2000 Mathematical Subject Classification: 65D30, 65D32, 41A15, 65D07 ... 30 years, the spline functions were intensive studied, constituting the foundation in find ..... Lemma 1.4.3 If f â Hn[a, b] and p is a integer number, p ⥠2, then .... Remark 1.4.4 If in the quadrature formula (1.44) we choose p a even number, then we.
”BABES ¸ -BOLYAI” UNIVERSITY OF CLUJ-NAPOCA FACULTY OF MATHEMATICS AND COMPUTER SCIENCE
Ana Maria Acu
Spline Functions and Approximation Formulas of Definite Integrals – ABSTRACT OF THE DOCTORAL THESIS –
Supervisor: Prof. dr. Petru Blaga
CLUJ-NAPOCA 2007
2
To ........................................................... We announce you that on..........................in hall.............................. of the ”Babe¸s-Bolyai” University, Cluj-Napoca, there will be held the thesis Spline Functions and Approximation Formulas of Definite Integrals, by Ana Maria Acu, to obtain the title of Doctor in Mathematics. We respectfully invite you to participate to this thesis delivery.
3
Cuprins
Short presentation of the thesis.................................................................4 Thesis references................................................................................................39
2000 Mathematical Subject Classification: 65D30, 65D32, 41A15, 65D07 Key Words and Phrases: quadrature formulas, spline functions, orthogonal polynomial, optimal quadrature, monospline
4
Short presentation of the thesis This thesis belong of Numerical Analysis and its object is the study of some quadrature formulae and the usage of spline functions in their study. The relation between monosplines and quadrature formulae was observed by Birkhoff in 1906, Peano in 1914 and Tschakaloff in 1939. This relation was described and exploited further by Nikolski in 1950. The name monospline was used for the first time by Schoenberg in 1965. In the last 30 years, the spline functions were intensive studied, constituting the foundation in find some optimal quadrature formulae. The first results about optimal quadrature formulae was obtained by A. Sard, L.S. Meyers and S.M. Nikolski. The present thesis has four chapters and 144 references, from these 12 papers are signed as sole authors and 2 as co-authors. The first chapter, entitled ”Monospline functions” has 5 sections. In first two sections are performed some general concepts about the polynomial spline functions and the quadrature formulae of the interpolator type, thus in section 1.3 we can to perform the relation between monospline and quadrature formulae. Let ∆ : −∞ =: x0 < x1 < x2 < · · · < xm < xm+1 := ∞ be fixed points on the real line R. Definition 1.1.1 The function s : R → R is called a spline function of degree n with knots {xk }m k=1 if: i)for each k = 0, · · · , m, the function s(x) coincides on (xk , xk+1 ) with a polynomial of degree not greater then n; ii) s ∈ C n−1 (R). Denote by Sn (∆) the class of all spline functions of degree n with knots ∆ = {xk }m k=1 . Theorem 1.1.1 The function s belongs to Sn (∆), if and only if it may be written in the form m n X X j aj x + ck (x − xk )n+ , s(x) = j=0
k=1
with some real coefficients {aj }nj=0 and {ck }m k=1 . Moreover, ck =
s(n) (xk + 0) − s(n) (xk − 0) , n!
for k = 1, 2, . . . , m. 5
Denote by z := (z1 , z2 , · · · , zm ) a vector which verifies for 1 ≤ i ≤ m, the relation 1 ≤ zi ≤ n. The vectorul z is called incidence vector. Definition 1.1.2 The function s : R → R is called a spline function of degree n, with knots {xi }m i=1 of multiplicities z1 , z2 , · · · , zm , if i)for each k = 0, . . . , m, the function s(x) coincides on (xk , xk+1 ) with a polynomial of degree not greater then n; ii) s ∈ C n−zk (xk−1 , xk+1 ) for k = 1, m. Denote by Sn (∆, z) the class of all spline functions of degree n with knots ∆ = {xi }m i=1 of multiplicities z = (z1 , z2 , · · · , zm ). Theorem 1.1.2 The function s belongs to Sn (∆, z), if and only if it may be written in the form n m zX k −1 X X s(x) = a j xj + ckν (x − xk )n−ν + , j=0
k=1 ν=0
zk −1 with some real coefficient {aj }nj=0 and {ckν }m k=1 ν=0 . Moreover,
ckν =
s(n−ν) (xk + 0) − s(n−ν) (xk − 0) . (n − ν)!
Definition 1.2.1 It is called a quadrature formula or formula of numerical integration, the following formula Z m X I[f ] = f (x)dλ(x) = Ai λi [f ] + Rm [f ],
R
i=0
where λi [f ], i = 0, m are the punctual (local) information relating to function f , which to integrate with respect to the measure dλ, Ai , i = 0, m are called the coefficients of the quadrature formula, and Rm [f ] is the remainder term. Definition 1.2.2 A quadrature formula has degree of exactness equal n, if Rm [e0 ] = 0 , Rm [e1 ] = 0 , · · · , Rm [en ] = 0, where ej (t) = tj . Moreover, if Rm [en+1 ] 6= 0 then the quadrature formula has degree of exactness effectively equal n. The class of quadrature formulae of the interpolator type is obtained by integration of interpolation formulae. For example, the quadrature formulae of the 6
Newton-Cotes type are the formulae of the interpolator type and they are obtained by integration of Lagrange interpolation formula, corresponding to integrable function f : [a, b] → R. We shall enumerate some quadrature formulae of the Newton-Cotes type: -the mid-point quadrature formula µ ¶ Z b (b − a)3 00 a+b f (x)dx = (b − a)f + f (ξ), a < ξ < b, 2 24 a -the trapezoid quadrature formula Z b (b − a)3 00 b−a [f (a) + f (b)] − f (ξ), a < ξ < b, f (x) = 2 12 a - Simpson0 s quadrature formula · µ ¶ ¸ Z b a+b (b − a)5 (4) b−a f (a) + 4f + f (b) − f (x) = f (ξ), a < ξ < b, 6 2 2880 a -Newton-Simpson0 s quadrature formula · µ ¶ µ ¶ ¸ Z b b−a 2a + b a + 2b f (x) = f (a) + 3f + 3f + f (b) 8 3 3 a (b − a)5 (4) − f (ξ), a < ξ < b. 6480 © ª We use the notation H n [a, b] := f ∈ C n−1 [a, b], f (n−1) absolutely continuous . Theorem 1.3.1 (Peano0 s theorem) Let L(f ) be an arbitrary linear functional defined £ ¤ in H n [a, b] such that the function K(t) := L (x − t)n−1 is integrable over [a, b]. Suppose + that L(p) = 0 for each polynomial p ∈ Pn−1 . Then Z b 1 L(f ) = K(t)f (n) (t)dt (n − 1)! a for each f ∈ H n [a, b]. Definition 1.3.1 It is called monospline of degree n with knots x1 < · · · < xm of multiplicities z1 , . . . , zm , respectively, a function of the form m z −1
n−1
k tn X j X X n−ν−1 M (t) = , + aj t + ckν (t − xk )+ n! j=0 k=1 ν=0
m zk −1 with real coefficients {aj }n−1 j=0 , {ckν }k=1 ν=0 and 1 ≤ zk ≤ n, k = 1, m.
Definition 1.3.2 It is caled generalised monospline of degree n with knots x1 < · · · < xm of multiplicities z1 , . . . , zm , respectively, a function of the form M (t) = v(t) +
n−1 X j=0
j
aj t +
m zX k −1 X k=1 ν=0
7
n−ν−1 ckν (t − xk )+ ,
m zk −1 th with real coefficients {aj }n−1 j=0 , {ckν }k=1 ν=0 and 1 ≤ zk ≤ n, k = 1, m, where v is the n integral of weight function w : [a, b] → R.
Let m z −1
n−1
(1.19)
k n−ν−1 (xk − t)+ (b − t)n X (b − t)n−j−1 X X M (t) = − Bj − akν n! (n − j − 1)! k=1 ν=0 (n − ν − 1)! j=0
m zk −1 be the monospline with real coefficients {Bj }n−1 j=0 and {akν }k=1 ν=0 . We choose the following quadrature formula
Z
b
(1.21)
f (x)dx = a
n−1 X
Aj f
(j)
(a) +
j=0
n−1 X
Bj f
(j)
(b) +
m zX k −1 X
akν f (ν) (xk ) + Rn [f ]
k=1 ν=0
j=0
which has degree of exactness equal n − 1, namely R[p] = 0 (∀) p ∈ Pn−1 . Between the monospline (1.19) and the quadrature formula (1.21) there is a connection, m zk −1 namely, the coefficients {Bj }n−1 j=0 and {akν }k=1 ν=0 of the quadrature formula are the same with the coefficients of monospline(1.19), Aj = (−1)n−j−1 M (n−j−1) (a), j = 0, n − 1 and the remainder term of quadrature formula has the representation: Z b M (t)f (n) (t)dt (∀) f ∈ H n [a, b]. Rn [f ] = a
In section 1.4, by using this connection between the monosplines and the quadrature formulae and by choosing conveniently the monospline we obtained some quadrature formulae. Some of them are the generalization of the trapezoid, mid-point and Simpson 0 s quadrature formulae. The results of this section were obtained in [3], [6], [8], [11]. Let (∆m )m∈N be a division of [a, b], ∆m : a = x0 < x1 < x2 < · · · < xm−1 < xm = b and (ξi )i=1,m a system of intermediate points, ξ1 < ξ2 < · · · < ξm , ξi ∈ [xi−1 , xi ], for i = 1, m. We choose the monospline of degree n with knots {xi }m−1 i=1 of multiplicities zi = n, i = 1, m − 1 n−1
(1.24)
m−1 n−1
n−k−1 (xi − t)+ (b − t)n−k−1 X X (b − t)n X − Ak,m − Ak,i M (t) = n! (n − k − 1)! (n − k − 1)! i=1 k=0 k=0
where (xi − ξi )k+1 − (xi − ξi+1 )k+1 , i = 1, m − 1, k = 0, n − 1, (k + 1)! (b − ξm )k+1 , k = 0, n − 1. = (−1)k (k + 1)!
Ak,i = (−1)k Ak,m
8
Lemma 1.4.1 The monospline, defined in (1.24) has the following representation n (ξi − t) , t ∈ [xi−1 , xi ) , i = 1, m − 1, n! n M (t) = (ξ − t) m , t ∈ [xm−1 , b]. n! By using connection between the monospline (1.24) and the quadrature formula Z
b
(1.24)
f (t)dt = a
m X n−1 X
Ak,i f (k) (xi ) + Rn [f ],
i=0 k=0
the following lemma can be obtained: Lemma 1.4.2 If f ∈ H n [a, b], then Z
b
(1.26)
f (t)dt = a
n−1 X m X
Ak,i f (k) (xi ) + Rn [f ]
k=0 i=0
where Z
b
n
(1.27)
Rn [f ] = (−1)
Kn (t)f (n) (t)dt,
a
n (t − ξi ) , t ∈ [xi−1 , xi ) , i = 1, m − 1, n! n Kn (t) = (t − ξm ) , t ∈ [xm−1 , b] n!
(1.28)
and for k = 0, n − 1 (xi − ξi )k+1 − (xi − ξi+1 )k+1 , i = 1, m − 1, (k + 1)! (a − ξ1 )k+1 Ak,0 = (−1)k+1 , (k + 1)! (b − ξm )k+1 Ak,m = (−1)k . (k + 1)! Ak,i = (−1)k
P. Cerone and S.S. Dragomir have studied in [41] the following quadrature formula: Z
b
(1.29) a
where
n−1 X
(b − a)k+1 (k) f (t)dt = [1 + (−1) ] k+1 f 2 (k + 1)! k=0
µ
k
(t − a)n , n! Kn (t) = (t − b)n , n! 9
·
a+b 2
¶
Z + (−1)
¸ a+b t ∈ a, , 2 µ a+b t∈ , b] . 2
b
n a
Kn (t)f (n) (t)dt
For n = 1 the formula (1.29) is the mid-point quadrature formula µ ¶ Z b Z b a+b − K1 (t)f 0 (t)dt. f (t)dt = (b − a)f 2 a a We can observe that the quadrature formula (1.29) can be obtained by choosing the division of [a,b]: a+b ∆ : a = x0 < x1 < x2 = b, x1 = , 2 the intermediate points · ¸ a+b ξ1 = a ∈ a, , 2 ·
¸ a+b ξ2 = b ∈ ,b . 2 and by using Lemma 1.4.2. A number of authors have considered generalizations of known and some new quadrature formulae. For example, generalization of the trapezoid, mid-point and Simpson 0 s quadrature formulae are considered by G.A.Anastassiou, P.Cerone, S.S.Dragomir, Lj.Dedi´ c, M.Mati´ c, Peˇ cari´ c, C.E.M. Pearce, N. Ujevi´ c and S. Varoˇ sanec in [21], [40], [41], [57], [103], [142]. In [142], Nenad Ujevi´ c considers for the quadrature formula · µ ¶ µ ¶¸ Z b b−a 3a + b a + 3b f (t)dt = (1.30) f +f + R[f ] 2 4 4 a the following generalization Z
b a
(1.31)
· µ ¶ µ ¶¸ b−a 3a + b a + 3b f (t)dt = f +f 2 4 4 · µ ¶ µ ¶¸ m X (b − a)2i+1 3a + b a + 3b (2i) (2i) +2 f +f + R [f ] 42i+1 (2i + 1)! 4 4 i=1 ·
¸ n−1 where m = , 2
Z
b
n
R [f ] = (−1)
a
1 (t − a)n , n! ¶n 1µ a+b Un (t) = , t− n! 2 1 n (t − b) , n! 10
Un (t)f (n) (t)dt, ·
¸ 3a + b t ∈ a, , 4 µ ¶ 3a + b a + 3b t∈ , , 4 4 · ¸ a + 3b t∈ ,b . 4
and f ∈ H n [a, b]. We can observe that the quadrature formula (1.31) can be obtained by choosing the division of [a,b]: ∆ : a = x0 < x1 < x2 < x3 = b, x1 =
3a + b a + 3b , x2 = , 4 4
the intermediate points ¸ 3a + b ξ1 = a ∈ a, , ¸ ·4 a+b 3a + b a + 3b , ξ2 = ∈ , 2· 4¸ 4 a + 3b ξ3 = b ∈ ,b , 4 ·
and by using Lemma 1.4.2. In [8], [6], [11], [3] we have studied another quadrature formulae and we have considered some generalizations of them. · ¸ hpi n−1 Let n ∈ N, p ∈ N, p ≥ 2, m = and q = . 2 2 Lemma 1.4.3 If f ∈ H n [a, b] and p is a integer number, p ≥ 2, then Z
b
(1.32) a
µ ¶ q m X (b − a)2k+1 X (2k) b−a f (t)dt = 2 f a + (2ν − 1) p2k+1 (2k + 1)! ν=1 p k=0 n−1 1 − (−1)p X (b − a)k+1 (k) + (−1)k k+1 f (b) + R[f ] , 2 p (k + 1)! k=0
·
¸ hpi n−1 , where m = ,q= 2 2 Z (1.33)
b
n
R [f ] = (−1)
Sn (t)f (n) (t)dt,
a
(1.34)
1 (t − a)n, n! µ ¶n 1 t−a−2ν b − a , n! p Sn (t) = µ ¶n 1 b−a , t − a − 2q n! p
¶ b−a t ∈ [a , a + , p · ¶ b−a b−a t ∈ a+(2ν −1) , a+(2ν +1) p p ν = 1, q − 1, q > 1, · ¸ b−a t ∈ a + (2q − 1) ,b . p
Remark 1.4.1 If in the quadrature formula (1.32) we choose p a even number, then we obtain a quadrature formula of open type. 11
In [8], [11] we have obtained some inequalities of the remainder term from the quadrature formula (1.32). Theorem 1.4.1 The monospline Sn (t), n > 1 of degree n, defined in (1.34), verifies Z
(1.35) (1.36) (1.37)
b
1 − (−1)p (b − a)n+1 · n+1 , if n is odd, Sn (t)dt = 2 p (n + 1)! a · ¸ Z b 1 − (−1)p (b − a)n+1 |Sn (t)| dt = 2q + , 2 pn+1 (n + 1)! a (b − a)n max |Sn (t)| = . t∈[a,b] pn n!
Remark 1.4.2 The quadrature formula (1.32) has degree of exactness equal n − 1, but if p is even number and n is a odd number, the quadrature formula has degree of exactness equal n. Theorem 1.4.2 If f ∈ H n [a, b] and there exist real numbers γn , Γn such that γn ≤ f (n) (t) ≤ Γn , t ∈ [a, b], then ¯ ¯¶¸ · µ (b−a)n+1 1−(−1)p Γn −γn ¯¯ Γn +γn ¯¯ (1.38) |R[f ]| ≤ q(Γn −γn )+ +¯ , if n is odd, (n+1)!pn+1 2 2 2 ¯ ¸ 1 − (−1)p (b − a)n+1 kf (n) k∞ , if n is even. |R[f ]| ≤ 2q + 2 pn+1 (n + 1)! ·
(1.39)
Theorem 1.4.3 Let f ∈ H n [a, b] and let n be odd. If there exist a real number γn such that γn ≤ f (n) (t), t ∈ [a, b], then (1.42)
· ¸ (b − a)n+1 |γn | 1 − (−1)p |R[f ]| ≤ · T n − γn + pn n! 2 (n + 1)p
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 , t ∈ [a, b] , then (1.43)
· ¸ (b − a)n+1 |Γn | 1 − (−1)p |R[f ]| ≤ · Γn − Tn + . pn n! 2 (n + 1)p
12
By using quadrature formula (1.32) for different values of p and n we have computed Z 0.35 Z π 4 2 2 −x x e dx, x2 sin xdx and we have compared these results with the dx, 2−4 x 0 0 0 exact values of integrals.
Z
1
Z
1
Exact value of
x2 e−x dx = 0.1606027942...
0
n 1 2 3 4 5
p=3 0.1757028738 0.1552651270 0.1598745395 0.1600637779 0.1605910028 Z
0.35
Exact value of 0
n 1 2 3 4 5
p=4 0.1571906174 0.1571906174 0.1604240313 0.1604240313 0.1606020357
p=3 -0.1772412433 -0.1766075370 -0.1768219019 -0.1768195368 -0.1768200279 Z
π 4
Exact value of
p=5 0.1657044559 0.1583468671 0.1605050909 0.1605296162 0.1606022363
p=6 0.1589834525 0.1589834525 0.1605669840 0.1605669840 0.1606027269
2 dx = −0.1768200201... x2 − 4 p=4 p=5 -0.1767015470 -0.1769719468 -0.1767015470 -0.1767438125 -0.1768198718 -0.1768202648 -0.1768198718 -0.1768199583 -0.1768200199 -0.1768200205
p=6 -0.1767672800 -0.1767672800 -0.1768199907 -0.1768199907 -0.1768200201
x2 sin xdx = 0.0887552845...
0
n 1 2 3 4 5
p=3 0.1234796102 0.07046819157 0.08907645590 0.08898362001 0.08875083243
p=4 0.07865454284 0.07865454284 0.08882454113 0.08882454113 0.08875514835 ·
n−1 Let n ∈ N, p ∈ N, p ≥ 1, m = 2
p=5 0.1013997005 0.08231558980 0.08879590523 0.08878387370 0.08875507821 ¸
·
p=6 0.08430599815 0.08430599815 0.08876887512 0.08876887512 0.08875527256
¸ p+1 and r = . 2
13
Lemma 1.4.4 If f ∈ H n [a, b] and p is a integer number, p ≥ 1, then µ ¶ Z b r−1 m X b−a (b − a)2k+1 X (2k) f f (t)dt = 2 a + 2ν (1.44) p2k+1 (2k + 1)! ν=1 p a k=0 µ ¶ n−1 X (b − a)k+1 1 + (−1)p (k) k (k) + f (a) + (−1) f (b) · + R[f ] , pk+1 (k + 1)! 2 k=0 · ¸ ¸ · p+1 n−1 ,r= , where m = 2 2 Z b n (1.45) R [f ] = (−1) Sn (t)f (n) (t)dt , a
¶n µ 1 b−a t− a−(2ν +1) , p n! (1.46)
Sn (t) =
· ¶ b−a b−a t ∈ a+2ν , a+(2ν +2) p p µ ¶n ν =·0, r − 2 , r > 2 , ¸ 1 b−a b−a , t ∈ a + (2r − 2) t − a − (2r − 1) ,b . n! p p
Remark 1.4.4 If in the quadrature formula (1.44) we choose p a even number, then we obtain a quadrature formula of close type. In [6], we have obtained some inequalities of the remainder term from the quadrature formula (1.44). Theorem 1.4.4 The monospline Sn (t), n > 1 of degree n, defined in(1.46), verifies Z b (−1)p − 1 (b − a)n+1 Sn (t)dt = (1.47) · n+1 , if n is odd, 2 p (n + 1)! a · ¸ Z b (−1)p − 1 (b − a)n+1 (1.48) |Sn (t)| dt = 2r + , 2 pn+1 (n + 1)! a (b − a)n (1.49) max |Sn (t)| = . t∈[a,b] pn n!
Remark 1.4.5 The quadrature formula (1.44) has degree of exactness equal n − 1, but if p is even number and n is a odd number, the quadrature formula has degree of exactness equal n. Theorem 1.4.5 If f ∈ H n [a, b] and there exist real numbers γn , Γn such that γn ≤ f (n) (t) ≤ Γn , t ∈ [a, b], then ¯ ¯¶¸ · µ (−1)p −1 Γn −γn ¯¯Γn +γn ¯¯ (b−a)n+1 r(Γn −γn)+ −¯ , if n is odd (1.50) |R[f ]| ≤ (n+1)!pn+1 2 2 2 ¯ 14
·
(1.51)
¸ (−1)p − 1 (b − a)n+1 |R[f ]| ≤ 2r + kf (n) k∞ , if n is even. n+1 2 p (n + 1)!
Theorem 1.4.6 Let f ∈ H n [a, b] and let n be odd. If there exist real number γn such that γn ≤ f (n) (t) , t ∈ [a, b], then · ¸ (b − a)n+1 1 − (−1)p |γn | (1.54) |R[f ]| ≤ T n − γn + · pn n! 2 (n + 1)p 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 , t ∈ [a, b] , then · ¸ (b − a)n+1 1 − (−1)p |Γn | (1.55) |R[f ]| ≤ Γ n − Tn + . · pn n! 2 (n + 1)p formula (1.44) for different values of p and n we have computed Z 1 By using quadrature Z 1 3 1 dx and (1+x2 ) 2 dx and we have compared these results with the exact values x 0 1+e 0 of integrals. Z
1
Exact value of 0
n 1 2 3 4 5
p=3 0.3928290875 0.3789401986 0.3798299450 0.3798942454 0.3798858922 Z
1
Exact value of
1 dx = 0.3798854936... 1 + ex p=4 p=5 0.3810056897 0.3845351435 0.3793373127 0.3795351435 0.3798736963 0.3798783337 0.3798882918 0.3798866670 0.3798855817 0.3798855115
p=6 0.3803813785 0.3796398776 0.3798831762 0.3798860593 0.3798855008
3
(1 + x2 ) 2 dx = 1.5679519622...
0
n 1 2 3 4 5
p=3 1.490670780 1.490670780 1.567398578 1.567398578 1.567953389
p=4 1.655878024 1.523295503 1.568643955 1.567780788 1.567950830
p=5 1.539833194 1.539833194 1.567880955 1.567880955 1.567952011
15
p=6 1.607144560 1.548218995 1.568088497 1.567917995 1.567951861
We studied the quadrature formulae with the weight function w(t) = (b − t)(t − a). These results were obtained in [5] and [14]. Let m ∈ N, m ≥ 2 and (∆m )m∈N be a division of [a, b], ∆m : a = x0 < x1 < x2 < · · · < xm−1 < xm = b and (ξi )i=1,m a system of intermediate points, ξ1 < ξ2 < · · · < ξm , ξi ∈ [xi−1 , xi ] , for i = 1, m. Let (1.56) n−1 m−1 n−1 X (t−xi )n−k−1 2 (t−a)n−k−1 X X (t−a)n+1 + n+2 Mn (t) = (b − a)− (t−a) + Ak,0 + Ak,i , (n+1)! (n + 2)! (n−k−1)! i=1 k=0 (n−k−1)! k=0
where (a − ξ1 )k+1 (a − ξ1 )k+2 (a − ξ1 )k+3 + (b − 2ξ1 + a) −2 , (k + 1)! (k + 2)! (k + 3)! (b − ξi+1 )(ξi+1 − a)(xi − ξi+1 )k+1 − (b − ξi )(ξi − a)(xi − ξi )k+1 = (k + 1)! k+2 (b − 2ξi+1 + a)(xi − ξi+1 ) − (b − 2ξi + a)(xi − ξi )k+2 + (k + 2)! k+3 (xi − ξi+1 ) − (xi − ξi )k+3 − 2 , (k + 3)! k = 0, n − 1, i = 1, m − 1
Ak,0 = (b − ξ1 )(ξ1 − a) Ak,i
be the generalized monospline of degree n. Lemma 1.4.5 The generalized monospline, defined in (1.56), has the representation (t − ξi )n+1 (t − ξi )n+2 (t−ξi )n + (b−2ξ + a) −2 , (b − ξ )(ξ − a) i i i n! (n + 1)! (n + 2)! t ∈ [xi−1 , xi ) , i = 1, m − 1 , (1.57) Mn (t) = n (t−ξm )n+1 (t−ξm )n+2 (t−ξm ) +(b−2ξm + a) −2 , (b−ξm )(ξm −a) n! (n + 1)! (n + 2)! t ∈ [x ,x ]. m−1
Let Z
b
(1.62)
w(t)f (t)dt = a
+
n−1 m−1 n−1 X XX (−1)k+1 Ak,0 f (k) (a) + (−1)k+1 Ak,i f (k) (xi ) i=1 k=0
k=0 n−1 X
(−1)k Ak,m f (k) (b) + R[f ],
k=0
16
m
be a quadrature formula, where w(t) = (b − t)(t − a). By using connection between the generalized monospline (1.56) and the quadrature n−1 m−1 formula (1.62), namely, the coefficients {Ak,i }k=0 i=0 of the quadrature formula are the (n−k−1) same with the coefficients of monospline (1.56), Ak,m = Mn (b), k = 0, n − 1 and the remainder term of quadrature formula has the representation Z b n R[f ] = (−1) Mn (t)f (n) (t)dt, f ∈ H n [a, b] a
we obtain the following lemma: Lemma 1.4.6 If f ∈ H n [a, b], then Z
b
(1.58)
w(t)f (t)dt = a
+
n−1 X
(−1)
k+1
Ak,0 f
(k)
(a) +
m−1 n−1 XX
(−1)k+1 Ak,i f (k) (xi )
i=1 k=0
k=0 n−1 X
(−1)k Ak,m f (k) (b) + R[f ],
k=0
where w(t) = (b − t)(t − a), (a − ξ1 )k+1 (a − ξ1 )k+2 (a − ξ1 )k+3 + (b − 2ξ1 + a) −2 , (k + 1)! (k + 2)! (k + 3)! (b − ξi+1 )(ξi+1 − a)(xi − ξi+1 )k+1 − (b − ξi )(ξi − a)(xi − ξi )k+1 (k + 1)! k+2 (b − 2ξi+1 + a)(xi − ξi+1 ) − (b − 2ξi + a)(xi − ξi )k+2 (k + 2)! k+3 (xi − ξi+1 ) − (xi − ξi )k+3 2 , (k + 3)! · ¸ (b − ξm )k+1 (b − ξm )k+2 (b − ξm )k+3 (b − ξm )(ξm − a) + (b − 2ξm + a) −2 , (k + 1)! (k + 2)! (k + 3)! k = 0, n − 1 , i = 1, m − 1,
Ak,0 = (b − ξ1 )(ξ1 − a) Ak,i = + − Ak,m =
Z
b
n
(1.59)
R[f ] = (−1)
Mn (t)f (n) (t)dt
a
and
(1.60)
Mn (t) =
(t − ξi )n+1 (t − ξi )n+2 (t − ξi )n + (b − 2ξ + a) − 2 , (b − ξ )(ξ − a) i i i n! (n + 1)! (n + 2)! t ∈ [x , x ) , i = 1, m − 1, i−1
i
(t−ξm )n (t−ξm )n+1 (t−ξm )n+2 (b−ξm )(ξm −a) +(b−2ξm + a) −2 , n! (n + 1)! (n + 2)! t ∈ [xm−1 , xm ] . 17
Remark 1.4.6 If we choose ξ1 = a and ξm = b, then the formula (1.58) is a quadrature formula of open type. In the case of equidistance nodes, namely, xi = a + 2ih, i = 0, m , ξi = a + (2i − 1)h, i = 1, m, b−a where h = , we have given some inequalities for the remainder term. In this case we 2m have the quadrature formula Z b n−1 m−1 n−1 X XX (1.63) w(t)f (t)dt = (−1)k+1 Ak,0 f (k) (a) + (−1)k+1 Ak,i f (k) (xi ) a
+
i=1 k=0
k=0 n−1 X
(−1)k Ak,m f (k) (b) + R[f ],
k=0
where w(t) = (b − t)(t − a), ½ ¾ k+3 2(m − 1) 2 k+1 h Ak,0 = (−1) 2m − 1 − − , (k + 1)! k+2 (k + 2)(k + 3) ª hk+3 © (2m − 2i − 1)(2i + 1)(−1)k+1 − (2m − 2i + 1)(2i − 1) Ak,i = (k + 1)! ª ª hk+3 © 2hk+3 © + (2m − 4i − 2)(−1)k − (2m − 4i + 2) − (−1)k+1 − 1 , (k + 2)! (k + 3)! ½ ¾ hk+3 2(m − 1) 2 Ak,m = 2m − 1 − − , (k + 1)! k+2 (k + 2)(k + 3) k = 0, n − 1, i = 1, m − 1. Z
b
n
(1.64)
R[f ] = (−1)
Mn (t)f (n) (t)dt
a
and (1.65)
( Mn (t) =
Pn,i (t), t ∈ [a + (2i − 2)h, a + 2ih) , i = 1, m − 1, Pn,m (t), t ∈ [a + (2m − 2)h, a + 2mh] ,
where Pn,i (t) = (2m − 2i + 1)(2i − 1) + (2m − 4i + 2) −
h2 [t − a − (2i − 1)h]n n!
h · [t − a − (2i − 1)h]n+1 (n + 1)!
2 [t − a − (2i − 1)h]n+2 , i = 1, m. (n + 2)! 18
Theorem 1.4.7 The generalized monospline of degree n, Mn (t), n > 1, defined in (1.65) verifies: Z b (1.66) Mn (t)dt = 0, if n is odd, a ½ 2 ¾ Z b 2mhn+3 2m + 1 2 (1.67) |Mn (t)| dt = · − , (n + 1)! 3 (n + 2)(n + 3) a ¸ n+2 · h n 2 m − , if m is even , n+2 n! (1.68) max |Mn (t)| = · ¸ t∈[a,b] 2 hn+2 2 m − , if m is odd. n! (n + 1)(n + 2)
Theorem 1.4.8 If f ∈ H n [a, b], n > 1 and there exist real numbers γn , Γn such that γn ≤ f (n) (t) ≤ Γn , t ∈ [a, b], then ½ ¾ mhn+3 (Γn − γn ) 2m2 + 1 2 (1.69) |R[f ]| ≤ − , if n is odd (n + 1)! 3 (n + 2)(n + 3) and (1.70)
2mhn+3 |R[f ]| ≤ (n + 1)!
½
2m2 + 1 2 − 3 (n + 2)(n + 3)
¾
° (n) ° °f ° , if n is even. ∞
Theorem 1.4.9 Let f ∈ H n [a, b], n > 1 and let n, m be odd numbers. If there exist a real number γn such that γn ≤ f (n) (t), then · ¸ 2mhn+3 2 2 (1.73) |R[f ]| ≤ m − · (Tn − γn ), n! (n + 1)(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 · ¸ 2mhn+3 2 2 (1.74) |R[f ]| ≤ m − · (Γn − Tn ). n! (n + 1)(n + 2)
Theorem 1.4.10 Let f ∈ H n [a, b], n > 1, n a odd number and let m be even. If there exist a real number γn such that γn ≤ f (n) (t), then · ¸ 2mhn+3 n 2 (1.75) |R[f ]| ≤ m − · (Tn − γn ), n! n+2 19
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 · ¸ 2mhn+3 n 2 |R[f ]| ≤ m − · (Γn − Tn ). n! n+2
(1.76)
In the section 1.5 we have obtained a property of the intermediate point from the quadrature formulae previous studied. These results were obtained in [9]. We have considered the quadrature formula (1.32). If f : [a, b] → R , f ∈ C n [a, b] and n is even, then for any x ∈ (a , b ] there is cx ∈ (a, x) such that Z
x
(1.84) a
n−1 [X [ p2 ] µ ¶ 2 ] 2k+1 X (x − a) x−a (2k) f a + (2ν − 1) f (t)dt =2 p2k+1 (2k + 1)! ν=1 p k=0
n−1 1 − (−1)p X (x − a)k+1 (k) (x − a)n+1 (n) + (−1)k k+1 f (x) + n f (cx ). 2 p (k + 1)! p (n + 1)! k=0
Theorem 1.5.8 If f ∈ C 2n+1 [a, b] , n is even and f (n+1) (a) 6= 0, then for the intermediate point cx that appears in formula (1.84), it follows: 1 , cx − a 2 lim = 1 n x→a x − a , + 2 2 2p (n + 2)
if p is even, if p is odd.
We have considered the quadrature formula (1.44). If f : [a, b] → R, f ∈ C n [a, b] and n is even, then for any x ∈ (a , b ] there is cx ∈ (a, x) such that Z
x
(1.86) a
n−1 [X µ ¶ r−1 2 ] (x − a)2k+1 X (2k) x−a f (t)dt = 2 f a + 2ν 2k+1 (2k + 1)! p p ν=1 k=0 µ ¶ n−1 X (x − a)k+1 1 + (−1)p (k) k (k) + f (a) + (−1) f (x) · pk+1 (k + 1)! 2 k=0
+
(x − a)n+1 (n) f (cx ), (n + 1)!pn
·
¸ p+1 where r = . 2 20
Theorem 1.5.9 If f ∈ C 2n+1 [a, b] , n is even and f (n+1) (a) 6= 0, then for the intermediate point cx that appears in formula (1.86), it follows: 1 , if p is even, cx − a 2 lim = n 1 x→a x − a , if p is odd. − 2 2 2p (n + 2) The second chapter, entitle ”Optimal quadrature formulas” has 2 sections. In section 2.1 are studied the optimal quadrature formulae in the sense of Sard. This quadrature formulae can be obtained by integration of the spline interpolation formula. In section 2.2 by using connection between monosplines and quadrature formulae are presented some general concept about the optimal quadrature formulae in the sense of Nikolski. Interesting results about the optimal quadrature formulae in the sense of Nikolski were obtained by G. Coman, I. Gˆansc˘a, G. Micula, in [45], [46], [47], [48], [49], [50], [51], [53], [52]. In the section 2.2, from the class of quadrature formulae studied in the first chapter, we have obtained the optimal quadrature formulae in the sense of the Nikolski. Let H be the class of sufficiently smooth functions f : [a, b] → R and we consider the following quadrature formula with degree of exactness equal n − 1 Z b m zX i −1 X (2.1) f (x)dx = Aki f (k) (xi ) + Rn [f ], a
i=0 k=0
where the nodes a ≤ x0 < x1 < · · · < xm ≤ b have the multiplicities zi , 1 ≤ zi ≤ n. The quadrature formula (2.1) is called optimal in the sense of Sard in the space H, if Em,n (H, A) = sup |Rn [f ]| f ∈H
zi −1 attains the minimum value with regard to A, where A = {Aki }m i=0 k=0 are the coefficients of quadrature formula. The quadrature formula (2.1) is called optimal in sense Nikolski in the space H, if
Em,n (H, A, X) = sup |Rn [f ]| f ∈H
zi −1 attains the minimum value with regard to A and X, where A = {Aki }m i=0 k=0 are the coefficients and X = (x0 , x1 , . . . , xm ) are the nodes of quadrature formula. We denote n o ° ° Wpn [a, b] := f ∈ C n−1 [a, b], f (n−1) absolutely continuous , °f (n) °p < ∞
with
½Z kf kp :=
b
p
¾ p1
|f (x)| dx a
kf k∞ := sup |f (x)| . x∈[a,b]
21
, for 1 ≤ p < ∞,
If f ∈ Wpn [a, b], by using Peano0 s theorem, the remainder term can be written Z
b
Rn [f ] =
Kn (t)f (n) (t)dt,
a
·
¸ (x − t)n−1 + where Kn (t) = Rn . (n − 1)! For remainder term we have the evaluation (2.2)
|Rn [f ]| ≤
£
where
Mn[p] [f ]
Z Mn[p] [f ]
b
= a
¤ p1
·Z
b
q
¸ 1q
|Kn (t)| dt
,
a
¯ (n) ¯p ¯f (t)¯ dt, 1 + 1 = 1, p q
with remark that in the cases p = 1 and p = ∞ this evaluation is |Rn [f ]| ≤ Mn[1] [f ] sup |Kn (t)| ,
(2.3)
t∈[a,b]
Z (2.4)
|Rn [f ]| ≤
where
Mn∞ [f ] Z
b
|Kn (t)| dt, a
¯ (n) ¯ ¯f (t)¯ dt, a ¯ ¯ Mn∞ [f ] = sup ¯f (n) (t)¯ . [1] Mn [f ]
b
=
t∈[a,b]
Let Gn,m be the class of quadrature formulae of the form (1.26). In the section 2.2 from the class of quadrature formulae Gn,m we have obtained the optimal quadrature formulae in the sense Nikolski. The quadrature formulae (1.26) is optimal in the sense of Nikolski in Wpn [a, b], p > 1, if
Z
b
|Kn (t)|q dt,
a
1 1 + =1 p q
attains the minimum value. The following results were obtained in [3]. Theorem 2.2.3 If f ∈ Wpn [a, b], p > 1, then quadrature formula of the form (1.26), optimal with regard to the error, is Z
b
f (x)dx = a
n−1 X m X
A∗k,i f (k) (x∗i ) + R∗n [f ],
k=0 i=0
22
where, for k = 0, n − 1 (b − a)k+1 , 2k+1 mk+1 (k + 1)! (b − a)k+1 A∗k,i = [1 + (−1)k ] k+1 k+1 , i = 1, m − 1, 2 m (k + 1)! (b − a)k+1 A∗k,m = (−1)k k+1 k+1 , 2 m (k + 1)! b−a x∗i = a + i, i = 1, m − 1, m A∗k,0 =
with (b − a)n |R∗n [f ]| ≤ · (2m)n n!
µ
b−a qn + 1
¶ 1q
£ ¤1 · Mn[p] [f ] p .
Theorem 2.2.4 The quadrature formula of the form (1.26) is optimal in the sense of Nikolski for p = ∞ if it has the coefficients and knots (b − a)k+1 , 2k+1 mk+1 (k + 1)! (b − a)k+1 , i = 1, m − 1, A∗k,i = [1 + (−1)k ] k+1 k+1 2 m (k + 1)! (b − a)k+1 A∗k,m = (−1)k k+1 k+1 , 2 m (k + 1)! b−a x∗i = a + i, i = 1, m − 1. m and there is for remainder term evaluation A∗k,0 =
|R∗n [f ]| ≤
(b − a)n+1 · Mn∞ [f ]. (n + 1)!(2m)n
For example, if f ∈ W21 [0, 1], then quadrature formula of the form (1.26), optimal with regard to the error, is " # Z 1 m−1 X µi¶ 1 f (x)dx = · f (0) + 2 f + f (1) + R∗1 [f ], 2m m 0 i=1 where |R∗1 [f ]| ≤
1 √ kf 0 k2 . 2m 3
For f ∈ W22 [0, 1] we have " # Z 1 m−1 X µi¶ 1 1 0 1 0 f (x)dx = · f (0) + 2 f (0) − f (1) + R∗2 [f ], f + f (1) + 2 2 2m m 8m 8m 0 i=1 where |R∗2 [f ]| ≤
1 √ kf 00 k2 . 2 8m 5 23
The third chapter, entitled ”B-spline functions” has 3 sections. In the first section are presented some general concepts about B-spline functions. Let x0 ≤ · · · ≤ xn+1 be arbitrary points in [a, b] such that x0 < xn+1 . Definition 3.1.1 The spline function B(x0 , · · · , xn+1 ; x) = (·−x)n+ [x0 , · · · , xn+1 ] is called a B-spline of degree n with knots x0 , · · · , xn+1 . Denote by (· − x)n+ [x0 , · · · , xn+1 ] the divided difference of the function (· − x)n+ at the points x0 , · · · , xn+1 . A property of B-spline it is: Z b 1 . B(x0 , · · · , xn+1 ; t)dt = n+1 a Given the sequence (finite or infinite) of points {xi }, such that · · · ≤ xi ≤ xi+1 ≤ · · · and xi < xi+n+1 for all i, we shall denote by Bi,n (x) the B-spline Bi,n (x) = (· − x)n+ [xi , · · · , xi+n+1 ] . The spline function Ni,n (x) = (xi+n+1 − xi )Bi,n (x) is called normalized B-spline. Theorem 3.1.2 Let a < xn+2 ≤ · · · ≤ xm < b be fixed points such that xi < xi+n+1 for all admissible i. Choose arbitrary 2n + 2 additional points x1 ≤ · · · xn+1 ≤ a and b ≤ xm+1 ≤ · · · ≤ xm+n+1 and define Bi,n (x) = B(xi , · · · , xi+n+1 ; x). The B-spline B1,n (x), · · · , Bm,n (x) constitute a basis for Sn (xn+2 , · · · , xm ) on [a, b]. The B-spline basis for the space Sn (xn+2 , · · · , xm ) was constructed by H.B. Curry and I.J. Schoenberg in [56]. Let {xi }m+n+1 be a given knot sequence such that x1 ≤ x2 ≤ · · · ≤ xn+1 ≤ a < xn+2 ≤ i=1 · · · ≤ xm < b < xm+1 ≤ · · · ≤ xm+n+1 . For each x and t in [a, b], n
(3.2)
(t − x) =
m X
ϕi,n (t)Ni,n (x),
i=1
where
( ϕi,n (t) :=
(t − xi+1 ) · · · (t − xi+n ) for n > 0 , 1 for n = 0.
For each 0 ≤ r ≤ n, (3.3)
r
x =
m X
αi,n Ni,n (x), x ∈ [a, b],
i=1
24
where
r! (n−r) ϕ (0), i = 1, m. n! i,n This result was given by Marsden in [88]. Another interesting proof can be seen in P.J. Barry, N. Dyn, R.N. Goldman and C.A. Micchelli in [26]. αi,n = (−1)r
In section 3.2 was presented a general construction of spline quasi-interpolants (see [86]). Interesting results about spline quasi-interpolants were obtained by P. Sablonni` ere in [109], [110], [111], [112], [113], T. Lyche and K. Morken in [86], D.Barrera, M.J. Ib´an ˜ ez, P. Sablonni` ere, D. Sbibih in [23], [24], [25], B.G. Lee, T. Lyche and L.L.Schumaker in [82]. Given a function f , the basic problem of spline approximation is to determine B-spline coefficients (ci )m i=1 such that m X Pf = ci Ni,n i=1
is a reasonable approximation to f . The basic challenge is therefore to devise a procedure for determining the B-spline coefficients. A method for determining the B-spline coefficients is presented in [86]. For the B-spline coefficients cj we use the following notation cj = λj (f ), because it depend on f . Suppose that n ≥ 2 and fix an integer i such that xi+n > xi+1 . To choose the subinterval [ai , bi ] = [xl , xl+1 ] of [xi+1 , xi+n ] and define the uniformly spaced points (3.8)
xi,k = ai +
k (bi − ai ), for k = 0, 1, · · · , n, n
in this interval. To define Pn f ∈ Sn (xn+2 , . . . , xm ) by Pn f (x) =
m X
λi (f )Ni,n (x), where
i=1
(3.9) λi (f ) =
n X
wi,k f (xi,k ).
k=0
The following Lemma ([86]) show how the coefficients (wi,k )nk=0 should be chosen so that Pn p = p for all p ∈ Pn . Lemma 3.2.2 Suppose that in (3.9) the functionals λi are given by λi (f ) = f (xi+1 ) if xi+n = xi+1 , while if xi+n > xi+1 we set wi,k = γi (pi,k ), k = 0, 1, . . . , n, 25
where γi (pi,k ) is the i-th B-spline coefficient of the polynomial pi,k (x) =
n Y
x − xi,j . xi,k − xi,j j=0,j6=k
Then the operator Pn defined in (3.9) satisfies Pn p = p for all p ∈ Pn . Lemma 3.2.3 Given a spline space Sn (xn+2 , . . . , xm ) and numbers v1 , . . . , vn . The i-th B-spline coefficient of the polynomial p(x) = (x − v1 ) . . . (x − vn ) can be written γi (p) =
1 n!
X Q (j1 ,...,jn )∈ n
(xi+j1 − v1 ) . . . (xi+jn − vn ),
Q where n is the set of all permutations of the integers 1, 2, . . . , n. By using the approximation by Taylor0 s polynomial can be obtained the following quasi-interpolant ([86]). Theorem 3.2.1(de Boor-Fix) Let r be an integer with 0 ≤ r ≤ n and let tj be a number in [xj , xj+n+1 ] for j = 1, . . . , m. Consider the quasi-interpolant (3.10)
Pn,r f =
m X
λj (f )Nj,n
j=1
where
r
1X (−1)k Dn−k ρj,n (tj )Dk f (tj ) λj (f ) = n k=0
and ρj,n (y) = (y − xj+1 ) · · · (y − xj+n ). Then Pn,r f = f for all f ∈ Pr and Pn,n f = f for all f ∈ Sn (xn+2 , . . . , xm ). By integration of spline approximation formula, in section 3.3, we have obtained a new quadrature formula. For the smooth function, we have given a estimation of remainder term. These results were obtained in [2]. If in Lemma 3.2.2 we choose n = 2 and x1 = x2 = x3 = a, xm+1 = xm+2 = xm+3 = b, we obtain the operator m X P2 f = λi (f )Ni,2 with i=1
(3.11)
1 λi (f ) = − f (xi+1 ) + 2f 2
µ
xi+1 + xi+2 2
¶
1 − f (xi+2 ) 2
which satisfy P2 p = p any p ∈ P2 . If we integrate the approximation formula of function f f (x) =
m X
λi (f )Ni,2 (x) + rm [f ]
i=1
26
to obtain the following quadrature formula with degree of exactness equal 2: · µ ¶ ¸ Z b m X xi+3 −xi 1 xi+1 +xi+2 1 − f (xi+1 )+2f − f (xi+2 ) +Rm [f ]. (3.12) f (x)dx= 3 2 2 2 a i=1 For m = 3 we have the Simpson0 s quadrature formula. We shall consider the equidistant nodes (xi )m i=4 from [a,b]. For m = 4 to obtain the quadrature formula · µ ¶ µ ¶ µ ¶¸ Z b 3a + b a+b a + 3b b−a 2f −f + 2f + R4 [f ], f (x)dx = 3 4 2 4 a and for m = 5 we have the quadrature formula · µ ¶ ¶ µ ¶ µ Z b 5 a+b b−a 4 5a + b 2a + b − f + 2f f (x)dx = f 3 3 6 6 3 2 a ¶ ¶¸ µ µ 5 4 a + 2b a + 5b − f + f + R5 [f ]. 6 3 3 6 We have studied the quadrature formula (3.12) for m ≥ 6 and for simplicity of calcu1 lations we choose a = 0 , b = 1 . If denote h = , we have xi = (i − 3)h, i = 4, m m−2 and the quadrature formula (3.12) can be written ( µ ¶ Z 1 m−3 X µ 2k − 1 ¶ m−4 X h 4 5 (3.13) f f (kh) f (x)dx = h f − f (h) + 2 h − 3 2 6 2 0 k=2 k=2 µ ¶¾ 5 2m − 5 4 − f ((m − 3)h) + f h + Rm [f ]. 6 3 2 The quadrature formula (3.13) has degree of exactness equal 3. The remainder term has the form: Z 1 1 K(t)f (4) (t)dt, where f ∈ H 4 [0, 1] Rm [f ] = 6 0 ( µ ¶3 4 £ ¤ (1 − t) 5 4 h 3 K(t) = Rm (· − t)+ = − (h − t)3+ (3.14) −h −t 4 3 2 6 + ¶3 m−4 m−3 X µ 2k − 1 X 5 (kh − t)3+ − ((m − 3)h − t)3+ +2 h−t − 2 6 + k=2 k=2 µ ¶3 ) 4 2m − 5 h−t . + 3 2 + Theorem 3.3.1 The monospline of degree 4, defined in (3.14) verifies (3.15)
K(t) = K(1 − t) (∀) t ∈ [0, 1],
(3.16)
K(t) ≥ 0 (∀) t ∈ [0, 1],
(3.17) (3.18)
h4 max K(t) = , t∈[0,1] 12 Z 1 1 29m − 88 K(t)dt = · . 480 (m − 2)5 0 27
˜ such that Theorem 3.3.2 If f ∈ H 4 [0, 1] and there exist real numbers m, ˜ M ˜ , then m ˜ ≤ f (4) (t) ≤ M 29m − 88 |Rm [f ]| ≤ 2880(m − 2)5
(3.25)
(
¯ ¯) ˜ +m ˜ −m M ˜ ¯¯ M ˜ ¯¯ +¯ ¯ . ¯ 2 ¯ 2
Theorem 3.3.3 Let f ∈ H 4 [0, 1]. If there exist a real number m ˜ such that (3) m ˜ ≤ f (t), then · ¸ 1 29m − 88 |Rm [f ]| ≤ · T −m ˜ + |m| ˜ , 72(m − 2)4 40(m − 2)
(3.28)
where T = f (3) (1) − f (3) (0). ˜ such that f (3) (t) ≤ M ˜ , t ∈ [0, 1], then If there exist a real number M · ¸ 29m − 88 ¯¯ ˜ ¯¯ 1 ˜ |Rm [f ]| ≤ · M −T + ¯M ¯ . 72(m − 2)4 40(m − 2)
(3.29)
By integration of approximation formula a function f and by choosing n=r=2, we obtained from (3.10) the following quadrature formula with degree of exactness equal 2 Z
b
(3.31) a
½ m X xj+3 − xj 1 f (x)dx = f (tj ) − [2tj − (xj+1 + xj+2 )] f 0 (tj ) 3 2 j=1 ¾ 1 00 + (tj − xj+1 )(tj − xj+2 )f (tj ) + Rm [f ]. 2
xj+1 + xj+2 and equidistant nodes (xi )m i=4 from the interval 2 1 , we have [a,b]. For simplicity of calculations we choose a = 0, b = 1. If denote h = m−2 xi = (i − 3)h, i = 4, m and the quadrature formula (3.32) can be written: We have considered tj =
Z
1
(3.32) 0
(
µ ¶ m−3 X µ 2k − 1 ¶ 2 µ 2m−5 ¶ 2 h 1 f (x)dx = h f (0)+ f h + f h + + f 3 3 2 2 3 2 k=2 " µ ¶ m−3 2 X µ 2k − 1 ¶ 1 h 2 00 h + f (1) − f h + f 00 3 8 3 2 2 k=2 µ ¶¸¾ 2 00 2m − 5 h + Rm [f ]. + f 3 2 28
The quadrature formula (3.32) has degree of exactness 3, and the remainder term has the form Z 1 1 Rm [f ] = K(t)f (4) (t)dt, where f ∈ H 4 [0, 1], 6 0 £ ¤ (3.33) K(t) = Rm (· − t)3+ ( µ ¶3 m−3 ¶3 X µ 2k − 1 (1 − t)4 2 h + = −h −t h−t 4 3 2 2 + + k=2 µ ¶3 2 2m − 5 1 + h−t + (1 − t)3 3 2 3 + " µ µ ¶ ¶ ¶ #) m−3 X µ 2k−1 3h2 2 h 2 2m−5 − −t + h−t + h−t . 4 3 2 2 3 2 + + + k=2 Theorem 3.3.4 The monospline of degree 4, defined in (3.33) verifies (3.34)
(3.35)
K(t) = K(1 − t) (∀) t ∈ [0, 1],
−
5 4 h4 h ≤ K(t) ≤ (∀) t ∈ [0, 1], 192 12 Z
1
K(t)dt =
(3.36) 0
1 43m − 136 · . 960 (m − 2)5
˜ = b − a , the quadrature formula (3.32) can be written By denoting h m−2 ( Ã ! m−3 µ ¶ Z b X ˜ 1 2 h 2k − 1 ˜ ˜ (3.37) f (x)dx = h f a+ h f (a) + f a + + 3 3 2 2 a k=2 " Ã ! ¶ µ ˜2 2 ˜ 2 1 h h 2m − 5 ˜ + f a+ h + f (b) − f 00 a + 3 2 3 8 3 2 ¶ µ ¶#) m−3 X µ 2k − 1 2 2m − 5 ˜ + f 00 a + ˜ + R˜m [f ]. + f 00 a + h h 2 3 2 k=2 The fourth chapter, entitled ”Spline functions and quadrature formulas of Gauss type” has 8 sections. In section 4.1 we recall some general concepts about the orthogonal polynomials and we use them in section 4.2. for to make a short exposure about the Gauss quadrature formulas. Let dλ be a nonnegative measure on the line R with compact or infinite support, for which all moments Z µk = tk dλ(t), k = 0, 1, . . .
R
29
exists and are finite, and µ0 > 0. Definition 4.1.2 The polynomial πm (t) = tm + am−1 tm−1 + · · · + a1 t + a0 , which satisfies the orthogonality conditions Z (4.3) [πm (t)]2s+1 tk dλ(t) = 0, k = 0, 1, . . . , m − 1,
R
is called s-orthogonal polynomial with respect to the measure dλ(t). Remark 4.1.1 For s=0 we have the case of the orthogonal polynomial. Let [a, b] be the support of the nonnegative measure, dλ(t) = w(t)dt, where w(t) is the weight function. The classic orthogonal polynomials are (α,β) Jm - Jacobi 0 s polynomial, (α) Lm - Laguerre 0 s polynomial, Hm - Hermite 0 s polynomial. Below are mentioned the intervales of orthogonality and corresponding weight orthogonal polynomials (α,β) Jm (α) Lm Hm
[a, b] [−1, 1] [0, ∞) (−∞, +∞)
w(t) remarks β (1 − t) (1 + t) α > −1, β > −1 e−t tα α > −1 −t2 e α
Definition 4.1.3 The polynomial πm,σ (t) =
m Y
(t − xk ),
k=1
which satisfies the orthogonality conditions Z Y m (4.5) (t − xi )2si +1 tk dλ(t) = 0, k = 0, 1, . . . , m − 1,
R i=1
is called σ-orthogonal polynomial with respect to the measure dλ(t). The quadrature formula Z (4.12)
R
f (t)dλ(t) =
2sk m X X
(i) AG ik f (xk ) + Rm [f ]
k=1 i=0
has the maximum degree of exactness r = 2
m X k=1
sk + 2m − 1 if and only if xk , k = 1, m
are the roots of σ-orthogonal polynomial of degree m with respect to the measure dλ(t). The quadrature formula (4.12) is called Chakalov-Popoviciu quadrature formula. 30
In the case sk = 0, k = 1, m the quadrature formula (4.12) is a quadrature formula of Gauss type. Let [a, b] be the support of the nonnegative measure dλ(t) = w(t)dt, where w(t) is the weight function. The quadrature formulas Z
b
(4.13)
f (t)dλ(t) = a
2sk m X X
(i)
Aik f (xk ) +
k=1 i=0
q pk X X
αik f (i) (yk ) + Rm,q [f ]
k=1 i=0
where yk ∈ R \ (a, b), k = 1, q are fixed, distinct nodes and outside of (a, b), and the nodes xk ∈ (a, b), k = 1, m will be determined such that the quadrature formula (4.13) to have maximal degree of exactness are called quadrature formulas of Gauss type with fixed nodes. The quadrature formulas Z
b
(4.14)
f (t)dλ(t) = a
p X
(i)
αi1 f (a) +
i=0
2sk m X X
(i) R AR ik f (xk ) + Rm,p [f ]
k=1 i=0
where xk ∈ (a, b), k = 1, m, −∞ < a < +∞, p ∈ N, with Pm RR m,p [f ] = 0, for f ∈ P2( k=1 sk +m)+p
and Z
b
(4.15)
f (t)dλ(t) = a
p X
(i)
α ˜ i1 f (b) +
i=0
2sk m X X
R ˜ R [f ], A˜ik f (i) (x˜k ) + R m,p
k=1 i=0
where x˜k ∈ (a, b), k = 1, m, −∞ < b < +∞, p ∈ N, with Pm ˜R R m,p [f ] = 0, for f ∈ P2( k=1 sk +m)+p ,
are called Chakalov-Popoviciu quadrature formulas of Radau type. The quadrature formulas Z
b
(4.16)
f (t)dλ(t) = a
p1 X i=0
αi1 f (i) (a)+
p2 X
αi2 f (i) (b)+
2sk m X X
ALik f (i) (xk )+RLm,p1 ,p2 [f ],
k=1 i=0
i=0
where xk ∈ (a, b), k = 1, m, −∞ < a < b < +∞, p1 , p2 ∈ N, with RLm,p1 ,p2 [f ] = 0 for f ∈ P2(Pm , k=1 sk +m)+p1 +p2 +1 are called Chakalov-Popoviciu quadrature formulas of Lobatto type. In [127], M.M. Spalevi´ c gives a method of approximating a function by a spline function and show that the problem has a unique solution if and only if certain Chakalov-Popoviciu quadrature formulae of Radau and Lobatto type exist corresponding to measures depending on considered function. 31
A spline function of degree n ≥ 2 with the distinct nodes {tk }m k=1 , tk ∈ (0, 1) and defects z = (z1 , z2 , . . . , zm ) can be written (4.19)
s(t) = pn (t) +
m zX k −1 X
n−i cki (tk − t)+ ,
k=1 i=0
where cki are real numbers and pn (t) is a polynomial with degree most equal n. Choosing zk = 2sk + 1, k = 1, m, the spline function (4.19) can be written (4.20)
s(t) = pn (t) +
2sk m X X
n−i cki (tk − t)+ .
k=1 i=0
In section 4.3, by using M.M. Spalevi´ c 0 s reasoning, we show that the below problem has a unique solution if and only if certain Chakalov-Popoviciu quadrature of Lobatto type exist coresponding to measures depending on f . Problem 3. Determine the spline function s defined in (4.20) such that Z
1
(4.26)
Z
1
j
t s(t)dt = 0
m X sk + m) − 1. t f (t)dt, j = 0, 1, . . . , 2( j
0
k=1
The following two results were obtained in [4]. Theorem 4.3.5 Let f ∈ C n+1 [0, 1]. There exists a unique spline function, defined in (−1)n+1 (n+1) (4.20), satisfying (4.26), if and only if the measure dλ(t) = f (t)dt admits a n! Chakalov-Popoviciu quadrature formula of Radau type Z
1
(4.27)
f (t)dλ(t) = 0
n X
α ˜ k,1 f
(k)
(1) +
2sk m X X
(i) ˜ R [f ], A˜R xk ) + R ik f (˜ m,n
k=1 i=0
k=0
where ˜ R [f ] = 0, R m,n
, for f ∈ P2(Pm k=1 sk +m)+n
with distinct real zero x˜k ∈ (0, 1), k = 1, m. The spline function (4.20) is given by tk = x˜k , k = 1, m, n! cki = A˜R , k = 1, m, i = 0, 2sk , (n − i)! ik where x˜k and A˜R ik are the nodes and coefficients, respective of Chakalov-Popoviciu quadrature formula of Radau type, and the polynomial pn (t) is given by n X ¤ (t − 1)k £ (−1)k n!˜ αn−k,1 + f (k) (1) . pn (t) = k! k=0
32
Theorem 4.3.6 Let f ∈ C n+1 [0, 1] and rx (t) = (t − x)n+ , 0 ≤ t ≤ 1. If the spline function s, defined in (4.20), satisfies the relations (4.26), then ˜R f (x) − s(x) = R m,n [rx ], 0 < x < 1, ˜ R [f ] is the remainder term in the Chakalov-Popoviciu quadrature formula of where R m,n Radau type (4.27). In section 4.4, by using M.M. Spalevi´ c 0 s reasoning from paper [127], we have given a method of approximating certaine function by monospline and we have showed that the problem has a unique solution if and only if certain Chakalov-Popoviciu quadrature formulas of Radau and Lobatto type exist corresponding to measures depending on the considered function. Let n−1
(n − 1)! (1 − t)n X Bk M (t) = − (1 − t)n−k−1 n! (n − k − 1)! k=0
(4.30)
−
m zX k −1 X
aik
k=1 i=0
(n − 1)! n−i−1 (tk − t)+ (n − i − 1)!
be the monospline function, where Bk and aik are real numbers. Choosing zk = 2sk + 1, k = 1, m, the monospline function (4.30) can be written n−1
(n − 1)! (1 − t)n X Bk M (t) = − (1 − t)n−k−1 n! (n − k − 1)! k=0
(4.31)
−
2sk m X X k=1 i=0
aik
(n − 1)! n−i−1 (tk − t)+ . (n − i − 1)!
The following results were obtained in [4]. Problem 1. Determine the monospline M defined in (4.31) such that Z
1
(4.32)
Z
1
j
t M (t)dt = 0
m X t f (t)dt, j = 0, 1, . . . , 2( sk + m) + n − 1. j
0
k=1
Theorem 4.4.1 Let f ∈ C n [0, 1]. There exist a unique monospline M , defined in (4.31), satisfying (4.32), if and only if the measure dλ(t) =
£ ¤ 1 1 + (−1)n−1 f (n) (t) dt (n − 1)! 33
admits a Chakalov-Popoviciu quadrature formula of Lobatto type Z
1
(4.33)
f (t)dλ(t) = 0
n−1 X
αk,1 f
(k)
(0)+
k=0
n−1 X
αk,2 f
(k)
(1)+
2sk m X X
ALik f (i) (xk )+RLm,n [f ],
k=1 i=0
k=0
where RLm,n [f ] = 0,
for f ∈ P2(Pm , k=1 sk +m)+2n−1
with distinct real zeros xk ∈ (0, 1), k = 1, m. The monospline (4.31) is given by tk = xk , k = 1, m, (−1)n−k (n−1−k) Bk = αk,2 + f (1) k = 0, n − 1, (n − 1)! aik = ALik , k = 1, m, i = 0, 2sk , where xk and ALik are the nodes and coefficients, respective of Chakalov-Popoviciu quadrature formula of Lobatto type. Theorem 4.4.2 Let f ∈ C n [0, 1] and rx (t) = (t − x)n−1 + , 0 ≤ t ≤ 1. If the monospline M, defined in (4.31), satisfies the relations (4.32), then M (x) − f (x) = RLm,n [rx ], 0 < x < 1, where RLm,n [f ] is the remainder term in the Chakalov-Popoviciu quadrature formula of Lobatto type(4.33). Problem 2. Determine the monospline M , defined in (4.31) such that Z
1
(4.38)
Z
1
j
t M (t)dt = 0
m X sk + m) − 1. t f (t)dt, j = 0, 1, . . . , 2( j
0
k=1
Theorem 4.4.3 Let f ∈ C n [0, 1]. There exist a unique monospline M, defined in (4.31), satisfying (4.38), if and only if the measure dλ(t) =
£ ¤ 1 1 + (−1)n−1 f (n) (t) dt (n − 1)!
admits a Chakalov-Popoviciu quadrature formula Radau type Z
1
(4.39)
f (t)dλ(t) = 0
n−1 X
α ˜ k,1 f
(k)
(1) +
2sk m X X
(i) ˜ R [f ], A˜R xk ) + R ik f (˜ m,n
k=1 i=0
k=0
where ˜ R [f ] = 0, for f ∈ P2(Pm s +m)+n−1 , R m,n k=1 k 34
with distinct real zeros x˜k ∈ (0, 1), k = 1, m. The monospline defined in (4.31), is given by tk = x˜k , k = 1, m, (−1)n−k (n−1−k) Bk = α ˜ k,1 + f (1), k = 0, n − 1, (n − 1)! aik = A˜R ik , k = 1, m, i = 0, 2sk , where x˜k and A˜R ik are the nodes and coefficients, respective of Chakalov-Popoviciu quadrature formula of Radau type. n−1 Theorem 4.4.4 Let f ∈ C n [0, 1] and rx (t) = (t − x)+ , 0 ≤ t ≤ 1. If the monospline M, defined in (4.31), satisfies the relations (4.38), then
˜ R [rx ], 0 < x < 1, M (x) − f (x) = R m,n ˜R where R m,n [f ] is the remainder term in the Chakalov-Popoviciu quadrature formula of Radau type (4.39). Problema 3. Determine the monospline M , defined in (4.31), such that M (j) (1) = f (j) (1), j = 0, n − 1,
(4.42) Z
1
(4.43)
Z
1
j
t M (t)dt = 0
m X sk + m) − 1. tj f (t)dt, j = 0, 1, . . . , 2(
0
k=1
Theorem 4.4.5 Let f ∈ C n [0, 1]. There exist a unique monospline M , defined in (4.31), satisfying (4.42) and (4.43), if and only if the measure £ ¤ 1 dλ(t) = 1 + (−1)n−1 f (n) (t) dt (n − 1)! admits a Chakalov-Popoviciu quadrature formula of Radau type Z 1 2sk n−1 m X X X (i) R (k) (4.44) AR f (t)dλ(t) = αk,1 f (0) + ik f (xk ) + Rm,n [f ], 0
k=1 i=0
k=0
where Pm RR m,n [f ] = 0, for f ∈ P2( k=1 sk +m)+n−1 ,
with distinct real zeros xk ∈ (0, 1), k = 1, m. The monospline monospline (4.31) is given by tk = xk , k = 1, m, (−1)n−k (n−1−k) f (1), k = 0, n − 1, Bk = (n − 1)! aik = AR ik , k = 1, m, i = 0, 2sk , 35
where xk and AR ik are the nodes and coefficients, respectively of Chakalov-Popoviciu quadrature formula of Radau type. Theorem 4.4.6 Let f ∈ C n [0, 1] and rx (t) = (t − x)n−1 + , 0 ≤ t ≤ 1. If the monospline M, defined in (4.31), satisfies the relations (4.42) and (4.43), then M (x) − f (x) = RR m,n [rx ], 0 < x < 1, where RR m,n [f ] is the remainder term in the Chakalov-Popoviciu quadrature formula of Radau type(4.44). In section 4.5 we have studied a property of the intermediate point from the quadrature formula of Gauss-Legendre type. This result was obtained in [7]. Let µ ¶ Z b m b−a (b − a) X b+a 1 ·f f (t)dt = xk + d h (0,0) i m + 1 k=0 (0,0) 2 2 a Jm (xk ) Jm+1 (x) dx x=xk (b − a)2m+3 [(m + 1)!]4 (2m+2) (4.51) + f (ξ), a < ξ < b (2m + 3)[(2m + 2)!]3 be the quadrature formula of Gauss-Legendre type. If f ∈ C 2m+2 [a, b], then for any x ∈ (a , b] there is cx ∈ (a, x) such that Z
x a
(4.52)
m
(x − a) X f (t)dt = m + 1 k=0
1 ·f h i d (0,0) (0,0) J (x) Jm (xk ) dx m+1 x=xk 2m+3 4 (x − a) [(m + 1)!] (2m+2) + f (cx ). (2m + 3)[(2m + 2)!]3
µ
x−a x+a xk + 2 2
¶
Theorem 4.5.2 If f ∈ C 2m+4 [a, b] and f (2m+3) (a) 6= 0, then for the intermediate point cx − a 1 cx , which appears in formula (4.52), we have lim = . x→a x − a 2 In section 4.6, by using quadrature formulas, we have given some new results in extremal problems with polynomials which are generalizations of some results of F. Locher and D. Acu. The following results were obtained in [1]. Theorem 4.6.8 For each polynomial p2m+1 (x) ≥ 0, x ∈ [−1, 1], of the degree 2m + 1 and with the dominant coefficient equal 1, the inequality Z1 w(x)p2m+1 (x)dx ≥ −1
36
1 ||Qm ||2 a2m
is valid, with equality only if p2m+1 (x) =
1 2 2 (x + 1)Qm (x) am
where Qm (x) is the polynomial of degree m, with the dominant coefficient am , from the system of orthogonal polynomials on the interval [−1, 1] referring to the weight w(x)(x+1). Theorem 4.6.9 For each polynomial p2m+1 (x) ≤ 0, x ∈ [−1, 1], of the degree 2m + 1 and with the dominant coefficient equal 1, the inequality Z1 w(x)p2m+1 (x)dx ≤ − −1
1 2 2 ||Qm || , am
is valid, with equality only if p2m+1 (x) = −
1 2 2 (1 − x)Qm (x), am
where Qm (x) is the polynomial of degree m, with the dominant coefficient am , from the system of orthogonal polynomials on the interval [−1, 1] referring to the weight (1 − x)w(x). Theorem 4.6.10 For each polynomial p2m+2 (x) ≤ 0, x ∈ [−1, 1], of the degree 2m + 2 and with the dominant coefficient equal 1, the inequality Z1 w(x)p2m+2 (x)dx ≤ − −1
1 2 2 ||Qm || am
is valid, with equality only if p2m+2 (x) =
(x2 − 1) 2 Qm (x) a2m
where Qm (x) is the polynomial of degree m with the dominant coefficient am , from the system of orthogonal polynomial on the interval [−1, 1] referring to the weight (x + 1)(1 − x)w(x). In section 4.7, by using quadrature formulas of Gauss-Lobatto type, we have given some new results in extremal problems with polynomials. These results obtained in [12], [13] are generalizations of some results of I. Popa ( [105], [106]). (α,β) We denote J˜n the Jacobi polynomial of degree n, normalized to have the leading coefficient equal 1. Theorem 4.7.2 If pn−1 ∈ Pn−1 be such that ¯ ¯ ¯ ¯ ˜(α+1,β+1) (xi )¯ , i = 1, n, |pn−1 (xi )| ≤ ¯Jn−1 37
(α,β) where the xi are the zeros of J˜n , then
Z
1
(4.78) −1
· 2n+α+β−2
2
h i2 (k+1) (1 − x)k+α (1 + x)k+β pn−1 (x) dx ≤
(n − 1)! Γ(2n + α + β + 1)
¸2 ·
Γ(n + α + β + k + 3)Γ(n + α + 1)Γ(n + β + 1) · Γ(n − k − 1)
·
1 (n + α + β + k + 3)(n − k − 2) (n − k − 1)(n + k + α + β + 2) − + + k+β+2 2(k + β + 2)(k + β + 3) 2(k + β + 1)(k + β + 2) ¸ (n + α + β + k + 3)(n − k − 2) (n − k − 1)(n + k + α + β + 2) 1 , − + k+α+2 2(k + α + 2)(k + α + 3) 2(k + α + 1)(k + α + 2)
(α+1,β+1) with equality for pn−1 = J˜n−1 , where k = 0, n − 2.
Theorem 4.7.3 If pn−1 ∈ Pn−1 be such that ¯ ¯ ¯ (α+1,β+1) ¯ |pn−1 (xi )| ≤ ¯J˜n−1 (xi )¯ , i = 1, n, (α,β) where xi are the zeros of J˜n , and r(x) = b(b − 2a)x2 + 2c(b − a)x + a2 + c2 is a real polynomial with 0 < a < b, |c| < b − a, b 6= 2a, then
Z
1
(4.79)
k+α
r(x)(1 − x)
k+β
(1 + x)
−1
·
(n − 1)! Γ(2n + α + β + 1)
h
i2
(k+1) pn−1 (x)
dx ≤
¸2
Γ(n + α + β + k + 3)Γ(n + α + 1)Γ(n + β + 1) · Γ(n − k − 1) ½ · 2(−b2 + 2ab + bc − ac) (n + α + β + k + 3)(n − k − 2) + 1− k+β+2 2(k + β + 3) ¸ (n − k − 1)(n + k + α + β + 2) (a − b + c)2 2(b2 − 2ab + bc − ac) + · − + 2(k + β + 1) k+β+2 k+α+2 · ¸ ¾ (n + α + β + k + 3)(n − k − 2) (n − k − 1)(n + k + α + β + 2) (a − b − c)2 + · 1− 2(k + α + 3) 2(k + α + 1) k+α+2 2n+α+β−2
2
·
(α+1,β+1) with equality for pn−1 = J˜n−1 , where k = 0, n − 2.
1 Remark 4.7.1 For α = β = − to obtain the results of I. Popa from papers [105] and 2 [106]. Let A : C[a, b] → R be a arbitrary linear functional. By using the values of function f on n points from interval [a, b], in section 4.8, we have given a approximation formula of linear functional A(f ). The approximation formula has degree of exactness equal 2n − 1. This result was obtained in [10]. 38
We consider zi , i = 1, n the roots of polynomial ¯ ¯ µ µn ¯ 0 µ1 . . . ¯ . .. . . .. .. ¯ .. . ¯ (4.80) Pn (z) = ¯ ¯ µn−1 µn . . . µ2n−1 ¯ ¯ 1 z ... zn where µj = A(ej ), ej (x) = xj , ¯ ¯ 1 1 ¯ ¯ ¯ z1 z2 ∆ = ¯¯ . .. . ¯ .. ¯ n−1 n−1 ¯ z1 z2 ¯ ¯ 1 ... ¯ ¯ ¯ z1 . . . ∆i = ¯¯ . .. . ¯ .. ¯ n−1 ¯ z1 ...
¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯
j = 0, 2n − 1 and we denote ¯ ¯ ¯ µ 1 ... 1 ... 1 ¯¯ ¯ 0 ¯ ¯ ¯ µ1 z2 . . . z n . . . zn ¯ .. .. .. .. .. ¯¯ , ∆1 = ¯¯ .. . . . . . ¯ ¯ . ¯ ¯ n−1 ¯ µn−1 z2 . . . znn−1 . . . znn−1 ¯ 1 zi−1 .. .
µ0 µ1 .. .
1 zi+1 .. .
... ... .. .
1 zn .. .
n−1 n−1 zi−1 µn−1 zi+1 . . . znn−1 ¯ ¯ ¯ 1 ¯ . . . 1 µ 0 ¯ ¯ ¯ ¯ ¯ z1 . . . zn−1 µ1 ¯ ¯ ∆n = ¯ . .. .. .. ¯¯ . . . . . ¯ ¯ . ¯ n−1 ¯ n−1 ¯ z1 . . . zn−1 µn−1 ¯
¯ ¯ ¯ ¯ ¯ ¯, ¯ ¯ ¯ ¯
¯ ¯ ¯ ¯ ¯ ¯ , i = 2, n − 1, ¯ ¯ ¯ ¯
Theorem 4.8.1 If the polynomial (4.80) has n real and distinct roots from interval [a, b], then (4.82)
A(f ) =
n X
ci f (zi ) + R[f ]
i=1
where ci =
∆i , i = 1, n, has degree of exactness equal 2n-1. ∆
39
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50
I take this opportunity to express my respectful gratitude and thanks to my supervisor Prof. univ. dr. Petru Blaga for the ideas, the support and his permanent advice and scrupulous guidance through my entire work on this thesis. Also, I want to remember that I received the help and support of great professors Gheorghe Micula and Alexandru Lupa¸s for realization of this thesis.
51
