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Bounds for the Chebyshev functional and applications to the weighted integral formulae
Applied Mathematics and Computation 268 (2015) 957–965
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Bounds for the Chebyshev functional and applications to the weighted integral formulae J. Peˇcaric´ a, M. Ribiˇcic´ Penava b,∗, A. Vukelic´ c a
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia c Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia b
a r t i c l e
i n f o
a b s t r a c t
MSC: 26D15 65D30 65D32
The aim of this paper is to provide some error estimates for the general weighted n-point quadrature formulae by using some inequalities for the Chebyshev functional. The above results are applied to obtain some new bounds for the Gauss–Chebyshev formulae of the first and the second kind.
Keywords: Chebyshev functional Grüss inequality Weighted quadrature formulae Gauss–Chebyshev formulae
© 2015 Elsevier Inc. All rights reserved.
1. Introduction Let us suppose f (r−1) is a continuous function of bounded variation on [a, b] for some r ≥ 1 and let w: [a, b] → [0, ∞) is some b probability density function, that is integrable function satisfying a w(t )dt = 1. In paper [11] authors have proved the following two weighted quadrature formulae of Euler type:
b a
w(t ) f (t )dt =
n k=1
−
r (b − a)i−1 Ak f (xk ) + i! i=1
(b − a)
r−1
r!
b a
b a
b a
w(t ) f (t )dt =
n k=1
−
∗
r−1 (b − a)i−1 Ak f (xk ) + i! i=1
(b − a)
r−1
r!
b a
b
a
Corresponding author. Tel.: +385 31224800. E-mail address: [email protected] (M. Ribiˇcic´ Penava).
http://dx.doi.org/10.1016/j.amc.2015.07.004 0096-3003/© 2015 Elsevier Inc. All rights reserved.
b
a
w(u)B∗r
and
w(t )Bi
u−t
b−a
a
b
t −a b−a
du −
w(t )Bi
n
dt −
Ak Bi
k=1
Ak Br
t −a
dt −
b−a
n k=1
Ak Bi
u − a u−t w(u) B∗r − Br du b−a
x − a k
x − t k ∗
k=1
b−a
n
b−a
b−a
f (i−1) (b) − f (i−1) (a)
d f (r−1) (t )
x − a k b−a
(1)
f (i−1) (b) − f (i−1) (a)
958
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
−
n
x − t k
Ak B∗r
b−a
k=1
− Br
x − a k
d f (r−1) (t ),
b−a
(2)
where nk=1 Ak = 1. If we put r = m + s, these formulae are exact for all polynomials of degree ≤ m − 1. The functions t→Bk (t), k ≥ 0, t ∈ R are Bernoulli polynomials, Bk = Bk (0), k ≥ 0 Bernoulli numbers, and t → B∗k (t ), k ≥ 0 are periodic functions of period 1, related to Bernoulli polynomials as B∗k (t ) = Bk (t ), 0 ≤ t < 1. More about Bernoulli polynomials, Bernoulli numbers and periodic functions B∗k can be found in [1]. For two real functions f, g: [a, b] → R, the Chebyshev functional [10] is defined by
1 b−a
C ( f, g) =
b
a
f (t )g(t )dt −
1
(b − a)
2
b
a
f (t )dt ·
b a
g(t )dt,
where f, g are such that f, g, f · g ∈ L1 [a, b]. The symbol Lp [a, b], 1 ≤ p < ∞, stands for the space of p-power integrable functions on b interval [a, b] equipped with the norm f p = ( a | f (t )| p dt )1/p . Further, L∞ [a, b] denotes the space of all essentially bounded functions on interval [a, b] with the norm f ∞ = ess supt∈[a,b] | f (t )|. For two bounded integrable functions f, g : [a, b] → R and α , β , γ and δ real numbers such that α ≤ f(t) ≤ β , and γ ≤ g(t) ≤ δ , for all t ∈ [a, b], the following inequality is well known as the Grüss inequality (see [10], p. 296)
1 4
|C ( f, g)| ≤ (β − α)(δ − γ ). Over the last decades many researchers have investigated inequalities related to the Chebyshev functional and their applications in Numerical analysis (see [3,4,7,10,14] and the references cited therein). Cerone and Dragomir [5] proved the following Grüss type inequalities: Theorem 1. Let f, g: [a, b] → R be two absolutely continuous functions on [a, b] with
2 2 (· − a)(b − ·) f , (· − a)(b − ·) g ∈ L1 [a, b],
then
|C ( f, g)|
b 2 1/2 1 1 1/2 ≤ √ [C ( f, f )] √ (t − a)(b − t ) g (t ) dt b−a a 2 b 2 1/2 b 2 1/2 1 ≤ t − a b − t f t dt · t − a b − t g t dt . ( )( ) () ( )( ) () 2(b − a) a a
(3)
√ The constants 1/ 2 and 1/2 are the best possible. Theorem 2. Assume that g: [a, b] → R is monotonic nondecreasing on [a, b] and f: [a, b] → R is absolutely continuous with f ∈ L∞ [a, b], then
1 f ∞ · 2(b − a)
|C ( f, g)| ≤
b a
(t − a)(b − t )dg(t ).
(4)
The constant 1/2 is the best possible. In this paper we obtain some new estimations of the reminder in the generalized weighted quadrature formulae (1) and (2) by using Theorems 1 and 2. On this way we generalized results from papers [8,9,13]. As special cases, some error estimates for the Gauss–Chebyshev formulae of the first and the second kind are derived. More about quadrature formulae and error estimations (from the point of view of inequality theory) can be found in monographs [2] and [6]. 2. Main results We introduce the following notation
Gr (t ) =
b a
w(u)B∗r
u−t b−a
du −
n
Ak B∗r
k=1
x − t k b−a
,
and
Fr (t ) =
a
b
u−t
w(u) B∗r
b−a
− Br
u − a b−a
du −
n k=1
x − t k
Ak B∗r
b−a
− Br
x − a k b−a
.
(5)
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
959
So, we can rewrite (5) in the following form
Fr (t ) = Gr (t ) − Gr (a).
(6)
Some new Grüss type inequality for the weighted quadrature formulae of Euler type follows.
2
Theorem 3. Let f: [a, b] → R be such that f(r) is an absolutely continuous function and f (r+1) ∈ L1 [a, b] for some r ≥ 1 and let w: b [a, b] → [0, ∞) is some probability density function, that is integrable function satisfying a w(t )dt = 1. Then the following equality holds
b a
w(t ) f (t )dt −
×
n
Ak f (xk ) −
i=1
k=1
b a
w(t )Bi
r (b − a)i−1 i!
t −a b−a
n
dt −
Ak Bi
x − a k b−a
k=1
f (i−1) (b) − f (i−1) (a)
= Kr ( f )
(7)
and the remainder Kr (f) satisfies the inequality
(b − a)r−1 |Kr ( f )| ≤ √ 2(r!) × −2
×
n
b a
Ak
a
b
w(u)B∗r
a
b a
k=1 b
w(u)
B∗2r
u − t 2 du
b−a
u − x k
b−a
2 (t − a)(b − t ) f (r+1) (t ) dt
dt + ( − 1)r−1
du + B2r
n
A2k
+
n n
Ak Al B∗2r
k=1 l=1,l =k
k=1
1/2
(b − a)(r!)2 (2r)!
.
x − x k l
1/2
b−a (8)
Proof. Applying Theorem 1 with Gr in place of f and f(r) in place of g we deduce
b b 1 b 1 ( r) ( r) b − a a Gr (t ) f (t )dt − (b − a)2 a Gr (t )dt · a f (t )dt b 2 1/2 1 1/2 ≤ , [C (Gr (·), Gr (·))] (t − a)(b − t ) f (r+1) (t ) dt a 2(b − a)
where
C (Gr (·), Gr (·)) =
1 b−a
b a
2 (Gr (t )) dt −
1
(b − a)2
b a
(9)
2 Gr (t )dt
.
By using properties of Bernoulli polynomials and periodic function B∗r we obtain
b a
B∗r
x−t b−a
dt =
x
Br
x−t
a
b−a
dt +
b
Br
x−t
x
b−a
+ 1 dt = 0, x ∈ [a, b]
which implies
b a
Gr (t )dt = 0.
(10)
Further, we compute
b a
(Gr (t )) dt = 2
b
b
w(u)
B∗r
u − t 2
du dt b−a b b n u − t ∗ xk − t −2 Ak w(u) B∗r Br dt du b−a b−a a a a
a
k=1
+
n k=1
+
A2k
b
a
n n k=1 l=1,l =k
x − t 2 k ∗ Br
b−a
Ak Al
b a
B∗r
dt
x − t x − t k l ∗ b−a
Br
b−a
dt
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J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
and using integration by parts we have
b a
B∗r
u − t x − t k ∗ Br
b−a
b−a
dt
b u−t r(r − 1) · · · 2 x −t ∗ B∗1 k B2r−1 dt b−a b−a (r + 1)(r + 2) · · · (2r − 1) a b u − x (r!)2 u−t k = ( − 1)r−1 −2 B∗2r dt + (b − a)B∗2r b−a b−a (2r)! a = ( − 1)r−1
= ( − 1)r−1
(b − a)(r!)2 ∗ u − xk B2r . b−a (2r)!
Similarly,
b
x − t 2 k ∗ Br
a
and
b
a
B∗r
b−a
dt = ( − 1)r−1
x − t x − t k l ∗ Br
b−a
b−a
(b − a)(r!)2 B2r (2r)!
dt = ( − 1)r−1
(b − a)(r!)2 ∗ xk − xl B2r . b−a (2r)!
Now, using (1) and (9), we deduce representation (7) and inequality (8). Remark 1. From (6) and (10) we obtain
b
a
and
b
a
Fr (t )dt =
b a
(Fr (t ))2 dt =
Gr (t )dt −
b a
b a
Gr (a)dt = −(b − a)Gr (a)
(Gr (t ))2 dt − 2Gr (a)
b a
Gr (t )dt + (b − a)(Gr (a))2 .
Then on similar way as in previous proof from (2) we derive representation (7) and bound (8), too. Throughout the paper, the symbol [f(k) ; a, b] stands for the divided difference of function f(k)
f (k) ; a, b =
f (k) (b) − f (k) (a) . b−a
Theorem 4. Let f: [a, b] → R be such that f(r) is an absolutely continuous function and f (r+1) ≥ 0 on [a, b]. Then representation (7) holds and the reminder Kr (f) satisfies the following inequality
(r−2)
(b − a)r−1 r−1) r−1) ( ( |Kr ( f )| ≤ ; a, b . Gr−1 (t ) ∞ f (a) + f (b) − 2 f 2(r − 1)!
(11)
Proof. Applying Theorem 2 with Gr in place of f and f(r) in place of g we obtain
b b 1 b 1 ( r) ( r) b − a a Gr (t ) f (t )dt − (b − a)2 a Gr (t )dt · a f (t )dt ≤
Since
a
b
r 2(b − a)
Gr−1 (t ) ∞ 2
b a
(t − a)(b − t ) f (r+1) (t )dt =
(t − a)(b − t ) f (r+1) (t )dt.
(12)
b
(2t − (a + b)) f (r) (t )dt
= (b − a) f (r−1) (b) + f (r−1) (a) − 2 f (r−2) (b) − f (r−2) (a) , a
using equality (7) and inequality (12), we deduce estimate (11). Remark 2. In non-weighted case (w(u) = 1/(b − a), u ∈ [a, b]), for n = 2, Ak = 1/2, x1 = x, x2 = 1 − x, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for Euler two-point quadrature formula considered in paper [13]. Further, in non-weighted case for n = 3, x1 = x, x2 = 1/2, x3 = 1 − x, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for three-point quadrature formula of Euler type established in paper [8] and, in non-weighted case for n = 4, x1 = 0, x2 = x, x3 = 1 − x, x4 = 1, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for four-point quadrature formula of Euler type established in paper [9].
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
961
Remark 3. In non-weighted case for n = 2, Ak = 1/2, x1 = 0, x2 = 1, r = 1, a = 0 and b = 1 we get inequality related to the trapezoid formula
1 1 1/2
2 1 f (t )dt − 1 f (0) + f (1) ≤ √ t ( 1 − t ) f ( t ) dt . · 2 6 0 2 0 For specific function, the polynomial of the second degree we obtain the same inequality as in paper [12] (see also [6], p. 48). Remark 4. Taking n = 3, x1 = 0, x2 = 1/2, x3 = 1, r = 1, a = 0 and b = 1 in non-weighted case of (8) we deduce the following Simpson-type rule
1 1 1/2 1
2 1 1 ≤ √ f (t )dt − f ( 0 ) + 4 f + f ( 1 ) t ( 1 − t ) f ( t ) dt . · 6 2 0 6 2 0 We are only using the assumption first derivation is absolutely continuous rather than the more restrictive assumption of a bounded fourth derivative. For specific function, the polynomial of the second degree we obtain the same inequality as in paper [15]. 3. Applications for the Gauss–Chebyshev formulae of the first kind of Euler type For the weight function w(t ) =
1 −1
π
√1
1−t 2
, t ∈ [−1, 1] the following Gauss–Chebyshev formulae hold
f (t ) dt = π Ak f (xk ) + Rn ( f ), √ 2 1−t k=1 n
(13)
where Ak = 1/n, k = 1, . . . , n, and xk , k = 1, . . . , n, are zeros of the Chebyshev polynomials of the first kind defined as
Tn (x) = cos (n arccos (x)). The polynomial Tn (x) has exactly n distinct zeros, all of which lie in the interval [−1, 1] and are given by
xk = cos
(2k − 1)π 2n
.
The error of approximation formula (13) is given by
Rn ( f ) =
π
22n−1 (2n)!
f (2n) (ξ ),
ξ ∈ ( − 1, 1).
Let us suppose f (m+s−1) is a continuous function of bounded variation on [−1, 1] for some (m + s) ≥ 1. Peˇcaric´ et al. [11] have established the following Gauss–Chebyshev formulae of the first kind of Euler type:
1 −1
1 n 2m+s−1 f (t ) π CG1 dt = f (xk ) + Tm+s ( f, n) + GCG1 (t, n)d f (m+s−1) (t ) √ n (m + s)! −1 m+s 1 − t2 k=1
(14)
1 n 2m+s−1 f (t ) π CG1 dt = f (xk ) + Tm+s−1 ( f, n) + F CG1 (t, n)d f (m+s−1) (t ), √ n ( m + s)! −1 m+s 1 − t2 k=1
(15)
and
1 −1
where CG1 Tm+s
s 2 j+m−1 × ( f, n) = (m + j)! j=0
GCG1 m+s (t, n) =
π n
n
B∗m+s
1
−1
x − t k
k=1
1
2
B j+m √ 1 − t2
−
1 −1
√
t + 1
1 1 − u2
2 B∗m+s
dt −
n π
n
u − t 2
B j+m
x + 1 k 2
k=1
f (m+ j−1) (1) − f (m+ j−1) ( − 1) ,
du
and CG1 Fm+s (t, n) =
n π
n
k=1
B∗m+s
x − t k 2
− Bm+s
x + 1 k 2
This formulae are exact for all polynomials of degree ≤ m − 1.
−
1 −1
1 u−t B∗m+s √ 2 1 − u2
− Bm+s
u + 1 2
du.
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J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
Theorem 5. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) s ≥ 1. Then the following equality holds
1
−1
∈ L1 [−1, 1] for some m +
n f (t ) π CG1 CG1 dt − f (xk ) − Tm+s ( f, n) = Km+s √ ( f, n) n 1 − t2 k=1
CG1 f, n satisfies the inequality and the remainder Km+s ( )
|
2
CG1 Km+s
( f, n)| ≤ √
2m+s−1
2(m + s)!
+
1
−1
1 −1
√
1
B∗m+s
1 − u2
(16)
u − t 2 2
du
dt
u − x n 2π 1 ((m + s)!)2 1 k − B∗2(m+s) du √ 2 n 2 (2(m + s))! 1 − u −1 k=1 x − x 1/2 n n π2 k l ∗
m+s−1 2
( − 1) π2
+
n
×
B2(m+s) +
1 −1
1 − t2
n2
k=1 l=1,l =k
B2(m+s)
2
1/2
2
f (m+s+1) (t ) dt
.
(17)
Proof. This is a special case of Theorem 3 for a = −1, b = 1, r = m + s, w(t ) =
π
√1
1−t 2
and Ak =
1 n.
Remark 5. For n = 1 and x1 = 0 we get one-point Gauss–Chebyshev formula of the first kind of Euler type
1
−1
f (t ) CG1 CG1 dt = π f (0) + Tm+s ( f, 1) + Km+s √ ( f, 1). 1 − t2
In a special case for m = 1 and s = 0 we obtain the following inequality
1 1 (2) 2 1/2 f (t ) 2 ≤ π −2 dt − π f ( 0 ) 1 − t f t dt . √ ( ) −1 1 − t 2 −1
Remark 6. For n = 2, x1 = −
1
−1
f (t ) π dt = √ 2 1 − t2
√ 2 2
and x2 =
√ f −
2 2
√
2 2
+f
we get two-point Gauss–Chebyshev formulae of the first kind of Euler type
√ 2 2
CG1 CG1 + Tm+s ( f, 2) + Km+s ( f, 2).
Specially for m1 < 4 and s = 0 we derive the following inequalities
1 √ √ 1 (m +1) 2 1/2 f (t ) π 2 2 CG1 2 1 , (t ) dt ≤ V2 (m1 , 0) · −1 1 − t f −1 √1 − t 2 dt − 2 f − 2 + f 2
where V2CG1 (1, 0) =
√
π 2−4 ≈ 0.470576, V CG1 2, 0 ≈ 0.0798678 and V CG1 3, 0 ≈ 0.0222603. ( ) ( ) 2 2 2
Remark 7. For n = 3, x1 = − type
1
−1
f (t )
π
dt = √ 3 1 − t2
√ 3 2 , x2
= 0 and x3 =
√
√ 3 f − 2
3 2
we get three-point Gauss–Chebyshev formulae of the first kind of Euler
√ + f (0) + f
3 2
CG1 CG1 + Tm+s ( f, 3) + Km+s ( f, 3).
In special cases for m1 < 6 and s = 0 we obtain inequalities
1 √ √ 1 (m +1) 2 1/2 3 f (t ) π 3 CG1 2 1 , (t ) dt ≤ V3 (m1 , 0) · −1 1 − t f −1 √1 − t 2 dt − 3 f − 2 + f (0) + f 2
where V3CG1 (1, 0) ≈ 0.307238, V3CG1 (2, 0) ≈ 0.0344436, V3CG1 (3, 0) ≈ 0.00549894, V3CG1 (4, 0) ≈ 0.00103907 and V3CG1 (5, 0) ≈ 0.000246304. Theorem 6. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) ≥ 0 on [−1, 1]. Then repreCG1 f, n satisfies the following inequality sentation (16) holds and the reminder Km+s ( ) CG1 |Km+s ( f, n)| ≤
CG1 2m+s−1 Gm+s−1 (t, n) ∞ (m + s − 1)!
f (m+s−1) (−1) + f (m+s−1) (1) (m+s−2) − f ; −1, 1 . 2
Proof. This is a special case of Theorem 4 for a = −1, b = 1, r = m + s, w(t ) =
π
√1
1−t 2
and Ak =
1 n.
(18)
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
963
4. Applications for the Gauss–Chebyshev formulae of the second kind of Euler type For the weight function w(t ) =
1
−1
1 − t 2 f (t )dt =
2
n π
2
√
1−t 2
π
, t ∈ [−1, 1] the following Gauss–Chebyshev formulae stand
Ak f (xk ) + Rn ( f ),
(19)
k=1
where Ak are given by
2 2 sin Ak = n+1
kπ , n+1
k = 1, . . . , n,
and xk , k = 1, . . . , n, are zeros of the Chebyshev polynomials of the second kind defined as
Un (x) =
sin [(n + 1) arccos (x)] . sin [arccos (x)]
The polynomial Un (x) has exactly n distinct zeros, all of which lie in the interval [−1, 1] and
xk = cos
kπ . n+1
The error of approximation formula (19) is given by
Rn ( f ) =
π
22n+1 (2n)!
f (2n) (ξ ),
ξ ∈ ( − 1, 1).
Let us suppose f (m+s−1) is a continuous function of bounded variation on [−1, 1] for some (m + s) ≥ 1. In [11] authors have established the following Gauss–Chebyshev formulae of the second kind of Euler type:
1
−1
1
− t2
f (t )dt =
n π
n+1
2
sin
k=1
kπ n+1
CG2 f (xk ) + Tm+s ( f, n) +
2m+s−1 (m + s)!
1
−1
(m+s−1) (t ) GCG2 m+s (t, n)d f
(20)
and
1
−1
1 − t 2 f (t )dt =
n π
n+1
2
sin
k=1
kπ n+1
CG2 f (xk ) + Tm+s−1 ( f, n) +
2m+s−1 (m + s)!
1
−1
CG2 Fm+s (t, n)d f (m+s−1) (t ),
where CG2 Tm+s ( f, n) =
1
s 2 j+m−1 × (m + j)! j=0
− GCG2 m+s
(t, n) =
n π
n+1
k=1
n π
n+1
2
sin
2
sin
k=1
−1
1 − t 2 B j+m
t + 1 2
kπ x +1 B j+m k n+1 2
kπ x −t B∗m+s k n+1 2
dt
−
f (m+ j−1) (1) − f (m+ j−1) ( − 1) ,
1
−1
1 − u2 B∗m+s
and CG2 Fm+s (t, n) =
n π
2
sin
kπ n+1
B∗m+s
x − t k
− Bm+s
x + 1 k
n+1 2 2 k=1 1 u − t u + 1 − 1 − u2 B∗m+s − Bm+s du. 2 2 −1
These formulae are exact for all polynomials of degree ≤ m − 1.
u − t 2
du
(21)
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J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
Theorem 7. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) s ≥ 1. Then the following equality holds
1
−1
1 − t 2 f (t )dt −
n π
n+1
sin
2
k=1
kπ n+1
|
( f, n)| ≤ √
2(m + s)!
+ +
+
×
1
1
−1
−1
2m+s−1
∈ L1 [−1, 1] for some m +
CG2 CG2 f (xk ) − Tm+s ( f, n) = Km+s ( f, n)
CG2 f, n satisfies the inequality and the remainder Km+s ( )
CG2 Km+s
2
1−
u2 B∗m+s
u − t 2 2
du
(22)
dt
1 u − x n 2π ((m + s)!)2 kπ 2 k − sin 1 − u2 B∗2(m+s) du n+1 n+1 2 (2(m + s))! −1 k=1 n kπ 4
m+s−1 2
( − 1)
π2
(n + 1)2 π2
sin
k=1 n n
(n + 1)2 1 −1
B2(m+s)
n+1
2
sin
k=1 l=1,l =k
1−t
2
kπ n+1
2 f (m+s+1) (t ) dt
2
sin
lπ x − xl B∗2(m+s) k n+1 2
1/2
1/2 .
(23)
Proof. This is a special case of Theorem 3 for a = −1, b = 1, r = m + s, w(t ) =
2
√
1−t 2
π
and Ak =
2 n+1
sin2
kπ n+1
.
Remark 8. For n = 1 and x1 = 0 we get one-point Gauss–Chebyshev formulae of second kind of Euler type
1
−1
1 − t 2 f (t )dt =
π 2
CG2 CG2 f (0) + Tm+s ( f, 1) + Km+s ( f, 1).
In a special case for m = 1 and s = 0 we derive the following inequality
1 1 (2) 2 1/2 π CG2 2 2 f (t )dt − f 1 − t 0 ≤ V 1, 0 1 − t f t dt , ( ) ( ) () 1 −1 2 −1
where V1CG2 (1, 0) ≈ 0.409931. Remark 9. For n = 2, x1 = − 21 and x2 =
1
−1
1 − t 2 f (t )dt =
1 2
π 1 4
f −
2
we get two-point Chebyshev–Gauss formulae of the second kind of Euler type
+f
1 2
CG2 CG2 + Tm+s ( f, 2) + Km+s ( f, 2).
In special cases for m1 < 4 and s = 0 we consider the following inequalities
1 1 1 (m +1) 2 1/2 π 1 2 2 1 ≤ V2CG2 (m1 , 0) 1 − t f t dt − f − + f 1 − t f t dt , ( ) ( ) −1 4 2 2 −1
where V2CG2 (1, 0) ≈ 0.227524, V2CG2 (2, 0) ≈ 0.0342287 and V2CG2 (3, 0) ≈ 0.0077412. Remark 10. Further, for n = 3, x1 = − Euler type
1
−1
1
− t2
f (t )dt =
π 8
√ 2 2 , x2
= 0 and x3 =
√ 2 f − 2
√
2 2
we get three-point Gauss–Chebyshev formulae of second kind of
√ + 2 f ( 0) + f
2 2
CG2 CG2 + Tm+s ( f, 3) + Km+s ( f, 3).
Specially for m1 < 6 and s = 0 we obtain the following inequalities
1 √ √ 1 (m +1) 2 1/2 2 π 2 CG2 2 2 1 , (t ) dt ≤ V3 (m1 , 0) −1 1 − t f −1 1 − t f (t )dt − 8 f − 2 + 2 f (0) + f 2
where V3CG2 (1, 0) ≈ 0.169113, V3CG2 (2, 0) ≈ 0.0174633, V3CG2 (3, 0) ≈ 0.00238238, V3CG2 (4, 0) ≈ 0.000382337 and V3CG2 (5, 0) ≈ 0.0000773027.
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
965
Theorem 8. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) ≥ 0 on [−1, 1]. Then repreCG2 f, n satisfies the following inequality sentation (22) holds and the reminder Km+s ( )
|
CG2 Km+s
( f, n)| ≤
2m+s−1 (m+s−1)!
GCG2 m+s−1 (t, n)
∞
f (m+s−1) (−1)+ f (m+s−1) (1) 2
−
f (m+s−2) ; −1, 1
Proof. This is a special case of Theorem 4 for a = −1, b = 1, r = m + s, w(t ) =
2
√
1−t 2
π
.
and Ak =
(24) 2 n+1
sin2
kπ n+1
.
Acknowledgment This work has been fully supported by Croatian Science Foundation under the project 5435. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulae, Graphs and Mathematical Tables, Applied Math. Series 55, National Bureau of Standards, 4th printing, Washington, 1965. ˇ ´ A. Civljak, ´ M. Ribiˇcic´ Penava, General Integral Identities and Related Inequalities, Element, Zagreb, 2013. [2] A. Aglic´ Aljinovic, S. Kovaˇc, J. Peˇcaric, ´ M. Matic, ´ J. Peˇcaric, ´ Some inequalities of Euler–Grüss type, Comput. Math. Appl. 41 (2001) 843–856. [3] Lj. Dedic, [4] S.S. Dragomir, Some Grüss-type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4 (2) (2003). Article 42. ˇ [5] P. Cerone, S.S. Dragomir, Some new Ostrowski-type bounds for the Cebyšev functional and applications, J. Math. Inequal. 8 (1) (2014) 159–170. ´ J. Peˇcaric, ´ I. Peric, ´ A. Vukelic, ´ Euler integral identity, quadrature formulae and error estimations, Element, Zagreb, 2011. [6] I. Franjic, ´ B. Tepeš, A Grüss-type inequality and its applications, J. Inequal. Appl. 3 (2005) 277–288. [7] S. Izumino, J. Peˇcaric, ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ New estimations of the remainder in three-point quadrature formulae of Euler type, J. Math. [8] M. Klariˇcic´ Bakula, J. Peˇcaric, Inequal. 9 (4) (2015) 1143–1156. ˇ ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ Some inequalities for the Cebyšev [9] M. Klariˇcic´ Bakula, J. Peˇcaric, functional and general four-point quadrature formulae of Euler type, Mat. Bilten (2015), accepted for publication. ´ J.E. Peˇcaric, ´ A.M. Fink, Classical and New inequalities in Analysis, Kluwer Academic, Dordrecht, 1993. [10] D.S. Mitrinovic, ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ Euler’s method for weighted integral formulae, Appl. Math. Comput. 206 (2008) 445–456. [11] J. Peˇcaric, ´ A. Vukelic, ´ Estimations of the error for two-point formula via pre-Grüss inequality, Gen. Math. 13 (2) (2005) 95–104. [12] J. Peˇcaric, ˇ ´ A. Vukelic, ´ Some inequalities for the Cebyšev [13] J. Peˇcaric, functional and Euler two-point formulae, J. Math. Inequal. 9 (4) (2015) 1195–1205. ´ A new generalization of Grüss inequality in inner product spaces, J. Math. Anal. Appl. 250 (2000) 494–511. [14] N. Ujevic, ´ Sharp inequalities of Simpson type and Ostrowski type, Comput. Math. Appl. 48 (2004) 145–151. [15] N. Ujevic,
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Bounds for the Chebyshev functional and applications to the weighted integral formulae J. Peˇcaric´ a, M. Ribiˇcic´ Penava b,∗, A. Vukelic´ c a
Faculty of Textile Technology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000 Osijek, Croatia c Faculty of Food Technology and Biotechnology, University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia b
a r t i c l e
i n f o
a b s t r a c t
MSC: 26D15 65D30 65D32
The aim of this paper is to provide some error estimates for the general weighted n-point quadrature formulae by using some inequalities for the Chebyshev functional. The above results are applied to obtain some new bounds for the Gauss–Chebyshev formulae of the first and the second kind.
Keywords: Chebyshev functional Grüss inequality Weighted quadrature formulae Gauss–Chebyshev formulae
© 2015 Elsevier Inc. All rights reserved.
1. Introduction Let us suppose f (r−1) is a continuous function of bounded variation on [a, b] for some r ≥ 1 and let w: [a, b] → [0, ∞) is some b probability density function, that is integrable function satisfying a w(t )dt = 1. In paper [11] authors have proved the following two weighted quadrature formulae of Euler type:
b a
w(t ) f (t )dt =
n k=1
−
r (b − a)i−1 Ak f (xk ) + i! i=1
(b − a)
r−1
r!
b a
b a
b a
w(t ) f (t )dt =
n k=1
−
∗
r−1 (b − a)i−1 Ak f (xk ) + i! i=1
(b − a)
r−1
r!
b a
b
a
Corresponding author. Tel.: +385 31224800. E-mail address: [email protected] (M. Ribiˇcic´ Penava).
http://dx.doi.org/10.1016/j.amc.2015.07.004 0096-3003/© 2015 Elsevier Inc. All rights reserved.
b
a
w(u)B∗r
and
w(t )Bi
u−t
b−a
a
b
t −a b−a
du −
w(t )Bi
n
dt −
Ak Bi
k=1
Ak Br
t −a
dt −
b−a
n k=1
Ak Bi
u − a u−t w(u) B∗r − Br du b−a
x − a k
x − t k ∗
k=1
b−a
n
b−a
b−a
f (i−1) (b) − f (i−1) (a)
d f (r−1) (t )
x − a k b−a
(1)
f (i−1) (b) − f (i−1) (a)
958
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
−
n
x − t k
Ak B∗r
b−a
k=1
− Br
x − a k
d f (r−1) (t ),
b−a
(2)
where nk=1 Ak = 1. If we put r = m + s, these formulae are exact for all polynomials of degree ≤ m − 1. The functions t→Bk (t), k ≥ 0, t ∈ R are Bernoulli polynomials, Bk = Bk (0), k ≥ 0 Bernoulli numbers, and t → B∗k (t ), k ≥ 0 are periodic functions of period 1, related to Bernoulli polynomials as B∗k (t ) = Bk (t ), 0 ≤ t < 1. More about Bernoulli polynomials, Bernoulli numbers and periodic functions B∗k can be found in [1]. For two real functions f, g: [a, b] → R, the Chebyshev functional [10] is defined by
1 b−a
C ( f, g) =
b
a
f (t )g(t )dt −
1
(b − a)
2
b
a
f (t )dt ·
b a
g(t )dt,
where f, g are such that f, g, f · g ∈ L1 [a, b]. The symbol Lp [a, b], 1 ≤ p < ∞, stands for the space of p-power integrable functions on b interval [a, b] equipped with the norm f p = ( a | f (t )| p dt )1/p . Further, L∞ [a, b] denotes the space of all essentially bounded functions on interval [a, b] with the norm f ∞ = ess supt∈[a,b] | f (t )|. For two bounded integrable functions f, g : [a, b] → R and α , β , γ and δ real numbers such that α ≤ f(t) ≤ β , and γ ≤ g(t) ≤ δ , for all t ∈ [a, b], the following inequality is well known as the Grüss inequality (see [10], p. 296)
1 4
|C ( f, g)| ≤ (β − α)(δ − γ ). Over the last decades many researchers have investigated inequalities related to the Chebyshev functional and their applications in Numerical analysis (see [3,4,7,10,14] and the references cited therein). Cerone and Dragomir [5] proved the following Grüss type inequalities: Theorem 1. Let f, g: [a, b] → R be two absolutely continuous functions on [a, b] with
2 2 (· − a)(b − ·) f , (· − a)(b − ·) g ∈ L1 [a, b],
then
|C ( f, g)|
b 2 1/2 1 1 1/2 ≤ √ [C ( f, f )] √ (t − a)(b − t ) g (t ) dt b−a a 2 b 2 1/2 b 2 1/2 1 ≤ t − a b − t f t dt · t − a b − t g t dt . ( )( ) () ( )( ) () 2(b − a) a a
(3)
√ The constants 1/ 2 and 1/2 are the best possible. Theorem 2. Assume that g: [a, b] → R is monotonic nondecreasing on [a, b] and f: [a, b] → R is absolutely continuous with f ∈ L∞ [a, b], then
1 f ∞ · 2(b − a)
|C ( f, g)| ≤
b a
(t − a)(b − t )dg(t ).
(4)
The constant 1/2 is the best possible. In this paper we obtain some new estimations of the reminder in the generalized weighted quadrature formulae (1) and (2) by using Theorems 1 and 2. On this way we generalized results from papers [8,9,13]. As special cases, some error estimates for the Gauss–Chebyshev formulae of the first and the second kind are derived. More about quadrature formulae and error estimations (from the point of view of inequality theory) can be found in monographs [2] and [6]. 2. Main results We introduce the following notation
Gr (t ) =
b a
w(u)B∗r
u−t b−a
du −
n
Ak B∗r
k=1
x − t k b−a
,
and
Fr (t ) =
a
b
u−t
w(u) B∗r
b−a
− Br
u − a b−a
du −
n k=1
x − t k
Ak B∗r
b−a
− Br
x − a k b−a
.
(5)
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
959
So, we can rewrite (5) in the following form
Fr (t ) = Gr (t ) − Gr (a).
(6)
Some new Grüss type inequality for the weighted quadrature formulae of Euler type follows.
2
Theorem 3. Let f: [a, b] → R be such that f(r) is an absolutely continuous function and f (r+1) ∈ L1 [a, b] for some r ≥ 1 and let w: b [a, b] → [0, ∞) is some probability density function, that is integrable function satisfying a w(t )dt = 1. Then the following equality holds
b a
w(t ) f (t )dt −
×
n
Ak f (xk ) −
i=1
k=1
b a
w(t )Bi
r (b − a)i−1 i!
t −a b−a
n
dt −
Ak Bi
x − a k b−a
k=1
f (i−1) (b) − f (i−1) (a)
= Kr ( f )
(7)
and the remainder Kr (f) satisfies the inequality
(b − a)r−1 |Kr ( f )| ≤ √ 2(r!) × −2
×
n
b a
Ak
a
b
w(u)B∗r
a
b a
k=1 b
w(u)
B∗2r
u − t 2 du
b−a
u − x k
b−a
2 (t − a)(b − t ) f (r+1) (t ) dt
dt + ( − 1)r−1
du + B2r
n
A2k
+
n n
Ak Al B∗2r
k=1 l=1,l =k
k=1
1/2
(b − a)(r!)2 (2r)!
.
x − x k l
1/2
b−a (8)
Proof. Applying Theorem 1 with Gr in place of f and f(r) in place of g we deduce
b b 1 b 1 ( r) ( r) b − a a Gr (t ) f (t )dt − (b − a)2 a Gr (t )dt · a f (t )dt b 2 1/2 1 1/2 ≤ , [C (Gr (·), Gr (·))] (t − a)(b − t ) f (r+1) (t ) dt a 2(b − a)
where
C (Gr (·), Gr (·)) =
1 b−a
b a
2 (Gr (t )) dt −
1
(b − a)2
b a
(9)
2 Gr (t )dt
.
By using properties of Bernoulli polynomials and periodic function B∗r we obtain
b a
B∗r
x−t b−a
dt =
x
Br
x−t
a
b−a
dt +
b
Br
x−t
x
b−a
+ 1 dt = 0, x ∈ [a, b]
which implies
b a
Gr (t )dt = 0.
(10)
Further, we compute
b a
(Gr (t )) dt = 2
b
b
w(u)
B∗r
u − t 2
du dt b−a b b n u − t ∗ xk − t −2 Ak w(u) B∗r Br dt du b−a b−a a a a
a
k=1
+
n k=1
+
A2k
b
a
n n k=1 l=1,l =k
x − t 2 k ∗ Br
b−a
Ak Al
b a
B∗r
dt
x − t x − t k l ∗ b−a
Br
b−a
dt
960
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
and using integration by parts we have
b a
B∗r
u − t x − t k ∗ Br
b−a
b−a
dt
b u−t r(r − 1) · · · 2 x −t ∗ B∗1 k B2r−1 dt b−a b−a (r + 1)(r + 2) · · · (2r − 1) a b u − x (r!)2 u−t k = ( − 1)r−1 −2 B∗2r dt + (b − a)B∗2r b−a b−a (2r)! a = ( − 1)r−1
= ( − 1)r−1
(b − a)(r!)2 ∗ u − xk B2r . b−a (2r)!
Similarly,
b
x − t 2 k ∗ Br
a
and
b
a
B∗r
b−a
dt = ( − 1)r−1
x − t x − t k l ∗ Br
b−a
b−a
(b − a)(r!)2 B2r (2r)!
dt = ( − 1)r−1
(b − a)(r!)2 ∗ xk − xl B2r . b−a (2r)!
Now, using (1) and (9), we deduce representation (7) and inequality (8). Remark 1. From (6) and (10) we obtain
b
a
and
b
a
Fr (t )dt =
b a
(Fr (t ))2 dt =
Gr (t )dt −
b a
b a
Gr (a)dt = −(b − a)Gr (a)
(Gr (t ))2 dt − 2Gr (a)
b a
Gr (t )dt + (b − a)(Gr (a))2 .
Then on similar way as in previous proof from (2) we derive representation (7) and bound (8), too. Throughout the paper, the symbol [f(k) ; a, b] stands for the divided difference of function f(k)
f (k) ; a, b =
f (k) (b) − f (k) (a) . b−a
Theorem 4. Let f: [a, b] → R be such that f(r) is an absolutely continuous function and f (r+1) ≥ 0 on [a, b]. Then representation (7) holds and the reminder Kr (f) satisfies the following inequality
(r−2)
(b − a)r−1 r−1) r−1) ( ( |Kr ( f )| ≤ ; a, b . Gr−1 (t ) ∞ f (a) + f (b) − 2 f 2(r − 1)!
(11)
Proof. Applying Theorem 2 with Gr in place of f and f(r) in place of g we obtain
b b 1 b 1 ( r) ( r) b − a a Gr (t ) f (t )dt − (b − a)2 a Gr (t )dt · a f (t )dt ≤
Since
a
b
r 2(b − a)
Gr−1 (t ) ∞ 2
b a
(t − a)(b − t ) f (r+1) (t )dt =
(t − a)(b − t ) f (r+1) (t )dt.
(12)
b
(2t − (a + b)) f (r) (t )dt
= (b − a) f (r−1) (b) + f (r−1) (a) − 2 f (r−2) (b) − f (r−2) (a) , a
using equality (7) and inequality (12), we deduce estimate (11). Remark 2. In non-weighted case (w(u) = 1/(b − a), u ∈ [a, b]), for n = 2, Ak = 1/2, x1 = x, x2 = 1 − x, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for Euler two-point quadrature formula considered in paper [13]. Further, in non-weighted case for n = 3, x1 = x, x2 = 1/2, x3 = 1 − x, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for three-point quadrature formula of Euler type established in paper [8] and, in non-weighted case for n = 4, x1 = 0, x2 = x, x3 = 1 − x, x4 = 1, a = 0 and b = 1 inequalities (8) and (11) reduce to the Grüss type inequalities for four-point quadrature formula of Euler type established in paper [9].
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
961
Remark 3. In non-weighted case for n = 2, Ak = 1/2, x1 = 0, x2 = 1, r = 1, a = 0 and b = 1 we get inequality related to the trapezoid formula
1 1 1/2
2 1 f (t )dt − 1 f (0) + f (1) ≤ √ t ( 1 − t ) f ( t ) dt . · 2 6 0 2 0 For specific function, the polynomial of the second degree we obtain the same inequality as in paper [12] (see also [6], p. 48). Remark 4. Taking n = 3, x1 = 0, x2 = 1/2, x3 = 1, r = 1, a = 0 and b = 1 in non-weighted case of (8) we deduce the following Simpson-type rule
1 1 1/2 1
2 1 1 ≤ √ f (t )dt − f ( 0 ) + 4 f + f ( 1 ) t ( 1 − t ) f ( t ) dt . · 6 2 0 6 2 0 We are only using the assumption first derivation is absolutely continuous rather than the more restrictive assumption of a bounded fourth derivative. For specific function, the polynomial of the second degree we obtain the same inequality as in paper [15]. 3. Applications for the Gauss–Chebyshev formulae of the first kind of Euler type For the weight function w(t ) =
1 −1
π
√1
1−t 2
, t ∈ [−1, 1] the following Gauss–Chebyshev formulae hold
f (t ) dt = π Ak f (xk ) + Rn ( f ), √ 2 1−t k=1 n
(13)
where Ak = 1/n, k = 1, . . . , n, and xk , k = 1, . . . , n, are zeros of the Chebyshev polynomials of the first kind defined as
Tn (x) = cos (n arccos (x)). The polynomial Tn (x) has exactly n distinct zeros, all of which lie in the interval [−1, 1] and are given by
xk = cos
(2k − 1)π 2n
.
The error of approximation formula (13) is given by
Rn ( f ) =
π
22n−1 (2n)!
f (2n) (ξ ),
ξ ∈ ( − 1, 1).
Let us suppose f (m+s−1) is a continuous function of bounded variation on [−1, 1] for some (m + s) ≥ 1. Peˇcaric´ et al. [11] have established the following Gauss–Chebyshev formulae of the first kind of Euler type:
1 −1
1 n 2m+s−1 f (t ) π CG1 dt = f (xk ) + Tm+s ( f, n) + GCG1 (t, n)d f (m+s−1) (t ) √ n (m + s)! −1 m+s 1 − t2 k=1
(14)
1 n 2m+s−1 f (t ) π CG1 dt = f (xk ) + Tm+s−1 ( f, n) + F CG1 (t, n)d f (m+s−1) (t ), √ n ( m + s)! −1 m+s 1 − t2 k=1
(15)
and
1 −1
where CG1 Tm+s
s 2 j+m−1 × ( f, n) = (m + j)! j=0
GCG1 m+s (t, n) =
π n
n
B∗m+s
1
−1
x − t k
k=1
1
2
B j+m √ 1 − t2
−
1 −1
√
t + 1
1 1 − u2
2 B∗m+s
dt −
n π
n
u − t 2
B j+m
x + 1 k 2
k=1
f (m+ j−1) (1) − f (m+ j−1) ( − 1) ,
du
and CG1 Fm+s (t, n) =
n π
n
k=1
B∗m+s
x − t k 2
− Bm+s
x + 1 k 2
This formulae are exact for all polynomials of degree ≤ m − 1.
−
1 −1
1 u−t B∗m+s √ 2 1 − u2
− Bm+s
u + 1 2
du.
962
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
Theorem 5. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) s ≥ 1. Then the following equality holds
1
−1
∈ L1 [−1, 1] for some m +
n f (t ) π CG1 CG1 dt − f (xk ) − Tm+s ( f, n) = Km+s √ ( f, n) n 1 − t2 k=1
CG1 f, n satisfies the inequality and the remainder Km+s ( )
|
2
CG1 Km+s
( f, n)| ≤ √
2m+s−1
2(m + s)!
+
1
−1
1 −1
√
1
B∗m+s
1 − u2
(16)
u − t 2 2
du
dt
u − x n 2π 1 ((m + s)!)2 1 k − B∗2(m+s) du √ 2 n 2 (2(m + s))! 1 − u −1 k=1 x − x 1/2 n n π2 k l ∗
m+s−1 2
( − 1) π2
+
n
×
B2(m+s) +
1 −1
1 − t2
n2
k=1 l=1,l =k
B2(m+s)
2
1/2
2
f (m+s+1) (t ) dt
.
(17)
Proof. This is a special case of Theorem 3 for a = −1, b = 1, r = m + s, w(t ) =
π
√1
1−t 2
and Ak =
1 n.
Remark 5. For n = 1 and x1 = 0 we get one-point Gauss–Chebyshev formula of the first kind of Euler type
1
−1
f (t ) CG1 CG1 dt = π f (0) + Tm+s ( f, 1) + Km+s √ ( f, 1). 1 − t2
In a special case for m = 1 and s = 0 we obtain the following inequality
1 1 (2) 2 1/2 f (t ) 2 ≤ π −2 dt − π f ( 0 ) 1 − t f t dt . √ ( ) −1 1 − t 2 −1
Remark 6. For n = 2, x1 = −
1
−1
f (t ) π dt = √ 2 1 − t2
√ 2 2
and x2 =
√ f −
2 2
√
2 2
+f
we get two-point Gauss–Chebyshev formulae of the first kind of Euler type
√ 2 2
CG1 CG1 + Tm+s ( f, 2) + Km+s ( f, 2).
Specially for m1 < 4 and s = 0 we derive the following inequalities
1 √ √ 1 (m +1) 2 1/2 f (t ) π 2 2 CG1 2 1 , (t ) dt ≤ V2 (m1 , 0) · −1 1 − t f −1 √1 − t 2 dt − 2 f − 2 + f 2
where V2CG1 (1, 0) =
√
π 2−4 ≈ 0.470576, V CG1 2, 0 ≈ 0.0798678 and V CG1 3, 0 ≈ 0.0222603. ( ) ( ) 2 2 2
Remark 7. For n = 3, x1 = − type
1
−1
f (t )
π
dt = √ 3 1 − t2
√ 3 2 , x2
= 0 and x3 =
√
√ 3 f − 2
3 2
we get three-point Gauss–Chebyshev formulae of the first kind of Euler
√ + f (0) + f
3 2
CG1 CG1 + Tm+s ( f, 3) + Km+s ( f, 3).
In special cases for m1 < 6 and s = 0 we obtain inequalities
1 √ √ 1 (m +1) 2 1/2 3 f (t ) π 3 CG1 2 1 , (t ) dt ≤ V3 (m1 , 0) · −1 1 − t f −1 √1 − t 2 dt − 3 f − 2 + f (0) + f 2
where V3CG1 (1, 0) ≈ 0.307238, V3CG1 (2, 0) ≈ 0.0344436, V3CG1 (3, 0) ≈ 0.00549894, V3CG1 (4, 0) ≈ 0.00103907 and V3CG1 (5, 0) ≈ 0.000246304. Theorem 6. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) ≥ 0 on [−1, 1]. Then repreCG1 f, n satisfies the following inequality sentation (16) holds and the reminder Km+s ( ) CG1 |Km+s ( f, n)| ≤
CG1 2m+s−1 Gm+s−1 (t, n) ∞ (m + s − 1)!
f (m+s−1) (−1) + f (m+s−1) (1) (m+s−2) − f ; −1, 1 . 2
Proof. This is a special case of Theorem 4 for a = −1, b = 1, r = m + s, w(t ) =
π
√1
1−t 2
and Ak =
1 n.
(18)
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
963
4. Applications for the Gauss–Chebyshev formulae of the second kind of Euler type For the weight function w(t ) =
1
−1
1 − t 2 f (t )dt =
2
n π
2
√
1−t 2
π
, t ∈ [−1, 1] the following Gauss–Chebyshev formulae stand
Ak f (xk ) + Rn ( f ),
(19)
k=1
where Ak are given by
2 2 sin Ak = n+1
kπ , n+1
k = 1, . . . , n,
and xk , k = 1, . . . , n, are zeros of the Chebyshev polynomials of the second kind defined as
Un (x) =
sin [(n + 1) arccos (x)] . sin [arccos (x)]
The polynomial Un (x) has exactly n distinct zeros, all of which lie in the interval [−1, 1] and
xk = cos
kπ . n+1
The error of approximation formula (19) is given by
Rn ( f ) =
π
22n+1 (2n)!
f (2n) (ξ ),
ξ ∈ ( − 1, 1).
Let us suppose f (m+s−1) is a continuous function of bounded variation on [−1, 1] for some (m + s) ≥ 1. In [11] authors have established the following Gauss–Chebyshev formulae of the second kind of Euler type:
1
−1
1
− t2
f (t )dt =
n π
n+1
2
sin
k=1
kπ n+1
CG2 f (xk ) + Tm+s ( f, n) +
2m+s−1 (m + s)!
1
−1
(m+s−1) (t ) GCG2 m+s (t, n)d f
(20)
and
1
−1
1 − t 2 f (t )dt =
n π
n+1
2
sin
k=1
kπ n+1
CG2 f (xk ) + Tm+s−1 ( f, n) +
2m+s−1 (m + s)!
1
−1
CG2 Fm+s (t, n)d f (m+s−1) (t ),
where CG2 Tm+s ( f, n) =
1
s 2 j+m−1 × (m + j)! j=0
− GCG2 m+s
(t, n) =
n π
n+1
k=1
n π
n+1
2
sin
2
sin
k=1
−1
1 − t 2 B j+m
t + 1 2
kπ x +1 B j+m k n+1 2
kπ x −t B∗m+s k n+1 2
dt
−
f (m+ j−1) (1) − f (m+ j−1) ( − 1) ,
1
−1
1 − u2 B∗m+s
and CG2 Fm+s (t, n) =
n π
2
sin
kπ n+1
B∗m+s
x − t k
− Bm+s
x + 1 k
n+1 2 2 k=1 1 u − t u + 1 − 1 − u2 B∗m+s − Bm+s du. 2 2 −1
These formulae are exact for all polynomials of degree ≤ m − 1.
u − t 2
du
(21)
964
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
Theorem 7. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) s ≥ 1. Then the following equality holds
1
−1
1 − t 2 f (t )dt −
n π
n+1
sin
2
k=1
kπ n+1
|
( f, n)| ≤ √
2(m + s)!
+ +
+
×
1
1
−1
−1
2m+s−1
∈ L1 [−1, 1] for some m +
CG2 CG2 f (xk ) − Tm+s ( f, n) = Km+s ( f, n)
CG2 f, n satisfies the inequality and the remainder Km+s ( )
CG2 Km+s
2
1−
u2 B∗m+s
u − t 2 2
du
(22)
dt
1 u − x n 2π ((m + s)!)2 kπ 2 k − sin 1 − u2 B∗2(m+s) du n+1 n+1 2 (2(m + s))! −1 k=1 n kπ 4
m+s−1 2
( − 1)
π2
(n + 1)2 π2
sin
k=1 n n
(n + 1)2 1 −1
B2(m+s)
n+1
2
sin
k=1 l=1,l =k
1−t
2
kπ n+1
2 f (m+s+1) (t ) dt
2
sin
lπ x − xl B∗2(m+s) k n+1 2
1/2
1/2 .
(23)
Proof. This is a special case of Theorem 3 for a = −1, b = 1, r = m + s, w(t ) =
2
√
1−t 2
π
and Ak =
2 n+1
sin2
kπ n+1
.
Remark 8. For n = 1 and x1 = 0 we get one-point Gauss–Chebyshev formulae of second kind of Euler type
1
−1
1 − t 2 f (t )dt =
π 2
CG2 CG2 f (0) + Tm+s ( f, 1) + Km+s ( f, 1).
In a special case for m = 1 and s = 0 we derive the following inequality
1 1 (2) 2 1/2 π CG2 2 2 f (t )dt − f 1 − t 0 ≤ V 1, 0 1 − t f t dt , ( ) ( ) () 1 −1 2 −1
where V1CG2 (1, 0) ≈ 0.409931. Remark 9. For n = 2, x1 = − 21 and x2 =
1
−1
1 − t 2 f (t )dt =
1 2
π 1 4
f −
2
we get two-point Chebyshev–Gauss formulae of the second kind of Euler type
+f
1 2
CG2 CG2 + Tm+s ( f, 2) + Km+s ( f, 2).
In special cases for m1 < 4 and s = 0 we consider the following inequalities
1 1 1 (m +1) 2 1/2 π 1 2 2 1 ≤ V2CG2 (m1 , 0) 1 − t f t dt − f − + f 1 − t f t dt , ( ) ( ) −1 4 2 2 −1
where V2CG2 (1, 0) ≈ 0.227524, V2CG2 (2, 0) ≈ 0.0342287 and V2CG2 (3, 0) ≈ 0.0077412. Remark 10. Further, for n = 3, x1 = − Euler type
1
−1
1
− t2
f (t )dt =
π 8
√ 2 2 , x2
= 0 and x3 =
√ 2 f − 2
√
2 2
we get three-point Gauss–Chebyshev formulae of second kind of
√ + 2 f ( 0) + f
2 2
CG2 CG2 + Tm+s ( f, 3) + Km+s ( f, 3).
Specially for m1 < 6 and s = 0 we obtain the following inequalities
1 √ √ 1 (m +1) 2 1/2 2 π 2 CG2 2 2 1 , (t ) dt ≤ V3 (m1 , 0) −1 1 − t f −1 1 − t f (t )dt − 8 f − 2 + 2 f (0) + f 2
where V3CG2 (1, 0) ≈ 0.169113, V3CG2 (2, 0) ≈ 0.0174633, V3CG2 (3, 0) ≈ 0.00238238, V3CG2 (4, 0) ≈ 0.000382337 and V3CG2 (5, 0) ≈ 0.0000773027.
J. Peˇcari´c et al. / Applied Mathematics and Computation 268 (2015) 957–965
965
Theorem 8. Let f : [−1, 1] → R be such that f (m+s) is an absolutely continuous function and f (m+s+1) ≥ 0 on [−1, 1]. Then repreCG2 f, n satisfies the following inequality sentation (22) holds and the reminder Km+s ( )
|
CG2 Km+s
( f, n)| ≤
2m+s−1 (m+s−1)!
GCG2 m+s−1 (t, n)
∞
f (m+s−1) (−1)+ f (m+s−1) (1) 2
−
f (m+s−2) ; −1, 1
Proof. This is a special case of Theorem 4 for a = −1, b = 1, r = m + s, w(t ) =
2
√
1−t 2
π
.
and Ak =
(24) 2 n+1
sin2
kπ n+1
.
Acknowledgment This work has been fully supported by Croatian Science Foundation under the project 5435. References [1] M. Abramowitz, I.A. Stegun (Eds.), Handbook of Mathematical Functions with Formulae, Graphs and Mathematical Tables, Applied Math. Series 55, National Bureau of Standards, 4th printing, Washington, 1965. ˇ ´ A. Civljak, ´ M. Ribiˇcic´ Penava, General Integral Identities and Related Inequalities, Element, Zagreb, 2013. [2] A. Aglic´ Aljinovic, S. Kovaˇc, J. Peˇcaric, ´ M. Matic, ´ J. Peˇcaric, ´ Some inequalities of Euler–Grüss type, Comput. Math. Appl. 41 (2001) 843–856. [3] Lj. Dedic, [4] S.S. Dragomir, Some Grüss-type inequalities in inner product spaces, J. Inequal. Pure Appl. Math. 4 (2) (2003). Article 42. ˇ [5] P. Cerone, S.S. Dragomir, Some new Ostrowski-type bounds for the Cebyšev functional and applications, J. Math. Inequal. 8 (1) (2014) 159–170. ´ J. Peˇcaric, ´ I. Peric, ´ A. Vukelic, ´ Euler integral identity, quadrature formulae and error estimations, Element, Zagreb, 2011. [6] I. Franjic, ´ B. Tepeš, A Grüss-type inequality and its applications, J. Inequal. Appl. 3 (2005) 277–288. [7] S. Izumino, J. Peˇcaric, ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ New estimations of the remainder in three-point quadrature formulae of Euler type, J. Math. [8] M. Klariˇcic´ Bakula, J. Peˇcaric, Inequal. 9 (4) (2015) 1143–1156. ˇ ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ Some inequalities for the Cebyšev [9] M. Klariˇcic´ Bakula, J. Peˇcaric, functional and general four-point quadrature formulae of Euler type, Mat. Bilten (2015), accepted for publication. ´ J.E. Peˇcaric, ´ A.M. Fink, Classical and New inequalities in Analysis, Kluwer Academic, Dordrecht, 1993. [10] D.S. Mitrinovic, ´ M. Ribiˇcic´ Penava, A. Vukelic, ´ Euler’s method for weighted integral formulae, Appl. Math. Comput. 206 (2008) 445–456. [11] J. Peˇcaric, ´ A. Vukelic, ´ Estimations of the error for two-point formula via pre-Grüss inequality, Gen. Math. 13 (2) (2005) 95–104. [12] J. Peˇcaric, ˇ ´ A. Vukelic, ´ Some inequalities for the Cebyšev [13] J. Peˇcaric, functional and Euler two-point formulae, J. Math. Inequal. 9 (4) (2015) 1195–1205. ´ A new generalization of Grüss inequality in inner product spaces, J. Math. Anal. Appl. 250 (2000) 494–511. [14] N. Ujevic, ´ Sharp inequalities of Simpson type and Ostrowski type, Comput. Math. Appl. 48 (2004) 145–151. [15] N. Ujevic,