This Nsearch was supporrted in part by the Mathematics Division of
the Air Force Office of Scientific Researah tmder Grant No. APOSR-68-1415.
THE L1 NORM OF THE APPROXIf4ATION.ERROR
FOR BERNSTEIN-TYPE POLYNOMIALS
by
WASSILY HOEFFDING
Department of Statistics
University of North Caro'tina at Chapel BiZZ
Institute
of
Statistics Mimeo Series NO. 679
ApJLU.., 1970
THE t 1 NORM OF THE APPROXIMATION ERROR FOR BERNSTEIN-TYPE POLYNOMIALS·
by
Wassily Hoeffding
Depa%'tment of Statisti.cs
Uni vem ty of North Caro Zina
Chapel Hi'LZ~ North Carolina 27514
1.
IntttOduction and statement of resu'Lts.
with the estimation of the
L
l
This paper is concemed
norm of the difference between a function of
bounded variation and an associated Bemste1n polynomial. and with the analogous problem for a Lebesgue integrable fUDction of bounded variation inside
(O~l).
A real-valued function defined in the opeD interval
said to be of bounded variation inside
in every closed sub-interval of
be denoted by
SU-.
(O,l)
(0,1).
(0.1)
is
if it is of bounded variation
The class of these functions will
To formulate some of the results we state the follow-
1ng lemma. which is a simple consequence of the well-known canonical representation of a function of bounded variation.
I.£MurA 1.
sented as
f · f -f ,
l 2
functions on
*
The function
(0,1).
where
f
is in
f
l
Moreover, if
and
BU·
f
2
f£8U·.
if and only if it can be repreare nondecreasing real-valued
the functions
f
l
and
f
2
This research UJaB supported in PQ.Pt by the Mathematics Division of the
Air Force Office Of Scientifio Research under Grant No. AFOSR-68-1415.
2
C:8Il
be
80
[x,y]
chosen that t for
0 < x < y < 1,
the total vanation of
is the sum of the total variations of
f
l
and
f
2 on
f
on
[s,y],
implying
If
f
is finite in the c:losed interval
Bemste1n polynomial of order n,
[0,1],
the associated
denoted in operator notation by
Bnf,
is defined by
II
Bnf(x)
(2)
•
I
f(1/n)Pn,i(x),
1-0
where
(3)
For
f
Lebesgue integrable on
polynomials
(0,1),
Pnf(x). txBrt+1F(X),
we shall use the modified Bernstein
where
F(x)
= l~f(Y)dy.
Explicitly (see
Lorentz [1], Chapter II),
n
(4)
Pnf(x)
•
I
1=0
For
If
f
ff. sV- ,
(i+l)/(u+l)
(D+l)
J
f(y)dy Pn,i(x).
i/{rt+l)
let
is represented 1n the form (1), we have
is nor<ecreasing,
The following four theorems will be proved.
J(f). J(f )+J(f ).
1
2
If
f
3
I.
TI£OREM
ation inside
(0,1),
1
Io
(7)
If
f
is a Lebesgue integrable function of bounded varithen
IF f(x) - f(x)ldx S C J(f),
n
n
where
(8)
The sign of equality in (7) holds if and only if
the intervals
THEOREM
2.
(O,a)
Let
and
f
(a,l),
2
a
lim
n+ClO
n~ J1 Ip
0
(0,1),
and such that the functions
f(x) - f(x)ldx
n
irrespective of whether J(f)
are imposed.
with the numerical constant
and
is finite or infinite.
that the latter is guaranteed only if
ffBV·
l
Then
J(f)
is sufficient for the
norm of the approximation error to be of order n 4".
beyond
f
= (2/~)~J(f),
Theorem 1 shows that the finiteness of
L
1
~(n+l) - l o r a . 1~(n+1)-1.
0
in representation (1) are Lebesgue integrable.
(9)
1s constant in each of
be a step function with finitely many steps in
every closed sup-interval of
f
where
f
J(f)
Theorem 2 implies
is finite when no restrictions
It also shows that the upper bound in (7),
(2/e)~
reduced to
(2/~)%,
is asymptotically
attained for every fixed step function of the specified type.
If
f
is nondecreasing, the condition
much stronger than square integrability of
that if
f
is nondecreasing,
conversely), and that
J(f) < ClO
J(f) < ClO
f.
is stronger, but not
Explicitly, it can be shown.
implies
l~f2(x){log(1+1f(x) I) }2+6 dx
l~f2(x)dx
< ClO
<
00
for some
(but not
6 > 0
4
implies
n
for
J(f) <
~
CD.
If
f
is nondecreasing and square integrable, we have
2
(10)
where
is an absolute constant.
C
The proof of (10) is sketched at the end
of Section 2.
If
is convex, (10) is true with
f
logo remcwed (as can be shown by
means of Jensen's inequality).
Concerning the Bernstein polynomials (2), Theorem 1 immediately implies the following.
If
F
1s the d1fference of two bounded, convex, ab-
solutely continuous functions on
[0,1]
and
J(F f )
1s finite, then
,l](B F-F) • o(n-~). (I am indebted to Professor G.G. Lorentz for this
n
rO
observation.) We also have the following two analogs of Theorems 1 and 2.
V4lt
TI-EOREM
(11)
3.
Let
f
Jol IBn f(x)
where
C
n
be of bounded variation 1n
- f(x)ldx
[0,11.
:s;
1s given by (8).
THEOREM 4.
Let
f
be a step function of bounded variation on
with finitely many steps 1n every closed sub-interval of
holds true with
P
n
replaced by
(0,1).
[0,1]
Then (9)
B •
n
The upper bound in (11) can not be replaced by
C n 4;J(f)
absolute constant, as the following example shows.
Let
os
0(0-
x < a, f(x). c if a < x s 1, where a •
n o n
calculation
Then
with
f(x)· b
1).
C an
if
By a simple
5
J(f)
2,
Proof of Theorem 1.
Ib-cl a~n
•
(1+0(1».
The modified Bemstein po1yaomial defined by
(4) may be written in the form
Pn
f(x) • Jl K(x,y)
On
(12)
fey) ely,
where
and
[u]
denotes the largest integer contained in
Jo1 Kn (x,y) dy
(14)
till
Io
1,
I
u.
K (x,y) dx •
n
We note that
1.
Let
Hn
(x,u) • Jl K(x,y)
un
(1S)
dye
A simple calculation shows that
where
cS (u)
(17)
n
..
(n+1)u - [(n+l)u],
n
(18)
Gn, k(X)
•
r
pn, i(X)
i=k
and
n, o(x)
G
till
1,
n, n+l(x).
G
0
1 k let)
= n JfX Pn-.-
o
for
0 < x < 1.
dt,
k - l, ••• ,n ,
6
Let
x£(O,l)
P f(x) - f(x)
be a continuity point of
f.
We have from (12) and (14)
II K (x,y){f(y)-f(x)}cly
•
non
. - IXo Kn (x,y) IXy df(u)dy + IIx Kn (x,y) J(Y df(u)dy
x
Since
!~u(x,Y)dY =- 1-Hn (x,u),
P f(x) - f(x)
(19)
•
-
we thus have
IX
(1-H (x,u»df(u) +
non
f1x
H (x,u)df(u).
n
Hence
1
Jo
Ip
f(x) - f(x)ldx
S
n
1
rx
(l-H (x,u»dlf(u)ldx + I J1 H (x.~)dlf(u)ldx
Io J n
0 x n
I
o
(20)
= J1o Dn (u)dlf(u) I,
where
(21)
Dn(u)
•
J1 (I-H (x,u»dx +fu H (x,u)dx.
u,
non
Therefore,
where
(23)
cu •
~up
u-t
OsuSl
(l-u)-~ D
(u).
n
From (15) and (14), it is easily seen that
(24)
D (u)
•
2 JU H (x,u)dx.
nOn
7
We now show that
For k
(n+l)u < k+l
S
(k. O,l, ••• ,n), we have
[(n+l)u]. k,
1-6 (u) • k+l-(n+l)u, and, by (16) and (18),
n
Henee it is sufficient to show that the function
g(u)
fUo Gn, k+l(x)dx +
•
(k+l-(n+l)u)
JruO pn, k(x)dx -
u(l-u)pn, k(u)
is identically zero.
It is easy to verify the identities
u(l-u)p'n, k(u)
up'n, k(u)
•
(k-nu)pn, k(u),
=
npn, k(u) - np n-1 , k(u) •
Hence
g'(u)
•
Gn, k+l(u) - (0+1)
JU0 Pn, k(x)dx +
(k+l-(n+l)u) pn, k(u)
- (1-2u) pn, k(u) - u(l-u) p'n, k(u)
Thus
gU(u). 0,
For
and since
g(O) = g'(O) = 0,
[(n+l)u]. k fixed,
its maximum at
u
= (k~)/(n+l).
J'(l-u)~pn, k(u)
identity (25) is proved.
= (n)uk+!!(l_u)n-k~
k
Hence, by (25),
attains
8
u-i (1-u)-~
(26)
Dn(u)
2u~ (1-u)~
•
Pn.k(u)
~ 2 (n) (k+:j')k+;- (n-k#)n-k~ (n+l) -n-1
k
•
cn (k).
say.
Now
k..4
n-k-l
(k..4) 2 (n-kJ )
2
n-k
2
2
•
k+l
k*
n-k.J.
(k~) 2(n_k~)
2
2
2
.-
-------------::-
P(k)
F(n-k-l)
•
where
P(k)
•
(k+f)
tt4
-k-!
2 (k+l) -1 (k4>
2.
It is readily seen that
:. logF(k)
is strictly positive for
log(k~)
1:1
k
~
O.
- log(k+t) - (k+l)-l
Hence
P(k)
is strictly increasing.
Therefore
max
(27)
O~k~n
c (k)
n
•
=
c (0)
n
c (n)
n
Also, the left-hand side of (26) is equal to
u
a
~(n+l)
-1
or
u .. l-1i'(n+l)
to the expression in (27).
-1
•
Thus
Cn.
•
2~ (~)~ (n+l)-n-l.
c (0)
n
if and only if
as defined by (23). is equal
By (20). equality in (22) can hold only if
f
~(n+l)-1 or l-~(n+l)-1. A direct
takes two values and the saltus is at
calculation shows that equality does hold in this case.
The inequality in (8) is easily verified. completing the proof.
We now indicate the proof of inequality (10).
D (u)
n
~
(2/e)lfn -lfulf (l-u)~.
Hence if
f
It has been shown that
is nondecreasing.
9
l-e
f
Dn(u)dlf(u)I
&
Integration by parts and application of Schwarz's inequality shows that the
right-hand side does not; exceed
0
for
< e S
If we set
1/3.
r~Dn (u)dl f(u)1
£ . (n+l)
-1
,
the remaining contribution to
is of smaller order of magnitude, and (10) follows from
(20) •
3.
Proof of Theorem Z.
given for a step function
with
(0,15)
IS < 1.
f
For convenience of notation, the proof will be
with finitely many steps in eveTY interval
In the general case of Theorem 2, the· proof requires
only trivial modifications.
It is irrelevant how
f
is defined at its
points of discontinuity, and we may assume that
if
j • 1,2, ...
,
(28)
tim aj
j+ClO
•
1.
Let
I (x, u)
=
H (x, u) - 1
I n (x,u)
•
Hn (x,u)
n
n
if
0 < u S x < 1,
(29)
and let
if
0 < x < u < 1 ,
m be a fixed positive integer.
tinuity point of
f,
By (19), if
xe (0,1)
is a con-
10
m
·r
Pnf(x) - f(x)
1
In(x,ai)(bi+l-bi) +
J
i-I
am+l-
I n (x,u)df(u).
Hence
JoSm Ip f(x)-f(x)ldx
(30)
n
lei
A + 6R,
•
n
n
s 11
where
8m
(31)
m
Ir
J
An ..
o
(32)
Rn
II
•
In(x,ai)(b1+1-bi)ldX,
i-I
I'm IIn(x,u)ldxdlf(u)l.
am+l- 0
From (16) and (18) we obtain by straightforward calculation
l (x-u)2 d H (x,u)
xn
Jo
Hence 1f 0 < x
H (x,u)
n
For u < x
(33)
<
S
S 3u(1-u)n-l ,
0 SuS 1.
u,
<
(u-x) -2
JX
o (u-y)
2 d H (y,u)
yn
S 3u(1-u) (u-x) -2 n-1 •
1, we have the same upper bound for I-Hn(X,u),
S 3u(1-u) (u-x) -2 n -1 ,
IIn(x,u) I
o<
x,
so that
u < 1.
From (32) and (33), we have
The last integral is finite since
grable. Hence R • O(n-l ) and
n
f 1 and £2
,
8m
(34)
in
fo Ipn f(x)
- f(x)ldx •
A + O(n-1).
n
(1) are Lebesgue inte-
11
Let
i ~ j-l,
a _
j l
< x < (a _ +8 ) /2.
j l
j
x
for
uniformly in
In (x,a ) • O(n- l )
i
Then, by (33),
i · 1, ••• ,m.
if
Benee
m
(35)
L
In (x,ai ) (bi+l-b i )
l
(Hn(x,aj_l>-l)(bj-bj_l) + O(n- )
•
i-l
if
for
j . 1, ••• ,m,
mifomly for
X! (0, am) •
j . 1, the first term on
(For
the right is zero.)
In a similar way, it is seen that
if
for
j . l, ... ,m,
uniformly for
x!(O,a ).
m
It follows that
(37)
A
n
•
Another application of (33) shows that if the upper limits of integration in the first sum in (37) are replaced by
in the second sum by
0,
then a term of order n- l
An·
L
j~
and the lower limits
is added.
1
m-l
(38)
1
Ibj+l-b j I{J
I
m
+ Ib-.L.l-b
IUT
aj
U-Bn (x,aj»dx + J Hn (X,aj)dX}
~
.
0
am H
J0
Henee
n
1
(x,a )dx + O(n- ).
m
':
(
,
II'
12
With (34), (21) and (24), we thus have
m-l
Joam Ipnf(x)-f(x)ldx
(39)
r
-
uf(O,l)
I
j-l
+
For
Dn(aj )Ibj+l-b j
~Dn (am) Ibm+l-bmI + O(n-1).
fixed, we have by (25) and Stirling's formula
D (u)
(2/lT);' n -t ulf (l-u)-t +
-
n
oen~).
Inserting this expression in (39) and recalling (28), we obtain
8m
fo
(40)
If
IPnf(x)-f(x)ldx
00,
J(f)·
J(f) <
n < x < 1,
and
Since
€.
and let
r~IPnf-£ldx
m,
Given a positive
ao.
/~X~(l-X)~dlf(u)1 <
(2/1T)~ n-~
Let
£,
f (x). f(x)-fl(x).
2
choose
if
/~IPn£l-flldx
I
l
Urn n-t
ao
0
IPnfl - flldx
0
= n(£)e(O,l)
T)
s nt
< x
so that
fl(x). fen)
if
Then
£1 has finitely many steps, (40) with
By Theorem l,
x~(l-x)%dlf(x)1
and (9) is proved in this ease.
fl(x). f(x)
differs from
n+
8m-l+
{fo
the integral on the right side of (40) may be made as
large as we please by choice of
Let
•
•
1~IPnf2-f2Idx.
by at most
f· f
l
implies
(~)lf J(fl )
•
13
Since
is arbitrary, these facts imply that (9) holds if
£
J(f) <
0>.
The
proof is complete.
4.
Proof of Theorems 3 and 4.
Since a funetion of bounded variation
can be represented in the form (1), and
we may assume, in the proof of Theorem 3, that
function of bounded variation.
f
(1+1)/(n+1)
I
cn, i Pn, l(x),
C
n
,l
..
(n+1)
i-O
i )
f(n+1
f
. f(y)dy.
f
1/(n+1)
is nondecreasing,
~
cn,i
<
-
£ (i+1)
n+1'
i
f(n+
1).!'
~
f(ni )
i 1
~ f(n+
+1)'
i
I:
O, ••• ,n.
and therefore
Hence
n
(41)
is a nondecreasing
By (4),
n
Since
~ I:~"'lJIBnfi-fildx,
JIBnf-fldx
l Ip lex) - B f(x)ldx
Jo
n
n
~ I
I Ic
1
o
n, i - £(1)
n Ipn, i(X)dx
i-O
i=O
•
Inequality (11) now follows from
V~[0,1](£)(n+1)
-1
•
14
1o n
1 laf(x) - f(x)ldx
1 Ipn o+n
1
S
0
f(x) - f(x)ldx
Ilia f(x) - P f(x)ldx
n
and Theorem 1.
The conditions of Theorem 4 imply those of Theorem 2, and Theorem 4
follows from (9) and (41).
Reference
1.
Lorentz, G.G., Bernstein PoZ.ynomiaZ8~
Toronto, 1953.
University of Toronto Press,