An. Şt. Univ. Ovidius Constanţa
Vol. 17(2), 2009, 181–190
TWO DIMENSIONAL DIVIDED
DIFFERENCES WITH MULTIPLE KNOTS
Ovidiu T. Pop and Dan Bărbosu
Abstract
The notion of two dimensional divided difference was introduced by
T. Popoviciu in 1934. Many properties of these differences were obtained
by D. V. Ionescu. Other properties of the mentioned differences were
obtained by the authors of the present paper.
The focus of the present paper is to establish properties of two dimensional differences in the case of multiple knots. First, we establish
some properties of the univariate divided differences with multiple knots:
a representation theorem using the determinants and a mean value theorem. Next, one proves the main results of the paper which are a representation theorem for the bivariate divided differences with multiple
knots and a mean-value theorem for this kind of divided differences.
1
Introduction
In this section let be N = {1, 2, . . . }, N0 = N ∪ {0}, m ∈ N0 , r0 , r1 , ..., rm ∈ N,
r0 + r1 + · · · + rm = M + 1, α = max{r0 − 1, r1 − 1, . . . , rm − 1}, I ⊂ R be
an interval and Dα (I) the set of all real functions f , α times differentiable
on I. If α = 0, we consider that D0 (I) = F (I) = {f |f : I → R}. Let
x0 , x1 , . . . , xm ∈ I be distinct knots.
The m-th order divided differences of f ∈ F (I) on the knots x0 , x1 , . . . , xm
is defined by
m
X
f (xk )
[x0 , x1 , . . . , xm ; f ] =
,
(1)
u′ (xk )
k=0
Key Words: divided differences; two dimensional divided differences with multiple knots.
Mathematics Subject Classification: 41A05, 41A63
Received: January, 2009
Accepted: September, 2009
181
182
Ovidiu T. Pop and Dan Bărbosu
where u(x) = (x − x0 )(x − x1 ) · . . . · (x − xm ).
In the following, the divided difference with multiple knots
x0 , x0 , . . . , x0 , x1 , x1 , . . . , x1 , . . . , xm , xm , . . . , xm ; f
|
{z
} |
{z
}
{z
}
|
r0 times
r1 times
(r )
(r )
rm times
(r )
will be denoted by [x0 0 , x1 1 , . . . , xm0 ; f ], where f ∈ Dα (I).
It is known that (see [3], [4]) or [6])
m)
(r0 ) (r1 )
(W f ) x0(r0 ) , x1(r1 ) , . . . , x(r
m
(rm )
,
x0 , x1 , . . . , xm ; f =
(r )
(r )
(r ) V x0 0 , x1 1 , . . . , xmm
where
(r0 )
(W f ) x0
1
0
x0
1
...
...
x0r0 −1
(r0 −1)xr00−2
0
0
...
(r0 −1)!
1
0
xm
1
...
...
0
0
...
xrmm −1
(rm −1)xrmm −2
...
(rm −1)!
=
(r1 )
, x1
(rm )
, . . . , xm
(2)
(3)
x0M −1
(M −1)x0M −2
...
...
...
...
...
...
...
−r0
(M −1)(M −2) · . . . · (M −r0 +1)xM
0
...
−rm
(M −1)(M −2) · . . . · (M −rm + 1)xM
m
M −1
xm
M −2
(M −1)xm
f (x0 )
f ′ (x0 )
(r −1)
f(x00)
f (xm )
f ′ (xm )
(r −1)
m
f(xm
)
and
(r )
(r )
m)
V x0 0 , x1 1 , . . . , x(r
m
1 x0
0 1
0 0
= 1 xm
0 1
0 0
...
...
...
...
...
...
x0r0 −1
(r0 −1)xr00 −2
...
(r0 −1)!
...
rm −1
xm
(rm−1 )xrmm −2
...
(rm −1)!
=
m rY
k −1
Y
k=0 i=0
...
...
...
...
...
...
i!
(4)
M−r0+1 M (M − 1) · . . . · (M −r0 +2)x0
M
xm
M −1
M xm
M−rm +1 M (M − 1) · . . . · (M −rm + 2)xm
Y
xM
0
−1
M xM
0
(xk − xs )rk ·rs
0≤s<k≤m
183
TWO DIMENSIONAL DIVIDED DIFFERENCES WITH MULTIPLE KNOTS
is the generalized determinant of Vandermonde.
If k ∈ {0, 1, . . . , m} and i ∈ {0, 1, . . . , rk − 1} we denote
(r )
(r )
m)
Vk,i x0 0 , x1 1 , . . . , x(r
m
1
0
= (−1)M +r0 +r1 +...+rk−1 +i 0
0
x0
0
0
0
(5)
−1
...
xM
0
...
M −i ...
(M −1)(M −2) · . . . · (M −i+1)xk
−i−2 ,
... (M −1)(M −2) · . . . · (M −i−1)xM
k
...
M −rm ... (M −1)(M −2) · . . . · (M −rm +1)xm
so the above determinant is obtained from (3) by elimination the line r0 + r1 +
· · · + rk−1 + i + 1 and the column M + 1.
Theorem 1 If f ∈ Dα (I), the identity
(r0 ) (r1 )
m)
x0 , x1 , . . . , x(r
m ;f
=
holds.
(6)
m rX
k −1
X
(i)
(r )
(r )
m)
f (xk )
Vk,i x0 0 , x1 1 , . . . , x(r
m
(r0 )
(r1 )
(rm )
V x0 , x1 , . . . , xm
k=0 i=0
1
Proof. Taking into account the relation (2), yields
(r0 ) (r1 )
m)
x0 , x1 , . . . , x(r
m ;f
1
·
=
(r0 )
(r1 )
(r ) V x0 , x1 , . . . , xmm
−1
1
x0
...
xM
0
...
M −r
0
0
... (M −1)(M −2) · . . . · (M − rk−1 + 1)xk−1 k−1
1
xk
...
xM−1
k
m −2
X
1
...
(M −1)xM
0
k
...
k=0 −rk
0
0
...
(M −1)(M −2) · . . . · (M − rk + 1)xM
k
1 xk+1 ...
xM
k+1
...
−rm
0
0
...
(M −1)(M −2) · . . . · (M − rm + 1)xM
m
0 f (xk ) f ′ (xk ) (rk −1) f(xk ) 0 0 0
and developing the above determinant from the column M + 1, we obtain the
relation (6).
184
Ovidiu T. Pop and Dan Bărbosu
Remark 1 If r0 = r1 = · · · = rm = 1 in Theorem 1 we get the relation (1).
Theorem 2 The following relation
(r )
(r )
k
m)
x0 0 , x1 1 , . . . , x(r
m ;x
(7)

if k ≤ M − 1

 0,
1,
if k = M
=


r0 x0 + r1 x1 + · · · + rm xm , if k = M + 1
holds, where x ∈ I.
Proof. From (3) the first and the second statement from (7) follow.
Taking relation (3) into account, we start from the equality
i
h
(r )
(r )
rm
r1
r0
m)
= 0.
x0 0 , x1 1 , . . . , x(r
m ; (x − x0 ) (x − x1 ) · . . . · (x − xm )
Then, it follows
h
i
(r )
(r )
M +1
m)
x0 0 , x1 1 , . . . , x(r
− s1 xM +s2 xM −1 + · · · +(−1)M +1 sM +1 = 0,
m ;x
where s1 = r0 x0 + s1 x1 + · · · + rm xm , hence
i
h
i
h
(r0 )
(r1 )
(r )
(r )
(rm )
M
M +1
m)
x
,
x
,
.
.
.
,
x
;
x
=
s
x0 0 , x1 1 , . . . , x(r
;
x
1
m
m
0
1
and the third statement of (7) follows.
Theorem 3 Let a = min{x0 , x1 , . . . , xm }, b = max{x0 , x1 , . . . , xm }, f ∈
C M −1 ([a, b]), exists f (M ) on (a, b) and f (lk ) (xk ) = 0, where lk ∈ {0, 1, . . . , rk −
1}, k ∈ {0, 1, . . . , m}. Then exists ξ ∈ (a, b) such that
!
h
i
r0 x0 +r1 x1 + · · · +rm xm
(r0 )
(r1 )
(rm )
x0 , x1 , . . . , xm ; f 1+(M +1)ξ −
(8)
(r )
(r )
(r ) V x0 0 , x1 1 , . . . , xmm
1 (M )
f
(ξ).
=
M!
Proof. Define the auxiliary function F : [a, b] → R by
(r )
(r )
m)
F (x) = (W f ) x, x0 0 , x1 1 , . . . , x(r
m
h
i (r )
(r )
(r0 )
(r1 )
(rm )
m)
− x0 0 , x1 1 , . . . , x(r
;
f
V
x,
x
,
x
,
.
.
.
,
x
,
m
m
0
1
TWO DIMENSIONAL DIVIDED DIFFERENCES WITH MULTIPLE KNOTS
185
(r )
(r )
(r )
for any x ∈ [a, b], where the first line in (W f ) x, x0 0 , x1 1 , . . . , xmm ,
(r )
(r )
(r )
is
1 x x2 . . . xM f (x),
respectively
V x, x0 0 , x1 1 , . . . , xmm
1 x x2 . . . xM xM +1 . On verifies immediately that F (lk ) (xk ) = 0, where
lk ∈ {0, 1, . . . , rk − 1}, k ∈ {0, 1, . . . , m}, so the function F has r0 + r1 + · · · +
rm = M + 1 roots. By the generalized Rolle Theorem, there exists a point
ξ ∈ (a, b) such that F (M ) (ξ) = 0. But
0 0 . . .
0
M ! f (M ) (x)
F (M ) (x) = 1 x0 . . . xM −1 xM
f (x0 ) 0
0
...
0
M ! (M +1)!x
h
i 0 0 . . .
(r )
(r )
m)
−1
+1 − x0 0 , x1 1 , . . . , x(r
xM
xM
xM
m ; f 1 x0 . . .
0
0
0
...
(r )
(r )
m)
= (−1)M +2 M !(W f ) x0 0 , x1 1 , . . . , x(r
m
i
h
(r )
(r )
(r )
(r )
m)
m)
+(−1)M +3 f (M ) (x)V x0 0 , x1 1 , . . . , x(r
− x0 0 , x1 1 , . . . , x(r
m
m ;f
(r )
(r )
m)
· (−1)M +2 M !(W xM +1 ) x0 0 , x1 1 , . . . , x(r
m
(r1 )
(r0 )
(rm )
M +3
+(−1)
(M + 1)!xV x0 , x1 , . . . , xm
and taking (2) and (7) into account, the relation (8) follows.
In [4] the following mean-value theorem for divided differences with multiple knots is proved.
Theorem 4 If a = min{x0 , x1 , . . . , xm }, b = max{x0 , x1 , . . . , xm },
f ∈ C M −1 ([a, b]) and f (M ) exists on (a, b), then there exists ξ ∈ (a, b) such
that
h
i
1 (M )
(r )
(r )
m)
x0 0 , x1 1 , . . . , x(r
f
(ξ).
(9)
m ;f =
M!
In the second section, using this theorem we shall give a mean-value theorem
for two dimensional divided differences with multiple knots (Theorem 6).
2
The definition of two dimensional divided differences
with multiple knots
In the following let m, n ∈ N0 , r0 , r1 , . . . , rm , q0 , q1 , . . . , qn ∈ N, r0 + r1 +
· · · + rm = M + 1, q0 + q1 + · · · + qn = N + 1, I, J ⊂ R intervals, I × J
186
Ovidiu T. Pop and Dan Bărbosu
be a bidimensional interval and Dα,β (I × J) the set of all real valued bi∂f i+j
variate functions f with the property that exits
on I × J, where
∂xi ∂y j
i ∈ {0, 1, . . . , α}, j ∈ {0, 1, . . . , β}, α = max{r0 − 1, r1 − 1, . . . , rm − 1} and
p = max{q0 − 1, q1 − 1, . . . , qn − 1}.
Let x0 , x1 , . . . , x
h m ∈ I, y0 , y1 , . . . , yn ∈ J ibe distinct knots. For y ∈
(r )
(r )
(r )
J, we denote by x0 0 , x1 1 , . . . , xmm ; f (x, y)
the parametric extension
x
of M -th order divided differences with multiple knots, equivalent the M th order divided differences of the function f (·, y) : I → R, y ∈ J, with
respect the knots x0 , x0 , . . . , x0 , x1 , x1 , . . . , x1 , . . . , xm , xm , . . . , xm is defined
{z
} |
{z
}
{z
}
|
|
r0 times
r1 times
rm times
by (see (6))
h
i
(r )
(r )
m)
x0 0 , x1 1 , . . . , x(r
=
m ; f (x, y)
x
·
M rX
k −1
X
k=0 i=0
1
V
(r )
(r )
(r )
x0 0 , x1 1 , . . . , xmm
∂if
(r )
(r )
m)
(xk , y).
Vk,i x0 0 , x1 1 , . . . , x(r
m
∂xi
(10)
In a similar way, the parametric extension of N -th order divided differences
of the function f (x, ∗); J → R, x ∈ I, with respect to the knots
y0 , y0 , . . . , y0 , y1 , y1 , . . . , y1 , . . . , yn , yn , . . . , yn
|
{z
} |
{z
}
{z
}
|
q0 times
q1 times
qn times
is defined by
=
1
(q0 )
V y0
(q1 )
, y1
h
(q0 )
y0
(q )
, . . . , yn n
(q1 )
, y1
i
, . . . , yn(qn ) ; f (x, y)
n qX
l −1
X
l=0 j=0
y
(11)
∂j f
(q ) (q )
(x, yl ).
Vl,j y0 0 , y1 1 , . . . , yn(qn )
∂y j
187
TWO DIMENSIONAL DIVIDED DIFFERENCES WITH MULTIPLE KNOTS
Theorem 5 The following equalities
h
h
i i
(q ) (q )
(r )
(r )
m)
y0 0 , y1 1 , . . . , yn(qn ) ; x0 0 , x1 1 , . . . , x(r
m ; f (x, y)
x y
h
i (r )
(r )
(q0 ) (q1 )
(qn )
m)
= x0 0 , x1 1 , . . . , x(r
;
y
,
y
,
.
.
.
,
y
;
f
(x,
y)
m
n
0
1
(12)
y x
n
m
l −1
k −1 qX
X X rX
1
(q ) (q )
(q )
V y0 0 , y1 1 , . . . , yn n k=0 l=0 i=0 j=0
V
∂ i+j f
(r )
(r )
(q ) (q )
m)
(xk , yl )
·Vk,i x0 0 , x1 1 , . . . , x(r
Vl,j y0 0 , y1 1 , . . . , yn(qn )
m
∂xi ∂y j
=
(r )
(r )
(r )
x0 0 , x1 1 , . . . , xmm
hold, where (x, y) ∈ I × J.
Proof. Taking into account (10) and (11), we have
h
(q0 )
y0
"
(q1 )
, y1
(q0 )
= y0
h
i i
(r )
(r )
m)
, . . . , yn(qn ) ; x0 0 , x1 1 , . . . , x(r
m ; f (x, y)
x y
(q1 )
, y1
, . . . , yn(qn ) ;
1
V
(r )
(r )
(r )
x0 0 , x1 1 , . . . , xmm
#
∂if
(r )
(r )
m)
· Vk,i x0 0 , x1 1 , . . . , x(r
(xk , y)
m
∂xi
·
m rX
k −1
X
Vk,i
k=0 i=0
(r )
m)
x0 0 , . . . , x(r
m
1
=
(r )
(r )
V x0 0 , . . . , xmm
·
1
(q0 )
V y0
(q1 )
, y1
(q )
, . . . , yn n
n qX
l −1
X
l=0 j=0
=
y
V
m rX
k −1
X
k=0 i=0
1
(r )
(r )
x0 0 , . . . , xmm
i
(q0 ) (q1 )
(qn ) ∂ f
y0 , y1 , . . . , yn ; i (xk , y)
∂x
y
m rX
k −1
X
k=0 i=0
(r )
m)
Vk,i x0 0 , . . . , x(r
m
∂ i+j f
(q ) (q )
(xk , yl ),
Vl,j y0 0 , y1 1 , . . . , yn(qn )
∂xi ∂y j
and (12) follows.
Definition 1 The (m, n)-th order divided difference of the function f ∈ Dα,β (I×
J) with respect to the distinct knots (xi , yj ) ∈ I × J, i ∈ {0, 1, . . . , m},
188
Ovidiu T. Pop and Dan Bărbosu
j ∈ {0, 1, . . . , n} is defined by
" (r0 ) (r1 )
(r )
x0 , x1 , . . . , xmm
(q0 )
y0
(q1 )
, y1
(q )
, . . . , yn n
;f
#
(13)
n rX
m X
l −1
k −1 qX
X
1
=
(r )
(r )
(r )
(q ) (q )
(q )
V x0 0 , x1 1 , . . . , xmm V y0 0 , y1 1 , . . . , yn n k=0 l=0 i=0 j=0
∂ i+j f
(q ) (q )
(r )
(r )
m)
Vl,j y0 0 , y1 1 , . . . , yn(qn )
Vk,i x0 0 , x1 1 , . . . , x(r
(xk , yl ).
m
∂xi ∂y j
In the paper [2], we give a mean-value theorem for two divided differences
with simple knots.
Let a, b, c, d be real numbers defined by a = min{x0 , x1 , . . . , xm }, b =
max{x0 , x1 , . . . , xm }, c = min{y0 , y1 , . . . , yn } and d = max{y0 , y1 , . . . , yn }.
∂M f
(·, y) exists on (a, b) for any y ∈
∂xM
∂M f
∂ M +N
[c, d],
(x, ∗) ∈ C N −1 ([c, d]) and
(x, ∗) exists on (c, d) for any
M
∂x
∂xM ∂xN
x ∈ (a, b), then exists (ξ, η) ∈ (a, b) × (c, d) such that
" (r0 ) (r1 )
#
(r )
x0 , x1 , . . . , xmm
∂ M +N f
1
(ξ, η),
(14)
;
f
=
(q ) (q )
(q )
M ! N ! ∂xM ∂y N
y 0 , y 1 , . . . , yn n
Theorem 6 If f (· , y) ∈ C M −1 ([a, b]),
0
1
where ”·” and ”∗” stand for the first and respectively second variable.
Proof. Taking into account (12) and (13) and applying the mean-value
theorem for one dimensional divided differences (see Theorem 4), there exist
ξ ∈ (a, b) and respectively η ∈ (c, d) such that
" (r0 ) (r1 )
#
(r )
x0 , x1 , . . . , xmm
;f
(q ) (q )
(q )
y0 0 , y1 1 , . . . , yn n
h
h
i i
(r )
(r )
(q ) (q )
m)
= y0 0 , y1 1 , . . . , yn(qn ) ; x0 0 , x1 1 , . . . , x(r
m ; f (x, y)
x y
M
1 ∂ f
(q ) (q )
= y0 0 , y1 1 , . . . , yn(qn ) ;
(ξ, y)
M ! ∂xM
y
M
1
∂
f
(q ) (q )
=
y 0 , y1 1 , . . . , yn(qn ) ; M (ξ, y)
M! 0
∂x
y
=
1
∂ M +N f
(ξ, η),
M ! N ! ∂xM ∂y N
TWO DIMENSIONAL DIVIDED DIFFERENCES WITH MULTIPLE KNOTS
189
so the equality (14) holds.
Remark 2 Because r0 , r1 , . . . , rm ∈ N and r0 + r1 + · · · + rm = M + 1, it
results that M ≥ m, so if m tends to ∞ it results that M also tends to ∞.
We consider a function f : [a, b] × [c, d] → R, f ∈ C ∞,∞ ([a, b] × [c, d]), f
possessing uniform bounded partial derivatives, so there exists M > 0 such
that
k+l
∂ f
(15)
∂xk ∂y l (x, y) ≤ M
for any (x, y) ∈ [a, b] × [c, d] and any (k, l) ∈ N0 × N0 .
Theorem 7 If (xm )m≥0 and (yn )n≥0 are sequences of distinct points from
[a, b], respectively [c, d], then
lim
m,n→∞
"
(r )
(r )
(r )
(q1 )
(q )
x0 0 , x1 1 , . . . , xmm
(q0 )
y0
, y1
, . . . , yn n
#
; f = 0.
(16)
Proof. Taking into account (14), (15) and Remark 2, relation (16) follows.
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[8] D. D. Stancu, Lectures and Exercises in Numerical Analysis, I, lito Univ. ”BabeşBolyai”, Cluj-Napoca, 1977 (in Romanian).
[9] D. D. Stancu, Gh. Coman, O. Agratini, Trı̂mbiţaş, R., Numerical Analysis and theory
of aproximation, I, Presa Univ. Clujeană, Cluj-Napoca, 2001 (in Romanian).
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Ovidiu T. Pop and Dan Bărbosu
(Ovidiu T. Pop) National College ”Mihai Eminescu”,
5 Mihai Eminescu Street,
440014 Satu Mare,
Romania,
E-mail: ovidiutiberiu@yahoo.com
(Dan Bărbosu) North University of Baia Mare,
Department of Mathematics and Computer Science
76 Victoriei,
430122 Baia Mare,
Romania,
E-mail: barbosudan@yahoo.com