General Mathematics Vol. 15, No. 1 (2007), 81–92
Monosplines and inequalities for the
remainder term of quadrature formulas
1
Ana Maria Acu
Dedicated to Professor Alexandru Lupaş on the occasion of his 65th
birthday
Abstract
In this paper we studied some quadrature formulas witch are obtained using connection between the monosplines and the quadrature
formulas. For the remainder term we give some inequalities.
2000 Mathematics Subject Classification: 26D15, 65D30
Key words and phrases: quadrature rule, numerical integration, error
bounds
1
Introduction
We denote
o
n
°
°
Wpn [a, b] := f ∈ C n−1 [a, b] , f (n−1) absolutely continuous , °f (n) °p < ∞
with
kf kp :=
½Z
b
p
|f (x)| dx
a
¾ p1
for 1 ≤ p < ∞
kf k∞ := supvraix∈[a,b] |f (x)| .
Definition 1.The function s(x) is called a spline function of degree n with
knots {xi }m−1
i=1 if a := x0 < x1 < · · · < xm−1 < xm := b and
i) for each i = 0, . . . , m − 1 , s(x) coincides on (xi , xi+1 ) with a polynomial of degree not greater then n;
ii) s(x), s′ (x), . . . , s(n−1) (x) are continuous functions on [a, b].
1
Received 18 December, 2006
Accepted for publication (in revised form) 13 January, 2007
81
82
Ana Maria Acu
Definition 2.Functions of the form
Mn (t) =
tn
+ sn−1 (t) ,
n!
where sn−1 (t) is a spline of degree n − 1 are called monosplines.
Let
n−1
m−1 n−1
n−k−1
(xi − t)+
(b − t)n−k−1 X X
(b − t)n X
−
−
Ak,m
Ak,i
(1) Mn (t) =
n!
(n − k − 1)!
(n − k − 1)!
i=1 k=0
k=0
be the monospline of degree n and let
(2)
Z
b
f (t)dt =
a
n−1
m X
X
Ak,i f (k) (xi ) + R[f ] ,
i=0 k=0
be the quadrature formula. Between the monospline (1) and the quadrature
n−1 m
formula (2) there is a connection, namely, the coefficients {Ak,i }k=0
i=1 of
the quadrature formula are the same with the coefficients of monospline
(n−k−1)
(1), Ak,0 = (−1)n−k−1 Mn
(a) , k = 0, n − 1 and the remainder term of
quadrature formula have the representation :
Z b
R[f ] =
Mn (t)f (n) (t)dt , f ∈ W1n [a, b].
a
Definition 3.Functions of the form
Mn (t) = v(t) + sn−1 (t) ,
where sn−1 (t) is a spline of degree n − 1 and v is the nth integral of weight
function w : [a, b] → R, are called generalized monosplines.
If we choose the weight functions w(t) = (b − t)(t − a), then we obtain the
generalized monosplines
(3) Mn (t) = (a−b)
n−1
X
(t−b)n+2
(b−t)n−k−1
(t−b)n+1
−2
+(−1)n−1 Ak,m
(n+1)!
(n+2)!
(n−k−1)!
k=0
+ (−1)n−1
m−1
n−1
XX
i=1 k=0
Ak,i
n−k−1
(xi − t)+
.
(n − k − 1)!
Monosplines and inequalities for the remainder term ...
83
Between the monospline (3) and the quadrature formula
Z b
m X
n−1
X
(4)
w(t)f (t)dt =
Ak,i f (k) (xi ) + R[f ] ,
a
i=0 k=0
n−1 m
there is a connection, namely, the coefficients {Ak,i }k=0
i=1 of the quadrature formula are the same with the coefficients of monospline (3), Ak,0 =
(n−k−1)
(−1)k+1 Mn
(a) , k = 0, n − 1 and the remainder term of quadrature
formula has the representation :
Z b
n
Mn (t)f (n) (t)dt , f ∈ W1n [a, b].
R[f ] = (−1)
a
In the paper [1], [2], [3], [4] and [5] are considered generalization of the
trapezoid, mid-point and Simpson′ s quadrature rule. For example, in [3] is
studied a generalization of the mid-point quadrature rule:
µ
¶
Z b
Z b
n−1
X
£
¤ (b−a)k+1 (k) a+b
n
k
Kn (t)f (n) (t)dt ,
+(−1)
f
f (t)dt =
1+(−1)
k+1
2 (k+1)!
2
a
a
k=0
where

·
¸
n
(t
−
a)
a
+
b


, t ∈ a,

n!
2
µ
Kn (t) =
n
a+b
(t − b)


, t∈
, b]

n!
2
We observe that for n = 1 we get the mid-point rule
¶ Z b
µ
Z b
a+b
−
K1 (t)f ′ (t)dt .
f (t)dt = (b − a)f
2
a
a
We will study the case of the quadrature formula with the weight function
w(t) = (b − t)(t − a).
2
Main results
Lemma 1. If f ∈ W1n [a, b], then
Z b
n−1
X
£
¤
(5)
w(t)f (t)dt =
(−1)k +1 ·
a
k=0
(b−a)k+3
f (k)
k+2
2 (k+1)!(k+3)
µ
¶
a+b
+R[f ] ,
2
84
Ana Maria Acu
where w(t) = (b − t)(t − a)
n
(6)
R[f ] = (−1)
Z
b
Mn (t)f (n) (t)dt
a
and

¶
n+1
n+2
(t
−
a)
(t
−
a)
a
+
b


−2
, t ∈ [a,
 (b − a)
(n + 1)!
(n + 2)!
2 ¸
·
.
(7) Mn (t) =
n+2
n+1
a+b
(t − b)
(t − b)


−2
, t∈
,b
 (a − b)
(n + 1)!
(n + 2)!
2
Proof. We denoting
(t − a)n+2
(t − a)n+1
−2
and
Pn (t) = (b − a)
(n + 1)!
(n + 2)!
(t − b)n+2
(t − b)n+1
−2
Qn (t) = (a − b)
(n + 1)!
(n + 2)!
we observe that successive integration by parts yields the relation
Z b
Z a+b
Z b
2
(n)
n
(n)
n
n
Mn (t)f (t)dt = (−1)
(−1)
Pn (t)f (t)dt+(−1)
Qn (t)f (n) (t)dt
a
= (−1)n
n
+(−1)
a+b
2
a
n−1
X
k=0
n−1
X
¯ a+b
(−1)n−1−k Pn(n−1−k) (t)f (k) (t)¯a 2 +
n−1−k
(−1)
k=0
¯b
Q(n−1−k)
(t)f (k) (t)¯ a+b
n
2
+
Z
Z
a+b
2
Pn(n) (t)f (t)dt
a
b
a+b
2
Q(n)
n (t)f (t)dt
¯ a+b
·
¸
n−1
X
(t − a)k+3 (k) ¯¯ 2
(t − a)k+2
k+1
(−1)
−2
=
(b − a)
f (t)¯
(k
+
2)!
(k
+
3)!
a
k=0
¯
·
¸
Z
n−1
b
b
X
(t−b)k+2
(t−b)k+3 (k) ¯¯
k+1
(−1)
+
(a−b)
−2
f (t)¯ + (b−t)(t−a)f (t)dt
(k+2)!
(k+3)!
a+b
a
k=0
2
µ
¶
Z b
n−1
X£
¤
(b − a)k+3
a+b
k
(k)
=−
(−1) + 1 · k+2
f
+
w(t)f (t)dt ,
2
(k
+
1)!(k
+
3)
2
a
k=0
namely
µ
¶
Z b
n−1
X
£
¤
(b − a)k+3
a+b
k
(k)
w(t)f (t)dt =
(−1) + 1 · k+2
f
2 (k + 1)!(k + 3)
2
a
k=0
Z b
+ (−1)n
Mn (t)f (n) (t)dt .
a
Monosplines and inequalities for the remainder term ...
85
Remark 1.To observe that the quadrature formula (5) it is the open type.
Remark 2.If for the generalized monospline (3) we consider the particular
£
¤
a+b
, x2 = b, Ak,2 = 0, Ak,1 = (−1)k + 1 ·
case m = 2, x0 = a, x1 =
2
(b − a)k+3
, k = 0, n − 1 , we will have
2k+2 (k + 1)!(k + 3)
(t − b)n+2
(t − b)n+1
−2
+
(n + 1)!
(n + 2)!
¶n−k−1
¶k+3 µ
µ
¶µ
n−1
¤
2(−1)n−1 X £
a+b
b−a
n+2
k
.
−t
(−1) + 1 (k+2)
k+3
(n + 2)! k=0
2
2
+
¶
a+b
, then
If t ∈ [a,
2
Mn (t) = (a − b)
Mn (t) = (a − b)
(t − b)n+1
(t − b)n+2 2(−1)n−1
−2
+
(n + 1)!
(n + 2)!
(n + 2)!
¸
·
b−a
b−a
n+1
n+1
n+2
n+2
(b−t) +(n + 2)
(a − t)
− (b − t)
+ (a − t)
· (n + 2)
2
2
= (b − a)
(t − a)n+1
(t − a)n+2
−2
.
(n + 1)!
(n + 2)!
¸
a+b
, b , then
If t ∈
2
·
(t − b)n+1
(t − b)n+2
Mn (t) = (a − b)
−2
.
(n + 1)!
(n + 2)!
Theorem 1. The generalized monospline of degree n , Mn (t), n > 1, defined
in (7), verifies
Z b
(8)
Mn (t)dt = 0 , if n is odd ,
a
Z b
(b − a)n+3
(9)
,
|Mn (t)| dt = n+1
2 (n + 1)!(n + 3)
a
(b − a)n+2
(10)
.
max |Mn (t)| = n+1
t∈[a,b]
2 n!(n + 2)
86
Ana Maria Acu
Proof. We have
Z b
Mn (t)dt =
a
Z
a+b
2
Pn (t)dt +
a
Z
b
a+b
2
Qn (t)dt =
·
¸¯ a+b ·
¸¯b
(t−a)n+2
(t−a)n+3 ¯¯ 2
(t−b)n+3 ¯¯
(t−b)n+2
= (b−a)
−2
−2
+ (a−b)
(n+2)!
(n+3)! ¯a
(n+2)!
(n+3)! ¯ a+b
2
= [1 + (−1)n ]
If n is odd, then
Z
(b − a)n+3
.
2n+2 (n + 1)!(n + 3)
b
Mn (t)dt = 0 .
a
Z
b
|Mn (t)| dt =
a+b
2
a
a+b
2
|Pn (t)| dt +
a
a
Z
Z
Z
b
a+b
2
|Qn (t)| dt =
¸ Z b ·
¸
·
(t−a)n+2
(b−t)n+2
(b−t)n+1
(t−a)n+1
dt+
dt
−2
−2
(b−a)
(b−a)
a+b
(n+1)!
(n+2)!
(n+1)!
(n+2)!
2
=
(b − a)n+3
.
2n+1 (n + 1)!(n + 3)
(
max |Mn (t)| = max
)
max |Pn (t)| , max |Qn (t)|
,b]
t∈[ a+b
2
¯
¶
¶¯¾
½ µ
µ
a + b ¯¯
a + b ¯¯
(b − a)n+2
, ¯Qn
.
= max Pn
=
¯
2
2
2n+1 n!(n + 2)
t∈[a,b]
t∈[a, a+b
2 ]
Theorem 2. If f ∈ W1n [a, b] , n > 1 and there exist numbers γn , Γn such
that γn ≤ f (n) (t) ≤ Γn , t ∈ [a, b], then
(11)
|R[f ]| ≤
Γn − γ n
(b − a)n+3
·
, if n is odd
2n+2
(n + 1)!(n + 3)
and
(12)
|R[f ]| ≤
° (n) °
(b − a)n+3
°f ° , if n is even.
∞
2n+1 (n + 1)!(n + 3)
Monosplines and inequalities for the remainder term ...
87
Proof. Let n be odd. Using relations (6) and (8) we can written
n
R[f ] = (−1)
Z
b
Mn (t)f
(n)
n
(t)dt = (−1)
Z
a
a
b
·
¸
γn + Γn
(n)
Mn (t) f (t) −
dt
2
such that we have
(13)
We also have
(14)
¯Z b
¯
¯
¯ (n)
γ
+
Γ
n
n
¯
|Mn (t)| dt .
|R[f ]| ≤ max ¯¯f (t) −
t∈[a,b]
2 ¯ a
¯
¯
¯ (n)
γn + Γn ¯¯ Γn − γn
¯
max f (t) −
≤
.
t∈[a,b] ¯
2 ¯
2
From (9), (13) and (14) we have
|R[f ]| ≤
(b − a)n+3
Γn − γ n
·
.
2n+2
(n + 1)!(n + 3)
Let n be even. Then we have
Z b
° (n) °
°
°
·
|Mn (t)| dt =
|R[f ]| ≤ f
∞
a
° (n) °
(b − a)n+3
°f ° .
∞
2n+1 (n + 1)!(n + 3)
Theorem 3. Let f ∈ W1n [a, b], n > 1 and let n be odd. If there exist a real
number γn such that γn ≤ f (n) (t), then
(15)
|R[f ]| ≤ (Tn − γn ) ·
where
Tn =
(b − a)n+3
2n+1 n!(n + 2)
f (n−1) (b) − f (n−1) (a)
.
b−a
If there exist a real number Γn such that f (n) (t) ≤ Γn , then
(16)
|R[f ]| ≤ (Γn − Tn ) ·
(b − a)n+3
.
2n+1 n!(n + 2)
88
Ana Maria Acu
Proof. Using relation (8) we can written
¯Z b
¯
¯
¯
¡ (n)
¢
|R[f ]| = ¯¯
f (t) − γn Mn (t)dt¯¯ .
a
From (10) we have
|R[f ]| ≤ max |Mn (t)| ·
t∈[a,b]
Z
a
b
¡
¢
f (n) (t) − γn dt
£ (n−1)
¤
(b−a)n+3
(b−a)
(n−1)
(Tn −γn) .
f
(b)−f
(a)−γ
(b−a)
=
n
2n+1 n!(n+2)
2n+1 n!(n+2)
In a similar way we can prove that (16) holds.
Now, we will study the case of the quadrature formula of close type,
with the weight function w(t) = (b − t)(t − a).
n+2
=
Lemma 2. If f ∈ W1n [a, b], then
¶k+3
Z b
n−1 µ
X
b−a
1
3k + 10
w(t)f (t)dt =
f (k) (a)
4
k!
(k
+
2)(k
+
3)
a
k=0
¶
µ
¶
µ
n−1
X£
¤ b−a k+3
1 3k+11 (k) a+b
k
f
(17)
(−1) +1
+
4
(k+1)!
k+3
2
k=0
µ
¶k+3
n−1
X
b−a
1
3k + 10
k
+
(−1)
f (k) (b) + R[f ] ,
4
k!
(k
+
2)(k
+
3)
k=0
where w(t) = (b − t)(t − a)
(18)
n
R[f ] = (−1)
Z
b
Mn (t)f (n) (t)dt
a
and
(19)
 µ ¶2 µ
¶n
µ
¶n+1
µ
¶n+2
b−a
2
3a+b
3a+b
3a+b
3 b−a


+
−
,
t−
t−
t−



n! 4
4
2(n+1)!
4
(n+2)!
4

¶


a+b


t ∈ [a,

µ ¶2 µ
¶n
µ
¶n+1
µ
¶2n+2
Mn (t) =

3 b−a
a−b
2
a+3b
a+3b
a+3b


+
−
,
t−
t−
t−


n! 4
4
2(n+1)!
4
(n+2)! · 4

¸



a+b


,b
t∈
2
Monosplines and inequalities for the remainder term ...
89
Proof. We denoting
µ ¶2 µ
¶n
µ
¶n+1
µ
¶n+2
3 b−a
b−a
2
3a+b
3a+b
3a+b
Pn (t) =
+
−
,
t−
t−
t−
n! µ 4 ¶ µ 4 ¶ 2(n+1)! µ 4 ¶
(n+2)! µ 4 ¶
2
n
n+1
n+2
a+3b
a+3b
a+3b
a−b
2
3 b−a
t−
t−
t−
+
−
,
Qn (t) =
n! 4
4
2(n+1)!
4
(n+2)!
4
we observe that successive integration by parts yields the relation
Z a+b
Z b
Z b
2
(n)
n
(n)
n
n
Qn (t)f (n)(t)dt
(−1)
Mn (t)f (t)dt = (−1)
Pn (t)f (t)dt+(−1)
a
= (−1)n
n
+(−1)
n−1
X
a+b
2
a
n−1
X
k=0
n−1
X
¯ a+b
(−1)n−1−k Pn(n−1−k) (t)f (k) (t)¯a 2 +
n−1−k
(−1)
k=0
"
¯b
Q(n−1−k)
(t)f (k) (t)¯ a+b
n
2
+
Z
Z
a+b
2
Pn(n) (t)f (t)dt
a
b
a+b
2
Q(n)
n (t)f (t)dt
µ
¶2 µ
¶k+1
b−a
3a + b
3
k+1
t−
(−1)
=
(k
+
1)!
4
4
k=0
¯ a+b
µ
¶k+2
µ
¶k+3 #
¯ 2
3a + b
3a + b
2
b−a
¯
t−
t−
f (k) (t)¯
−
+
¯
2(k + 2)!
4
(k + 3)!
4
a
"
µ
¶2 µ
¶k+1
µ
¶k+2
n−1
X
a+3b
a+3b
b−a
a−b
3
k+1
t−
t−
+
(−1)
+
(k+1)!
4
4
2(k+2)!
4
k=0
¯
#
b
µ
¶k+3
Z b
¯
a + 3b
2
¯
(k)
t−
f (t)¯
−
+
(b − t)(t − a)f (t)dt =
¯ a+b
(k + 3)!
4
a
2
¶k+3
n−1 µ
X
b−a
1
3k + 10
−
f (k) (a)
4
k!
(k
+
2)(k
+
3)
k=0
¶
µ
¶
µ
n−1
X£
¤ b − a k+3
1
3k + 11 (k) a + b
k
f
−
(−1) + 1
4
(k
+
1)!
k
+
3
2
k=0
¶k+3
µ
Z b
n−1
X
1
3k + 10
b−a
(k)
k
f (b) +
w(t)f (t)dt .
−
(−1)
4
k! (k + 2)(k + 3)
a
k=0
Theorem 4. The generalized monospline of degree n , Mn (t), n > 1, defined
90
Ana Maria Acu
in (19), verifies
Z b
(20)
Mn (t)dt = 0 , if n is odd ,
a
µ
¶n+3
Z b
b−a
4
|Mn (t)| dt =
(21)
(3n2 + 15n + 16) ,
4
(n
+
3)!
a
¶n+2
µ
1
b−a
(22)
(3n2 + 11n + 8) .
max |Mn (t)| =
t∈[a,b]
4
(n + 2)!
Proof. We have
Z b
Mn (t)dt =
a
Pn (t)dt +
a
n
= [1 + (−1) ]
Z
If n is odd, then
a+b
2
Z
b
µ
b−a
4
¶n+3
Z
b
a+b
2
Qn (t)dt =
2
(3n2 + 15n + 16) .
(n + 3)!
Mn (t)dt = 0 .
a
Z
b
|Mn (t)| dt =
Z
a+b
2
|Pn (t)| dt +
a
a
=
µ
b−a
4
¶n+3
max |Mn (t)| = max
Z
b
a+b
2
|Qn (t)| dt
4
(3n2 + 15n + 16) .
(n + 3)!
(
)
max |Pn (t)| , max |Qn (t)|
,b]
t∈[ a+b
2
¶ ¯ µ
¶¯¾ µ
¶n+2
½ µ
1
a+b ¯¯
b−a
a + b ¯¯
(3n2 + 11n + 8) .
, ¯Qn
= max Pn
=
¯
2
2
4
(n + 2)!
t∈[a,b]
t∈[a, a+b
2 ]
Theorem 5. If f ∈ W1n [a, b] , n > 1 and there exist numbers γn , Γn such
that γn ≤ f (n) (t) ≤ Γn , t ∈ [a, b], then
µ
¶n+3
4
Γn − γ n b − a
(3n2 + 15n + 16) , if n is odd
(23) |R[f ]| ≤
2
4
(n + 3)!
and
(24) |R[f ]| ≤
µ
b−a
4
¶n+3
°
°
4
(3n2 + 5n + 16) °f (n) °∞ , if n is even.
(n + 3)!
Monosplines and inequalities for the remainder term ...
91
Proof. If n is odd, then we can written
Γn − γ n
|R[f ]| ≤
2
Z
a
b
µ
Γn −γn
|Mn (t)| dt =
2
b−a
4
¶n+3
4
(3n2+15n+16) .
(n+3)!
Let n be even. Then we have
¶n+3
µ
Z b
°
°
° (n) °
4
b−a
(3n2+15n+16) °f (n) °∞ .
|R[f ]| ≤ °f °∞ ·
|Mn (t)| dt =
4
(n+3)!
a
Theorem 6. Let f ∈ W1n [a, b], n > 1 and let n be odd. If there exist a real
number γn such that γn ≤ f (n) (t), then
(25)
|R[f ]| ≤ (Tn − γn ) ·
µ
b−a
4
¶n+3
·
4
· (3n2 + 11n + 8)
(n + 2)!
where
Tn =
f (n−1) (b) − f (n−1) (a)
.
b−a
If there exist a real number Γn such that f (n) (t) ≤ Γn , then
(26)
|R[f ]| ≤ (Γn − Tn ) ·
µ
b−a
4
¶n+3
·
4
· (3n2 + 11n + 8) .
(n + 2)!
Proof. We have
|R[f ]| ≤ max |Mn (t)| ·
t∈[a,b]
=
µ
b−a
4
¶n+2
Z
a
b
¡ (n)
¢
f (t) − γn dt
£
¤
1
(3n2 + 11n + 8) f (n−1) (b)−f (n−1) (a)−γn (b−a)
(n + 2)!
= (Tn − γn ) ·
µ
b−a
4
¶n+3
·
4
· (3n2 + 11n + 8) .
(n + 2)!
In a similar way we can prove that (26) holds.
92
Ana Maria Acu
References
[1] G.A.Anastassiou,
Ostrowski Type Inequalities,
Math.Soc.,Vol 123,No 12,(1995),3775-3781.
Proc.
Amer.
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Point of View, Handbook of Analytic-Computational Methods in Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York,
(2000),135-200.
[3] P. Cerone and S.S.Dragomir, Trapezoidal-type Rules from an Inequalities Point of View, Handbook of Analytic-Computational Methods in
Applied Mathematics, Editor: G. Anastassiou, CRC Press, New York,
(2000),65-134.
[4] Lj. Dedić, M. Matić and J.Pečarić, On Euler trapezoid formulae, Appl.
Math. Comput.,123(2001),37-62.
[5] C.E.M. Pearce, J.Pečarić, N. Ujević and S. Varošanec, Generalizations
of some inequalities of Ostrowski-Grüss type, Math. Inequal. Appl., 3(1),
(2000), 25-34.
[6] Nenad Ujević, Error Inequalities for a Generalized Quadrature Rule ,
General Mathematics, Vol. 13, No. 4(2005), 51-64.
University ”Lucian Blaga” of Sibiu
Department of Mathematics
Str. Dr. I. Rat.iu, No. 5-7
550012 - Sibiu, Romania
e-mail: acuana77@yahoo.com