Annals of the University of Craiova, Mathematics and Computer Science Series Volume 37(1), 2010, Pages 104–109 ISSN: 1223-6934
On the Hermite interpolation polynomial ˘ rbosu Ovidiu T. Pop and Dan Ba
Abstract. The Newton form for the Hermite interpolation polynomial using the divided differences with multiple knots is proved. Using this representation, sufficient conditions for the convergence of the sequence of Hermite interpolation polynomials are established. One extends this way a result obtained by M. Ivan, regarding to sufficient conditions for the uniform convergence of the sequence of Lagrange polynomials. 2010 Mathematics Subject Classification. 41A05; 65D05. Key words and phrases. Hermite interpolation polynomial, divided difference with multiple knots, uniform convergence.
1. Introduction G. Faber [2] proved that there exists a function f ∈ C[0, 1] for which the sequence of Lagrange polynomials (Lm f )m≥0 doesn’t converges uniformly to f on [0, 1]. I. Muntean [7] generalized the result of G. Faber. M. Ivan [5] established sufficient conditions for the uniform convergence of the sequence of the Lagrange polynomials (Lm f )m≥0 associated the function f ∈ C[a, b]. The focus of the present paper is to establish sufficient conditions for the uniform convergence of the sequence of Hermite polynomials. First, let us to introduce the following notations: N = {1, 2, . . . }, N0 = N ∪ {0}, m ∈ N0 , r0 , r1 , . . . , rm ∈ N, r0 + r1 + · · · + rm = n + 1, α = max{r0 − 1, r1 − 1, . . . , rm − 1}. Next, if I ⊆ R is an interval then Dα (I) denotes the set of real valued functions α-times differentiable on I. If α = 0, then D0 (I) = F(I) = {f : I → R}. Let x0 , x1 , . . . , xm ∈ I be distinct knots. If f ∈ Dα (I), the divided difference with multiple knots £ ¤ x0 , x0 , . . . , x0 , x1 , x1 , . . . , x1 , . . . , xm , xm , . . . , xm ; f | {z } | {z } {z } | r0 times (r )
(r )
r1 times
rm times
(r )
will be denoted by [x0 0 , x1 1 , . . . , xmm ; f ]. It is known (see [4], [5], [7], [9]) that the divided difference with multiple knots can be represented under the form ¡ 0 ) (r1 ) (rm ) ¢ £ (r0 ) (r1 ) ¤ (W f ) x(r 0 , x1 , . . . , xm (rm ) x0 , x1 , . . . , xm ; f = , (1) ¡ (r ) (r ) (r ) ¢ V x0 0 , x1 1 , . . . , xmm Received January 11, 2010. Revision received February 14, 2010. 104
ON THE HERMITE INTERPOLATION POLYNOMIAL
105
¡ (r ) (r ) ¢ m) (W f ) x0 0 , x1 1 , . . . , x(r m
(2)
where
¯ ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ 0 ¯ ¯ = ¯. . . ¯ ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ 0
x0 1 ... 0 ... xm 1 ... 0
x20 . . . 2x0 . . . ... 0... ... x2m . . . 2xm . . . ... 0...
and
xn−1 0 (n−1)xn−2 0 ... 0 (n−1)(n−2) · . . . · (n−r0 +1)xn−r 0 ... xn−1 m (n−1)xn−2 m ... m (n−1)(n−2) · . . . · (n−rm +1)xn−r m
¯ f (x0 ) ¯¯ f 0 (x0 ) ¯¯ . . . ¯¯ (r0−1) ¯ f(x0 ) ¯ ¯ ... ¯ ¯ f (xm ) ¯ ¯ f 0 (xm ) ¯ ¯ ... ¯ (rm −1) ¯¯ f(xm )
¡ (r ) (r ) ¢ m) V x0 0 , x1 1 , . . . , x(r m ¯ ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ 0 ¯ = ¯¯. . . ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ 0
x0 1 ... 0 ... xm 1 ... 0
x20 . . . 2x0 . . . ... 0... ... x2m . . . 2xm . . . ... 0...
(3)
¯ ¯ xn0 ¯ n−1 ¯ nx0 ¯ ¯ ... ¯ n−r0 +1 ¯ n(n − 1) · . . . · (n − r0 + 2)x0 ¯ ¯. ... ¯ n ¯ xm ¯ n−1 ¯ nxm ¯ ¯ ... ¯ n−rm +1 ¯ n(n − 1) · . . . · (n − rm + 2)xm
In [4], the following mean-value theorem for divided differences with multiple knots was proved: Theorem 1.1. Let a, b ∈ R be given such that a < b. If f ∈ C n−1 [a, b] and exists f (n) on (a, b), then there exists ξ ∈ (a, b) such that the following i h 1 (n) (r ) (r ) m) x0 0 , x1 1 , . . . , x(r f (ξ) (4) m ;f = n! holds. 2. The Hermite interpolation polynomial In 1878 Charles Hermite proved that for f ∈ D(α) (I) there exists a unique polyno¡ (r ) (r ) (r ) ¢ mial of degree at most n, denoted (Hn f ) x0 0 , x1 1 , . . . , xmm (x) and called Hermite interpolation polynomial, such that the following conditions a0 . . .
a0
+a1 xk +a2 x2k + . . . a1 +2a2 xk + . . . ... k ∈ {0, 1, ..., m} +a1 x+a2 x2 + . . .
are verified.
r −1
+ark −1 xkk + . . . +(rk −1)ark−1 xrk−2 + . . . ... (rk −1)!ark −1 + ...
+an xn k = f (xk ) = f 0 (xk ) +nan xn−1 k ... (r −1) n−r +1 +n(n−1)...(n−rk +2)an xk k = f(xk ) k
(r0 )
+an xn = (Hn f )(x0
(r1 )
, x1
(r
)
, ..., xmm )(x) (5)
˘ O.T. POP AND D. BARBOSU
106
Note that (5) is a linear system in the unknowns a0 , a1 , . . . , an and it has solutions if and only if the following ¯ ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ ¯ 0 ¯ ¯ 1
x0 1 ... 0
x20 2x0 ... 0
... ... ... ...
xn 0 nxn−1 0 ... −rm+1 n(n−1) · . . . · (n−rm +2)xn m
x
x2
...
xn
¯ ¯ ¯ ¯ ¯ ¯ ¯=0 ¯ ¯ ¯ (r ) (r ) (Hn f )(x0 0 , ..., xmm )(x)¯ (6) f (x0 ) f 0 (x0 ) ... (rm −1) f(xm )
holds. But (6) can be expressed in the equivalent form
¯ ¯ 1 ¯ ¯ 0 ¯ + ¯¯. . . ¯ 0 ¯ ¯ 1
x0 1 ... 0 x
¯ ¯ 1 ¯ ¯ 0 ¯ ¯. . . ¯ ¯ ¯ 0 ¯ ¯ 1
x0 1 ... 0 x
x20 2x0 ... 0 x2
... ... ... ... ...
x20 2x0 ... 0 x2
... ... ... ... ...
xn 0 nxn−1 0 ... n−rm+1 n(n−1) · . . . · (n−rm +2)xm n x xn 0 nxn−1 0
m +1 n(n−1) · . . . · (n−rm +2)xn−r m n x
¯ f (x0 ) ¯¯ f 0 (x0 ) ¯¯ . . . ¯¯ (rm −1) ¯ f(x ) ¯ m ¯ ¯ 0
¯ ¯ ¯ ¯ ¯ ¯ = 0. ¯ ¯ 0 ¯ (rm ) (r1 ) (r0 ) ¯ (Hn f )(x0 , x1 , ..., xm )(x) 0 0
From the above identity, it follows
¡ (r ) (r ) (r ) ¢ ¢ ¡ (r ) (r ) (U f ) x0 0 , x1 1 , . . . , xmm (x) m) (x) = − (Hn f ) x0 0 , x1 1 , . . . , x(r ¡ (r ) (r ) m (r ) ¢ V x0 0 , x1 1 , . . . , xmm
(7)
¡ (r ) (r ) ¢ m) (U f ) x0 0 , x1 1 , . . . , x(r (x) m
(8)
where
¯ ¯ 1 ¯ ¯ 0 ¯ = ¯¯. . . ¯ 0 ¯ ¯ 1
x0 1 ... 0 x
x20 2x0 ... 0 x2
... ... ... ... ...
xn0 nxn−1 0 ... n(n − 1) · . . . · (n − rm + 2)xn−rm +1 xn
¯ f (x0 ) ¯ ¯ f 0 (x0 ) ¯ ¯ ... ¯. ¯ (rm −1 ¯ f(xm ) ¯ 0 ¯
Taking (7) and (8) into account, yields that the coefficient of xn in the Hermite polynomial is ¡ (r ) (r ) (r ) ¢ (W f ) x0 0 , x1 1 , . . . , xmm , ¡ (r ) (r ) (r ) ¢ V x0 0 , x1 1 , . . . , xmm £ (r ) (r ) ¤ (r ) i.e. it is x0 0 , x1 1 , . . . , xmm ; f .
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3. The Hermite interpolatory divided difference formula In the following we use the ideas of M. Ivan [5], for the construction of Newton’s interpolatory divided difference formula. Let Q(x) the polynomial defined by ¢ ¡ (r ) (r ) m) (x) Q(x) = (Hn f ) x0 0 , x1 1 , . . . , x(r m £ ¤ (r ) r0 r1 rm−1 rm −1 (r0 ) m) x0 , x1 1 , ..., x(r −(x−x0 ) (x−x1 ) · . . . · (x−xm−1 ) (x−xm ) m ;f . The polynomial Q(x) has the degree at most (n − 1) and it satisfies the relations Q(ik ) (xk ) = 0, ik ∈ {0, 1, . . . , rk − 1}, k ∈ {0, 1, . . . , m − 1}; Q(i) (xm ) = 0, i ∈ {0, 1, . . . , rm − 2}. Taking the uniqueness of the Hermite interpolation polynomial into account, from the above identities it follows ¡ (r ) (r ) ¢ m −1) Q(x) = (Hn−1 f ) x0 0 , x1 1 , . . . , x(r (x). m This way is obtained the following recurrence formula ¡ (r ) (r ) ¢ m) (Hn f ) x0 0 , x1 1 , . . . , x(r (x) (9) m ¡ (r0 ) (r1 ) ¢ m −1) = (Hn−1 f ) x0 , x1 , . . . , x(r (x) m £ ¤ (r ) r0 r1 rm−1 rm −1 (r0 ) m) +(x−x0 ) (x−x1 ) · . . . · (x−xm−1 ) (x−xm ) x0 , x1 1 , ..., x(r m ;f . From (9), yields
£ (2) ¤ (2) (H2 f )(x0 )(x) = (H1 f )(x0 )(x) + (x − x0 ) x0 ; f , £ (3) ¤ (2) (3) (H3 f )(x0 )(x) = (H2 f )(x0 )(x) + (x − x0 )2 x0 ; f , ... ¡ (r ) (r ) ¢ (rm−1 ) m −1) (Hn−1 f ) x0 0 , x1 1 , . . . , xm−1 , x(r (x) m ¢ ¡ (r0 ) (r1 ) (rm−1 ) m −2) , x(r (x) = (Hn−2 f ) x0 , x1 , . . . , xm−1 m + (x−x0 )r0 (x−x1 )r1 · . . . · (x−xm−1 )rm−1 (x−xm )rm −2 £ (r ) (r ) ¤ (rm−1 ) m −1) · x0 0 , x1 1 , . . . , xm−1 , x(r ;f , m ¡ (r ) (r ) ¢ m) (Hn f ) x0 0 , x1 1 , . . . , x(r (x) m ¡ (r0 ) (r1 ) ¢ (rm−1 ) m −1) = (Hn−1 f ) x0 , x1 , . . . , xm−1 , x(r (x) m + (x − x0 )r0 (x − x1 )r1 · . . . · (x − xm−1 )rm−1 (x − xm )rm −1 £ (r ) (r ) ¤ m) · x0 0 , x1 1 , . . . , x(r m ;f .
Adding the above identities, it follows: Theorem 3.1. The Hermite interpolation polynomial can be represented under the form rX 0 −1 ¡ (r ) (r ) ¢ m) (Hn f ) x0 0 , x1 1 , . . . , x(r (x) = (x − x0 )i f (i) (x0 ) m i=0
+
m X
(x−x0 )r0 (x−x1 )r1 ...(x−xk−1 )rk−1
k=1
rk X i=1
£ (r ) (r ) ¤ (i) (x−xk )i−1 x0 0 , x1 1 , ..., xk ; f .
(10)
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˘ O.T. POP AND D. BARBOSU
Remark 3.1. For r0 = r1 = · · · = rm = 1, from (10) follows the Newton interpolation divided difference formula, so (10) can be called the Newton form of the Hermite interpolation polynomial. 4. Uniform approximation via Hermite interpolation In [5], M. Ivan establish sufficient conditions for the uniform convergence of the sequence of Lagrange interpolation polynomials. Using his ideas, we shall gave sufficient conditions for the uniform convergence of the Hermite interpolation polynomials. First, let us to recall the following Theorem 4.1. [5] Suppose that f ∈ C n [a, b] and f (n+1) exists on (a, b). Then, there exists ξ ∈ (a, b) such that ¡ (r ) (r ) ¢ m) f (x) − (Hn f ) x0 0 , x1 1 , . . . , x(r (x) (11) m = (x − x0 )r0 (x − x1 )r1 . . . (x − xm )rm
f (n+1) (ξ) . (n + 1)!
Next, using the ideas from [5], suppose that f ∈ C ∞ [a, b] possesses uniform bounded derivatives such that there exists M > 0 so that |f (k) (x)| ≤ M,
(12)
for any x ∈ [a, b] and any k ∈ N0 . Remark 4.1. Because r0 , r1 , . . . , rm ∈ N and r0 + r1 + · · · + rm = n + 1, it follows n ≥ m, so if m tends to infinity then n also tends to infinity. Now, we are ready to prove the main result of this section which is the following Theorem 4.2. Suppose that (xm )m≥0 is a sequence of distinct knots, xm ∈ [a, b] for any m ∈ N0 . Then the inequality ¯ (b − a)n+1 ¯ ¢ ¡ (r ) (r ) ¯ ¯ m) (x) − f (x) M (13) ¯≤ ¯(Hn f ) x0 0 , x1 1 , . . . , x(r m (n + 1)! holds for any x ∈ [a, b], any m ∈ N0 and the sequence ³ ¡ (r ) (r ) ¢ ´ m) (x) (Hn f ) x0 0 , x1 1 , . . . , x(r m
n≥0
converges to f , uniformly on [a, b]. Proof. Taking (11) and (12) into account, yields ¯ ¯ ¡ (r ) (r ) ¢ ¯ ¯ m) (x) − f (x) ¯(Hn f ) x0 0 , x1 1 , . . . , x(r ¯ m ¯ ¯ 1 ¯ (n+1) ¯ (b−a)n+1 |(x−x0 )r0 (x−x1 )r1 · . . . · (x−xm )rm | ¯f(ξ) ¯ ≤ M, = (n+1)! (n+1)! (b − a)n+1 i.e. (13) holds. Next, because lim = 0, from (13) it follows that the n→∞ (n + 1)! ³ ´ ¡ (r0 ) (r1 ) ¢ (r ) sequence (Hn f ) x0 , x1 , . . . , xmm (x) converges to f , uniformly on [a, b]. n≥0
¤
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References [1] Gh. Coman, Analiz˘ a Numeric˘ a, Ed. Libris, Cluj-Napoca, 1995. ¨ [2] G. Faber, Uber die interpolatorische Darstellung stetiger Funktionen, Jahrb. Deut. Math. Verein 23 (1914), 190–200. adulescu, Singular Elliptic Problems. Bifurcation and Asymptotic Analysis, [3] M. Ghergu and V. R˘ Oxford Lecture Series in Mathematics and its Applications 37 (2008). [4] D. V. Ionescu, Diferent¸e divizate, Ed. Academiei, Bucure¸sti, 1978. [5] M. Ivan, Elements of Interpolation Theory, Mediamira Science Publisher, Cluj-Napoca, 2004. aly, V. R˘ adulescu and C. Varga,Variational Principles in Mathematical Physics, Ge[6] A. Krist´ ometry and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, Encyclopedia of Mathematics and its Applications 136 (2010). [7] I. Muntean, Functional Analysis, Univ. ”Babe¸s-Bolyai” Cluj-Napoca, 2002. [8] I. P˘ av˘ aloiu and N. Pop, Interpolare ¸si aplicat¸ii, Ed. Risoprint, Cluj-Napoca, 2005. arbosu, Two dimensional divided differences with multiple knots, An. St. [9] O. T. Pop and D. B˘ Univ. Ovidius Constant¸a 17 (2009), NO. 2, 181–190. a numeric˘ a ¸si teoria [10] D. D. Stancu, Gh. Coman, O. Agratini and R. Trˆımbit¸a¸s, Analiz˘ aproxim˘ arii, I, Presa Univ. Clujean˘ a, Cluj-Napoca, 2001. (Ovidiu T. Pop) National College ”Mihai Eminescu” Mihai Eminescu Street, No. 5, Satu Mare 440014, Romania E-mail address:
[email protected] (Dan B˘ arbosu) North University of Baia Mare Department of Mathematics and Computer Science Victoriei 76, 430122 Baia Mare, Romania E-mail address:
[email protected]
