- Email: [email protected]
Symmetric multivariate Chebyshev polynomials
Applied Mathematics and Computation 187 (2007) 530–534 www.elsevier.com/locate/amc
Symmetric multivariate Chebyshev polynomials Luis Verde-Star Departamento de Matema´ticas, Universidad Auto´noma Metropolitana, Mexico D.F. 09340, Mexico
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract We introduce a general method for the symmetrization of univariate polynomials and use it to construct symmetric polynomials in r + 1 variables that generalize the classical Chebyshev polynomials of the first kind. We show that, on the set [1, 1]r+1, such polynomials provide the best uniform approximations of the complete symmetric polynomials hk(x0, x1, . . ., xr) by means of symmetric polynomials of total degree less than n, where k is a partition of n. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Symmetric polynomials; Multivariate Chebyshev polynomials; Divided differences; Best uniform approximation
1. Introduction Univariate Chebyshev polynomials play a very important role in interpolation and approximation theory and several areas of numerical analysis and combinatorics. There are numerous papers dealing with many different generalizations of the Chebyshev polynomials to the case of several real or complex variables. See, for example [1,2,4,6,8,10,13]. Such generalizations are usually obtained by emphasizing some aspect of the univariate Chebyshev polynomials which should be essentially preserved in the multivariate version sought. Some of the most important properties of the Chebyshev polynomials are related to extreme problems. For example, the (normalized) monic nth degree Chebyshev polynomial of the first kind has the minimum uniform norm on the interval [1, 1] among all monic polynomials of degree n. This property has been used to obtain multivariate generalizations. See [7,8,10,11,13]. In the present paper, we introduce a method to obtain from a univariate polynomial p(t) a polynomial P(x0, x1, . . ., xr) that is symmetric in the variables xi. Our construction uses divided differences and gives explicit expressions for the multivariate polynomials. We apply the method to the family of Chebyshev polynomials of the first kind, and obtain a family of symmetric multivariate Chebyshev polynomials that forms a basis for the algebra of symmetric polynomials in r + 1 indeterminates. We also show that the multivariate Chebyshev polynomials have norm minimization properties.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.003
L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
531
The construction of the multivariate polynomials can be easily implemented in the Maple software package, since it includes a package that handles Chebyshev and other orthogonal polynomials in a direct way. The graphs of the two variable symmetric polynomials can be visualized using the graphics package in Maple. There are several important aspects that are not considered in this paper, such as the orthogonality, the connections with difference and differential equations, the interpolation properties, and the relationships with other kinds of generalized multivariate Chebyshev polynomials. 2. Symmetrization of univariate polynomials We give here some basic definitions about symmetric polynomials following Stanley’s book [14], but we only deal with symmetric functions of a fixed finite set of indeterminates, and not with formal power series in an infinite number of indeterminates, as is customary in the theory of symmetric functions. Let r be a fixed nonnegative integer and let x0, x1, . . ., xr be indeterminates that commute with each other. For each positive integer k define the polynomial: X k k hk ðx0 ; x1 ; . . . ; xr Þ ¼ x00 x11 xkr r ; ð2:1Þ where the sum runs over all vectors (k0, k1, . . ., kr) with nonnegative integer coordinates such that k0 + k1 + + kr = k. Note that h0 1 and that hk is a symmetric polynomial in theindeterminates xi with total degree equal to k. Note also that the number of summands in (2.1) is equal to kþr . r AP partition k of a nonnegative integer n is a sequence ðk1 ; k2 ; . . . ; kk Þ 2 Nk satisfying k1 P k2 P P kk and ki = n. We identify k with the infinite sequence (k1, k2, . . ., kk, 0, 0, . . .). We let Par(n) denote the set of all partitions of n. If k is in Par(n) we write jkj = n. Note that Par(0) consists of the empty partition. Let Par denote the union of the sets Par(n) for n = 0, 1, 2, . . . Define the complete homogeneous symmetric polynomial hk for k in Par by hk ¼ hk1 hk2 hk3 if k = (k0, k1, . . .). Let us note that hk has total degree equal to jkj. The polynomials hk with k in Par(n) generate a vector space over the real numbers that we denote by Kn. The vector space direct sum of the Kn, for n 2 N, is denoted by K. If f 2 Kn and g 2 Km then it is clear that fg 2 Kn+m and therefore K has the structure of a graded algebra over R. Let P be the vector space of all the polynomials in t with real coefficients and let Pn denote the subspace of polynomials with degree at most equal to n. For r in N define the linear map symr : P ! K by: symr tk ¼
hk ðx0 ; x1 ; . . . ; xr Þ ; kþr r
k 2 N:
ð2:2Þ
We call symr the r-symmetrizer map. Note that symr sends Pn into the subspace K0 K1 K2 Kn of K. Define the diagonal substitution map diagt : K ! P by: diagt gðx0 ; x1 ; . . . ; xr Þ ¼ gðt; t; . . . ; tÞ;
g 2 K:
ð2:3Þ
Note that diagt is linear and multiplicative and kþr k diagt hk ðx0 ; x1 ; . . . ; xr Þ ¼ t ; k 2 N: r Therefore diagt symrtk = tk, for k 2 N, that is, diagt is a left inverse for symr and diagt symr is the identity map on P. We will give a simple analytical interpretation of the operator symr. For this purpose we need some basic properties of divided differences. For the theory of divided differences we refer the reader to [3,5,15]. For j P 0 we let D(x0, x1, . . ., xj) denote the divided difference operator that acts on functions of t and we use the notation
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L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
f ½x0 ; x1 ; . . . ; xj ¼ Dðx0 ; x1 ; . . . ; xj Þf ðtÞ: The operator D(x0, x1, . . ., xj) is defined recursively as follows: D(x0)f(t) = f(x0), f ðx0 Þ f ðx1 Þ Dðx0 ; x1 Þf ðtÞ ¼ ¼ f ½x0 ; x1 ; x0 x1 and Dðx0 ; x1 ; . . . ; xj ; xjþ1 Þf ðtÞ ¼
f ½x0 ; x1 ; . . . ; xj1 ; xj f ½x0 ; x1 ; . . . ; xj1 ; xjþ1 : xj xjþ1
Using the basic identity m1 xm y m X ¼ xi y m1i ; xy i¼0
m P 1;
repeatedly, it is easy to show that X i i i Dðx0 ; x1 ; . . . ; xj Þtm ¼ x00 x11 xjj ;
m P j;
where the sum runs over all the vectors (i0, i1, . . ., ij) in Njþ1 such that i0 + i1 + + ij = m j. Therefore Dðx0 ; x1 ; . . . ; xj Þtm ¼ hmj ðx0 ; x1 ; . . . ; xj Þ;
m P j;
ð2:4Þ
and D(x0, x1, . . ., xj)tm = 0 if m < j. We can consider symr as a composition of two linear operators as follows. Let the linear map Intr : P ! P be defined by Z Z Z tkþr k ; Intr t ¼ r tk dt ¼ kþr r where the indefinite integration is repeated r times, taking the integration constants equal to zero. Then, in view of (2.4) and (2.2) we have symr tk ¼ Dðx0 ; x1 ; . . . ; xr ÞIntr tk ;
k 2 N:
ð2:5Þ
Therefore, symr is the composition of the map Intr and the divided difference operator D(x0, x1, . . ., xr). We need the following basic property of divided differences in order to prove our first theorem. Lemma 1. Let f(t) be an r times continuously differentiable function on the interval [a, b] and let x0, x1, . . ., xr be points in [a, b], distinct or not. Then there exists n 2 [mini xi, maxi xi] such that f ½x0 ; x1 ; . . . ; xr ¼
f ðrÞ ðnÞ : r!
For a proof of this Lemma see [3, Theorem 2.5]. If A is a closed and bounded subset of Rm and f is a continuous function defined on A we write kf kA ¼ max jf ðuÞj: u2A
Theorem 1. Let p(t) be a polynomial and let I = [a, b] be a bounded and closed interval. Then ksymr pkI rþ1 ¼ kpkI : Proof. Let n be the degree of p and let q(t) = Intrp(t). From the definition of Intr we see that q is a polynomial of degree n + r and q(r)(t)/r! = p(t) for all real t. Let (x0, x1, . . ., xr) 2 Ir+1. Then, by (2.5) we have symr pðx0 ; x1 ; . . . ; xr Þ ¼ Dðx0 ; x1 ; . . . ; xr ÞIntr pðtÞ ¼ Dðx0 ; x1 ; . . . ; xr ÞqðtÞ:
L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
533
By Lemma 1 there exists n 2 [mini xi, maxi xi] I such that symr pðx0 ; x1 ; . . . ; xr Þ ¼ qðrÞ ðnÞ=r! ¼ pðnÞ: Therefore ksymr ; pkI rþ1 6 kpkI . Since diagtsymr is the identity map on P, we have symr pðt; t; . . . ; tÞ ¼ diagt symr pðx0 ; x1 ; . . . ; xr Þ ¼ pðtÞ; and this completes the proof of the theorem.
t 2 I;
h
3. Symmetric Chebyshev polynomials In this section we apply the symmetrization map symr to the classical Chebyshev polynomials of the first kind to obtain a family of symmetric polynomials with interesting approximation properties. We follow the notation of Rivlin’s book [12]. See also [9]. The Chebyshev polynomials of the first kind are defined by T n ðtÞ ¼ cosðn arccos hÞ;
n P 0;
where t = cosh. The degree of Tn is n and if n P 1 the leading coefficient of Tn equals 2n1. The normalized monic polynomial 21nTn is denoted by Te n . Note that T 0 ¼ Te 0 1. Let I denote the interval [1, 1]. It is clear that kTnkI = 1 for n P 0. One of the most important properties of the polynomials Tn is the following theorem. Theorem 2. If p is a monic polynomial of degree n P 1 then kpkI P k Te n kI ¼ 21n ; with equality only if p ¼ Te n . For a proof see Rivlin [12, Theorem 2.1]. We will use the following formulas in our development. See [12, p. 5]. 1 T m ðxÞT n ðxÞ ¼ T mþn ðxÞ þ T jmnj ðxÞ ; m; n P 0; 2 Z 1 T nþ1 ðxÞ T n1 ðxÞ T n ðxÞ dx ¼ þ C; n P 2: 2 nþ1 n1
ð3:1Þ ð3:2Þ
Recall that r is a fixed nonnegative integer. We define S n ðx0 ; x1 ; . . . ; xr Þ ¼ symr T n ;
n P 0:
We call Sn the symmetric Chebyshev polynomial of degree n. Note that Sn is a linear combination of the com. All the coefplete symmetric polynomials hk, for k = 0,1, . . ., n, with the coefficient of hn equal to 2n1 = nþr r ficients of Sn are easily found, using the coefficients of Tn. For example, for r = 2 we have 1 S 2 ¼ h2 h0 ; 3
2 S 3 ¼ h3 h1 ; 5
S4 ¼
8 4 h4 h2 þ h0 : 15 3
Note that, by Theorem 1, we have kS n kI rþ1 ¼ kT n kI ¼ 1: Let k = (k1, k2, . . ., kk, 0, 0, . . .) be a partition of length k of a positive integer. Write (3.1) in the form T mþn ðxÞ ¼ 2T m ðxÞT n ðxÞ T mn ðxÞ;
m P n:
ð3:3Þ
By repeated application of this formula we obtain T jkj ¼ 2k1 T k1 T k2 T kk
k X j¼2
2kj T rj T kjþ1 T kjþ2 T kk ;
ð3:4Þ
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where rj = k1 + k2 + + kj1 kj, for 2 6 j 6 k. Note that the first term in the right-hand side of (3.4) has degree equal to jkj, and the summand with index j has degree equal to jkj 2kj. For k = (k1, k2, . . ., kk, 0, 0, . . .) we define the symmetric Chebyshev polynomial Sk, in the variables x0, x1, . . ., xr, by S k ¼ 2k1 S k1 S k2 S kk
k X
2kj S rj S kjþ1 S kjþ2 S kk :
ð3:5Þ
j¼2
Note that Sk is a symmetric polynomial of total degree jkj, and its homogeneous part of degree jkj is 2jkj1
hk1 hk 2 hkk : k1 þ r k2 þ r kk þ r r
r
ð3:6Þ
r
Using induction on the number k of parts of k, some properties of divided differences, and properties of the Chebyshev polynomials, we can show that kS k kI rþ1 ¼ 1 for all partitions k in Par. Since diagtSn = Tn(t) for n P 0, and diagt is linear and multiplicative, using (3.4) and (3.5) we obtain diagt S k ¼ T jkj ðtÞ;
k 2 Par:
ð3:7Þ
Therefore jSk(x0, x1, . . ., xr)j attains its maximum value at points on the diagonal D = {(t, t, . . ., t):t 2 I} of the cube Ir+1. Theorem 3. Let k be a partition of length k of the positive integer n and let P be a symmetric polynomial of degree n = jkj such that its homogeneous part of degree n equals the expression in (3.6). Then kP kI rþ1 P kS k kI rþ1 :
ð3:8Þ
Proof. Let Q(t) = diagtP(x0, x1, . . ., xr). Then, by the hypothesis and the definition of diagt, Q(t) is a univariate polynomial of degree n with leading coefficient equal to 2n1. Therefore, by Theorem 2 we have kQkI P kT n kI ¼ kdiagt S k kI ¼ kS k kI rþ1 ; and hence kP kI rþ1 P kS k kI rþ1 ; and this completes the proof.
h
References [1] W. Bajguz, On generalization of Tchebycheff polynomials, Integ. Transform. Spec. Funct. 9 (2000) 91–98. [2] R.J. Beerends, Chebyshev polynomials in several variables and the radial part of the Laplace–Beltrami operator, Trans. Am. Math. Soc. 328 (1991) 779–814. [3] S.D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, third ed., McGraw-Hill, New York, 1980. [4] G. Dattoli, Integral transforms and Chebyshev-like polynomials, Appl. Math. Comput. 148 (2004) 225–234. [5] C. de Boor, A Practical Guide to Splines, revised ed., Applied Mathematical Sciences, vol. 27, Springer, New York, 2001. [6] F. Hazama, Twists and generalized Chebyshev polynomials. I and II, JP, J. Algebra Number Theory Appl. 2 (2002) 61–82, 83–100. [7] A. Kroo´, Chebyshev-type extremal problems for multivariate polynomials on convex bodies, East J. Approx. 5 (1999) 211–221. [8] K.H. Kwon, D.W. Lee, Tchebycheff polynomials on a disk, J. Comput. Appl. Math. 162 (2004) 359–363. [9] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC Press, London/Boca Raton, FL, 2003. [10] D.J. Newman, Yuan Xu, Tchebycheff polynomials on a triangular region, Constr. Approx. 9 (1993) 543–546. [11] S. Re´ve´sz, Uniqueness of multivariate Chebyshev-type extremal polynomials for convex bodies, East J. Approx. 7 (2001) 205–240. [12] T.J. Rivlin, Chebyshev polynomials. From Approximation Theory to Algebra and Number Theory, second ed., Pure and Applied Mathematics, Wiley, New York, 1990. [13] T.J. Rivlin, Some remarks on multivariate Chebyshev polynomials. Approximation theory and applications, Comput. Math. Appl. 33 (1997) 223–228. [14] R.P. Stanley, Enumerative Combinatorics, vol. II, Cambridge University Press, Cambridge, 1999 (Chapter 7). [15] L. Verde-Star, Divided differences and their applications, in: Approximation and Optimization, vol. I, Proceedings of the International Conference on Approximation and Optimization, Cluj-Napoca, Romania, 1996, pp. 121–138.
Symmetric multivariate Chebyshev polynomials Luis Verde-Star Departamento de Matema´ticas, Universidad Auto´noma Metropolitana, Mexico D.F. 09340, Mexico
Dedicated to Professor H.M. Srivastava on the occasion of his 65th birthday
Abstract We introduce a general method for the symmetrization of univariate polynomials and use it to construct symmetric polynomials in r + 1 variables that generalize the classical Chebyshev polynomials of the first kind. We show that, on the set [1, 1]r+1, such polynomials provide the best uniform approximations of the complete symmetric polynomials hk(x0, x1, . . ., xr) by means of symmetric polynomials of total degree less than n, where k is a partition of n. Ó 2006 Elsevier Inc. All rights reserved. Keywords: Symmetric polynomials; Multivariate Chebyshev polynomials; Divided differences; Best uniform approximation
1. Introduction Univariate Chebyshev polynomials play a very important role in interpolation and approximation theory and several areas of numerical analysis and combinatorics. There are numerous papers dealing with many different generalizations of the Chebyshev polynomials to the case of several real or complex variables. See, for example [1,2,4,6,8,10,13]. Such generalizations are usually obtained by emphasizing some aspect of the univariate Chebyshev polynomials which should be essentially preserved in the multivariate version sought. Some of the most important properties of the Chebyshev polynomials are related to extreme problems. For example, the (normalized) monic nth degree Chebyshev polynomial of the first kind has the minimum uniform norm on the interval [1, 1] among all monic polynomials of degree n. This property has been used to obtain multivariate generalizations. See [7,8,10,11,13]. In the present paper, we introduce a method to obtain from a univariate polynomial p(t) a polynomial P(x0, x1, . . ., xr) that is symmetric in the variables xi. Our construction uses divided differences and gives explicit expressions for the multivariate polynomials. We apply the method to the family of Chebyshev polynomials of the first kind, and obtain a family of symmetric multivariate Chebyshev polynomials that forms a basis for the algebra of symmetric polynomials in r + 1 indeterminates. We also show that the multivariate Chebyshev polynomials have norm minimization properties.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.09.003
L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
531
The construction of the multivariate polynomials can be easily implemented in the Maple software package, since it includes a package that handles Chebyshev and other orthogonal polynomials in a direct way. The graphs of the two variable symmetric polynomials can be visualized using the graphics package in Maple. There are several important aspects that are not considered in this paper, such as the orthogonality, the connections with difference and differential equations, the interpolation properties, and the relationships with other kinds of generalized multivariate Chebyshev polynomials. 2. Symmetrization of univariate polynomials We give here some basic definitions about symmetric polynomials following Stanley’s book [14], but we only deal with symmetric functions of a fixed finite set of indeterminates, and not with formal power series in an infinite number of indeterminates, as is customary in the theory of symmetric functions. Let r be a fixed nonnegative integer and let x0, x1, . . ., xr be indeterminates that commute with each other. For each positive integer k define the polynomial: X k k hk ðx0 ; x1 ; . . . ; xr Þ ¼ x00 x11 xkr r ; ð2:1Þ where the sum runs over all vectors (k0, k1, . . ., kr) with nonnegative integer coordinates such that k0 + k1 + + kr = k. Note that h0 1 and that hk is a symmetric polynomial in theindeterminates xi with total degree equal to k. Note also that the number of summands in (2.1) is equal to kþr . r AP partition k of a nonnegative integer n is a sequence ðk1 ; k2 ; . . . ; kk Þ 2 Nk satisfying k1 P k2 P P kk and ki = n. We identify k with the infinite sequence (k1, k2, . . ., kk, 0, 0, . . .). We let Par(n) denote the set of all partitions of n. If k is in Par(n) we write jkj = n. Note that Par(0) consists of the empty partition. Let Par denote the union of the sets Par(n) for n = 0, 1, 2, . . . Define the complete homogeneous symmetric polynomial hk for k in Par by hk ¼ hk1 hk2 hk3 if k = (k0, k1, . . .). Let us note that hk has total degree equal to jkj. The polynomials hk with k in Par(n) generate a vector space over the real numbers that we denote by Kn. The vector space direct sum of the Kn, for n 2 N, is denoted by K. If f 2 Kn and g 2 Km then it is clear that fg 2 Kn+m and therefore K has the structure of a graded algebra over R. Let P be the vector space of all the polynomials in t with real coefficients and let Pn denote the subspace of polynomials with degree at most equal to n. For r in N define the linear map symr : P ! K by: symr tk ¼
hk ðx0 ; x1 ; . . . ; xr Þ ; kþr r
k 2 N:
ð2:2Þ
We call symr the r-symmetrizer map. Note that symr sends Pn into the subspace K0 K1 K2 Kn of K. Define the diagonal substitution map diagt : K ! P by: diagt gðx0 ; x1 ; . . . ; xr Þ ¼ gðt; t; . . . ; tÞ;
g 2 K:
ð2:3Þ
Note that diagt is linear and multiplicative and kþr k diagt hk ðx0 ; x1 ; . . . ; xr Þ ¼ t ; k 2 N: r Therefore diagt symrtk = tk, for k 2 N, that is, diagt is a left inverse for symr and diagt symr is the identity map on P. We will give a simple analytical interpretation of the operator symr. For this purpose we need some basic properties of divided differences. For the theory of divided differences we refer the reader to [3,5,15]. For j P 0 we let D(x0, x1, . . ., xj) denote the divided difference operator that acts on functions of t and we use the notation
532
L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
f ½x0 ; x1 ; . . . ; xj ¼ Dðx0 ; x1 ; . . . ; xj Þf ðtÞ: The operator D(x0, x1, . . ., xj) is defined recursively as follows: D(x0)f(t) = f(x0), f ðx0 Þ f ðx1 Þ Dðx0 ; x1 Þf ðtÞ ¼ ¼ f ½x0 ; x1 ; x0 x1 and Dðx0 ; x1 ; . . . ; xj ; xjþ1 Þf ðtÞ ¼
f ½x0 ; x1 ; . . . ; xj1 ; xj f ½x0 ; x1 ; . . . ; xj1 ; xjþ1 : xj xjþ1
Using the basic identity m1 xm y m X ¼ xi y m1i ; xy i¼0
m P 1;
repeatedly, it is easy to show that X i i i Dðx0 ; x1 ; . . . ; xj Þtm ¼ x00 x11 xjj ;
m P j;
where the sum runs over all the vectors (i0, i1, . . ., ij) in Njþ1 such that i0 + i1 + + ij = m j. Therefore Dðx0 ; x1 ; . . . ; xj Þtm ¼ hmj ðx0 ; x1 ; . . . ; xj Þ;
m P j;
ð2:4Þ
and D(x0, x1, . . ., xj)tm = 0 if m < j. We can consider symr as a composition of two linear operators as follows. Let the linear map Intr : P ! P be defined by Z Z Z tkþr k ; Intr t ¼ r tk dt ¼ kþr r where the indefinite integration is repeated r times, taking the integration constants equal to zero. Then, in view of (2.4) and (2.2) we have symr tk ¼ Dðx0 ; x1 ; . . . ; xr ÞIntr tk ;
k 2 N:
ð2:5Þ
Therefore, symr is the composition of the map Intr and the divided difference operator D(x0, x1, . . ., xr). We need the following basic property of divided differences in order to prove our first theorem. Lemma 1. Let f(t) be an r times continuously differentiable function on the interval [a, b] and let x0, x1, . . ., xr be points in [a, b], distinct or not. Then there exists n 2 [mini xi, maxi xi] such that f ½x0 ; x1 ; . . . ; xr ¼
f ðrÞ ðnÞ : r!
For a proof of this Lemma see [3, Theorem 2.5]. If A is a closed and bounded subset of Rm and f is a continuous function defined on A we write kf kA ¼ max jf ðuÞj: u2A
Theorem 1. Let p(t) be a polynomial and let I = [a, b] be a bounded and closed interval. Then ksymr pkI rþ1 ¼ kpkI : Proof. Let n be the degree of p and let q(t) = Intrp(t). From the definition of Intr we see that q is a polynomial of degree n + r and q(r)(t)/r! = p(t) for all real t. Let (x0, x1, . . ., xr) 2 Ir+1. Then, by (2.5) we have symr pðx0 ; x1 ; . . . ; xr Þ ¼ Dðx0 ; x1 ; . . . ; xr ÞIntr pðtÞ ¼ Dðx0 ; x1 ; . . . ; xr ÞqðtÞ:
L. Verde-Star / Applied Mathematics and Computation 187 (2007) 530–534
533
By Lemma 1 there exists n 2 [mini xi, maxi xi] I such that symr pðx0 ; x1 ; . . . ; xr Þ ¼ qðrÞ ðnÞ=r! ¼ pðnÞ: Therefore ksymr ; pkI rþ1 6 kpkI . Since diagtsymr is the identity map on P, we have symr pðt; t; . . . ; tÞ ¼ diagt symr pðx0 ; x1 ; . . . ; xr Þ ¼ pðtÞ; and this completes the proof of the theorem.
t 2 I;
h
3. Symmetric Chebyshev polynomials In this section we apply the symmetrization map symr to the classical Chebyshev polynomials of the first kind to obtain a family of symmetric polynomials with interesting approximation properties. We follow the notation of Rivlin’s book [12]. See also [9]. The Chebyshev polynomials of the first kind are defined by T n ðtÞ ¼ cosðn arccos hÞ;
n P 0;
where t = cosh. The degree of Tn is n and if n P 1 the leading coefficient of Tn equals 2n1. The normalized monic polynomial 21nTn is denoted by Te n . Note that T 0 ¼ Te 0 1. Let I denote the interval [1, 1]. It is clear that kTnkI = 1 for n P 0. One of the most important properties of the polynomials Tn is the following theorem. Theorem 2. If p is a monic polynomial of degree n P 1 then kpkI P k Te n kI ¼ 21n ; with equality only if p ¼ Te n . For a proof see Rivlin [12, Theorem 2.1]. We will use the following formulas in our development. See [12, p. 5]. 1 T m ðxÞT n ðxÞ ¼ T mþn ðxÞ þ T jmnj ðxÞ ; m; n P 0; 2 Z 1 T nþ1 ðxÞ T n1 ðxÞ T n ðxÞ dx ¼ þ C; n P 2: 2 nþ1 n1
ð3:1Þ ð3:2Þ
Recall that r is a fixed nonnegative integer. We define S n ðx0 ; x1 ; . . . ; xr Þ ¼ symr T n ;
n P 0:
We call Sn the symmetric Chebyshev polynomial of degree n. Note that Sn is a linear combination of the com. All the coefplete symmetric polynomials hk, for k = 0,1, . . ., n, with the coefficient of hn equal to 2n1 = nþr r ficients of Sn are easily found, using the coefficients of Tn. For example, for r = 2 we have 1 S 2 ¼ h2 h0 ; 3
2 S 3 ¼ h3 h1 ; 5
S4 ¼
8 4 h4 h2 þ h0 : 15 3
Note that, by Theorem 1, we have kS n kI rþ1 ¼ kT n kI ¼ 1: Let k = (k1, k2, . . ., kk, 0, 0, . . .) be a partition of length k of a positive integer. Write (3.1) in the form T mþn ðxÞ ¼ 2T m ðxÞT n ðxÞ T mn ðxÞ;
m P n:
ð3:3Þ
By repeated application of this formula we obtain T jkj ¼ 2k1 T k1 T k2 T kk
k X j¼2
2kj T rj T kjþ1 T kjþ2 T kk ;
ð3:4Þ
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where rj = k1 + k2 + + kj1 kj, for 2 6 j 6 k. Note that the first term in the right-hand side of (3.4) has degree equal to jkj, and the summand with index j has degree equal to jkj 2kj. For k = (k1, k2, . . ., kk, 0, 0, . . .) we define the symmetric Chebyshev polynomial Sk, in the variables x0, x1, . . ., xr, by S k ¼ 2k1 S k1 S k2 S kk
k X
2kj S rj S kjþ1 S kjþ2 S kk :
ð3:5Þ
j¼2
Note that Sk is a symmetric polynomial of total degree jkj, and its homogeneous part of degree jkj is 2jkj1
hk1 hk 2 hkk : k1 þ r k2 þ r kk þ r r
r
ð3:6Þ
r
Using induction on the number k of parts of k, some properties of divided differences, and properties of the Chebyshev polynomials, we can show that kS k kI rþ1 ¼ 1 for all partitions k in Par. Since diagtSn = Tn(t) for n P 0, and diagt is linear and multiplicative, using (3.4) and (3.5) we obtain diagt S k ¼ T jkj ðtÞ;
k 2 Par:
ð3:7Þ
Therefore jSk(x0, x1, . . ., xr)j attains its maximum value at points on the diagonal D = {(t, t, . . ., t):t 2 I} of the cube Ir+1. Theorem 3. Let k be a partition of length k of the positive integer n and let P be a symmetric polynomial of degree n = jkj such that its homogeneous part of degree n equals the expression in (3.6). Then kP kI rþ1 P kS k kI rþ1 :
ð3:8Þ
Proof. Let Q(t) = diagtP(x0, x1, . . ., xr). Then, by the hypothesis and the definition of diagt, Q(t) is a univariate polynomial of degree n with leading coefficient equal to 2n1. Therefore, by Theorem 2 we have kQkI P kT n kI ¼ kdiagt S k kI ¼ kS k kI rþ1 ; and hence kP kI rþ1 P kS k kI rþ1 ; and this completes the proof.
h
References [1] W. Bajguz, On generalization of Tchebycheff polynomials, Integ. Transform. Spec. Funct. 9 (2000) 91–98. [2] R.J. Beerends, Chebyshev polynomials in several variables and the radial part of the Laplace–Beltrami operator, Trans. Am. Math. Soc. 328 (1991) 779–814. [3] S.D. Conte, C. de Boor, Elementary Numerical Analysis: An Algorithmic Approach, third ed., McGraw-Hill, New York, 1980. [4] G. Dattoli, Integral transforms and Chebyshev-like polynomials, Appl. Math. Comput. 148 (2004) 225–234. [5] C. de Boor, A Practical Guide to Splines, revised ed., Applied Mathematical Sciences, vol. 27, Springer, New York, 2001. [6] F. Hazama, Twists and generalized Chebyshev polynomials. I and II, JP, J. Algebra Number Theory Appl. 2 (2002) 61–82, 83–100. [7] A. Kroo´, Chebyshev-type extremal problems for multivariate polynomials on convex bodies, East J. Approx. 5 (1999) 211–221. [8] K.H. Kwon, D.W. Lee, Tchebycheff polynomials on a disk, J. Comput. Appl. Math. 162 (2004) 359–363. [9] J.C. Mason, D.C. Handscomb, Chebyshev Polynomials, Chapman & Hall/CRC Press, London/Boca Raton, FL, 2003. [10] D.J. Newman, Yuan Xu, Tchebycheff polynomials on a triangular region, Constr. Approx. 9 (1993) 543–546. [11] S. Re´ve´sz, Uniqueness of multivariate Chebyshev-type extremal polynomials for convex bodies, East J. Approx. 7 (2001) 205–240. [12] T.J. Rivlin, Chebyshev polynomials. From Approximation Theory to Algebra and Number Theory, second ed., Pure and Applied Mathematics, Wiley, New York, 1990. [13] T.J. Rivlin, Some remarks on multivariate Chebyshev polynomials. Approximation theory and applications, Comput. Math. Appl. 33 (1997) 223–228. [14] R.P. Stanley, Enumerative Combinatorics, vol. II, Cambridge University Press, Cambridge, 1999 (Chapter 7). [15] L. Verde-Star, Divided differences and their applications, in: Approximation and Optimization, vol. I, Proceedings of the International Conference on Approximation and Optimization, Cluj-Napoca, Romania, 1996, pp. 121–138.