Math. Zeitschr. 106, 170-174 (1968)

On a Generalized Bernstein Polynomial of Jakimovski and Leviatan

BRUCE WOOD*

1. Introduction

The Bernstein Polynomial defined by

B~(f, x)= x"(1-x)m-"f m '

where 0_<x_< 1 and the domain of f includes [0, 1], has been generalized in many ways. In a recent paper [1] Jakimovski and Leviatan have given a generalization L,, (f, x) which is generated by a sequence of functions, each of which is analytic on A = {z: [z[ < 1 }. Under certain conditions on the generating sequence, they establish results concerning the pointwise and uniform con- vergence of {Lm(f, x)} to f(x) when f is a real-valued function of the real variable x (see [I], Theorem 2.1).

It is the purpose of this note to consider the uniform convergence of {Lm (f, z)} to f(z) when f is a complex-valued function of the complex variable z. When f is a certain analytic function and the generating sequence satisfies rather general conditions, it will be shown in section 2 that {L~ (f, z)} converges to f(z) uniformly on closed subsets of A = {z: [z[ < 1}.

In the sequel let ek (z) = z k for k = 0, 1, 2 , . . . .

2. The Operators and a Convergence Theorem

Let {g,,} be a sequence of functions, each of which is analytic fin the disk I z[ < R, R > 1, and such that {gm} converges to a function g uniformly in some disk [z[ < r for some r, 1 < r < R. Define the polynomials {~(m) (x)} by the equation

gm(u)(l +u) x= n = O

With every funct ionf defined on [0, 11 associate for m= 1, 2, 3, ... the operator

L,.(f, z) 1 ~ ( - _, ~ m _ . . - n - for z~z], - - 1 ) m - n s ( m ) ( 1)z"(1-z ) ' - " f g(z-1).:o

For gm(U)=g(u)=--l, 0<Re(z )<1 and hn(z)=0, the operator Lm(fz ) is the m-th. Bernstein polynomial B m ( f z).

* This research has been supported in part by a National Science Foundation Traineeship. This paper is a portion of the author's doctoral dissertation written at Lehigh University in

1966-67 under the direction of Professor J. P. King,

On a Generalized Bernstein Polynomial

Theorem 2.1. Let the series ~ ak be absolutely convergent. Let k = 0

f(z)-=- ~ ak Z k for z~A. k=0

Let A be a closed subset of A. Suppose

r > 2 and

Then g ( z - 1)~0 for z~A.

limoo L ~ (f, z )=f(z)

171

(2.1)

(2.2)

uniformly on A.

The proof of Theorem 2.1 depends upon the following lemmas.

Lemma 2.2. I f conditions (2.1) and (2.2) are satisfied then there exist con- stants Ks, s=0 , 1,'..., such that

IJ~ ) (e s, O) L p[ <Ks , (2.3)

for all nonnegative integers m and p.

Proof. Let o o

gin(Z)= ~ (-- 1) n amn Z n for m~Z. n = 0

It follows from Lemma4.4 of [1] that

X S m

Lm (es, x ) - g (x-- 1) n~o am n (1 -- X)"

1 ~s ~ Qski(x ) -q g ( x - 1 ) k = l i :o ,~i

m k n . . . ( n - i + l ) a , , ~ ( l - x ) n-i,

where 0___x_<l and the (2ski(X) are polynomials in x which do not depend on m. Let

~ ( z ) = 1 m g ( z - 1 ~ , =~o amn(1- z)"'

1 h(z ) - g(z) and film(Z)= n . . . ( n - - i + l ) % , ( 1 - - z ) ~-', i=0, 1 . . . . . s.

n = i

Then am(Z) is analytic on zJ by (2.2). From (2.1), the definition of {gin} and an easy computation, it follows that

limo ~m(Z)= 1 uniformly on zl.

Thus there exists a constant H such that I~m (t) l _-< H for I tl = 1 and m e Z. Also,

d i (0. i 4: s --'xSlx=~ i ~ s (, i=s.

172 B. W o o d :

Therefore /~) (e~, O)

p!

and

1 a/p a

= _~,~.-'(0)

( p - st!

=IT~,~_ =~ ~ ~(t)t ~-'~+ ~dt 1

< H ,

1 d v r ,t~t

P * ~ 1 d p -J

~-~--o\j/ = ~=omk

d p

pt

h~JJ(-1) ~ 1 1 dP-i r176 + ~ j! ~ (p_j)! dx~-~ ,,~k~ , j = o k = l i= �9

p -- i d I /

a x i

for all m and p in Z. Now s is fixed and Qs~i(x) is a polynomial in x which depends on s. Also,

g~)(O)(z- fl~(z)--- ~ n ( n - 1)... ( n - i + 1)

, = i n !

It follows from (2.1), the definition of {gin} and an easy computation ~'~hat

l im/~ (z )=g0~(z - 1 ) , i = 0 , t , 2 . . . . , s

uniformly on zi. Thus, by a similar calculation there exist constants H~, i= 0, 1, ..., s, such that

i (P-J)~

for all nonnegative integers p - j and m. Consider the series

Jh(J~( - 1)1

j = 0

First, g(z) is analytic on f l= {z: !z+ II < 1} by (2.1). Next, Iz+ 1/< 1 implies that g (z),t:0 by (2.2). Thus h(z) is analytic on (L Therefore

,i=o J! = j=o J! I - 2 - ( - l ) J ; < ~ 1 7 6

on a Generalized Bernstein Polynomial 173

Finally IS~)(e.O) N H + ~ fh(J)(- 1)1 s 1 k

j=O J"

for all m and p in Z.

The next two lemmas are elementary and the proofs are omitted.

L e m m a 2.3.

( - 1)",-" :~m)_,,(- n - 1) = k_~ ~ ; k a m k . (2.4)

L e m m a 2.4. Let z~A. Then /) k

for all nonnegative integers i, n and k.

L e m m a 2.5. Let I I - z { < r , (2.6)

zEA, (2.7) and

g(z- 1),0. (2.8) Then there exists a constant B such that

[L",(ei,z)L<B for all m and i in Z.

Proof It follows from (2.4), (2.5) and (2.7) that

1 m " - " / m - k , { } g~)(0) (~_)i ~oZ"(1 - z)",-" 2 ( - 1 )k -~ - - . r ILm(ei, z)I=lg(z_l)[ , k : O \ n /

1 ~ Ig~)(o)l < - 2-' k! II-zlk.. = I f ( z - 1)l k=o

It is easy to see, using (2.6) and the definition of g",, that

lim ~ Ig~)(~ [1-zlk- - ~---Ig(~'(0)l [1-zL k=O k! ",~oo k:0 k!

This completes the proof.

L e m m a 2.6. Let (2.1) and (2.2) be satisfied. Then

~i+mooLm(ei, z)= z' for each ieZ and each zeA.

Proof It follows from Lemma 2.2 that there exist constants Ki, i = 0, 1, 2, . . . , such that

I t~ (e , , 0) 1 - - k! ---<Ki

t74 B. Wood: On a Generalized Bernstein Polynomial

for all nonnegative integers m and k. Thus the series

g~(ei, O) zk ~=o k!

is uniformly convergent in m for each i e Z and each zezl. By Lemma 4.5 of [13,

E~(e,, O) {1, k = i ,!im k! - O, k:4= i "

Thus

tim Lm (ei, z) = lim k ! m -'+ ~ I"1 ~ ~ k : O

Z i

for each i e Z and each zeA .

Proof of Theorem 2.1. For each m and each zez~.

Lm(f, z)= ~, ak Lm(ek, Z). (2.9) k=0

Lemmas 2.5, 2.6 and the absolute convergence of ~ a~ now imply that {L,,(f, z)} k=0

converges to f ( z ) for each fixed zeA. If follows from (2.9), the absolute con-

~Lm~f, )} is vergence of a k and the proof of Lemma 2.5 that the sequence s , k=0

uniformly bounded on closed subsets of A. Also Lm(f, ) is analytic on A for each m. The result now follows from Vitalrs Theorem.

Reference 1. Jakimovski, A., and D. Leviatan: Generalized Bernstein polynomials. Math. Z. 93, 416-~,26

(1966).

Dr. Bruce Wood Lehigh University and The University of Arizona Department of Mathematics Tucson, Arizona 85721 (USA)

(Received April 5, 1968)