APPROXIMATION OF A MONOTONE FUNCTION
BY BERNSTEm POLYNOMIALS
by
Wassily Hoeffding
University of North Carolina
Institute of Statistics Mimeo Series No. 581
May 1968
This research was supported by the Mathematics Division
of the Air Force Office of Scientific Research Grant No.
AF-AFOSR-68-1415.
DEPARTMENT OF STATISTICS
University of North Carolina
Chapel Hill, N. C.
APPROXIMA.TION OF A MONOTONE FUNCTION
BY BERNSTEIN POLYNOMIALS*
For a real-valued function f on [O,lJ the Bernstein polynomial of order n
is defined as B;(X) = ~=O f(i/n)(~) xi (1 - x)n-i.
For f Lebesgue-integrable
on (0,1) we consider the polynomials P;(x) =~ I{+l (x), where F(x) =
~xf(Y)dY.
Explicitly (see Lorentz [2J, Chapter II),
n
(1)
L
P; (x) =
(n+l)
i=O
r
(i+l)/(n+l)
f(y)dy
(~)
xi (l_X)n-i •
i/(n+l)
In this note the following result is proved.
There is a numerical constant C such that if f is monotone and
Theorem.
Lebesgue integrable on (0,1), then
I P;(x)
- f(x) Idx
~
1
Cn -i
JI
f(x) - a Ix-, (l-x) -l dx
o
for n = 1,2, ••• , where a is an arbitrary real number.
Remark 1.
The theorem implies that if F is convex (or concave) and
absolutely continuous on [O,lJ, and F' (x) x-i(l-X) -i is integrable, then
var(O,l)(~ -
F) = o(n-i,.
(I am indebted to Professor G.G. Lorentz for this.
observation.)
Remark 2.
The theorem shows that if f is monotone and
-r.nis research was supported in part by the Mathematics Division of the
Air Force Office of Scientific Research.
2
1
J
!f(x) I x-l (I-x) -~dx <
00,
o
then
1
J Ip~(x)
(4)
- f(x) Idx = o(n-i).
o
By Lemma 1 below, (3) implies square integrability of the monotone fUnction f.
The converse does not hold.
The author does not know whether square inte-
grability is sufficient for
(4). The following two propositions show that
a slightly stronger condition than square integrability implies (3), and that
a slightly weaker condition is not enough for (4) to hold.
Proposition 1.
(5)
If f is monotone and
J
:r2(x) (log(l +
I rex) I
»2+8 dx < ..
o
for some 8 > 0, then (3) holds.
For every 8 > 0 there is a monotone fUnction f such that
Proposition 2.
J
1
r2(x) (log(l + !f(x)
(6)
I)
-1_8 dx <
00
o
and (4) does not hold.
The proof of the theorem is preceded by three lemmas.
Lemma 1.
{J
If f in monotone, then
l
1
(7)
o
r2(x)dx
y~
J
1
!f(x) !x- l (l-x)-l dx.
0
We may assume that f in nondecreasing.
o~
Then there is a number b,
b ~ 1, such that f(x) ~ 0 for x < b and f(x)
2: 0 for x > b. If
3
b < x < Y < 1, then f(x) (l-x) -i S fey) (l-y) -;, hence
1
s
f(x)(l-x) i
J
J
1
J
f(y)(l-y) -;dY S
x
f(y)y -'i(l_y) -tdy.
Therefore
b
1 1 1
J
:e2(x)dx
=
b
b
{J If(x) Ix-i(l-X) -~
b
1
An analogous inequality holds with
(7).
imply
~~
f(x) (l-x) -tf(x) (l-x)
J
b
replaced by
b
Lemma 2.
J.
r.
The two inequalities
0
There is a numerical constant C such that if f is nondecreasl
ing,
n-l
(8)
L{f(~:i)
-
f(n~l)
}
(~)i
1
(1-
~)t s C1J If(x) IX-;(l-X)-iwe.
i=l
0
Denote the sum on the left of (8) by S and let hex)
n
Sn = Ii=l f(i/(n+l) (h«i-l)/n) - h(i/n»).
Ih r (x) I
1
~ ~-l(l-X) -~
= xi(l_x)t.
It is easy to show (using
that Ih( (i-l)/n)-h(i/n)l
S n -l(n~l) -;(1_ n~l) -t ,
-< i -< n.
Hence
(9)
Sn
~ n-1 ~=l If(n~l)l(n~l) -;
(1 -
n~l) -to
Due to the m:monicity of f, (8) is readily obtained from (9).
Lemma 3.
For 0 < x < 1,
n
(10)
l5~)
x j (l_X)n- j
~
(*) -i (1 - *) -n+i xi(l_x)n-i, nx :s i :s n-l,
j=i
L
i
(11)
j=O
(~)
xj(l-x)n- j
:s (~)-i(l
-
~)-n+i
Then
xi(l_X)n-i, 1:S i
S
nx.
4
Inequalities (10) and (11) follow, for instance, from Theorem 1 in [1],
where further references are given.
Proof of the Theorem.
It is sufficient to prove (2) for f nondecreasing
and a = 0 (since the left side does not change if f is replaced by f + const).
Let f i = f(i/(n+1», 1 ~ i ~ n. Since f is nondecreasing,
n-l
1
n
i
P~(x) ~ Lfi+1 (~) x (l_X)n-i + (n+1)
f(y)dy x
i=O
n/(n+l)
n-l
n
J
= fl +
I
(fi +l - f i )
i=l
I
(~)xj(l-X)n-j-fnXn+(n+l)
j=i
1
J
n/(n+l)
If we majorize the sum ~=i by 1 for i ~ [nx] and apply (10) to the terms
wi th i ~ [nx] + 1, we obtain
n-l
f(
'\
(
)(i)-i(
i)-n+i i(
)n-i
Pn x) - f[nx]+l ~ ~
fi+l-f i
1x I-x
i=[nx]+l
n
(12)
n
r
1
- f
n
xn + (n+1)
n
f(y)dy x .
n/(n+l)
and, applying (11) to the terms with i
pf(x)_ f[ ] >
n
nx +1 (13)
_[~
L
i=l
~
[nxl, thAt.
_f.)(~)-i(l_ !.)-n+ixi(l_x)l•
n
(f.
J.+l J. n
l/(n+l)
+ (n+l)
J
o
f(y)dy (l_x)n.
.A._
n
f(y)dy x •
f (:i.-x{
1
5
Note that the sum o'f the last two terms in (12) is nonnegative, so that
the right side o'f (12) is nonnegative.
nonpositive.
Similarly, the right side o'f (13) is
Hence it 'follows from (12) and (13) that
n-1
'\ (f + -'f 1toi)-i( 1- i)-n+i
n xi( 1-x)n-i
IPnf( x ) -f[nxJ+1 I ~ ~
i 1 i
)n
+ f 1 (1-x
i=l
(14)
1/(n+1)
J
-(n+1)
o
1
J
n
'f(y)dy(l-X)n-'f x +(n+1)
'f(y)dy x
n
n/(n+1)
By Sterling t s formula,
1
(15)
J
xi (l_X)n-idx = i! (~:i
: ~ c2n-i(~)i+l(1-*)n-i+i,
1
~
i
~
n-1,
o
where C is a numerical constant. Integrating both sides of (14), applying
2
(15), and noting that 'f ~ f n, we obtain
1
1
J IP~(x)
o
-
i
'f[nxJ+l1dx ~ c 2n1/(n+1)
('fi+1-fi)(~)i(1- ~)i
-J
1
1
l'f [nx]+l • 'f(x) Idx
o
~J ('f(:~1)
{l
- 'f(:1)}dx
0 1'.j(n+1)
n+1
=n
'f(x)dx -
n (n+1)
(18)
'f(y)dy.
n/(n+1)
1
J
J
'f(y)dy +
o
(17)
I
i=l
(16)
Also,
n-1
J
..
d(x)dx
J' .
0
r
1
+ 3{
'f(X)dx n!<n+1)
l/(n+1)
J
o
'f (x) dx}
n
6
By Schwarz's inequality, each of the last two integrals is
~
l
{J :e2(x)dxJ· It now follows from (18) and Lemmas 1 and 2
1
n-l
that in-
o
equality (2) holds with C
= CI C2+ 6.
Proof of Proposition 1.
Let 8
>
For definiteness assume that f is nondecreasing'
0,
g(u) = (1+U)
2
..2+&
(log(l+u) J
•
The function g is positive, increasing and convex on
K
J
=
g( If'(x)
I>
dx
is finite under condition (5).
(O,~),
and
Let b s [0,1] be such
o
that f(x) < 0 for x < b and f(x) > 0 for x > b.
Ir(x) I
= --rex)
implies
J
If(y) Idy)
~
x
l
x--
o
J
g( If(y) I)dy
b
b
If(x) IX-t(l-X)-iax
o
~J
b
The analogous inequality with
J
If(x) Ix-l(l-X)-iax
o
and
g-l(Kx- l )x-1-(l_X)-iax.
0
1
and Kx-l replaced by
J
and K(l_x)-l is
b
Hence we have
1
J
1< g-l(Kx- l )
-
o
obtained in a similar way_
~ x-IK.
0
Since g is nondecreasing, we obtain If(x)
J
Then
::: x-I !Ir(y) , ; Since g is convex, Jensen's inequality
o
x
g(x-1
Let 0 < x < b.
~
1
2
J
g-l(Kx-l)x-i(l_X)-twc.
0
Since g-l is increasing and positive, the integral on the right converges if
1
J
o
g-lex-l)x-idx does, 'Which is equivalent to the convergence of
7
00
,(U g(u) -3/2 gf (u)du.
The latter integral converges since the integrand is
1
bounded by const. u -l( log u )-l--!a
2.
Proof of Proposition 2.
First note that if f is nondecreasing, the
integrals in (1) are nondecreasing as i increases.
r
Hence
1
p~ (x) :s
(n+l)
f(y)dy = cn' say, for 0
n/(n+l)
f(a -) < c < f(a +).
n
n
- n1
(19)j1P~(X)-f(X) Idx
o
:s x :s 1.
Then f(x) > pf(x) for a < x < 1.
- n
n
1
Hence
1
?J (f(X)-P~(x)}dx ?J
= (l-x)-l(-lOg (l_~}r,
r
1
f(x)dx-(l-an)(n+l)
an'ln
Now let f(x)
Choose an so that
J
f(x)dx.
n/(n+l)
> o. Clearly f is nondecreasing
With f so defined, the integral in (6) converges if
on (0,1).
1
J( )
I-x -1 (- log ()}2r-l-&
I-x
dx converges, which is true for 2r < 8.
As
l
e .. 0+,
1
J
f(x)dx - 2el
(-
log e)r.
l-e
- 2l l (log n)r as n " 0 0 . A straightforward
n
2
the root a of f(a ) = c satisfies 1 - a - c- (10g
n
n
n
n
n
Hence c
calculation shows that
c2 )2r - in-I.
n
These
facts im;ply that the last lower bound in (19) is asymptotically equal to
in -l
(log n)r, Which completes the proof.
8
REFERENCES
1.
Wassily Hoeffding, Probability inequalities for sums of bounded random
variables, J. Amer. Statist. Assoc. 58 (1963), 13-30.
2.
G.G. Lorentz,
Bernstein polynomials,
Toronto, 1953.
University of Toronto Press,