Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....We give two further applications: 1. Summability of orthogonal series in generalized harmonics. The study of generalized spherical harmonics associated with a finite reflection group and a multiplicity function k 0 was one of the starting points of Dunkl s theory in [D3] and has been extended in [X2] and [X3] Many results for classical spherical harmonics carry over to these spherical k harmonics, where harmonizity is now meant with respect to Delta k . In particular, there is a natural decomposition of P N n j S N Gamma1 into subspaces of k spherical harmonics, which are orthogonal ....

....harmonics carry over to these spherical k harmonics, where harmonizity is now meant with respect to Delta k . In particular, there is a natural decomposition of P N n j S N Gamma1 into subspaces of k spherical harmonics, which are orthogonal in L 2 (S N Gamma1 ; w k (x)dx) In [X2], Ces aro summability of generalized Fourier expansions with respect to an orthonormal basis of spherical k harmonics is studied. Recall that a sequence fs n g n2Z is called Ces aro summable of order ffi to s , for short, C; ffi) summable to s , if 1 Gamma n ffi n Delta n X k=0 n ....

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Xu, Y., Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups. Proc. Amer. Math. Soc., to appear.

....of the Fourier transform) are introduced. Most results are known (see [D1, D2, D3, dJ, R3, O1] and presented for the convenience of the reader only. Formally new results are the injectivity of the Dunkl transform of measures and L evy s continuity theorem. Moreover, based on a result of Xu [X], we identify the Dunkl transform of radially symmetric functions on R N in terms of a classical Hankel transform on [0; 1[ Section 3 is devoted to Dunkl s Laplacian, which generates a one parameter semigroup of Markov kernels on R N . This semigroup may be considered as a generalization of ....

....k (x)dx) are again radial, and that Dunkl transforms can be computed via associated classical Hankel transforms. This result is not obvious, as the weight w k is usually invariant only under the reflection group W . Our proof is based on the explicit integration of the operator V over spheres in [X]. Before doing this, we recapitulate some facts about Hankel transforms: 2.7. The Hankel transform. For ff Gamma1=2 , define the measure ff on [0; 1[ by d ff (r) 2 ff Gamma(ff 1) Gamma1 r 2ff 1 dr: The Hankel transform H ff of order ff on L 1 ( 0; 1[ ff ) is then ....

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Xu, Y.: Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups. Proc. Amer. Math. Soc. 125, 2963--2973 (1997).

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Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

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Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....are orthogonal with respect to h d are studied by Dunkl ( 13, 14] see [15] and the references therein) They are called h harmonics, since they satisfy many properties that are similar to those of ordinary harmonics. In particular, summability of h harmonic expansions have been studied in [26, 18, 27] and weighted approximation theory by polynomials in L ) has been developed in [29, 30] For the usual harmonic analysis and approximation on the sphere, see [9, 17, 19, 22] and the references therein. The study in the weighted case often becomes more dicult, since the orthogonal group acts ....

....algebra generated by the partial derivatives and the one generated by Dunkl s operators. This operator, V , is linear and it is determined uniquely by V P n P n ; V 1 = 1; D i V = V i ; 1 i d 1: The compact formula of the reproducing kernel for H ) is given by ([26]) x; y) n V [C n (h ; yi) x) 2.2) where, and throughout this paper, we x the value of as : 2 with = v : 2.3) The function C n (t) is the standard Gegenbauer polynomial, orthogonal with respect to the weight function w (t) 1 t 1=2 ....

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Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....h d are called h harmonics, they are de ned and studied by Dunkl ( 7, 8] see [9] and the references therein) The h harmonics satisfy many properties that are similar to those of ordinary harmonics. In particular, results on summability of the h harmonic expansions have been developed in [23, 13, 27, 29]. While some of the results can be derived using methods similar to those used for ordinary harmonics, others become much more dicult to establish, largely due to the fact that the orthogonal group acts transitively on the sphere S but a re ection group does not. In order to understand the ....

....algebra generated by the partial derivatives and the one generated by Dunkl s operators. The intertwining operator V is a linear operator determined uniquely by V Pn Pn ; V 1 = 1; D i V = V i ; 1 i d 1: The compact formula of the reproducing kernel for H ) is given by ([23]) 2.4) Pn (h ; x; y) V [C n (h ; yi) x) where C n is the usual Gegenbauer polynomial of degree n. If all v = 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 ....

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Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....(f; h ; x) f(y)P n (h ; x; y)h (y)d (y) f P n (h ) x) 4.1) It is not hard to see that the reproducing kernel is independent of the choice of the particular bases. In fact, for h harmonics, the kernel enjoys a compact formula in terms of the intertwining operator [51], n j j 1 (d 1) 2 j j 1 (d 1) 2 V [C (j j 1 (d 1) 2) n (hx; i) y) 4.2) 10 where x; y 2 S and C ( n is the Gegenbauer polynomial of degree n. Here and in the following the reader may want to keep in mind that if h (x) 1, then the h harmonics are just the classical ....

....the limit relation lim 0 c f(t) 1 t dt = f(1) f( 1) 2: In this case, we can write down the reproducing kernel P n (h ; x; y) explicitly. Although a closed form of the intertwining operator is not known in general, its average over the sphere can be computed as shown in [51]. Theorem 4.1 Let h be de ned as in (2.3) associated to a re ection group. Let V be the intertwining operator. Then V f(x)h (x)d = A f(x) 1 jxj j j 1 1 dx (4.4) for f 2 , where A is a constant that can be determined by setting f(x) 1. The equation (4.4) is not ....

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Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc., 125 (1997), 2963-2973.

....operator de ned by V P n P n ; V 1 = 1; D i V = V i ; 1 i d 1: The intertwining operator allows us to draw analogues between the theory of ordinary harmonics and the theory of h harmonics. For example, the reproducing kernel P n ( of the space H ) is given in terms of V as ([6, 16]) n (x; y) n j j 1 (d 1) 2 j j 1 (d 1) 2 V [C (j j 1 (d 1) 2) For = 0, V = id, the h harmonics become the usual harmonic polynomials. For the general theory and many important properties of h harmonics, we refer to [4 7] and the references there. In the following we let ....

....V f(hx; i) is a function of the same type and is closely related to the expansion of f in Gegenbauer polynomials. For = 0, this shows that the expansion of f(hx; i) in the ordinary harmonics is again a function of hx; i. The convergence of the Fourier expansion in h harmonics is discussed in [16]. 3. Funk Hecke formula for orthogonal polynomials on B We follow the study in [17] about the relation between orthogonal polynomials on B and those on S . The discussion in [17] holds for rather general weight functions, we shall restrict to the weight function W ; in (1.2) Let n ....

Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....The operator V is a linear operator uniquely de ned by ( 5] V Pn Pn ; V 1 = 1; D i V = V i ; i = 1; 2; 7) where Pn denotes the class of homogeneous polynomials of degree n. No closed formula of V is known at the moment. A general formula for the reproducing kernel P n of H n states that ([5,7]) n (x; y) Y n (y) Y n (y) n 2 2 2 2 [V C n (h ; yi) x) jyj = jxj = 1; 8) where x = r(cos ; sin ) and y = r(cos ; sin ) In particular, by the de nition of h harmonics in this case, we have 2n (x; y) D n (cos 2 )D sin 2 sin 2 D n 1 (cos 2 )D n ....

Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc., 125 (1997), 2963-2973. 8

....helps us to draw a parallel between the theory of h harmonics and that of ordinary harmonics. The intertwining operator V is the unique linear operator de ned by V Pn Pn ; V 1 = 1; D i V = V i ; 1 i d: For h harmonics, the reproducing kernel Pn (h ) enjoys a compact formula ([7, 26]) Pn (h ; x; y) n (m 2) 2 (m 2) 2 V [C ( m 2) 2) n (hx; i) y) x; y 2 S ; 2 8) where = P m i and C n is the Gegenbauer polynomial of degree n (see [22, p. 80] where the notation P n is used) If = 0, h harmonics become ordinary harmonics, we have V = ....

....group G and a multiplicity function . Let f 2 L ) resp. C(S ) for 1 p 1 [resp. p = 1] Then the expansion of f as the Fourier series with respect to h is (C; summable in L ) with 1 p 1 [resp. p = 1] provided (d 2) 2. Proof. The case p = 1 is proved in [26] under the condition that the intertwining operator V is positive, which was conjectured by Dunkl and proved since then by R osler in [17] The Ces aro (C; means of the Fourier orthogonal series with respect to h are given by f(y)K (y)d d ; where the kernel K ) is the ....

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Y. Xu. Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....of V . Lemma 2.6. The intertwining operator V satisfy the property that V [f( ax) V [f(a( x) where a is a nonzero scalar and f is a polynomial. Using the intertwining operator, the reproducing kernel P n (h ) in (2. 27) can be written in terms of the Gegenbauer polynomials ([29]) n (d 1) 2 (d 1) 2 ( d 1) 2) n (hx; i) y) where x; y 2 S . This formula has been used to study summability of h harmonic expansions in [29] Making use of the connection between orthogonal polynomials on B and h harmonics, we can then derive a compact formula ....

....intertwining operator, the reproducing kernel P n (h ) in (2.27) can be written in terms of the Gegenbauer polynomials ( 29] n (d 1) 2 (d 1) 2 ( d 1) 2) n (hx; i) y) where x; y 2 S . This formula has been used to study summability of h harmonic expansions in [29]. Making use of the connection between orthogonal polynomials on B and h harmonics, we can then derive a compact formula for the reproducing kernel of V n (W ; This kernel is de ned by ; y) where fP ; g is an orthonormal basis of V n (W ; It is easy to see that ....

Y. Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....the kernel functions for h harmonic expansions are no longer in such simple form. However, some results can still be proved as before without dealing with the re ection symmetry, although such results are usually weaker. This is illustrated by the norm convergence of the Ces aro means, studied in [22, 26, 12] and also by a result on almost everywhere convergence of the de la Vall ee Poussin means from a somewhat di erent point of view. In this paper, we will use the de la Vall ee Poussin means of the h harmonic expansions as a motivation. The approximation behavior of these means can be described by ....

....by the partial derivatives and the one generated by Dunkl s operators. The intertwining operator V is a linear operator determined uniquely by V Pn Pn ; V 1 = 1; D i V = V i ; 1 i d: The compact formula of the reproducing kernel to h for any nite re ection group is given by ([22]) n (d 2) 2 (d 2) 2 V [C n (h ; yi) x) 2.1) If all v = 0, V becomes the identity operator and the above formula becomes Pn (x; y) n (d 2) 2 (d 2) 2 n (hx; yi) 2.2) the right hand is the so called zonal harmonic. However, an explicit formula of V is known ....

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Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....the proof of Theorem 5.3 (the inequality) and the proof of Theorem 5.2 (the equal sign) of [9] by taking = 0 there. There is an easier and much more general proof in the framework of weight functions invariant under a re ection group, which essentially comes down to an integration formula ([8]) for the intertwining operator in Dunkl s theory of h harmonics; see [2] Theorem 1.1 follows from the above theorem upon using (2.3) and the estimate of the integral of jP n j in [6, 7.34.1) p. 173] For the case of 0, however, the above theorem no longer holds. In fact, we have jKn (w ....

Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

....for the product Jacobi LECTURE NOTES ON ORTHOGONAL POLYNOMIALS OF SEVERAL VARIABLE 43 polynomials, 47] and [21] for h harmonics expansions and expansions on the unit ball and on the simplex. The integration formula of the intertwining operator and its application to the summability appears in [40]. The topic is still in its initial stage, apart from the problems on the growth rate of the partial sums, many questions such as those on L p and almost everywhere convergence have not been studied. ....

Yuan Xu, Integration of the intertwining operator for h-harmonic polynomials associated to re ection groups, Proc. Amer. Math. Soc. 125 (1997), 2963-2973.

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