- Email: [email protected]
Chebyshev's maximum principle in several variables
ARTICLE IN PRESS
Journal of Approximation Theory 123 (2003) 276–279 http://www.elsevier.com/locate/jat
Chebyshev’s maximum principle in several variables$ Ren-hong Wang and Zhou Heng Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China Received 14 November 2002; accepted in revised form 20 May 2003 Communicated by Paul Nevaı´
Abstract In this short note, we discuss the Chebyshev’s maximum principle in several variables. We show some analogous maximum formulae for the common zeros in some special cases. It can be regarded as the extension of the univariate case. r 2003 Elsevier Inc. All rights reserved. Keywords: Multivariate orthonormal polynomials; Reproducing kernel
Let o be a nonnegative weight function on ðN; NÞ for which xn oðxÞAL1 ðN; NÞ ðn ¼ 0; 1; yÞ: We construct the sequence of orthonormal polynomials pn ðo; xÞ ¼ gn ðoÞxn þ ? þ ðgn ðoÞ40Þ; satisfying Z N pm ðo; xÞpn ðo; xÞoðxÞ dx ¼ dmn : N
It is well known that all zeros xkn of pn ðoÞ are real and simple. Let us denote by Xn ðoÞ the greatest zero of pn ðoÞ: The sequence fXn ðoÞg is increasing, and, by virtue of a result of Chebyshev, (see [2–5]), RN xp2n1 ðxÞoðxÞ dx RN Xn ðoÞ ¼ max ðn ¼ 1; 2; yÞ; ð1Þ N 2 pn1 AP1n1 ;pn1 a0 N pn1 ðxÞoðxÞ dx $
Supported by the National Natural Science Foundation of China (No. 10271022). Corresponding author. E-mail address: [email protected] (Z. Heng).
0021-9045/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9045(03)00115-1
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
277
where P1n1 denotes the subspace of polynomials whose degree is not larger than n 1: It is the so-called Chebyshev’s maximum principle. The essence of relation (1) lies in the fact that the Gauss cubature formula for one variable always exists. But it is not always true for several variables. In the following, we will discuss the analogous relationship as (1) in the multivariate case under some conditions. For convenience, some basic notations and results for several variables are presented here. Let Pd be the space of all polynomials in d variables, Pdn the subspace of polynomials of total degree not larger than n in d variables. Let L be a positive definite linear functional acting on Pd ; that is, 8pAPd ;
Lðp2 Þ40;
pa0:
Let fPna gjaj¼n be a sequence of orthonormal polynomials with respect to P P k k L; Kn ðx; yÞ ¼ nk¼0 jaj¼k Pa ðxÞPa ðyÞ the nth reproducing kernel. Definition 1 (Dunkl and Yuan Xu [1]). A cubature formula of degree 2n 1 with dim Pdn1 nodes is called a Gaussian cubature. Definition 2. A positive definite linear functional L acting on Pd is called Gaussian, if fPna gjaj¼n has N ¼ dim Pdn1 common zeros. Lemma 1 (Dunkl and Yuan Xu [1]). Let L be a positive definite linear functional. Then L admits a Gaussian cubature formula of degree 2n 1 if and only if fPna gjaj¼n has N ¼ dim Pdn1 common zeros. In what follows, we shall always assume L is Gaussian, and Hn ¼ fXin ¼ are the mutually different common zeros of fPna gjaj¼n :
ðxni1 ; y; xnid ÞgN i¼1
Theorem 1. Let lðx1 ; y; xd Þ be an arbitrary linear function with respect to fxi gdi¼1 : Then maxflðXin Þ;
Xin AHn g ¼
max
pn1 APdn1 ;pn1 a0
LðlðxÞp2n1 ðxÞÞ : Lðp2n1 ðxÞÞ
Proof. Since L is Gaussian, we have
LðPðxÞÞ ¼
N X i¼1
rni PðXin Þ;
rni 40 ði ¼ 1; y; NÞ;
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
278
for all PðxÞAPd2n1 : Therefore, for arbitrary pn1 APdn1 ; LðlðxÞp2n1 ðxÞÞ ¼
N X
rni lðXin Þp2n1 ðXin Þ
i¼1
p maxflðXin Þ; Xin AHn gLðp2n1 ðxÞÞ; that is, maxflðXin Þ; Xin AHn gX
max
pn1 APdn1 ;pn1 a0
LðlðxÞp2n1 ðxÞÞ : Lðp2n1 ðxÞÞ
ð2Þ
On the other hand, let lk ðxÞ ¼
Kn1 ðx; Xkn Þ ; k ¼ 1; y; N; Kn1 ðXkn ; Xkn Þ
which is meaningful by Kn1 ðx; yÞ being always positive. Then, we have degðlk ðxÞÞ ¼ n 1;
lk ðXjn Þ ¼ dk;j ;
1pk; jpN:
Choosing k to be the index which satisfies lðXkn Þ ¼ maxflðXin Þ; Xin AHn g; it is easy to see that maxflðXin Þ; Xin AHn g ¼ lðXkn Þ ¼
LðlðxÞlk2 ðxÞÞ : Lðlk2 ðxÞÞ
ð3Þ
Combining (2) and (3), we can obtain the result immediately.
&
Remark 1. By Theorem 1, it is obvious that the sequence fmaxflðXin Þ; Xin AHn ggN n¼1 is nondecreasing. As some special cases of Theorem 1, we have the following. Corollary 1. maxfxnij ; Xin AHg ¼
max
pn1 APdn1 ;pn1 a0
Corollary 2. ( ) d X n n max xij ; Xi AHn ¼ j¼1
Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
max
pn1 APdn1 ;pn1 a0
Lð
Pd
j ¼ 1; y; d:
xj p2n1 ðxÞÞ : Lðp2n1 ðxÞÞ j¼1
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
Corollary 3.
(
maxfjxnij j; Xin AHn g
¼ max
max
pn1 APdn1 ;pn1 a0
max
pn1 APdn1 ;pn1 a0
Corollary 4. ( ) d X n n jxij j; Xi AHn ¼ max j¼1
Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
) Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
( max
ej ¼71;j¼1;y;d
279
max
pn1 APdn1 ;pn1 a0
Lð
j ¼ 1; 2; y; d:
Pd
j¼1
ej xj p2n1 ðxÞÞ
)
Lðp2n1 ðxÞÞ
Remark 2. It is worth noting that the Gaussian cubature formulae in several variables rarely exist although they indeed exist in some special cases.
References [1] C.F. Dunkl, Yuan Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge, 2001. [2] G. Freud, On the greatest zero of an orthogonal polynomial, I, Acta Sci. Math. (Szeged) 34 (1973) 91–97. [3] G. Freud, On the greatest zero of an orthogonal polynomial, II, Acta Sci. Math. (Szeged) 36 (1974) 49–54. [4] G. Freud, On estimations of the greatest zeros of an orthogonal polynomials, Acta Math. Acad. Sci. Hungar 25 (1974) 99–107. [5] G. Freud, On the greatest zero of an orthogonal polynomial, J. Approx. Theory 46 (1986) 16–24.
:
Journal of Approximation Theory 123 (2003) 276–279 http://www.elsevier.com/locate/jat
Chebyshev’s maximum principle in several variables$ Ren-hong Wang and Zhou Heng Institute of Mathematical Sciences, Dalian University of Technology, Dalian 116024, People’s Republic of China Received 14 November 2002; accepted in revised form 20 May 2003 Communicated by Paul Nevaı´
Abstract In this short note, we discuss the Chebyshev’s maximum principle in several variables. We show some analogous maximum formulae for the common zeros in some special cases. It can be regarded as the extension of the univariate case. r 2003 Elsevier Inc. All rights reserved. Keywords: Multivariate orthonormal polynomials; Reproducing kernel
Let o be a nonnegative weight function on ðN; NÞ for which xn oðxÞAL1 ðN; NÞ ðn ¼ 0; 1; yÞ: We construct the sequence of orthonormal polynomials pn ðo; xÞ ¼ gn ðoÞxn þ ? þ ðgn ðoÞ40Þ; satisfying Z N pm ðo; xÞpn ðo; xÞoðxÞ dx ¼ dmn : N
It is well known that all zeros xkn of pn ðoÞ are real and simple. Let us denote by Xn ðoÞ the greatest zero of pn ðoÞ: The sequence fXn ðoÞg is increasing, and, by virtue of a result of Chebyshev, (see [2–5]), RN xp2n1 ðxÞoðxÞ dx RN Xn ðoÞ ¼ max ðn ¼ 1; 2; yÞ; ð1Þ N 2 pn1 AP1n1 ;pn1 a0 N pn1 ðxÞoðxÞ dx $
Supported by the National Natural Science Foundation of China (No. 10271022). Corresponding author. E-mail address: [email protected] (Z. Heng).
0021-9045/03/$ - see front matter r 2003 Elsevier Inc. All rights reserved. doi:10.1016/S0021-9045(03)00115-1
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
277
where P1n1 denotes the subspace of polynomials whose degree is not larger than n 1: It is the so-called Chebyshev’s maximum principle. The essence of relation (1) lies in the fact that the Gauss cubature formula for one variable always exists. But it is not always true for several variables. In the following, we will discuss the analogous relationship as (1) in the multivariate case under some conditions. For convenience, some basic notations and results for several variables are presented here. Let Pd be the space of all polynomials in d variables, Pdn the subspace of polynomials of total degree not larger than n in d variables. Let L be a positive definite linear functional acting on Pd ; that is, 8pAPd ;
Lðp2 Þ40;
pa0:
Let fPna gjaj¼n be a sequence of orthonormal polynomials with respect to P P k k L; Kn ðx; yÞ ¼ nk¼0 jaj¼k Pa ðxÞPa ðyÞ the nth reproducing kernel. Definition 1 (Dunkl and Yuan Xu [1]). A cubature formula of degree 2n 1 with dim Pdn1 nodes is called a Gaussian cubature. Definition 2. A positive definite linear functional L acting on Pd is called Gaussian, if fPna gjaj¼n has N ¼ dim Pdn1 common zeros. Lemma 1 (Dunkl and Yuan Xu [1]). Let L be a positive definite linear functional. Then L admits a Gaussian cubature formula of degree 2n 1 if and only if fPna gjaj¼n has N ¼ dim Pdn1 common zeros. In what follows, we shall always assume L is Gaussian, and Hn ¼ fXin ¼ are the mutually different common zeros of fPna gjaj¼n :
ðxni1 ; y; xnid ÞgN i¼1
Theorem 1. Let lðx1 ; y; xd Þ be an arbitrary linear function with respect to fxi gdi¼1 : Then maxflðXin Þ;
Xin AHn g ¼
max
pn1 APdn1 ;pn1 a0
LðlðxÞp2n1 ðxÞÞ : Lðp2n1 ðxÞÞ
Proof. Since L is Gaussian, we have
LðPðxÞÞ ¼
N X i¼1
rni PðXin Þ;
rni 40 ði ¼ 1; y; NÞ;
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
278
for all PðxÞAPd2n1 : Therefore, for arbitrary pn1 APdn1 ; LðlðxÞp2n1 ðxÞÞ ¼
N X
rni lðXin Þp2n1 ðXin Þ
i¼1
p maxflðXin Þ; Xin AHn gLðp2n1 ðxÞÞ; that is, maxflðXin Þ; Xin AHn gX
max
pn1 APdn1 ;pn1 a0
LðlðxÞp2n1 ðxÞÞ : Lðp2n1 ðxÞÞ
ð2Þ
On the other hand, let lk ðxÞ ¼
Kn1 ðx; Xkn Þ ; k ¼ 1; y; N; Kn1 ðXkn ; Xkn Þ
which is meaningful by Kn1 ðx; yÞ being always positive. Then, we have degðlk ðxÞÞ ¼ n 1;
lk ðXjn Þ ¼ dk;j ;
1pk; jpN:
Choosing k to be the index which satisfies lðXkn Þ ¼ maxflðXin Þ; Xin AHn g; it is easy to see that maxflðXin Þ; Xin AHn g ¼ lðXkn Þ ¼
LðlðxÞlk2 ðxÞÞ : Lðlk2 ðxÞÞ
ð3Þ
Combining (2) and (3), we can obtain the result immediately.
&
Remark 1. By Theorem 1, it is obvious that the sequence fmaxflðXin Þ; Xin AHn ggN n¼1 is nondecreasing. As some special cases of Theorem 1, we have the following. Corollary 1. maxfxnij ; Xin AHg ¼
max
pn1 APdn1 ;pn1 a0
Corollary 2. ( ) d X n n max xij ; Xi AHn ¼ j¼1
Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
max
pn1 APdn1 ;pn1 a0
Lð
Pd
j ¼ 1; y; d:
xj p2n1 ðxÞÞ : Lðp2n1 ðxÞÞ j¼1
ARTICLE IN PRESS R.-h. Wang, Z. Heng / Journal of Approximation Theory 123 (2003) 276–279
Corollary 3.
(
maxfjxnij j; Xin AHn g
¼ max
max
pn1 APdn1 ;pn1 a0
max
pn1 APdn1 ;pn1 a0
Corollary 4. ( ) d X n n jxij j; Xi AHn ¼ max j¼1
Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
) Lðxj p2n1 ðxÞÞ ; Lðp2n1 ðxÞÞ
( max
ej ¼71;j¼1;y;d
279
max
pn1 APdn1 ;pn1 a0
Lð
j ¼ 1; 2; y; d:
Pd
j¼1
ej xj p2n1 ðxÞÞ
)
Lðp2n1 ðxÞÞ
Remark 2. It is worth noting that the Gaussian cubature formulae in several variables rarely exist although they indeed exist in some special cases.
References [1] C.F. Dunkl, Yuan Xu, Orthogonal Polynomials of Several Variables, Cambridge University Press, Cambridge, 2001. [2] G. Freud, On the greatest zero of an orthogonal polynomial, I, Acta Sci. Math. (Szeged) 34 (1973) 91–97. [3] G. Freud, On the greatest zero of an orthogonal polynomial, II, Acta Sci. Math. (Szeged) 36 (1974) 49–54. [4] G. Freud, On estimations of the greatest zeros of an orthogonal polynomials, Acta Math. Acad. Sci. Hungar 25 (1974) 99–107. [5] G. Freud, On the greatest zero of an orthogonal polynomial, J. Approx. Theory 46 (1986) 16–24.
: