Evaluation of the Remainder Term in
Approximation
Formulas by
Bernstein Polynomials
By D. D. Stancu
1. Introduction. In this paper [7] we deal with the evaluation of the remainder
term in approximation formulas, for functions of one and two variables, by means of
Bernstein polynomials.
First, we give a representation of the remainder by definite integrals. Thus, we
find the representation (3.4) for the remainder of formula (2.2) and the representation (7.3) for the one corresponding to the formula (6.1). The consequences of these
representations are formulas (5.3), (5.4), and (8.2), (8.3), respectively.
Then, we extend Aramä's [1] formula (9.1) to two variables and obtain the
representation (9.2). In order to establish this formula, we resorted to a method
employed by us, some years ago, in the evaluation of the remainder term of certain
interpolation formulas [5]. In the last part we give some numerical approximation
formulas which result from formulas (2.2) and (6.1).
2. The Remainder Rm (f; x) of Bernstein's
Approximation Formula. Let
f(x) be a real function defined on the interval [0, 1] with its second derivative Rintegrable on this interval.
The problem is to find an integral expression for the remainder of the approximation formula on the interval [0, 1] of the function f(x) by means of the corresponding Bernstein polynomial
in
(2.1)
Bmif;x)
U'
= £p».i(a;)/
1=0
where
Pm.iix)
= ( . ) xlil
— x)m~\
The expression
RM;x)
=/(*)
- Bm(J;x)
is called the remainder term of the approximation
(2.2)
formula
fix) = Bm(f; x) + Rm(f; x),
by the Bernstein polynomial (2.1).
Since
m
Bm(í;x)
Received
January
= £ pm,i(x)
,:=o
i
= 1,
26, 1962. Presented
Bmix;x)
to the American
m
= — £ ipm,iix)
m ¿=o
Mathematical
22, 1962.
270
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= x,
Society,
February
EVALUATION
OF THE REMAINDER
271
TERM
we have
Rmil; x) = Rm(x; x) = 0,
that is, the remainder term Rm(f; x) vanishes for any polynomial of the first
degree.
3. The Integral Representation
of Rm if; x). In order to obtain
result we shall use the auxiliary function
g(f;x) = Jof{x-
the desired
t)f"it)dt.
Since the second derivative of this function is equal to fix),
the function
Hx) = fix) — gif; x) represents a polynomial of the first degree. Taking this into
account, as well as the linearity of the remainder, we may write
Rmih; x) = hix) - Bmih; x) = fix) - g(f; x) - Bm(f - g; x)
= fix) - g(f; x) - Bm(f; x) + Bm(g; x) = 0
hence
(3.1)
Rm(f; x) = Rm(g; x).
Next, we turn to finding a convenient
evaluation
oí Rm(g;x).
We have
Rmig;x) = f ix - t)f"it) dt - £ PmAx) T (- - t)f"it) dt.
Jo
¿=o
Jo
\m
)
After certain calculations we obtain
i
£
¿_i
k_
Vm.iix) i"1 (Jo
\m
- t)f"(t)
dt = £
/
i-1
Vm,iix) £
i"1 (-
k=i Jk-i
\m
- t)f"it)
/
dt
j_
= j=i£ Jj-iP l~£?».,»(*)
(-t)f"it)]dt.
\_s=.j
\m
)
J
m
Therefore,
Rmig;x) = Í ix - í)/"(0dí
(3.2)
¿
- £
i=i Jj-i
£ Pb..(«) (-
s=j
\m
- t)f"it)
/
dt =
where, supposing that
m
1^ x á -
m
(fc = 1, 2, • ■• , m),
we have
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Jo
Vro(x;0r(0
dt,
272
D. D. STANCU
x — t — £
(-t)
pm,,(x),
if t e R—-,î-\
|_ m
mj
x —t
(3.3)
£(-t)pm,,ix),
s=k \m
/
and 1 g 3 Ú k - 1
if
í €-,x\
|_
to
J
<pm(x;t) =
- E (- - i) pm,,(aO, if iÇ Le,s=k \m
I
L mJ
- E (- - 0 pm,.U),
if t G ^—^, -
L m
It follows from (3.1) and (3.2) that the remainder
may be represented by the integral as follows
Rmif;x)
(3.4)
and fc + 1 < j <
toJ
of Bernstein's
formula
TO.
(2.2)
= [ <pmix;t)f"it) dt,
Jo
where <pmix;t) is the kernel, defined for 0 ^ x, t ^ 1 by (3.3).
Thus, formula (2.2) becomes
/(*) = Bmif;x) + f Vnix;t)f"it) dt.
(3.5)
Jo
4. Properties
of the Kernel <pm(x,t). Considering the identities
j—1
m
3= 0
8—j
J—1
£
7ft
Sp,KlS(.T.)
+
s=0
£
Spm,s(-^)
=
TOX,
s=j
we deduce that
x -
í — £
(-
ÍTy \TO
— t) pm,s(x)
/
= x — í — - (tos
to \
+
t (l
-
\
— £
£pm,s(.T))
s=0
spm,s(:c)
So
=
/
Thus, we have
<pm(.r; 0 = —£
(4.1)
(i — — ) Pm.six),
8=o \
to/
""j for« €
(4.2)
for t €
?»
-, -
TOJ
and
1 á j á fc - 1, while
(pm(a;;0 = —E ( * — -) PmAx),
*=o \
to/
[V '"I■
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-£
/
(i
,-0
\
)
-
-)
TO/
pm,,(a;).
EVALUATION
OF THE REMAINDER
273
TERM
It follows from (3.3), (4.1) and (4.2) that <pmix; t) ^
0 on the square
D: 0 ^ x, t ^ 1. Hence, y = <pit) = <pmix;t) represents, for fixed x, a polygonal continuous line which joins points (0, 0) and (0, 1) and is situated beneath the x-axis.
5. Other Evaluations of Rm(f; x). Since <pm(x;t) does not change sign on the
integration interval [0, 1], the first law of the mean may be applied to (3.5) and
we obtain
(5.1)
Rmif;x)
= u2(/)
/ <pm(x;t) dt,
inf f(t)
£ p.2Íf) g sup fit).
Jo
where
[0,1]
[0,1]
Because (pm(x; t) does not depend onf(x),
let us replace the function/(.-e)
in the
formula
f(x) = Bm(f;x) + ßiif)
/ <pmix;t) dt
Jo
by x ; we then obtain
x2 = Bmix2; x) + 2 I <pm(x;0 dt.
Jo
Since
o
/ 2
Bmix ;x)
\
1 'V1 -2
i
= — ¿^ i Pm.iix)
TO ¿=0
\
2 i xi\
= x +
— x)
TO
it follows that
(5.2)
¡\mix;t)dt
Jo
Thus, Bernstein's
approximation
(5.3)
In the particular
= -X(1
~X).
2to
formula (3.5) becomes
fix) = Bmif; x) - X{12~ X) mif).
case when/"ix)
(5.4)
is continuous on [0, 1] this may also be written as
fix) = Bmif; x) - X(\~X)
/"(£),
where 0 < ? < 1.
Remarks. 1. Using the hypothesis that the second derivative/"(z)
is continuous
on [0, 1], E. Voronowskaja [6] has established the asymptotic formula
(5.5)
fix) = Bm(f, x) - Xi\~X)n*)
* In fact, this formula
holds if /(x) is bounded
+ ~ *
on [0, 1] and if f"{x) exists at a point x of
[0, 1] (see [4] or [3]).
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274
ü. D. STANCU
where e,„ —*O, as m —* <*>.There is a close analogy between this formula and for-
mula (5.4).
2. If we take into account
the representation
of the Bernstein
polynomials
given by M. Kac [2]
= / f[amix; t)] al.
Bmif;x)
Jo
where
í
m
amix; 0 = - £
TO j=l
<Piix; t),
and
faix; t)
1
0
if
if
0 ^ t ^ x
i<iâl,
<Pi(x; t)
1
0
if
if
0 g t â x2 and
x2 < t ^ x and
x < t < x 4- a;(l —a;)
a; 4- a;(l — x) < í g 1,
formula (3.5) permits us to give the representation
/(*)
= [
Jo
\f[am(x;t)]+<pm(x;t)f"(t)\
dt.
6. The Remainder Rm,n(f; x, y) in the Case of Approximation by the Bernstein
Polynomials Bm¡n(f; x, y). Let fix, y) be a real function defined on the square
o:Oái|í=
1, let us suppose that it has for (x, ¡/) ffl
/"2 >/»'2>/iv i and that they are ß-integrable on D.
We may consider the approximation
(6.1)
the partial derivatives
formula
fix; y) = Bm,n(f; x, y) + Rm,n(f; x, y),
where
m
(6.2)
&,..(/;
x, y) = £
n
£
¿=o j=o
/
pm.iix)pnJiy)f
■
(-,-),
\to
-\
n/
are Bernstein polynomials of two variables of the (to, n)th degree. The remainder
Rm.nif; x, y) is, by definition, the difference between the function f(x, y) and the
polynomial Bm,n(f; x, y).
7. The Integral Representation of Rm,„(f; x, y). In order to obtain an integral
expression of the remainder, we might use the result established for functions of
one variable. Indeed, owing to formula (3.5), we have
(7.1)
f(x, y) = £ Vm.iix)fI- ,y)+
¿=0
where pm(x; t) is defined in (3.3).
\TO
/
[ <pmix;
t)fUt,y) dt,
Jo
Let us now expand f(i/m,
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y), also using the
EVALUATION OF THE REMAINDER TERM
275
result of the first part of this paper. We obtain
f(Lv)~Í:
\m
/
3=o
VnÀv)f(i,
3-)+ Jof My; z)ft (, z)/ dz,
\to n/
\to
where i>niy; z) has a definition analogous
to <pmix:t).
Thus, (7.1) becomes
m
ti
/
•
«\
/(*, v) = £¿=0 ;E= 0 pm.iix)p«Ay)f(\TO
-, nj-)
(7.2)
+ / <pm(x;t)f"iit, y) dt + / My; z) £ pm,iix)f"i (— , z) dz
Jo
Owing to formula
Jo
i=0
\TO
/
(3.5) we have
£ pmÂ*)&(L^=
f=0
\TO
/
/?•(*,t) - f <pmix;
DfMit, z) dt,
Jo
and formula (7.2) is reduced to formula (6.1) with the following integral expression
of the remainder
Rm.nif;x,y)
= / vmix; t)f'Mt, y) dt + j ¡f>niy;z)f"*ix; z) dz
Jo
Jo
(7.3)
, ,
/ / <Pmix;t)fniy; z)fi"Mt, z) dt dz.
-
Jo Jo
8. A Consequence of the Integral Representation of the Remainder Rm,nif; x, y).
If we take into account the fact that the functions <pmix; t) and 4>niy; z) do not
change sign if x, y, t, and z belong to the interval [0, 1], we may apply the first law
of the mean and obtain
Rm.nif; x, y) = m,oif) / Vm(x; t) dt 4- noAf)
/ <p„iy;z) dz
Jo
Jo
(8.1)
( ,
-
M2.2Í/) / / (pmix; t)\¡iniy; z) dt dz,
Jo Jo
where
iaífffiix,
D
y) ^ m.kif) ^ sup fitfk)(x, y).
D
Taking into account (5.2) and a formula analogous to this, we obtain the following expression for the remainder of formula (6.1) :
r.
/,
Rm,n(f;x,y)
(8.2)
\
= -
x(l
— x)
, ...
iHjtif) -
y(\
— y)
2n
,„,
W>.l(/)
_ .f(l - x)yjl
vmn
- y)
, ,
If we suppose that the partial derivatives which appear in (7.3) are continuous
on D, we may obtain from (8.2) an expression analogous to that in (5.4), which
corresponds to the case of one variable. In order to allow only two unknown num-
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276
D. D. STANCU
bers £ and 77from the interval [0, 1], we shall proceed in the following way. From
(7.3) we have
Rm.nif;x,y) = I <Pmix;
t) f'Mt,y) - J i/miy;s)fi"Mt, z) dz dt
+ 1 iniy;z)f"^ix,z) dz = f"*i£,y) —J ^niy;z)fx"Xi,z) dz J <pmix;t)
dt
+ / $niy;z)fz*ix,z)
(8.3)
+
Xil
dz --
Jo
— x)
2to
.11 ,
s
■fx*ik,y)
x( \ —x) /f t«iy;z)fx*l*iè,z)dz + / \¡/niy;z)f'éiix,z) dz
2to
x(l
Jo
Jo
— x) .1/ .
•. .
¡\iy;
t) \f"Ax, z) + X(\mx) &vMï,z)~\ dz.
= - *^Ù.fUis,y) + [C(x,v)+ ^¿r^v(£>2/)]
¡Jn(y;z)dz.
Finally we obtain the following evaluation :
d
ix
\
xil
— x)
„,, ,.
Rm,»if;x,y) = -~^—fxÁ&v)
■.
2to
y il
— y)
.11 ,
v
--—7^-íAx^>
2n
xil - x)yi\ - y) ,
fxvii,^),
4tow.
where £ and r¡ belong to the interval [0, 1] and have the same values in both terms
in which each appears.
9. The Expression of Rm,n(f; x, y) by Two-Dimensional
Divided Differences.
By using the method of Section 7 we may also extend to two variables the following
formula, given by 0. Aramä [1], for the remainder of formula (2.2).
RM;x) = -*(1„, x)[h,k,u;f],
it:
(9.1)
where h , U , t3 are points of [0, 1] and [k,k ,h ; f] represents the divided difference
of function /(x) on the points U (i = 1, 2, 3).
Let us suppose in this case that fix, y) is continuous on the domain D. Owing
to Aramä 's formula we have
fix,y) = £ Pm.iix)f (— ,y) --\ti,h,tz;f
i=o
\m
)
to
|_
Ji
,
where í¿ are points from [0, 1], while the divided difference refers to variable
Then
fß,y)
\m
/
= 3=o
±pn,iy)fß,j-)-y-^\to
n;
y)
Z\ ,Zi , Z3
where Z\, Zi , z3 are points from [0, 1].
Thus,
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\TO
/
x.
EVALUATION OF THE REMAINDER TERM
f(x,y)
= £
£
i=0 j=0
277
pm.iix)p: nAy)f(-J-)
\to n;
.t(1 — a:)
;f\x - yil „
TO
E PmÂx)\zi,Z2,za;f(^,yj\
n
t= 0
|_
\TO
/ _\y
.
But, on the other hand we have
£
Pm.iix)
Zi,Zi,z3;f
(¿»V.
=£
y
Zx,zi,z%; pm,iix)f f ^ , y )\
i=0
= \zi,Z2,23;
|_
£
Pm,iix)f
\~,y)
;=0
\TO
/
and
_
/ 1
\
'V ( 1 _
E Pm.iix) f [ — , y ) = fix,y)
¡=0
\TO
T* 1
+ -[íi,Í2,ís;/]x
/
ÍW
If we take into account that
[¿i, U , h ; [zi , z-i, Zz ; f]v]x = [zi , z%, z3 ; [t\, U , h ; f]x]y =
Zi , 23
;./'
this being the two-dimensional divided difference, we obtain formula (6.1) with the
following expression for the remainder :
Rm.nif; x, y) = --—
[ii ,h,k
(9.2)
;f]x - --—
k , z2 , z3 ;/]„
a;(l - x)yi\ l -
y) pi, fe, U 1
U,«2,23 J'
10. Applications. Using the approximation
formulas (2.2) and (6.1), one may
construct certain formulas for numerical differentiation
and integration of functions of one and two variables.
Thus, for instance, we have the numerical differentiation formulas
/'(0)
~['(s)-
/"(0) = to(to - 1) [/(|)
- 2/(l)
and the numerical integration
-¿>"(ö'
/(0)
+/(0)]
°<*<wrl
- (l - L)r(Ö,
0 < £ < 2TO-1
formulas
f/U) Ae= m_L-¿/(±)
_ 12m
J-/"(£),
+ 1 ¿=o \ra/
Jo
(m)
LI **>*dxdy
=(ro+ik+T)
5 s'fe»)
- ii^(^
- li^'^
where £, £ , 17,1? belong to the interval (0, 1).
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- nL^(^}'
278
D. D. STANCU
The quadrature
seen in [4].
formula (10.1) without an expression for the remainder maybe
University
of Wisconsin
Madison, Wisconsin, and
University
of Cluj
Rumania
1. O. Aramä, "On the properties on monotonicity
of the sequence of interpolation
polynomials of S. N. Bernstein and their application to the study of approximation
of functions,"
Mathematica (Cluj), v. 2 (25), 1960, p. 25-40 (Russian).
2. M. Kac,
"Une remarque
sur les polynômes
de M. S. Bernstein,"
Studia
Math.,
v. 7,
1938, p. 49-51.
3. G. G. Lorentz,
4. I. P. Natanson,
5. D. D. Stancu,
variables,"
Bernstein polynomials, Toronto, 1953.
Constructive theory of Functions, Moscow, 1949 (Russian).
"Considerations
on polynomial interpolation
of functions
of several
Bull. Univ. Babes-Bolyai (Cluj), v. 1, 1957, p. 43-82 (Rumanian).
6. E. Voronowskaja,
"Détermination
de la forme asymptotique
functions par les polynômes de M. Bernstein,"
d'approximation
des
C. R. Acad. Sei. URSS, A., 1932, p. 79-85.
(Russian).
7. Dimitrib
Stancu,
"On the remainder in the approximation
formalae by Bernstein's
polynomials,"
Abstract 588-2, Notices, American Mathematical
Society, v. 9, n. 1, 1962, p. 26.
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