Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....[D3] Th. 5.1) V k p(x) c k Z 1 Gamma1 f(xt) 1 Gamma t) k Gamma1 (1 t) k dt with c k = Gamma(k 1=2) Gamma(1=2) Gamma(k) 1.2) 2. The direct product case, associated with the reflection group Z N 2 on R N ; here a closed form of the intertwining operator was determined in [X1]. 3. The case of the symmetric group S 3 on R 3 , which has been studied in [D5] In [D3] the intertwining operator V k is, for k 0 , extended to a bounded linear operator on a suitably normed algebra of series of homogeneous polynomials on the unit ball. To allow a more convenient formulation ....

Xu, Y., Orthogonal polynomials for a family of product weight functions on the spheres. Canad. J. Math. 49 (1997), 175--192.

....The term labeled by i in the product has A i = P j i (k j ff j ) N Gammai 2 Gamma 1. By Definition 8 both the Jacobi polynomials in q i have the same indices (A i ; k i ffl i Gamma 1 2 ) and this makes the integral zero. This basis of polynomials on the sphere was also used in Xu[12]. In Theorem 2.1[2] it was shown that the adjoint of T i in H is a scalar multiple of x i Gamma jxj 2 T i , where : N Gamma 4 2 P v2R k v 2 P N i=1 x i x i , which is of course constant on each P n (the scalar multiple is 2 symbolically) Thus S i : x i T i Gamma ....

.... [7] and a hypergroup structure was recently obtained by Koornwinder and Schwartz [10] with the simplex being the domain of orthogonality) The problem of Poisson kernels and Cesaro summability of expansions in fp(ff; x)g for the general case (arbitrary parity of ff i ) was investigated by Xu [12]. This paper has shown how Maple can be used to prove identities in the algebra generated by Dunkl operators, multiplication by coordinate functions and group translation, essentially in polynomial arithmetic. This technique allowed a conceptual proof of a complete orthogonal decomposition of the ....

Xu Y. (1997). Orthogonal polynomials for a family of product weight functions on the spheres; Canadian J. Math. 49, 175-192.

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Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.

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Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

....0, V becomes the identity operator and the right hand is the so called zonal harmonic. However, an explicit formula of V is known only in the case of symmetric group S 3 for three variables and in the case of the abelian group Z 2 . In the latter case, V is an integral operator given by ([8, 22]) 2.5) V f(x) c [ 1;1] d 1 f(x 1 t 1 ; x d 1 t d 1 ) 1 t i ) 1 t i ) i 1 dt; where c denotes the constant c = b 1 : b d 1 and b r = dt. If some i = 0, then the formula holds under the limit relation 0 b f(t) 1 t) dt = f(1) f( ....

Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

....of ordinary harmonics to the h harmonics. The operator V is the unique linear operator de ned by V P n P n ; V 1 = 1; D i V = V i ; 1 i d: It is also proved in [39] that V is a positive operator. The closed form of V is known, however, only in the case of abelian group Z 2 ([14,50]) and the symmetric group S 3 ( 15] For further properties and results of h harmonics, we refer to [11 15] and the references there. Examples of re ection invariant weight functions. The group Z 2 is one of the simplest re ection groups. The weight function invariant under Z 2 is h ; ....

....related to the h harmonics associated to h ; in (2.4) in particular, the classical orthogonal polynomials on B are related to h harmonics associated to h (y) jy d 1 j . Compact formulae of an orthonormal basis of these polynomials on B can be obtained accordingly from formulae in [50]. Moreover, the second order partial di erential equation satis ed by the classical orthogonal polynomials can be derived from the h Laplacian by a simple change of variables ( 59] 3.2 Orthogonal polynomials on balls and on simplices Let W (x) W (x d ) be a weight function de ned on ....

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Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math., 49 (1997), 175-192.

....from the dominant convergence theorem that the desired result holds for f . If = 0, then V = id and the theorem reduces to Theorem A. At the time of this writing, the formula of the intertwining operator V is known only in the case of symmetric group of order 3 ( 7] and the group Z 2 ([6, 15]) In the latter case, we consider the associated weight function h given in (1.1) and we have that the intertwining operator is given by (2.5) V f(x) f(t 1 x 1 ; t d 1 x d 1 ) w i 1=2 (1 t i ) 1 t This formula also holds if some of i = 0, in such a case we take ....

....) 1 t This formula also holds if some of i = 0, in such a case we take the formula under the limit relation (2. 6) lim 0 w 1=2 f(t) 1 t dt = f(1) f( 1) 2: Moreover, in the case of Z 2 , an orthonormal basis for H ) can be given in terms of Jacobi polynomials (cf. [15]) The condition that f is continuous on [ 1; 1] can be relaxed if V has an integral formula. For example, in the case of h in (1.1) we can use (2.5) to show that f integrable is sucient for (2.3) to hold. As an application of the formula (2.3) let us consider the expansion of V f(hx; yi) ....

Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.

....as an integral of the Jacobi polynomial of the same index. There is another product formula of Jacobi polynomials due to Dijksma and Koornwinder [2] which expresses the product as an integral of a Gegenbauer polynomial (see (1) below) This wrong product formula has been used recently in [6] to study the orthogonal polynomials associated to a family of product weight function on the sphere in R d , a situation that serves as an example for Dunkl s theory of h harmonics associated with the re ection groups. The purpose of this note is to prove yet another product formula for Jacobi ....

....D n is the orthonormal polynomial of degree n with respect to the weight function w (x) w ; 1 x jxj 2 ; 1 x 1; 1=2; where w ; is the normalization constant so that the integral of w on [ 1; 1] is 1. In terms of the classical Jacobi polynomials P n we have ([6]) 2n (cos ) c n ( P 2 ; 2n 1 (cos ) c n ( 1) 1=2 xP 2 ; where c n ( is a normalization constant, c n ( 2 ) 2 ) 1) 1=2 (2n ) n ) n 1) n 2 ) n 5 We note that if = 0, then D ( 0) n = ....

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Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math., 49 (1997), 175-192.

....symbol. Furthermore, for 2 N and a; b 2 R, we write a b = a 1 b; a m b) 2.1. Monomial h harmonics. First we recall relevant part of the theory of h harmonics; see [4, 5, 7] and the reference therein. We shall restrict ourself to the case of h de ned in (1. 1) see also [15]. Let H ) denote the space of homogeneous orthogonal polynomials of degree n with respect to h . If all i = 0, then H ) is just the space of the ordinary harmonics. It is known that dim H ) dim P n dim P n 2 = n d 2 The elements of H ) are called ....

.... N 0 and j j = n, Y (x; j j (x j =r j ) Y (x; x d 1 Y ed (x; e d 1 ) in which A ; d 1 1=2) j j (d 1) 2) A ed ; ed 1 and [A ; j j (a j j ) j C j (1) The formulae given above is a reformulation of the basis given in [15] (also [7, p. 198] where the formulae are given in spherical coordinates, which corresponds to x j =r j = cos d 1 j , 1 j d, and are given in terms of the normalized generalized Gegenbauer polynomials e C n (t) 1=hn )C n (t) the normalization constant hn is given by h n = C n ....

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Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

....of ; as they are will become clear in Corollary 2.7 below, so is the reason that h ; is de ned on R instead of R . The explicit formula of the intertwining operator V is not known for the weight function in (2.10) For the weight function (2. 9) however, such a formula is given by [7, 25], V f(x) 1;1] d 1 f(t 1 x 1 ; t d 1 x d 1 ) where c = 1= R 1 dt for 0 and we let d 1 = This compact formula allows us to state the following identity: Corollary 2 7. For the weight function W ; x) C ; where i 0 and 0, de ned on B , ....

....or orthogonal polynomials associated with re ection invariant weight functions, closed formulae for orthonormal bases are not known for weight functions other than those in (2.9) For the weight function in (2. 9) an orthonormal basis can be given in terms of the Jacobi polynomials (see [8, 25]) and the intertwining operator is given by (2.11) The explicit formula allows us to prove: Theorem 3 5. Let W ; x) C ; resp. C(B ) for 1 p 1 [resp. p = 1] Then the expansion of f as the Fourier orthogonal series with respect to W ; is (C; summable in L ) ....

Y. Xu. Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

....of C n ) From (2.19) we can also de ne C n by a generating function ( 28, 2.12) When 0, n reduces to the Gegenbauer polynomials C n . The polynomials C n are orthogonal but not normalized with respect to w . We denote the corresponding orthonormal polynomials by n (In [28] the notation D n is used) These polynomials can be written in terms of Jacobi polynomials; we have 2n (x) d n ( P ; 1) 2.20a) 2n 1 (x) d n ( 1; 1 1=2 xP 2 ; 1) 2.20b) where d n ( 2n ) n ) n ....

....by j (2.24) 1 jx j j j 1 =2 1 jx d 1 j is de ned by (2. 25) A ( d 1=2) d 1) 2) These formulae can be veri ed using the product nature of the integral, or they can be derived from the compact formulae in [28] for the h harmonics associated to Z 2 , making use of the correspondence in Theorem 2.1. In using the formulae in [28] we need to make the substitution = d 1 , j = k j 1 k j and cos d j = x j = We should also point out that the notation in [28] is de ned in the reversed ....

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Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.

....and b r = Z 1 r 1 1 If some i = 0, then the formula holds under the limit relation lim 0 b f(t) 1 t) 1 dt = f(1) f( 1) 2: This leads to an explicit formula for the reproducing kernel; more precisely, for the weight function h in (1. 2) we have the compact formula ([21]) x; y) c n j j (d 2) 2 j j (d 2) 2 (2.4) 1;1] d n (x 1 y 1 t 1 : x d y d t d ) where j j = 1 d . One important property of the intertwining operator is that it is positive ( 16] that is, V p 0 if p 0. We will need another important ....

Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

....are introduced and studied by Dunkl in a number of papers; see [6, 7, 8] and the references in [10] A good reference for re ection groups is [13] The account of the theory of h harmonics given in [10] is self contained. The case of the product weight function in the section 3. 2 is studied in [39], while the monomial basis contained in the subsection 3.2.3 is new [50] Section 4, 5 and 6: The relation between orthogonal polynomials with respect to (1 kxk 2 ) m 1) 2 on B d and spherical harmonics on S d m can be traced back to the work of Hermite, Didon, Appell and Kamp e de ....

Yuan Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canad. J. Math. 49 (1997), 175-192.

No context found.

Xu, Y. (1997). Orthogonal polynomials for a family of product weight functions on the spheres. Can J Math 49, 175-192.

No context found.

Xu, Y. (1997). Orthogonal polynomials for a family of product weight functions on the spheres. Can J Math 49, 175-192.

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