BERNSTEIN POLYNOMIALS FOR FUNCTIONS
OF TWO VARIABLES OF CLASS C<»
EDWARD H. KINGSLEY1
Introduction.
Let <¡>{x,y) be a continuous
function of the real
variables x and y, where x and y are in the closed region R: O^x^ 1,
O^y^l.
The Bernstein polynomial
Bmn(x, y) associated
with the
function 4>(x, y) is defined as
(1)
Bmn(x, y) = ¿j 2-, <t>[— ' —)K.p(x)\m,,
p=o 5=0
\ n
where X„,„(x) = C«lPx*(l -*)»-»,
m /
(y)
Xm,8(y) = Cm,qy"(i -y)m~<>.
A function <¡>{x,y) is said to be of class C(*>for x and y in i?, if the
partial derivatives of order 1,2, ■ ■ • , k of 4>(x, y) exist and are continuous. We shall use the notation
,(••.*-••)
$
=
dw<t>(x, y)
., ■. ^_-
and, for brevity, shall omit functional arguments
whenever possible.
It is the purpose of this paper to prove the
Theorem.
(t = 0, 1, • • • , *)
from expressions
If 4>(x, y) is of class C(*>for x and y in R, then
hm
Bmn
= <j>
mtn= »
and the convergence is uniform in R.
This theorem, for functions of one variable and for k = 0, was
proved by S. Bernstein
[l],2 and again for functions of one variable
but for arbitrary
k, the theorem was proved by S. Wigert [2]. The
process of extending the results of Bernstein and Wigert to functions
of two variables of class C(W introduces aspects which are of interest.
Preliminary
results.
We shall make use of the relations
n
(2)
2^ X„,p(x)= 1,
_
p-o
Received by the editors December 12, 1949.
1 This is part of a thesis written under the direction of Professor E. D. Hellinger for
the degree M.S. at Northwestern University, June 1949.
8 Numbers in brackets refer to the bibliography at the end of the paper.
64
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65
BERNSTEIN POLYNOMIALS
23 (nx —pyK,p(x) = nx(l — x)
(8)
[3, p. 152].
3>—0
If we define, for k^O, i = 0, 1,
= Z£H)
*,
Í+(*-«)
c^í.oCV-rt.^í
g+(*-*-|3)N
m
■
a=0 0=0
then, by mathematical
induction,
the following two lemmas can be
established.
Lemma 1. If O^i^k,
i^n, k^m, x and y in R, then the kth partial
derivatives of the Bernstein polynomials
B
n—i m—k+i
n\m\
(¿,fc—
i)
(1) are given by
. .
(n — i) \(m — k + i) ! p=o 4=0
Lemma 2. IfO^i^k,
0¿p¿n
—i, O^q^m —k+i, and if <j>(x,
y) is
of class C{k)for x and y in R, then there exist two real numbers £ = £(/>),
7=7(3) SU£h^t
«,*-<)
Ap.q
0<£<1,
=
0<7<1
and such that
,k-i)/p
1
{i,k-i)
/ + £i q + y(k — Ï)
. , . <f>
I
n'mk~l
n
\
m
)•
The next lemma is basic for the proof of the theorem,
Lemma 3. For fixed x and y in R and for fixed positive integers M
and N, let d be an arbitrary positive number and let a(p, q) be a quantity
dependent upon p and q and such that
(4)
I a(p, q)I á xi for
(5)
I a(p, q) I = 7T2for
P_ :S d
N
and
M
P_ > d and \y -
I
N
M
úd,
> d.
Furthermore, assume that it is possible to split off from a(p, q) terms
a'{p) independent of q or terms b'{q) independent of p, that is,
a(p, q) = a"(p, q) + a'(p) = b"(p, q) + b'(q)
such that
(6)
(7)
P ^ d and
I cf'ip) I g *3 for]
I a'(p, g) I á 7T4for
N
x—
N
ûd,
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y-
M
>d,
66
E. H. KINGSLEY
b"(p, q)\£n
(8)
for
X —
N
> d and
y-
M
-a,
y' M £d,
b'(q) | è ire for
(9)
P
[February
then
N
M
E E a(P<°)^N,p^M.q
p—0 s=0
(10)
g
Proof.
1T1 +
1T4 +
7T2(M + 2V)
7TS
8MNd*
4Md2
T6 +
T5
4Nd*
Consider the inequality
AT M
E 2 a(í> ?)Xar.pX.af,4
p=.oj—o
â
E
E
|x-p/W|Sd
+
(ID
£
SI
|z-j>/AT|>d
IV—s/Af|>ti
+
E
E
52 +
I «"(#.?)+ «'(#)I xWl,+ xj,..
| v—g/3f |>d
E
|i-p/iV|>d
= Í1 +
a(/".9) I Xjv.pXif.,
E
|l-p/AT|gd
+
I <*(/»>
9)IXiv.pXjf,«
||f-s/.W|_d
I*"(#,?)+»'(«)IXw.pXj,.,
|»-j/Jf|Si
Î3 +
Í4,
then
Si á
Tl
¿j
\x-p/N\Éd
N
¿-I
^N.p^M.q
(by (4))
|ji-a/Af|ád
M
(by (3))
è Ti ¿_i 2-i XjV.pXjlf.j = Tl
p=0 j=0
and hence SiHti.
The inequalities
in (5) imply the inequality
M\Mx - pY + N2(My - q)*
2M*N'di
> 1
and by (5) again
I a(p, q)\ úifi<
Vi
2M2NW
{M\Nx - pY + N\My - q)2\,
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67
BERNSTEIN POLYNOMIALS
thus
N
T2
52 =
M
¿ ¿ [M\Nx
2M2iV2d2pTo£5
T2
2MNd2
-p)2+
NKMy - ç)2}Xv.PX^.t
(by (3) and (2))
[Mï(1 - x) + Ny(l - y)]
and since
max [Ma;(l — x) + iVy(l — y)J, =->M + N
we have
t2(M + tf)
52 á
8MNd*
In similar fashion, using (6), (7), (8), and (9), we obtain
1T6
7T3
53 =
Collecting
5T4 +
B
(i,*—»)
4Md2
these estimates
Proof of the theorem.
*
54 á
7T6 +
in (11) we obtain
4iV<f2
(10).
By Lemmas 1 and 2, we have the inequality
(»■*—•)
n^imrk+i
I
n\m{
p=o s=o I
\
(n — i) \{m - k + i) \nimh-i)
(..*-<)(p + £i
? + t(¿ - *)\
\
m
n_¡ m-k+i
n
{
. ti)
p-o s=o I
(i,k-i)
— 4>
,p
\
+
."m— k+ «.a
/
£i
q +
}
_
m
n
(X, y) > Xn-t,pXm-t+t,
¿(¿
i)
)
= Si + S2
We first estimate S2. Let 0(i'*-°(x, y) =^(x, y), then since \p(x, y)
is continuous for x and y in R, it is uniformly continuous and
bounded on R. That is, for an arbitrary positive number €, there
exists a number ¿(e) >0 such that if \x —X\\ ^d(e)/2 and \y—yi\
i=d(e)/2, where Xi and yx are in R, then
(12)
| *(*!, yx) - *(*, y) | g e/6
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68
E. H. KINGSLEY
[February
and
(13)
| t(x, y) I è L<*>,
for x and y in R,
where Lik) = max *,y£JJ | \¡/(x, y) \.
If p, n, i are positive integers such that O^p^n
real number such that 0<£<1,
then
P + ti
(14)
n —i
and if m, q, k, i are positive integers such that
and y a real number such that 0 <y < 1, then
q + y(k - i)
(15)
—i, and if £ is a
m —k + i
m
<
0 ^ q ^ m —(k —i),
k —i
Now if x is fixed and p is such that O^p^n—
1 and |x —p/(n —i)\
èd(e)/2, then by (14)
x —
d{¿)
P+ &
and if we choose
iVlt > 2i/d(«) + *■
then for n>Nu
¿/« < d(e)/2
and
x —
(16)
In the same manner,
P+ Ü
if « > Ni«.
gd(e)
choose
2(¿ - i)
Mu >(*-»)
+
¿(e)
then for fixed y and 5 such that O^q^m—k+iand
^d(e)/2, we obtain from (15)
ç + y(k - i)
(17)
m
If we choose Xi=(p + ¡-i)/n, yi = (q+y(k
d(e)
—i))/m
(16), (17) we have
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|y —q/(m —k+i)\
if «>
Mu-
in (12), then from
'95']
p + H q + y(k —i)
(18)
if n>Nu,
69
BERNSTEIN POLYNOMIALS
\
) - H*,y)
n
m>Mu,
\x-(p+&)/n\£dÇe)/2,
\y-(q+y(k~i))/m\
Sd(«)/2.
If we now let:
An_,- indicate summation for all p such that \x —p/(n —i) £d(e)/2,
A£_4 indicate summation for all p such that \x—p/(n —i) >d(e)/2.
Am_i+i indicate summation
for all q suchthat
\y— q/(m —k+i)
èd(e)/2,
à'm-t+t indicate
summation
for all q such that
\y —q/(m~k
+ i)
>d(t)/2,
then S2 may be written
as
P + É* q + y(k- *)>
A-_,'
¿n-i
A-.L
Am-*+ ■J I
m
\
- lK*. y) Xn-f.j.Xm.t+i,,
P + ¿i g + t(* + £ s \*t*-±
¡»n-»"
*)\
m
ûm-*+i
— ${x, y) K-i,p\m-k+i,(
+S
an-»
p + £¿
E |#(i±
')
9 + y(* - »)>
f»
am-i+«
./* + a y\i) Xn_,-.pxm_(
— w I-'
(19)
\
+ £
X
mm-'
«
/I
y ) - iK*. y) ^n-i,P\m~k+i,t
+ L r LÄ+Ü,î±3ÎLiS)
- $ I x, -)
x„_f p"m—k+i,q
+ z z L(,,i±*i^>)
Ai
I'
»»_i .• A„_i..,;
ûm-t+i
\
m
/
- iK*>
y) Xn—ftpXm_m.jie.
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70
E. H. KINGSLEY
[February
If we use (13) and (18) and let
<p +
ÍP
+ £i q + y(k - i)
a(P, q) = Ir-(\
m
n
iK*.y),
)-
d = ¿(e)/2,
Xl = 7T4 = 7T6 = i/6,
X2 = T3 = X5 = 2Z/4),
then the hypotheses
52^-
e
of Lemma 3 are satisfied and (19) becomes
3¿<*> / 1
(-H[¿U]2 \n — i
+
To estímate
,),
m — k + i/
for n>Nu,m>Mu.
Si, we note that there exist two numbers
Nlf>i,
Mit
>k —i such that if n > N2t, m > Mit, then
n\m\
(» — i)\{m — k + i)\nimk~i
<
6L<"
Thus
SiS-^-E
ti-^'l
(p + & q + y{k-i)
E *(—
p-o
«-
»Î
I x„_,-,
phm~k+i
,q
n—£ m—fc+i
==— E
6 p_0
E Xn-i.pXm-jt+i,,
by (13)
5=0
and hence
Si á e/6.
Using these estimates
of Si and S2 we have
(20) \bL >-*<' I^-^ + ttttt;
-:+-nHm — k +
[d(e)Y
■\_n — i
for n>Nu,
Nu; m>Mu,
M2l.
Let
(21)
Nu > i +:
(22)
M3« > (* - f) +
18L<*>
and take
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ij
1951]
BERNSTEINPOLYNOMIALS
71
N. = max (Nu, N2„ Ntt),
M, = max (Mu, M2„ Ms,).
In (21) let »>iV« and such that
3¿<*>
[d(e)]>(n-i)
€
<—
6
and in (22) let m>M, and such that
3LW
[d(t)]2(m-
Thus, for n>N,
and m>M„
k+i)
t
< —
6
(20) becomes
IiSt4>- ¿^
I< «
and the theorem is proved.
Bibliography
1. S. Bernstein,Soc. Math. Charkow (2) vol. 13 (1912-1913).
2. S. Wigert, Sur l'approximation
par polynômes, des fonctions continues, Arkiv
för Mathematik, Astronomi och Fysik vol. 22B (1932) pp. 1-4.
3. D. V. Widder, The Laplace transform, Princeton, 1946.
Roosevelt
College
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