A Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion

Transcription

1 Applied Mathematical Sciences, Vol, 207, no 6, HIKARI Ltd, wwwm-hikaricom A Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion Koichiro Shimada School of Science and Technology, Kwansei Gakuin University, Japan Shota Taguchi Kyoto Prefectural kumiyama High School, Japan Kazuaki Kitahara School of Science and Technology, Kwansei Gakuin University, Japan Copyright c 207 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited Abstract As is well known, if a function is analytic on an interval, then the function on a subinterval is expressed as the Taylor expansion about each point in the interval Furthermore, possibility of Taylor expansions of functions about two or three point has been studied as much useful expressions in many areas of mathematical analysis In this note, for given positive integers n, m, we show possibility of two point Taylor expansions of functions about two points, with multiplicity weight (n, m Mathematics Subject Classification: 4A58, 4A05, 4A0 Keywords: Taylor expansion, Hermite interpolating polynomials Introduction As is well known, polynomial approximation has a long history and has established the foundation of approximation theory In particular, interpolations by

2 308 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara polynomials play a very important part of polynomial approximation Before showing the purpose of this note, we briefly study Hermite interpolating polynomials Let A be an infinite subset of the real line R whose interior is not empty and let f be a real-valued function on A For any set X consisting of (n distinct points x 0,, x n in the interior of A and for any sequence of positive integers k 0,, k n, if f is sufficiently differentiable at x 0,, x n, then there exists a unique approximating polynomial p f,x(k0,,k n(x to f which is of degree at most m(= k 0 k n satisfying that p (j f,x(k 0,,k n (x i = f (j (x i, 0 i n, 0 j k i The points x 0,, x n and the polynomial p f,x(k0,,k n(x are called nodes and the Hermite interpolating polynomial to f at x 0,, x n with multiplicities k 0,, k n, respectively It is well known that for a set X consisting of one node x 0 with multiplicity n, the Hermite interpolating polynomial p f,x(n to f is the Taylor polynomial of f about x 0, that is, p f,x(n (x = f(x 0 f (x 0! (x x 0 f (n (x 0 (x x 0 n (n! Moreover, if f is infinitely differentiable at x 0 and if there exists a positive number ρ such that f(x = lim n p f,x(n (x for all x (x 0 ρ, x 0 ρ, then it is said that f has the Taylor expansion of f about x 0 on (x 0 ρ, x 0 ρ Hence, we make the following definition Definition Let f be a real-valued function on a subset A of the real line R whose interior is not empty X = {x 0,, x m } a set of m distinct nodes in the interior of A such that f is infinitely differentiable at x 0,, x m For given positive integers w 0,, w m, if lim p f,x(w 0 n,,w m n(x = f(x for all x A, n then we say that f has the m point Taylor expansion about x 0,, x m with multiplicity weight (w 0,, w m on A As is well known, if a function is analytic on an interval, then the function on a subinterval is expressed as the one point Taylor expansion about a point in the interval Furthermore, possibility of Taylor expansions of functions about two or three points has also been studied as much useful expressions in many areas of mathematical analysis Some representations of p f,x(n,,n (x are seen in Daivis [] and we can see the theory of m point Taylor expansion in the

3 A note on multiplicity weight of nodes of two point Taylor expansion 309 complex plane in Walsh [0] López and Temme [7, 8] study how m point Taylor expansions in the complex plane can be used in obtaining uniform asymptotic expansions of contour integrals Now we show previous results by Kitahara et al [4, 5, 6] about possibility of two point Taylor expansions of functions about two points, Theorem 2 (Kitahara, Chiyonobu and Tsukamoto [4, Theorem] Let f be a function on R, which is expressed as { p(x x [0, f(x = q(x x (, 0, where p and q are polynomials of degree at most n Let p f,{,}(l,l, l N be the Hermite interpolating polynomials to f at, with multiplicities l, l Then, f has the two point Taylor expansion about, with multiplicity weight (, on ( 2, 0 ( 0, 2, that is, ( lim p f,{,}(l,l(x = f(x for all x ( 2, 0 0, 2 l Moreover, if p(0 = q(0, then f has the two point Taylor expansion about, with multiplicity weight (, on ( 2, 2, that is, ( lim p f,{,}(l,l(x = f(x for all x 2, 2 l Theorem 3 (Kitahara, Yamada and Fujiwara [6, Theorem 3] Let f be a real-valued function on R which is expressed as { C x [0, f(x = C 2 x (, 0, where C and C 2 are real numbers Let p f,{,}(l,l, l N be the Hermite interpolating polynomials to f at, with multiplicities l, l Then, it holds that p f,{,}(l,l (0 = C C 2, l N 2 Theorem 4 (Kitahara and Okuno [5, Theorem 2] Let f be a function on R, which is expressed as { [ p(x x 3 f(x =, q(x x (,, 3 where p and q are polynomials of degree at most n Let p f,{,}(2l,l, l N be the Hermite interpolating polynomials to f at, with multiplicities 2l, l Let α be the real number with α < and (α 2 (α = 32 and β the real 27

4 3020 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara number with β > and (β 2 (β = Then, for each x ( α, 3 ( 3, β, there exists a positive number C p f,{,}(2l,l (x f(x C l for all l N, that is, f has the two point Taylor expansion about, with multiplicity weight (2, on ( ( α, 3, β Moreover, if p ( ( 3 3 = q 3, then there exists a positive number C such that ( ( p f,{,}(2l,l f C 3 3 l, l N, that is, f has the two point Taylor expansion about, with multiplicity weight (2, on (α, β The purpose of this note is to show generalizations of Theorem 3 and 4 Theorem 5 Let m, n be positive integers Let δ be a real number with δ > and δ 2 a real number with δ 2 > (, where is the point which divides the interval [, ] in the ratio n : m Let f be a piecewise analytic function { [ p(x x f(x =, q(x x (, such that f is equal to an analytic function p on [, which has the Taylor expansion of p about on ( δ 2, δ 2, and f is equal to an analytic function q on (, which has the Taylor expansion of q about on ( δ, δ Let p f,{,}(nl,ml, l N be the Hermite interpolating polynomials to f at, with multiplicities nl, ml Let α be the real number with α < and (α n (α m 2 nn mm = and β the real number with β > and (β n (β m = hold: ( For each x [ α, that 2 n n m m ( ( Then, the following propositions (, β], there exists a positive number C such pf,{,}(nl,ml (x f(x C l for all l N, that is, f has the two point Taylor expansion about, with multiplicity weight (n, m on [ ( α,, β] (2 For any real numbers a, b with α < a < < b < β, the sequence of functions {p f,{,}(nl,ml } l N uniformly converges to f on [α, a] [b, β] (3 If p ( ( = q, then there exists a positive number C such that ( ( n m n m p f,{,}(nl,ml f C l, l N,

5 A note on multiplicity weight of nodes of two point Taylor expansion 302 that is, f has the two point Taylor expansion about, with multiplicity weight (n, m on [α, β] (4 If p and q are constants such that p(x = C, q(x = C 2, respectively, then there exists a positive number C such that ( n m p f,{,}(nl,ml C C 2 2 C, l N l 2 Preliminaries In this section, we review the definition of divided differences and give two necessary propositions Definition 2 Let x 0 x n be a list of nodes In the list of nodes, only distinct nodes z 0,, z j appear and each node z i, i = 0,, j is just appeared k i times Let f be sufficiently differentiable at z 0,, z j Let p be the Hermite interpolating polynomials to f at z 0,, z j with multiplicities k 0,, k j Then, we call the coefficient of x n of the polynomial p is called the n- th order divided difference of f at x 0,, x n and it is denoted by f[x 0,, x n ] To make sure of multiplicities, we express for the divided difference f[x 0,, x n ] f[z 0,, z j ; k 0,, k j ] From Theorem 3 in Kincaid and Cheney[3, p following 356], we easily have the Proposition 22 Let x 0 x n be a list of nodes and let f be a real-valued function on an interval [a, b] which is sufficiently differentiable at x 0,, x n If p is the Hermite interpolating polynomial of f at x 0,, x n, then f(x p(x = f[x 0,, x n, x](x x 0 (x x (x x n, x [a, b] Proposition 23 (Kincaid and Cheney [3, p 372] Let z 0,, z j be a list of distinct nodes and k 0,, k j positive integers Let x 0,, x n be a list of nodes which satisfy that each node z i, i = 0,, j is just appeared k i times like this: (x 0,, x n = (z 0,, z }{{} 0,, z j,, z j }{{} k 0 k j If a function f is sufficiently differentiable at z 0,, z j, then the divided differences of f obey the following recursive formula: f[x,, x k ] f[x 0,, x k ] (x k x 0 x k x 0 f[x 0,, x k ] =, k = 0,, n f (k (x 0 (x k = x 0 k!

6 3022 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara From Ex 3 in Davis [, p 37], we obtain the following Proposition 24 Let a, b be distinct nodes and m, n positive integers Let f be a sufficiently differentiable function at a, b A, B are functions defined by A(x = f(x f(x, B(x = (x b m (x a n Then, the polynomial p f,{a,b}(n,m (x is expressed as m p f,{a,b}(n,m (x = (x a n k=0 B (k (b k! n (x b k (x b m k=0 A (k (a (x a k k! Proposition 25 (Durrett [2, p 37] Let X, X 2, be iid with EX i = 0, EXi 2 = σ 2, and E X i 3 = ρ < Then, it holds that ( P X X N σ x x e x2 2 dx 3ρ N 2π σ 3 N 3 Proof of Theorem 5 To show Theorem 5, we need to prepare two propositions Proposition 3 Let M, N be positive integers Let f be a real-valued function on R which is sufficiently differentiable at, Then, the following inequality holds: f[, t, ; N,, M] ( N ( NM M 2 NM k= ( 2N N M k f[, t; k, ] M k= Proof First, we show that for any positive integers M, N, ( k 2M f[t, ;, k] N M f[, t, ; N,, M] = N k= ( M 2 NM k M k= ( M k 2 NM k ( N M (k M ( N M (k N f[, t; k, ] f[t, ;, k] ( We prove this by induction Suppose that N = M = Then we have f[, t, ;,, ] = f[t, ;, ] f[, t;, ], 2

7 A note on multiplicity weight of nodes of two point Taylor expansion 3023 which is equal to the right hand formula of ( Next, under the condition that ( hold for N = and M = m, we consider the case N =, M = m We obtain f[, t, ;,, m ] f[t, ;, m ] f[, t, ;,, m] = 2 = 2 f[t, ;, m ] ( ( m m m m m ( ( m k m (k 2 2 m k 0 k= ( = ( m (m 2 f[, t;, ] 2 (m m m ( (m k 2 (m k k= f[t, ;, k] ( (m (k 0 f[, t,, ] f[t, ;, k], which is equal to the right hand formula of ( Hence, in an analogous way to the above, we show that ( hold for the cases that N =, M is any positive integer or the cases that N is any positive integer, M = Finally, under the condition that ( hold for the cases N M m n, we consider the case N = n, M = m From this assumption, we get the following equality: f[, t, ; n,, m] f[, t, ; n,, m] f[, t, ; n,, m ] = 2 = n ( ( m (k f[, t; k, ] 2 2 k m 2 k= 2 2 m k= n k= m k= ( m k 2 k ( m 2 k ( m k 2 k ( (k n ( (k m 2 ( (k n f[t, ;, k] f[, t; k, ] f[t, ;, k]

8 3024 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara Arranging the expression, we have f[, t, ; n,, m] = n (( ( ( m (k (k f[, t; k, ] 2 2 k m m 2 k= m (( ( ( m k (k (k f[t, ;, k] 2 2 k n n k= ( ( m m (m f[t, ;, m] 2 2 m n ( ( m (n f[, t; n, ] 2 2 (n m 2 n ( ( m n m (k = f[, t; k, ] 2 nm k m k= m k= ( m k 2 nm k ( n m (k n f[t, ;, k], which is equal to the right hand formula of ( In an analogous way to the above, we show that ( hold for the cases that N M m n Hence, we have shown the validity of ( Furthermore, since it holds that ( ( NM M = NM ( N, NM ( N k ( M NM NM (k M for k =,, N and ( M k ( NM (k for k =,, M, we have ( NM N NM f[, t, ; N,, M] 2 NM N k= 2 NM ( NM M 2 NM N ( N M (k 2 k f[, t; k, ] M M k= ( N k= ( N M (k 2 k f[t, ;, k] N ( k 2N f[, t; k, ] N M M k= ( k 2M f[t, ;, k] N M Proposition 32 Let m, n be positive integers Let δ be a real number with δ > and δ 2 a real number with δ 2 > (, where

9 A note on multiplicity weight of nodes of two point Taylor expansion 3025 is the point which divides the interval [, ] in the ratio n : m Let f be a piecewise analytic function { [ p(x x f(x =, q(x x (,, such that f is equal to an analytic function p on [, which has the Taylor expansion of p about on ( δ 2, δ 2, and f is equal to an analytic function q on (, which has the Taylor expansion of q about on ( δ, δ Let α be the real number with α < and (α n (α m 2 nn mm = ( and β the real number with β > and (β n (β m = Then, the following hold: (i For each t [α, β]\ { } (, there exist numbers C, C 2, r 0 < r < 2n, ( r 2 0 < r2 < 2m such that f[, t; i, ] C r, i i N, and f[t, ;, i] C 2 r2, i i N (ii If p ( ( = q, for each t [α, β], there exist numbers C, C 2, ( ( r 0 < r < 2n, r2 0 < r2 < 2m such that f[, t; i, ] C r, i i N, and f[t, ;, i] C 2 r2, i i N 2 n n m m ( Proof Since the proof of (ii can be reduced to that of (i, we prove (i And we only show f[, t; i, ] C r, i i N because f[t, ;, i] C 2 r2, i i N are analogously shown Let R, R 2 be real numbers with δ > R > 2n and δ 2 > R 2 > 2m From the assumption, q has the Taylor expansion of q about on [ R, R ], q(x = q (j ( (x j, x [ R, R ] Hence, there exists a positive integer N such that q (j ( R j <, j N And we have q (j ( < R j, j N ( Now, we consider estimations of f[, t; i, ] for the cases that ( t and (2 t (, β] [α, Case ( Since f(t = q(t, t [α,, by using Proposition 22 for t,

10 3026 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara we obtain f[, t; i, ] = For t =, since = = ( i f(t (t i ( i q(t (t i (t i j=i q (j ( (t j q (j ( (t j q (j ( (t j = f[, t; i, ] = f[ ; i ] = q(i (, i! q (ij ( (t j (i the equality stated above also holds Noting that R > max{α (, ( }, t < R and from ( for each positive integer i with i N, we have f[, t; i, ] ( R q (ij ( (i t j i ( j t < R 2n R ( From the definition of R, it follows that 0 < < R 2n Case (2 Since f(t = p(t, t (, β], by using Proposition 22 we have ( i q (j ( f[, t; i, ] = p(t (t j (t i Since p is continuous on [ R 2, R 2 ]( (, β], putting we have M = max p(x, x [ R 2,R 2 ] f[, t; i, ] p(t (t i (t i i ( i i M t (t i q (j ( (t j R i q (j ( (t j

11 A note on multiplicity weight of nodes of two point Taylor expansion 3027 To estimate i q (j (, (t i (t j we consider the cases that (a t (, R ] and (b t ( R, β] Case (2-a Since q has the Taylor expansion of q about on ( δ, δ and the sequence of functions { N q (j ( (t j } N 0 is uniformly bounded on [ R, R ], there exists a positive number M 2 such that N q (j ( (t j < M 2, N {0,, 2, }, t [ R, R ] Easily seeing that we get (t i i q (j ( ( i (t j M 2, t f[, t; i, ] (M M 2 ( i t Since t ( 2n, R ], 0 < < hold t 2n (2-b For each positive integer i with i N, noticing that t (R, β], we have i q (j ( (t j (t i (t i N Therefore, we get f[, t; k i, ] N q (j ( (t j ( q (j i ( (β j R N M (t i i j=n ( t R j ( i t R R q (j ( (β j t R ( As in seen in the case (, R satisfies 0 < R < 2n Consequently, for each t [α, β] \ { }, there exist C and r (0 < r < 2n such that f[, t; i, ] C r i, i N, R i

12 3028 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara which leads to the validity of (i Now we are in position to prove Theorem 5 Proof of Theorem 5 ( Since we easily see that (t n (t m 2 n n m m, t [α, β], ( from Proposition 22, for each t [α, β], we have f(t p f,{,}(nl,ml (t = f[, t, ; nl,, ml] (t n (t m l ( 2 n n m m l f[, t, ; nl,, ml] ( On the other hand, by using Proposition 3, Proposition 32 and Stirling s formula, } there exists a positive number C 3 satisfying that for each t [α, β] \ { f[, t, ; nl,, M] ( (l ( nl ( k ml 2n ml f[, t; k, ] 2 (l ( (l ml 2 (l ( (l ml 2 (l k= ( nl C 3 2 (l l k= ( k 2n C r k C 2n r ( ( n n m m ml k= k= ( 2m C 2 2m r 2 l C 2n r ( k 2m f[t, ;, k] k C 2 r k C 2 2m r 2 Since C, C 2, r, r 2 are considered as functions on [α, β] \ { }, putting C (t C(t = C 3 2n r (t C 2 (t 2m, r 2(t

13 A note on multiplicity weight of nodes of two point Taylor expansion 3029 we obtain for each t [α, β] \ { }, f(t p f,{,}(nl,ml (t C(t ( ( l ( 2 n n m m l 2 n n m m ( = C(t l We can prove Theorem 5 (3 similarly to Theorem 5 ( (2 We show C(t is bounded on [α, a] [b, β] by proving the following functions are bounded on [α, a] [b, β] (i C (t (ii 2n r (t (iii C 2 (t (iv 2m r 2(t From the proof of Proposition 32 (i, the functions C, C 2, r, r 2 are expressed as follows: [ c, t α, n m ( ] n m, R C (t =, c t, t ( R, β] R [, t α, n m ( R, β] R r (t = ( ], n m t, t, R [ c, t R 2, n m ( ] n m, β C 2 (t = r 2 (t = c R 2 t where c is a constant t R 2, t [α, R 2 ( ] n m, t [α, R 2, β [, t R 2, n m,, l

14 3030 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara Therefore, let a, a 2 be the real numbers with { 0 < a < min b n m, δ 2n { n m 0 < a 2 < min a, δ 2 2m by putting R = 2n a, R 2 = 2m a 2, we can see that functions (i, (ii, (iii and (iv are bounded on [α, a] [b, β] (4 Taguchi [9] already proved this proposition Here we show a proof of (4 by Proposition 25 that the standard normal distribution can be approximated by negative binomial distributions Without loss of generality, we can assume that { [ x f(x =, 0 x (, }, }, From Proposition 24, p f,{,}(nl,ml (x is expressed as follows: We get ml p f,{,}(nl,ml (x = (x nl = ml k=0 ( n m p f,{,}(nl,ml = k=0 ( (z nl (k z= (x k k! ( ( nl ( k nl k x x k 2 2 ml k=0 ( k nl k ( nl ( k n m On the other hand, let X, X 2, be independent geometric random variables, where X i has parameter p Then, we have EX i = p, V (X p i = p p 2 Therefore, from Proposition 25 there exists a positive number C such that P N ( X p k p x p x e x2 2 dx N k= 2π C N p 2 for all x R and for all N =, 2, binomial distribution N B(N, p, ( N P X k x = k= k {j Z 0jx} Since N k= X k obeys the negative ( k N p N ( p k k

15 A note on multiplicity weight of nodes of two point Taylor expansion 303 Hence, there exists a positive number C such that ( k N p N ( p k x k ( k Z 2π p 0k N p 2 x N( p p e x2 2 dx C N for all x R and for all N =, 2, Putting N = nl, p = n, x = n, for each l =, 2, we obtain (ml ( p p 2 N( p N x p = ml, and p f,{,}(nl,ml ( n m 2π n (ml e x2 2 dx C nl Therefore, for each l =, 2, we have ( n m p f,{,}(nl,ml 2 C 0 e x2 2 dx nl 2π 2π C n nl 2π (ml ( C = n n 2π (m l This completes the proof of Theorem 5 n (ml e x2 2 dx References [] P J Davis, Interpolation & Approximation, Dover, New York, 975 [2] R Durrett, Probability: Theory and Examples, 4th edition, Cambridge University Press, [3] D Kincaid and E W Cheney, Numerical Analysis, 2nd Ed, Brooks/Cole Publishing Company, New York, 996

16 3032 Koichiro Shimada, Shota Taguchi and Kazuaki Kitahara [4] K Kitahara, T Chiyonobu and H Tsukamoto, A note on two point Taylor expansion, International Journal of Pure and Applied Mathematics, 75 (202, no 3, [5] K Kitahara, T Okuno, A note on two point Taylor expansion III, International Journal of Modeling and Optimization, 4 (204, no 4, [6] K Kitahara, T Yamada and K Fujiwara, A note on two point Taylor expansion II, International Journal of Pure and Applied Mathematics, 86 (203, [7] J L López and N M Temme, Two-point Taylor expansion of analytic functions, Studies in Applied Mathematics, 09 (2002, [8] J L López and N M Temme, Multi-point Taylor expansion of analytic functions, Trans Am Math Soc, 356 (2004, [9] S Taguchi, A note on Two Point Taylor Expansion of Heaviside Function, Master s Thesis, Kwansei Gakuin University, 205 [0] J L Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, American Mathematical Society, Providence, 969 Received: October 30, 207; Published: Devember 5, 207