Received 31/05/11 - RGMIA
Received 31/05/11
A Three Point Quadrature Rule for Functions of Bounded Variation and Applications S.S. Dragomir1;2 and E. Momoniat1 Abstract. A three point quadrature rule approximating the Riemann integral for a function of bounded variation f by a linear combination with real coef…cients of the values f (a) ; f (x) and f (b) with x 2 [a; b] whose sum equalizes b a is given. Applications for special means inequalities and in establishing a priory error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families are provided as well.
1. Introduction In 1999, see [7, Proposition 2] or [11, p. 11], S.S. Dragomir has obtained the following bound for the three point approximation of the Riemann integral Rb f (t) dt a Z b (1.1) f (t) dt ( a) f (a) ( ) f (x) (b ) f (b) a
1 (b 4 +
1 2
where a [a; b] while
1 2
x
a+b + 2 x+b 2
a+x 2 x
b _
a) +
a+x + 2 b _
x+b 2
(f )
a
b and f : [a; b] ! R is a function of bounded variation on
(f ) denotes the total variation of f on [a; b] :
a
For = a and = b; we get from (1.1) the following Ostrowski’s type inequality …rstly obtained in 1999 in [7] Z b b a+b _ 1 (1.2) (b a) + x (f ) f (t) dt (b a) f (x) 2 2 a a for all x 2 [a; b] : The constant
1 2
cannot be replaced by a smaller quantity.
1991 Mathematics Subject Classi…cation. 41A51, 26D15, 47A63, 47A99. Key words and phrases. Three point rules, Quadratures, Integral inequalities, Special means, Selfadjoint operators in Hilbert Spaces, Spectral families. 1
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
2
The best inequality one can get from (1.2) is the following midpoint inequality Z b b _ a+b 1 f (t) dt (b a) f (1.3) (b a) (f ) : 2 2 a a
Here the constant 12 is also best. For some recent Ostrowski’s type inequalities, see [12], [13], [14], [15], [16], [17], [18] and the references therein. If = x = , then we get from (1.1) the following generalized trapezoidal inequality also obtained in 1999 [7, Proposition 1] Z b (1.4) f (t) dt (x a) f (a) (b x) f (b) a
1 (b 2
a) + x
a+b 2
b _
(f )
a
for all x 2 [a; b] : The constant 21 cannot be replaced by a smaller quantity. The best inequality one can get from (1.2) is the following trapezoidal inequality Z b b _ 1 f (a) + f (b) (1.5) f (t) dt (b a) (b a) (f ) : 2 2 a a Here the constant 21 is also best. For recent results on trapezoidal inequality, see [1], [3], [4], [10] and the references therein. Now, if we take x = a+b 2 in (1.1), then we get the inequality Z b a+b (1.6) f (t) dt ( a) f (a) ( )f (b ) f (b) 2 a 1 (b 4 +
1 2
a) +
1 2
3a + b 4
3a + b + 4 a + 3b 4
a + 3b 4 b _
(f )
a
a+b b: The best inequality one can obtain from (1.6), as pointed for a 2 out by Cerone and Dragomir in [11, p. 202], is obtained for = 3a+b and = a+3b 4 4 and has the form Z b b _ b a a+b f (a) + f (b) 1 (1.7) f (t) dt f + (b a) (f ) : 2 2 2 4 a a
The constant 41 is best possible in (1.7). For other three point quadrature rules with positive coe¢ cients see [2], [5], [6], [9] and the references therein. We observe that the three point quadrature formula ( a) f (a)+( ) f (x)+ Rb (b ) f (b) approximating the Riemann integral a f (t) dt has nonnegative coef…cients since a x b: The sum of these coe¢ cients is (b a) : It is therefore natural to put the more general question of approximating Rb f (t) dt by a linear combination with real coe¢ cients of the values f (a) ; f (x) a
A THREE POINT QUADRATURE RULE AND APPLICATIONS
3
and f (b) with x 2 [a; b] whose sum equalize the same (b a) : Some results that address this question for functions of bounded variation are presented below. Out of these results, some are applied for special means inequalities and in establishing a priory error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families.
2. The Results The …rst result is: Theorem 1. Let f : [a; b] ! C be a function of bounded variation on [a; b] and ; ; be real numbers with + + = 6 0: Then for any x 2 [a; b] we have the inequalities f (a) + f (x) + f (b) + +
(2.1)
max
+ max := B
+
+
1 b
; ;
x
a
;
Z
1 b
1 b
a
x
a
b
f (t) dt
a
( + )a + b + +
a + ( + )b ; + +
+
x _
b _
+
(f )
a
(f )
x
(a; b; x)
where (2.2)
B
; ;
b _
(a; b; x)
(f ) max
+
a
1 b
a
(a; b; x)
"
x
;
+
+
+
( + )a + b 1 ; x + + b a
; a + ( + )b + +
and (2.3)
B
; ;
b x _ 1_ (f ) + (f ) 2 a a
max + max
and
b _
+ 1 b
a
+ x
b _ x
;
1
(f )
#
x
b a a + ( + )b ; + +
(f ) denotes the total variation of f on [a; b] :
a
As a particular case of interest we have:
( + )a + b + + +
+
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
4
Corollary 1. With the assumptions of Theorem 1 we have the inequalities f (a) + f +
(2.4)
a+b 2
+ f (b)
+
1 b
a
Z
1 max fj + 2j + + j
:= C
f (t) dt
a a+b 2
1 max f2 j j ; j + 2j + + j +
b
_
jg j ; 2 j jg
(f )
a
b _
(f )
a+b 2
(a; b)
; ;
where C
(2.5)
; ;
1 2j + + j
(a; b)
max f2 j j ; 2 j j ; j +
j;j +
jg
b _
(f ) ;
a
and (2.6)
C
2
1 4 2j + + j 2
(a; b)
; ;
b 1_
a+b 2
(f ) +
a
(2.7)
1
f (x)
b
a
Z
=
b
x b
f (t) dt
a
b _
(f )
a
[max f2 j j ; j + Remark 1. We observe that, if get the Ostrowski type inequality
_
a+b 2
jg + max fj +
3
(f ) 5
j ; 2 j jg] :
= 0 in the inequality in (2.1), then we
a a
x _
(f ) +
a
b b
x a
b _
(f )
x
for any x 2 [a; b]. Since (x
a)
x _
(f ) + (b
x)
a
b _
1 (b 2
(f )
x
a) + x
b _
a+b 2
(f )
a
and (x
a)
x _ a
(f ) + (b
x)
b _ x
(f )
"
b x _ 1_ (f ) + (f ) 2 a a
then we get from (2.7) the known result (1.2) and " b Z b x _ 1 1_ f (t) dt (f ) + (2.8) f (x) (f ) b a a 2 a a for all x 2 [a; b] :
b _
#
(b
(f )
#
(f )
x
b _ x
a)
A THREE POINT QUADRATURE RULE AND APPLICATIONS
5
If v 2 [a; b] is a median point in the sense of bounded variation for the function v b _ _ f on [a; b] ; namely (f ) = (f ) ; then we also have a
v
1
f (v)
(2.9)
b
a
Remark 2. We notice that if (2.10)
+
f (a) + f (b) + 2
Z
b
1_ (f ) : 2 a
b
f (t) dt
a
= 2 and +
f
=
in (2.4), then Z b 1 f (t) dt b a a
a+b 2
b
1 max fj j ; j jg _ (f ) 2 j + j a
where ; 2 R with + 6= 0: In particular, for = = 1 we get from (2.10) the known inequality (1.7). If we take in (2.10) = 2 and = 1; then we get " # Z b b a+b 1_ 1 f (a) + f (b) 1 f + f (t) dt (f ) (2.11) 2 2 2 b a a 2 a The inequality (2.11) is sharp. For the choice = 2; = 4 we get from (2.10) that " # Z b 1 1 f (a) + f (b) a+b + (2.12) f (t) dt f 2 2 b a a 2 The inequality (2.12) is sharp.
b
1_ (f ) : 2 a
3. Proofs We use the Montgomery type identity established in [8] to write for the functions of bounded variation f : [a; b] ! C that (3.1)
f (x) =
1 b
a
Z
b
f (t) dt +
a
1 b
a
Z
b
K (t; x) df (t)
a
for any x 2 [a; b], where the second integral is taken in the Riemann-Stieltjes sense and the kernel K is de…ned as K : [a; b] [a; b] ! R with 8 < t a for a t x; K (t; x) = : t b for x < t b: Writing the representation for a and b we have Z b Z b 1 1 f (t) dt + (t (3.2) f (a) = b a a b a a and
(3.3)
f (b) =
1 b
a
Z
a
b
f (t) dt +
1 b
a
Z
a
b) df (t)
b
(t
a) df (t) :
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
6
Now, if we multiply (3.2) with ; (3.1) with , (3.3) with ; add the obtained equalities and divide the sum with + + 6= 0 we deduce the more general three point representation f (a) + f (x) + f (b) 1 = + + b a
(3.4)
for any x 2 [a; b] ; where the kernel K K
; ;
(t; x) =
8 > < t > : t
Z
b
f (t) dt +
a
; ;
: [a; b]
( + )a+ b + + a+( + )b + +
1 b
a
Z
b
K
; ;
(t; x) df (t)
a
[a; b] ! R is given by for a
t
x;
for x < t
b:
It is well known that if p : [c; d] ! C is a continuous function and v : [c; d] ! C Rd is of bounded variation, then the Riemann-Stieltjes integral c p (t) dv (t) exists and the following inequality holds Z
(3.5)
d
p (t) dv (t)
c
where
d _
max jp (t)j
t2[c;d]
d _
(v)
c
(v) denotes the total variation of v on [c; d] :
c
Utilizing (3.4) and (3.7), we have successively (3.6)
Z b 1 f (a) + f (x) + f (b) f (t) dt + + b a a Z x 1 ( + )a + b df (t) t b a a + + Z b 1 a + ( + )b + t df (t) b a x + + 1 b +
a t2[a;x] 1
b
max t
max t
a t2[x;b]
x ( + )a + b _ (f ) + + a b a + ( + )b _ (f ) + + x
for any x 2 [a; b] ; which is an inequality of interest in itself. Further, we observe that for any c a real number we have the equality (3.7)
max jt
t2[a;b]
cj = max fjc
aj ; jb
cjg :
Indeed, if c < a then maxt2[a;b] jt cj = b c and max fjc aj ; jb cjg = b c; if and if c 2 [a; b] then maxt2[a;b] jt cj = max fc a; b cg = 12 (b a) + c a+b 2 c > b then maxt2[a;b] jt cj = c a and max fjc aj ; jb cjg = c a:
A THREE POINT QUADRATURE RULE AND APPLICATIONS
7
Now, on making use of (3.7) we have (3.8)
max t
t2[a;x]
= max = max
( + )a + b + + ( + )a + b a ; x + + +
+
(b
a) ; x
( + )a + b + + ( + )a + b + +
and (3.9)
max t
t2[x;b]
= max = max
a+( + + + a+( x + a+( x +
)b + )b a + ( + )b ; b + + + + )b ; (b a) : + + +
Therefore (3.6), (3.8) and (3.9) produce the inequality in (2.1). The inequalities (2.2) and (2.3) follow by the elementary fact that my + nz
(m + n) max fy; zg
where m; n; y; z are nonnegative real numbers. Now, in order to prove the sharpness of the inequality (2.11), assume that there exists a C > 0 such that " # Z b b _ f (a) + f (b) 1 a+b 1 (3.10) f + f (t) dt C (f ) 2 2 2 b a a a holds for any function of bounded variation and on any interval [a; b]. If we choose f : [a; b] ! R, 8 < 1 if t = a; 0 if t 2 (a; b) f (t) = : 1 if t = b
which is of bounded variation on [a; b] then we get from (3.10) that 1 2C, which proves the sharpness of the inequality. Similarly, in order to prove the sharpness the inequality (2.12), if we assume that there exists a constant D > 0 such that " # Z b b _ 1 f (a) + f (b) 1 a+b + f (t) dt D (f ) : (3.11) f 2 2 2 b a a a If we consider the function f : [a; b] ! R given by 8 < 1 if t 2 [a; a+b 2 ); f (t) = : 1 if t 2 a+b 2 ;b
then f is of bounded variation on [a; b] ; and by (3.11) we get 1 that D 12 :
2D which implies
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
8
4. A Compounding Rule We consider the following partition of the interval [a; b] ;
n
: a = x0 < x1 < ::: < xn
1
< xn = b:
De…ne hk := xk+1 xk , 0 k n 1 and ( n ) = max fhk : 0 k n 1g the norm of the partition n : In order to exemplify how we can use the above results in order to produce Rb compounding quadrature rules to approximate the integral a f (t) dt; we consider for ; 2 R with + 6= 0; the two parameters family of three point quadrature rules
(4.1)
Tn (f;
n ; ; ) :=
+
+
+
n X1 k=0
n X1 k=0
f
f (xk ) + f (xk+1 ) hk 2 xk + xk+1 2
hk :
We notice that the family of quadrature rules (4.1) contain the trapezoid rule ( = 0), the midpoint rule ( = 0), the Simpson rule ( = 1; = 2) and the arithmetic mean of the trapezoid and midpoint rules ( = 1; = 1). The following proposition provides a priory error bounds in approximating the Rb integral a f (t) dt of the bounded variation f by the compounding quadrature rule Tn (f; n ; ; ) :
and
Proposition 1. Let f : [a; b] ! C be a function of bounded variation on [a; b] ; be real numbers with + 6= 0: Then
(4.2)
Z
b
f (t) dt = Tn (f;
n;
; ) + Rn (f;
n;
; )
a
and the remainder Rn (f;
(4.3)
jRn (f;
n;
n;
; ) satis…es the bounds
; )j
xk+1 n 1 _ 1 max fj j ; j jg X hk (f ) 2 j + j x k=0
1 max fj j ; j jg ( 2 j + j
k
n)
b _ a
(f ) :
A THREE POINT QUADRATURE RULE AND APPLICATIONS
9
Proof. Utilizing the generalized triangle inequality and (2.10) we have successively that jRn (f;
n;
n X1 Z
=
k=0
+ n X1 k=0
+
; )j f (t) dt
xk
n X1
f
k=0
Z
n X1
xk+1
+
xk + xk+1 2
k=0
hk
xk+1
f (t) dt
xk
n X1 k=0
f
xk + xk+1 2
f (xk ) + f (xk+1 ) hk 2
+ hk
xk+1 n 1 _ 1 max fj j ; j jg X hk (f ) 2 j + j x k=0
f (xk ) + f (xk+1 ) hk 2
k
1 max fj j ; j jg ( 2 j + j
and the proof is complete.
n)
b _
(f ) ;
a
5. Applications for Special Means It is well-known that, if f : [a; b] ! R is a convex function, then the celebrated Hermite-Hadamard inequality state that Z b a+b 1 f (a) + f (b) (5.1) f f (t) dt : 2 b a a 2 Utilizing this fact and the inequalities (2.11) and (2.12) we can state the following result: Proposition 2. Assume that f : [a; b] ! R is a convex function on [a; b]. Then " # Z b b f (a) + f (b) 1 a+b 1 1_ f + f (t) dt (5.2) 0 (f ) 2 2 2 b a a 2 a
and
(5.3)
f=
0
" # Z b 1 1 f (a) + f (b) + f (t) dt 2 2 b a a
f
a+b 2
b
1_ (f ) : 2 a
The case for concave functions g is similar by applying these inequalities for g: Let us recall the following means: a) The arithmetic mean A (a; b) :=
a+b ; a; b > 0; 2
b) The geometric mean G (a; b) :=
p
ab; a; b
0;
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
10
c) The harmonic mean H (a; b) :=
1 a
2 ; a; b > 0; + 1b
d) The identric mean 8 > 1 > < e I (a; b) := > > : a
1
bb aa
b
if if
e) The logarithmic mean 8 b > < ln b L (a; b) := > : a
f) The p logarithmic mean 8 > bp+1 ap+1 > < (p + 1) (b a) Lp (a; b) := > > : a
a
a ln a
if if
b 6= a
; a; b > 0
b=a
b 6= a
; a; b > 0
b=a
1 p
if if
b 6= a; p 2 Rn f 1; 0g
; a; b > 0:
b=a
It is well known that, if L 1 := L and L0 := I, then the function R 3p ! Lp is monotonically strictly increasing. In particular, we have H (a; b)
G (a; b)
L (a; b)
I (a; b)
A (a; b) :
Now, if we consider the power function f : [a; b] (0; 1) ! R given by f (t) = tp then we observe that for p 2 ( 1; 0) [ [1; 1) the function is convex while for p 2 (0; 1) the function is concave. Now, if we apply the inequality (5.2) for the convex function f (t) = tp we can state that 1 p (5.4) 0 A (ap ; bp ) A (a; b) + Lpp (a; b) 2 8 p b ap if p 1 1 < 2 : p a bp if p 2 ( 1; 0) n f 1g : In the case of concave functions, the same inequality (5.2) produces the inequality 1 p 1 p A (a; b) + Lpp (a; b) A (ap ; bp ) (b 2 2 Now, if we consider the convex function f : [a; b] f (t) = 1t , then by (5.2) we also have
(5.5)
0
ap ) if p 2 (0; 1) : (0; 1) ! R given by
1 1 b a A 1 (a; b) + L 1 (a; b) : 2 2 ab Moreover the inequality (5.2) applied for the concave function f : [a; b] R given by f (t) = ln t produces the result (5.6)
0
0
H
1
(a; b)
1 [ln A (a; b) + ln I (a; b)] 2
ln G (a; b)
1 ln 2
b a
(0; 1) !
A THREE POINT QUADRATURE RULE AND APPLICATIONS
11
which is equivalent with (5.7)
r
p
A (a; b) I (a; b) G (a; b)
1
b : a
Similar results can be obtained if one uses the inequality (5.3), however the details are left to the interested reader. 6. Applications for Selfadjoint Operators in Hilbert Spaces Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U ) included in the interval [m; M ] for some real numbers m < M and let fE g be its spectral family. It is well known that we have the following spectral representation in terms of the Riemann-Stieltjes integral : Z M (6.1) U= dE ; m 0
which in terms of vectors can be written as Z M (6.2) hU x; yi = d hE x; yi ; m 0
for any x; y 2 H: The function gx;y ( ) := hE x; yi is of bounded variation on the interval [m; M ] and gx;y (m
0) = 0 and gx;y (M ) = hx; yi
for any x; y 2 H: It is also well known that gx ( ) := hE x; xi is monotonic nondecreasing and right continuous on [m; M ]. We can state and prove now the following result concerning the numerical approximation of a selfadjoint operator on the complex Hilbert space (H; h:; :i) : Theorem 2. Let A be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (A) included in the interval [m; M ] for some real numbers m < M and let fE g be its spectral family. We consider the following partition of the interval [m; M ] n
:m=
0
A Three Point Quadrature Rule for Functions of Bounded Variation and Applications S.S. Dragomir1;2 and E. Momoniat1 Abstract. A three point quadrature rule approximating the Riemann integral for a function of bounded variation f by a linear combination with real coef…cients of the values f (a) ; f (x) and f (b) with x 2 [a; b] whose sum equalizes b a is given. Applications for special means inequalities and in establishing a priory error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families are provided as well.
1. Introduction In 1999, see [7, Proposition 2] or [11, p. 11], S.S. Dragomir has obtained the following bound for the three point approximation of the Riemann integral Rb f (t) dt a Z b (1.1) f (t) dt ( a) f (a) ( ) f (x) (b ) f (b) a
1 (b 4 +
1 2
where a [a; b] while
1 2
x
a+b + 2 x+b 2
a+x 2 x
b _
a) +
a+x + 2 b _
x+b 2
(f )
a
b and f : [a; b] ! R is a function of bounded variation on
(f ) denotes the total variation of f on [a; b] :
a
For = a and = b; we get from (1.1) the following Ostrowski’s type inequality …rstly obtained in 1999 in [7] Z b b a+b _ 1 (1.2) (b a) + x (f ) f (t) dt (b a) f (x) 2 2 a a for all x 2 [a; b] : The constant
1 2
cannot be replaced by a smaller quantity.
1991 Mathematics Subject Classi…cation. 41A51, 26D15, 47A63, 47A99. Key words and phrases. Three point rules, Quadratures, Integral inequalities, Special means, Selfadjoint operators in Hilbert Spaces, Spectral families. 1
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
2
The best inequality one can get from (1.2) is the following midpoint inequality Z b b _ a+b 1 f (t) dt (b a) f (1.3) (b a) (f ) : 2 2 a a
Here the constant 12 is also best. For some recent Ostrowski’s type inequalities, see [12], [13], [14], [15], [16], [17], [18] and the references therein. If = x = , then we get from (1.1) the following generalized trapezoidal inequality also obtained in 1999 [7, Proposition 1] Z b (1.4) f (t) dt (x a) f (a) (b x) f (b) a
1 (b 2
a) + x
a+b 2
b _
(f )
a
for all x 2 [a; b] : The constant 21 cannot be replaced by a smaller quantity. The best inequality one can get from (1.2) is the following trapezoidal inequality Z b b _ 1 f (a) + f (b) (1.5) f (t) dt (b a) (b a) (f ) : 2 2 a a Here the constant 21 is also best. For recent results on trapezoidal inequality, see [1], [3], [4], [10] and the references therein. Now, if we take x = a+b 2 in (1.1), then we get the inequality Z b a+b (1.6) f (t) dt ( a) f (a) ( )f (b ) f (b) 2 a 1 (b 4 +
1 2
a) +
1 2
3a + b 4
3a + b + 4 a + 3b 4
a + 3b 4 b _
(f )
a
a+b b: The best inequality one can obtain from (1.6), as pointed for a 2 out by Cerone and Dragomir in [11, p. 202], is obtained for = 3a+b and = a+3b 4 4 and has the form Z b b _ b a a+b f (a) + f (b) 1 (1.7) f (t) dt f + (b a) (f ) : 2 2 2 4 a a
The constant 41 is best possible in (1.7). For other three point quadrature rules with positive coe¢ cients see [2], [5], [6], [9] and the references therein. We observe that the three point quadrature formula ( a) f (a)+( ) f (x)+ Rb (b ) f (b) approximating the Riemann integral a f (t) dt has nonnegative coef…cients since a x b: The sum of these coe¢ cients is (b a) : It is therefore natural to put the more general question of approximating Rb f (t) dt by a linear combination with real coe¢ cients of the values f (a) ; f (x) a
A THREE POINT QUADRATURE RULE AND APPLICATIONS
3
and f (b) with x 2 [a; b] whose sum equalize the same (b a) : Some results that address this question for functions of bounded variation are presented below. Out of these results, some are applied for special means inequalities and in establishing a priory error bounds for the approximation of selfadjoint operators in Hilbert spaces by spectral families.
2. The Results The …rst result is: Theorem 1. Let f : [a; b] ! C be a function of bounded variation on [a; b] and ; ; be real numbers with + + = 6 0: Then for any x 2 [a; b] we have the inequalities f (a) + f (x) + f (b) + +
(2.1)
max
+ max := B
+
+
1 b
; ;
x
a
;
Z
1 b
1 b
a
x
a
b
f (t) dt
a
( + )a + b + +
a + ( + )b ; + +
+
x _
b _
+
(f )
a
(f )
x
(a; b; x)
where (2.2)
B
; ;
b _
(a; b; x)
(f ) max
+
a
1 b
a
(a; b; x)
"
x
;
+
+
+
( + )a + b 1 ; x + + b a
; a + ( + )b + +
and (2.3)
B
; ;
b x _ 1_ (f ) + (f ) 2 a a
max + max
and
b _
+ 1 b
a
+ x
b _ x
;
1
(f )
#
x
b a a + ( + )b ; + +
(f ) denotes the total variation of f on [a; b] :
a
As a particular case of interest we have:
( + )a + b + + +
+
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
4
Corollary 1. With the assumptions of Theorem 1 we have the inequalities f (a) + f +
(2.4)
a+b 2
+ f (b)
+
1 b
a
Z
1 max fj + 2j + + j
:= C
f (t) dt
a a+b 2
1 max f2 j j ; j + 2j + + j +
b
_
jg j ; 2 j jg
(f )
a
b _
(f )
a+b 2
(a; b)
; ;
where C
(2.5)
; ;
1 2j + + j
(a; b)
max f2 j j ; 2 j j ; j +
j;j +
jg
b _
(f ) ;
a
and (2.6)
C
2
1 4 2j + + j 2
(a; b)
; ;
b 1_
a+b 2
(f ) +
a
(2.7)
1
f (x)
b
a
Z
=
b
x b
f (t) dt
a
b _
(f )
a
[max f2 j j ; j + Remark 1. We observe that, if get the Ostrowski type inequality
_
a+b 2
jg + max fj +
3
(f ) 5
j ; 2 j jg] :
= 0 in the inequality in (2.1), then we
a a
x _
(f ) +
a
b b
x a
b _
(f )
x
for any x 2 [a; b]. Since (x
a)
x _
(f ) + (b
x)
a
b _
1 (b 2
(f )
x
a) + x
b _
a+b 2
(f )
a
and (x
a)
x _ a
(f ) + (b
x)
b _ x
(f )
"
b x _ 1_ (f ) + (f ) 2 a a
then we get from (2.7) the known result (1.2) and " b Z b x _ 1 1_ f (t) dt (f ) + (2.8) f (x) (f ) b a a 2 a a for all x 2 [a; b] :
b _
#
(b
(f )
#
(f )
x
b _ x
a)
A THREE POINT QUADRATURE RULE AND APPLICATIONS
5
If v 2 [a; b] is a median point in the sense of bounded variation for the function v b _ _ f on [a; b] ; namely (f ) = (f ) ; then we also have a
v
1
f (v)
(2.9)
b
a
Remark 2. We notice that if (2.10)
+
f (a) + f (b) + 2
Z
b
1_ (f ) : 2 a
b
f (t) dt
a
= 2 and +
f
=
in (2.4), then Z b 1 f (t) dt b a a
a+b 2
b
1 max fj j ; j jg _ (f ) 2 j + j a
where ; 2 R with + 6= 0: In particular, for = = 1 we get from (2.10) the known inequality (1.7). If we take in (2.10) = 2 and = 1; then we get " # Z b b a+b 1_ 1 f (a) + f (b) 1 f + f (t) dt (f ) (2.11) 2 2 2 b a a 2 a The inequality (2.11) is sharp. For the choice = 2; = 4 we get from (2.10) that " # Z b 1 1 f (a) + f (b) a+b + (2.12) f (t) dt f 2 2 b a a 2 The inequality (2.12) is sharp.
b
1_ (f ) : 2 a
3. Proofs We use the Montgomery type identity established in [8] to write for the functions of bounded variation f : [a; b] ! C that (3.1)
f (x) =
1 b
a
Z
b
f (t) dt +
a
1 b
a
Z
b
K (t; x) df (t)
a
for any x 2 [a; b], where the second integral is taken in the Riemann-Stieltjes sense and the kernel K is de…ned as K : [a; b] [a; b] ! R with 8 < t a for a t x; K (t; x) = : t b for x < t b: Writing the representation for a and b we have Z b Z b 1 1 f (t) dt + (t (3.2) f (a) = b a a b a a and
(3.3)
f (b) =
1 b
a
Z
a
b
f (t) dt +
1 b
a
Z
a
b) df (t)
b
(t
a) df (t) :
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
6
Now, if we multiply (3.2) with ; (3.1) with , (3.3) with ; add the obtained equalities and divide the sum with + + 6= 0 we deduce the more general three point representation f (a) + f (x) + f (b) 1 = + + b a
(3.4)
for any x 2 [a; b] ; where the kernel K K
; ;
(t; x) =
8 > < t > : t
Z
b
f (t) dt +
a
; ;
: [a; b]
( + )a+ b + + a+( + )b + +
1 b
a
Z
b
K
; ;
(t; x) df (t)
a
[a; b] ! R is given by for a
t
x;
for x < t
b:
It is well known that if p : [c; d] ! C is a continuous function and v : [c; d] ! C Rd is of bounded variation, then the Riemann-Stieltjes integral c p (t) dv (t) exists and the following inequality holds Z
(3.5)
d
p (t) dv (t)
c
where
d _
max jp (t)j
t2[c;d]
d _
(v)
c
(v) denotes the total variation of v on [c; d] :
c
Utilizing (3.4) and (3.7), we have successively (3.6)
Z b 1 f (a) + f (x) + f (b) f (t) dt + + b a a Z x 1 ( + )a + b df (t) t b a a + + Z b 1 a + ( + )b + t df (t) b a x + + 1 b +
a t2[a;x] 1
b
max t
max t
a t2[x;b]
x ( + )a + b _ (f ) + + a b a + ( + )b _ (f ) + + x
for any x 2 [a; b] ; which is an inequality of interest in itself. Further, we observe that for any c a real number we have the equality (3.7)
max jt
t2[a;b]
cj = max fjc
aj ; jb
cjg :
Indeed, if c < a then maxt2[a;b] jt cj = b c and max fjc aj ; jb cjg = b c; if and if c 2 [a; b] then maxt2[a;b] jt cj = max fc a; b cg = 12 (b a) + c a+b 2 c > b then maxt2[a;b] jt cj = c a and max fjc aj ; jb cjg = c a:
A THREE POINT QUADRATURE RULE AND APPLICATIONS
7
Now, on making use of (3.7) we have (3.8)
max t
t2[a;x]
= max = max
( + )a + b + + ( + )a + b a ; x + + +
+
(b
a) ; x
( + )a + b + + ( + )a + b + +
and (3.9)
max t
t2[x;b]
= max = max
a+( + + + a+( x + a+( x +
)b + )b a + ( + )b ; b + + + + )b ; (b a) : + + +
Therefore (3.6), (3.8) and (3.9) produce the inequality in (2.1). The inequalities (2.2) and (2.3) follow by the elementary fact that my + nz
(m + n) max fy; zg
where m; n; y; z are nonnegative real numbers. Now, in order to prove the sharpness of the inequality (2.11), assume that there exists a C > 0 such that " # Z b b _ f (a) + f (b) 1 a+b 1 (3.10) f + f (t) dt C (f ) 2 2 2 b a a a holds for any function of bounded variation and on any interval [a; b]. If we choose f : [a; b] ! R, 8 < 1 if t = a; 0 if t 2 (a; b) f (t) = : 1 if t = b
which is of bounded variation on [a; b] then we get from (3.10) that 1 2C, which proves the sharpness of the inequality. Similarly, in order to prove the sharpness the inequality (2.12), if we assume that there exists a constant D > 0 such that " # Z b b _ 1 f (a) + f (b) 1 a+b + f (t) dt D (f ) : (3.11) f 2 2 2 b a a a If we consider the function f : [a; b] ! R given by 8 < 1 if t 2 [a; a+b 2 ); f (t) = : 1 if t 2 a+b 2 ;b
then f is of bounded variation on [a; b] ; and by (3.11) we get 1 that D 12 :
2D which implies
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
8
4. A Compounding Rule We consider the following partition of the interval [a; b] ;
n
: a = x0 < x1 < ::: < xn
1
< xn = b:
De…ne hk := xk+1 xk , 0 k n 1 and ( n ) = max fhk : 0 k n 1g the norm of the partition n : In order to exemplify how we can use the above results in order to produce Rb compounding quadrature rules to approximate the integral a f (t) dt; we consider for ; 2 R with + 6= 0; the two parameters family of three point quadrature rules
(4.1)
Tn (f;
n ; ; ) :=
+
+
+
n X1 k=0
n X1 k=0
f
f (xk ) + f (xk+1 ) hk 2 xk + xk+1 2
hk :
We notice that the family of quadrature rules (4.1) contain the trapezoid rule ( = 0), the midpoint rule ( = 0), the Simpson rule ( = 1; = 2) and the arithmetic mean of the trapezoid and midpoint rules ( = 1; = 1). The following proposition provides a priory error bounds in approximating the Rb integral a f (t) dt of the bounded variation f by the compounding quadrature rule Tn (f; n ; ; ) :
and
Proposition 1. Let f : [a; b] ! C be a function of bounded variation on [a; b] ; be real numbers with + 6= 0: Then
(4.2)
Z
b
f (t) dt = Tn (f;
n;
; ) + Rn (f;
n;
; )
a
and the remainder Rn (f;
(4.3)
jRn (f;
n;
n;
; ) satis…es the bounds
; )j
xk+1 n 1 _ 1 max fj j ; j jg X hk (f ) 2 j + j x k=0
1 max fj j ; j jg ( 2 j + j
k
n)
b _ a
(f ) :
A THREE POINT QUADRATURE RULE AND APPLICATIONS
9
Proof. Utilizing the generalized triangle inequality and (2.10) we have successively that jRn (f;
n;
n X1 Z
=
k=0
+ n X1 k=0
+
; )j f (t) dt
xk
n X1
f
k=0
Z
n X1
xk+1
+
xk + xk+1 2
k=0
hk
xk+1
f (t) dt
xk
n X1 k=0
f
xk + xk+1 2
f (xk ) + f (xk+1 ) hk 2
+ hk
xk+1 n 1 _ 1 max fj j ; j jg X hk (f ) 2 j + j x k=0
f (xk ) + f (xk+1 ) hk 2
k
1 max fj j ; j jg ( 2 j + j
and the proof is complete.
n)
b _
(f ) ;
a
5. Applications for Special Means It is well-known that, if f : [a; b] ! R is a convex function, then the celebrated Hermite-Hadamard inequality state that Z b a+b 1 f (a) + f (b) (5.1) f f (t) dt : 2 b a a 2 Utilizing this fact and the inequalities (2.11) and (2.12) we can state the following result: Proposition 2. Assume that f : [a; b] ! R is a convex function on [a; b]. Then " # Z b b f (a) + f (b) 1 a+b 1 1_ f + f (t) dt (5.2) 0 (f ) 2 2 2 b a a 2 a
and
(5.3)
f=
0
" # Z b 1 1 f (a) + f (b) + f (t) dt 2 2 b a a
f
a+b 2
b
1_ (f ) : 2 a
The case for concave functions g is similar by applying these inequalities for g: Let us recall the following means: a) The arithmetic mean A (a; b) :=
a+b ; a; b > 0; 2
b) The geometric mean G (a; b) :=
p
ab; a; b
0;
S.S. DRAGOM IR 1;2 AND E. M OM ONIAT 1
10
c) The harmonic mean H (a; b) :=
1 a
2 ; a; b > 0; + 1b
d) The identric mean 8 > 1 > < e I (a; b) := > > : a
1
bb aa
b
if if
e) The logarithmic mean 8 b > < ln b L (a; b) := > : a
f) The p logarithmic mean 8 > bp+1 ap+1 > < (p + 1) (b a) Lp (a; b) := > > : a
a
a ln a
if if
b 6= a
; a; b > 0
b=a
b 6= a
; a; b > 0
b=a
1 p
if if
b 6= a; p 2 Rn f 1; 0g
; a; b > 0:
b=a
It is well known that, if L 1 := L and L0 := I, then the function R 3p ! Lp is monotonically strictly increasing. In particular, we have H (a; b)
G (a; b)
L (a; b)
I (a; b)
A (a; b) :
Now, if we consider the power function f : [a; b] (0; 1) ! R given by f (t) = tp then we observe that for p 2 ( 1; 0) [ [1; 1) the function is convex while for p 2 (0; 1) the function is concave. Now, if we apply the inequality (5.2) for the convex function f (t) = tp we can state that 1 p (5.4) 0 A (ap ; bp ) A (a; b) + Lpp (a; b) 2 8 p b ap if p 1 1 < 2 : p a bp if p 2 ( 1; 0) n f 1g : In the case of concave functions, the same inequality (5.2) produces the inequality 1 p 1 p A (a; b) + Lpp (a; b) A (ap ; bp ) (b 2 2 Now, if we consider the convex function f : [a; b] f (t) = 1t , then by (5.2) we also have
(5.5)
0
ap ) if p 2 (0; 1) : (0; 1) ! R given by
1 1 b a A 1 (a; b) + L 1 (a; b) : 2 2 ab Moreover the inequality (5.2) applied for the concave function f : [a; b] R given by f (t) = ln t produces the result (5.6)
0
0
H
1
(a; b)
1 [ln A (a; b) + ln I (a; b)] 2
ln G (a; b)
1 ln 2
b a
(0; 1) !
A THREE POINT QUADRATURE RULE AND APPLICATIONS
11
which is equivalent with (5.7)
r
p
A (a; b) I (a; b) G (a; b)
1
b : a
Similar results can be obtained if one uses the inequality (5.3), however the details are left to the interested reader. 6. Applications for Selfadjoint Operators in Hilbert Spaces Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U ) included in the interval [m; M ] for some real numbers m < M and let fE g be its spectral family. It is well known that we have the following spectral representation in terms of the Riemann-Stieltjes integral : Z M (6.1) U= dE ; m 0
which in terms of vectors can be written as Z M (6.2) hU x; yi = d hE x; yi ; m 0
for any x; y 2 H: The function gx;y ( ) := hE x; yi is of bounded variation on the interval [m; M ] and gx;y (m
0) = 0 and gx;y (M ) = hx; yi
for any x; y 2 H: It is also well known that gx ( ) := hE x; xi is monotonic nondecreasing and right continuous on [m; M ]. We can state and prove now the following result concerning the numerical approximation of a selfadjoint operator on the complex Hilbert space (H; h:; :i) : Theorem 2. Let A be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (A) included in the interval [m; M ] for some real numbers m < M and let fE g be its spectral family. We consider the following partition of the interval [m; M ] n
:m=
0
