

|fc/ii­z|»<S \k/n­x\»õ

k

(*) |A://Í—Jc| means |—— x\.

1 v

^ n2s å2s ■ 2 , (k — nx)*' . Pn,k,m(x,y)

Otsk+man \k/n­x\»ô

< Pn,2s(x,y) A (s) . ns A (s)

^ n2s _ ¿2s ""­ n2s _ ¿25 ~ ns _ ¿2s ' Ci­Q·'1·

For 5 = 1 , since Pra,2(x, y) = P«,2(x) = nx(l — x), (2.2.3)

yields

(2.2.4) > Pn,k,m(x,y) < \ ,9 <

«2 ¿2 4„,52·

0<5Α.·+»«ίκ

Now for a fixed /c > 0, we can find a positive integer s and a

real number a. in the open interval (0, 1/2) satisfying the relation

5 ( 1 - 2 « ) = k. Then for δ = η~α (2.2.3) implies :

(2.2.5) V

^

ç J&_ = £ψ

0sík+m<n \k/n­x\sn­a

By starting from the moment On,2s(x, y) similar results can

of course be obtained where m and y take the place of k and x.

III. MAIN THEOREMS

We state first a known result ([1] p. 51).

THEOREM 3.1

Let ƒ and Bnf be defined in 5. Then

(3.1.1) lim Bn[f(x, y)] = f(x, y)

in every point (x, y) of S where ƒ is continuous. The convergence is uniform iff is continuous in S.

DEFINITIONS

1) For a function /'defined in S, we use the following oscillation (<5i > 0, ¿a > 0)

(3.1.2) ω(δχ,δ2)= sup \f(x',y')­f(x,y)\.

\x­x'\<õl,\y­y'\<õi (x,y)eS,(x',y')e,S

2) We now denote as in [1] p. 20 Λ(χι, X2, δ) as the greatest integer contained in |xi — X2I δ > 0. Similar definition for

¿(yi, y2, δ). We are now able to prove the following estimation theorem.

THEOREM 3.2

Let ƒ and Bnfbe defined in S. Then for every point (x, y) of S

we have :

(3.2.1) \Bn[f(x,y)} -f(x,y)\<¿. ω(«­ι/2,«­i/a)

Proof

We have by our definitions and by (1.1.2) (3.2.2) \Bn[f(x,y)]­f(x,y)\*

Σ

-Μ τ ' i r ) ~

f(

'

ï ■ P

>

>™(

' y)

0s=fc+m=s?i \

Any interval k -, χ can be divided into / ftx,<5ii i subintervals the length of each being less than δι. Let

λ ft x, ­5:, ) ! Ν and call xjv­i, Xjv­2, ..., xi the interior subdivision points. In the same way let

/W-pjMa)

ί

and call yni-i, yivr-z, ■ .-,yi, the interior subdivision points of an

' m

interval

/; y . Let us now take te case N^ M. (The case Ν < M

would be similar). We then have :

k m n,' n

f

_{(TT' TT'}

_{' ' "}

₊

+ \f(XN-l,yM-l) — f{XN-2,yM-2)\ + ... +

I AXN-M+I, yi) — f(xN-M, y)\ + \f{xN-M,y) — f(xN-M-i,y)\ +

+ ... +

\f(xi,y)-Ax,y)\,

where some terms can vanish.

This means that in every case (/V> M or N < M) we have :

(3.2.3)

_{' Ί τ · τ ΐ "}

V

-

max A [­, x, oil .Æ \ λ I—,/ 7?7 y, δ2

+ 1

/

[%-,χ.δλ,

λΚ

,δ

Let Afc,m(x, y) = max (3.2.2) and (3.2.3) imply :

(3.2.4) \Bn[f(x, y)] ­ f(x, y)\ < ω(όι, ¿a) .

• ω((5ι,ό2)

. Then

XT _{[1 +λίι,τη(χ,γ)] ■ Pn,k,m(x,y)}

¡ζ ω ( a i , á2) .

< ω ( ¿ i , á2) ■

<

1 + 2 λ>ε,ιη(χ, y) ­ Pn,k,m{x, f)

1 + ' ¿2 · Pn,k,m(x, y) +

Osk+m^n 1

\k/n­x\i>õ1 m

ι V l " /

+ _O^sk+m^n / ãõ—^­ · Pn,k,m(x, y)

\m/n­y\»ô2

Ò2

< ω (ói, á2) .

ζ ω (ai, á2) .

1 Η rõ-íõ ■ Λι,2(*> jO + -^TT · Ô?1,2(X, ^ )

7 72á2 η2 δ2

2

4ηδ\ ' 4/1 ó | j

If we now let ái = á2 = ?7­1/2, we get (3.2.1), q.e.d.

Remark :

This theorem implies as a particular case the 2n d part of

theorem 3.1.

We now give the asymptotic formula mentioned in the intro­ duction and its proof.

THEOREM 3.3

Let ƒ be defined and bounded in S. In each point (x, y) of S

where ƒ has a 2nd-order differential we have

(3.3.1)

BnlAx, y)i

=

Αχ, y) +

\~

■

0

+

v ( l — v) d2f xy d2f Qn , ,· n

+ \ ■ ã­9 ■ ã a + ' w h e r e l l m 0« = °­

Ifall the 2W!Z­order partial derivatives off are continuous in S,

then :

lim ρΜ = 0, uniformly in S.

?¡.->co

Proof :

d2f d2f

By our assumptions . _ = . , and

.(3.3,) ^ . ï ) _ ^ .

Λ

+ (£_,). £ + ( * - , ) . £ +

+HH'-£f

+

ifc-')'-£

+

+ (Η·(γ');^

+

+

„ ( £ ,

£ , x j , ) . ( í ­

χ ) .

( £ ­ , ) ,

where

(3.3.3) lim ΦΙ = 0; lim Φ2 = 0; lim ç?3 = 0. k/n­>x m/n­>y fk/n­>x

{m/n^y

If we recall the identities

/ Pn.iix, y) = ßii,i(x, 7) = 0 ;

(3.3.4) j ¿ V ( x , 7) = WJC(1 ­ x ) ; ß«,2(x, y) = ny(l ­y);

i (k — rix) . [m — ny) . pn,k,m(x, y) = —nxy

O í í + f f i O l

it follows from (1.1.4) and (3.3.2) that :

(3.3.5)

B

[f(x,y))

= /(χ,

y) +

\~

. 0

+

y{\­y) d2f xy d2f

+ \ .­~4 ■ ä­4­ + rn, where

2« oy2 n dxõy

(3.3.6) 2 Pn,k,m(x,y) ■

0 s k+ìn si n

ι-m \2 (m \

τ~η

-

[τ'

>η

+

Τ-ή-ίτ-ή·*

(f

- x j . <ρι ί

n J

m \'

—. χ, y)

+

Since ƒ and hence Bufare bounded in 5 and the partial deriva­

tives are finite at the point (x, y), we have \<pi\ ¡ξ H,i= 1,2,3,

k

uniformly when ­ and — vary in S. We also have

n n χ ^ 1 and m

\~h~

ε > 0, there is a δ > 0 depending only on ε and (x, y) such that

\<Pi\ < ε, t 1, 2, 3, whenever χ < á and /;; _{y < δ.} All

this and the identities (3.3.4) imply :

\rn\ < ­õ · [nx(l ­x) + ny{\ —y) + nxy] +

3 / / ■ Í 2 Pn,k,m(x,y) +

\0s;k+m^n Otk+msin

\k/n­x\»o \mln­y\isô

2 Pn,k,m(x,y)\

ak+m&n I

Let δ = n a, 0 < α < ­ , α fixed; then ε depends on n and

lim ε(«) = 0.

By now using (2.2.5) with k > 1 we get

\>'n\ < Φ ) _ _ Ι _6 / / C (g. /Q = 4 w

where lim C(») = 0. This proves the first part of the theorem.

m-*oo

When the 2ncl­order partial derivatives are continuous everywhere

in 5, and hence bounded, the number H can be chosen the same everywhere in S. On the other hand, given ε > 0, we can now also choose the same δ for all points (x, y) of S. This implies that the estimation of rn becomes uniform in S, and so proves the 2n d part

of the theorem.

General remarle

All the results of this paper can easily be generalized to func-tions of n variables (n — 1, 2, 3, ...).

June 1961

REFERENCES

f1] LORENTZ G.G., "Bernstein Polynomials", Toronto 1953.

[2] POPOVICIU T., «Sur l'approximation des fonctions convexes d'ordre

supérieur». Mathematica — 10 — 1935, pp. 49-54.

[3] WORONOWSKAJA E. W., «Détermination de la forme asymptotique

d'appro-ximation des fonctions par les polynômes de Bernstein». Comptes Rendus Acad. Sci. U.S.S.R. — 1932, pp. 79-85.