有限反射不变测度下的调和分析
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摘要
本文主要研究在有限反射群(Coxeter群)下不变测度的调和分析。C．Dunkl自1988年以来的一系列工作开创了研究与反射对称和根系有关的多元特殊函数的有效途径，对数学中的多个领域产生了重要影响，也为调和分析带来了一个新的研究领域。其一，Dunkl理论把欧氏空间R~d上和球面S~(d-1)上关于在正交群下不变的Lebesgue测度的Fourier分析推广到带有在有限反射群(作为正交群的子群)下的不变测度的情况；其二，Dunkl理论又是单变量的Hankel变换、超球展开和Jacobi展开的高维推广；其三，在一些特殊参数下，Dunkl核函数的群不变部分(广义Bessel函数)就是欧几里德型对称空间上的球面函数，Dunkl理论对其有着很大的促进作用。这一方面反映了Dunkl理论的广泛性应用前景，另一方面反映了Dunkl理论具有巨大的研究潜力和可能性。
本文选择Dunkl理论中一些重要的调和分析问题作为研究课题，取得了许多有重要价值的结果，其中包括：
·研究了Dunkl变换的Bochner-Riesz平均B_R~α(f；x)和相应的球形和算子B~α，确定了它们在L~P空间上有界性的指标范围，这个范围对R~d上的径向函数来说是充分必要的。通过建立Dunkl变换的球面(L~p，L~2)限制定理，证明了对L~p(R~d，h_k~2dx)中具有紧支集的函数来说，该指标范围也是充分的。
·研究了Dunkl变换的Poisson积分、共轭Poisson积分以及相应的Riesz变换和一类奇异核函数的Dunkl变换。给出了Poisson积分的极大函数估计，并证明对反射群G=Z_2~d，相应的Poisson积分是几乎处处收敛的；通过给出R~d上关于反射不变测度w_k(x)dx局部可积函数作为缓增广义函数的适当解释，给出一类奇异核函数的Dunkl变换表示；利用Dunkl算子引入Cauchy-Riemann方程组，定义了相应的共轭调和函数系(广义Stein-Weiss系)，由此给出共轭Poisson积分的表述，并研究了其基本性质。在此基础上，引入主值意义下的Riesz变换，并对群G=Z_2~d的情况证明了这样的Riesz变换与共轭Poisson积分的边值在一定意义下是一致的。
·研究了关于反射不变测度的广义Hermite展开的收敛和求和问题，确定了R~d上径向函数的广义Hermite展开的Riesz平均S_R~δ(f；x)按L~p范数一致有界的充要条件。这一结果推广了Thangavelu关于通常多变量Hermite展开的工作。
·借助微分-差分技术建立了一类(一元)Sturm-Liouville算子的测不准原理。作为应用，我们得到了关于Jacobi展开，广义Hermite展开和Laguerre展开的测不准不等式，并分别证明由此得到的这些测不准关系中的常数是最优的。
The purpose of this dissertation is to study the harmonic analysis associated with measures which are invariant under finite reflection groups. This subject began with a series of papers of C. Dunkl, which built up the framework for a theory of special functions and integral transforms in several variables related with finite reflection groups. We select some basic and important problems in harmonic analysis associated with Dunkl's theory, which are
(1) The Bochner-Riesz means of the Dunkl transform;
(2) The Poisson integral and Riesz Transforms of the Dunkl transform;
(3) The Hermite expansions associate with measures invariant under finite reflection groups;
(4) Uncertainty principles for Sturm-Liouville operators.
本文选择Dunkl理论中一些重要的调和分析问题作为研究课题，取得了许多有重要价值的结果，其中包括：
·研究了Dunkl变换的Bochner-Riesz平均B_R~α(f；x)和相应的球形和算子B~α，确定了它们在L~P空间上有界性的指标范围，这个范围对R~d上的径向函数来说是充分必要的。通过建立Dunkl变换的球面(L~p，L~2)限制定理，证明了对L~p(R~d，h_k~2dx)中具有紧支集的函数来说，该指标范围也是充分的。
·研究了Dunkl变换的Poisson积分、共轭Poisson积分以及相应的Riesz变换和一类奇异核函数的Dunkl变换。给出了Poisson积分的极大函数估计，并证明对反射群G=Z_2~d，相应的Poisson积分是几乎处处收敛的；通过给出R~d上关于反射不变测度w_k(x)dx局部可积函数作为缓增广义函数的适当解释，给出一类奇异核函数的Dunkl变换表示；利用Dunkl算子引入Cauchy-Riemann方程组，定义了相应的共轭调和函数系(广义Stein-Weiss系)，由此给出共轭Poisson积分的表述，并研究了其基本性质。在此基础上，引入主值意义下的Riesz变换，并对群G=Z_2~d的情况证明了这样的Riesz变换与共轭Poisson积分的边值在一定意义下是一致的。
·研究了关于反射不变测度的广义Hermite展开的收敛和求和问题，确定了R~d上径向函数的广义Hermite展开的Riesz平均S_R~δ(f；x)按L~p范数一致有界的充要条件。这一结果推广了Thangavelu关于通常多变量Hermite展开的工作。
·借助微分-差分技术建立了一类(一元)Sturm-Liouville算子的测不准原理。作为应用，我们得到了关于Jacobi展开，广义Hermite展开和Laguerre展开的测不准不等式，并分别证明由此得到的这些测不准关系中的常数是最优的。
The purpose of this dissertation is to study the harmonic analysis associated with measures which are invariant under finite reflection groups. This subject began with a series of papers of C. Dunkl, which built up the framework for a theory of special functions and integral transforms in several variables related with finite reflection groups. We select some basic and important problems in harmonic analysis associated with Dunkl's theory, which are
(1) The Bochner-Riesz means of the Dunkl transform;
(2) The Poisson integral and Riesz Transforms of the Dunkl transform;
(3) The Hermite expansions associate with measures invariant under finite reflection groups;
(4) Uncertainty principles for Sturm-Liouville operators.
引文
[Bre] E. Breitenberger, Uncertainty measures and uncertainty relations for angle observables, Found. Phys. 15(1985), 353-364.
[CN] P. Carruthers and M. M. Nieto, Coherent states and the number-phase uncertainty relation, Phys. Rev. Lett. 14(1965), 387-389.
[CS] L. Carleson and P. Sjlin, Oscillatory integrals and multiplier problem for the disc,Stadia Math., 44(1972)3,287-299.
[Chi] T.S. Chihara, "An introduction to orthogonal polynomials," Gordon and Breach, New York, 1978.
[DS] D.L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(1989), 906-931.
[Du1] C.F.Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
[Du2] C.F.Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. Vol.311(1989), 167-183.
[Du3] C.F.Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with Dihedral Symmerty, J. Math. Anal. Appl. 143(1989), 459-470.
[Du4] C.F.Dunkl, Integral kernels with reflection group invariance , Can. J. Math. Vol.43 (1991),1213-1227.
[Du5] C.F. Dunkl, Hankel transforms associated to finite reflection groups, in "Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications (Tampa, 1991)", Contemp. Math. 138(1992),123-138.
[Du6] C.F.Dunkl, Intertwinging operators associated to the group S3, Trans. Amer. Math. Soc. Vol.347(1995), 3347-3374.
[DX] C.F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge Univ. Press, 2001.
[GM] E.Gorlich and C.Markett, A convolution structure for Laguerre series, Indag. Math. 44(1982),161-171.
[He] G.J. Heckman, Dunkl operators (Séminaire Bourbaki 828, 1996-97), Astérisque 245(1997), 223-236.
[Her] C.Herz, On the mean inversion of Fourier and Hankel transforms,Proc. Nat. Acad. Sci. U.S.A. 40(1954),996-999.
[Hu] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, 1990.
[Jeu] M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113(1993), 147-162.
[Ky] P.K.Kythe, Fundamental solutions for differential operators and applications, Birkhuser Boston Basel Berlin 1996.
[Li] Zh.-K. Li, Conjugate Jacobi series and conjugate functions, J.Approx. Theory,86 (1996) 179-196.
[LL1] Zh.-K. Li and L.-M. Liu, An uncertainty principle for Jacobi expansions, J. Math. Anal. Appl., 286 (2003), 652-663.
[LL2] Zh.-K.Li and L.-M.Liu, Harmonic analysis associated with finite reflection groups, in "Proc. of Symposium on Harmonic Analysis" (Tokyo, 2003).
[LL3] Zh.-K.Li and L.-M.Liu, Uncertainty principles for Sturm-Liouville operators, Constructive Approximation, (2004) to appear.
[LL4] L.-M. Liu and Zh.-K. Li, Harmonic analysis on extended Jacobi expansions: An application of Dunkl's theory, in "Analysis, Combinatorics and Computing," Edited by T. He, P. Shiue and Zh. Li, Nova Sci. Publ., Inc., New York, 2002, pp. 319-340.
[LW] 陆善镇、王昆扬，Bochner-Riesz平均，北京师范大学出版社(1988)．
[LX] Zh.-K. Li and Y. Xu, Summability of orthogonal expansions of several variables, J. Approx. Theory 122(2003), 267-333.
[Ma] C. Markett, Mean Cesaro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8(1982),19-37.
[MT] H. Mejjaoli and K. Trimèche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transforms and Special Functions, 12(2001), 279-302.
[Mo] M.A. Mourou, Transmution operators associated with a Dunkl type differential- difference operator on the real line and centain of their applitions, Integral Transforms and Special Functions, 12, No.1 (2001),, 77-88.
[NW] F.J. Narcowich and J. D. Ward, Nonstationary wavelets on the m-sphere for scattered date, Appl. Cornput. Harmonic Anal. 3(1996), 324-336.
[Op] E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Cornpositio Math. 85(1993), 333-373.
[Rose] M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In "Operator Theory: Advances and Applications," Vol.73, Birkhuser Verlag, Basel, 1994, pp.369-396.
[Rs1] M. Rsler, Bessel-type signed hypergroups on R, in "Probability measures on groups and related structures Ⅺ", Edited by H. Heyer and A. Mukherjea, World Scientific, Singapore, 1995, pp. 292-304.
[Rs2] M. Rsler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun. Math. Phys. 192(1998), 519-542.
[Rs3] M. Rsler, Positivity of Dunkl's intertwining operator, Duke Math. J. 98(1999), 445- 463.
[Rs4] M. Rsler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355(2003), 2413-2438.
[Rs5] M. Rsler, Dunkl operators: theory and applications, in "Orthogonal Polynomials and Special Functions," Lecture Notes in Math. 1817, Edited by E. Koelink and W. Van Assche, Springer-Verlag, Berlin, 2003, pp. 93-135.
[RV1] M. Rsler and M. Voit, An uncertainty principle for Ultraspherical expansions, J. Math. Anal. Appl. 209(1997), 624-634.
[RV2] M. Rsler and M. Voit, An uncertainty principle for Hankel transforms, Proc. Amer. Math. Soc. 127(1999), 183-194.
[SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press,1971.
[Str] R.S. Strichartz, Uncertainty principles in harmonic analysis, J. Funct. Anal. 84(1989), 97-114.
[Sz] G.Szeg, Orthogonal Polynomials, Amer.Math.Soc. Colloq. Publ. 23,4th ed., Amer. Math. Soc..Providence. RI. ,1975.
[Th1] S. Thangavelu, Hermite expansions on R~n for radial function, Proc. Amer. Math. Soc, Vol. 118, No. 4, 1993.
[Th2] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Princeton, New Jersey, 1993.
[Tr1] K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms and Special Functions, 12(2001), 349-374.
[Tr2] K. Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms and Special Functions 13(2002), 17-38.
[Xu1] Y.Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups,Proc. Amer. Math. Soc. Vol.125(1997), 2963-2973.
[Xu2] Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. 49(1997), 175-192.
[Xu3] Y.Xu, Intertwining operator and h-harmonic associated with reflection groups, Can. J. Math. Vol.50(1),(1998),193-209.
[Xu4] Y. Xu, Generalized classical orthogonal polynomials on the ball and on the simplex, Constr. Approx. 17(2001), 383-412.
[Xu5] Y. Xu, Almost everywhere convergence of orthogonal expansions of several variables, preprint.
[Xu6] Y. Xu, Weighted approximation of functions on the unit sphere, preprint.
[We] G.V.Welland, Norm convergence of Riesz-Bochner means for radial function, Can. J. Math., 27 (1975) No.1,176-185.
[CN] P. Carruthers and M. M. Nieto, Coherent states and the number-phase uncertainty relation, Phys. Rev. Lett. 14(1965), 387-389.
[CS] L. Carleson and P. Sjlin, Oscillatory integrals and multiplier problem for the disc,Stadia Math., 44(1972)3,287-299.
[Chi] T.S. Chihara, "An introduction to orthogonal polynomials," Gordon and Breach, New York, 1978.
[DS] D.L. Donoho and P. B. Stark, Uncertainty principles and signal recovery, SIAM J. Appl. Math. 49(1989), 906-931.
[Du1] C.F.Dunkl, Reflection groups and orthogonal polynomials on the sphere, Math. Z. 197 (1988), 33-60.
[Du2] C.F.Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. Vol.311(1989), 167-183.
[Du3] C.F.Dunkl, Poisson and Cauchy kernels for orthogonal polynomials with Dihedral Symmerty, J. Math. Anal. Appl. 143(1989), 459-470.
[Du4] C.F.Dunkl, Integral kernels with reflection group invariance , Can. J. Math. Vol.43 (1991),1213-1227.
[Du5] C.F. Dunkl, Hankel transforms associated to finite reflection groups, in "Proc. of the special session on hypergeometric functions on domains of positivity, Jack polynomials and applications (Tampa, 1991)", Contemp. Math. 138(1992),123-138.
[Du6] C.F.Dunkl, Intertwinging operators associated to the group S3, Trans. Amer. Math. Soc. Vol.347(1995), 3347-3374.
[DX] C.F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge Univ. Press, 2001.
[GM] E.Gorlich and C.Markett, A convolution structure for Laguerre series, Indag. Math. 44(1982),161-171.
[He] G.J. Heckman, Dunkl operators (Séminaire Bourbaki 828, 1996-97), Astérisque 245(1997), 223-236.
[Her] C.Herz, On the mean inversion of Fourier and Hankel transforms,Proc. Nat. Acad. Sci. U.S.A. 40(1954),996-999.
[Hu] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Univ. Press, 1990.
[Jeu] M.F.E. de Jeu, The Dunkl transform, Invent. Math. 113(1993), 147-162.
[Ky] P.K.Kythe, Fundamental solutions for differential operators and applications, Birkhuser Boston Basel Berlin 1996.
[Li] Zh.-K. Li, Conjugate Jacobi series and conjugate functions, J.Approx. Theory,86 (1996) 179-196.
[LL1] Zh.-K. Li and L.-M. Liu, An uncertainty principle for Jacobi expansions, J. Math. Anal. Appl., 286 (2003), 652-663.
[LL2] Zh.-K.Li and L.-M.Liu, Harmonic analysis associated with finite reflection groups, in "Proc. of Symposium on Harmonic Analysis" (Tokyo, 2003).
[LL3] Zh.-K.Li and L.-M.Liu, Uncertainty principles for Sturm-Liouville operators, Constructive Approximation, (2004) to appear.
[LL4] L.-M. Liu and Zh.-K. Li, Harmonic analysis on extended Jacobi expansions: An application of Dunkl's theory, in "Analysis, Combinatorics and Computing," Edited by T. He, P. Shiue and Zh. Li, Nova Sci. Publ., Inc., New York, 2002, pp. 319-340.
[LW] 陆善镇、王昆扬，Bochner-Riesz平均，北京师范大学出版社(1988)．
[LX] Zh.-K. Li and Y. Xu, Summability of orthogonal expansions of several variables, J. Approx. Theory 122(2003), 267-333.
[Ma] C. Markett, Mean Cesaro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8(1982),19-37.
[MT] H. Mejjaoli and K. Trimèche, On a mean value property associated with the Dunkl Laplacian operator and applications, Integral Transforms and Special Functions, 12(2001), 279-302.
[Mo] M.A. Mourou, Transmution operators associated with a Dunkl type differential- difference operator on the real line and centain of their applitions, Integral Transforms and Special Functions, 12, No.1 (2001),, 77-88.
[NW] F.J. Narcowich and J. D. Ward, Nonstationary wavelets on the m-sphere for scattered date, Appl. Cornput. Harmonic Anal. 3(1996), 324-336.
[Op] E.M. Opdam, Dunkl operators, Bessel functions and the discriminant of a finite Coxeter group, Cornpositio Math. 85(1993), 333-373.
[Rose] M. Rosenblum, Generalized Hermite polynomials and the Bose-like oscillator calculus, In "Operator Theory: Advances and Applications," Vol.73, Birkhuser Verlag, Basel, 1994, pp.369-396.
[Rs1] M. Rsler, Bessel-type signed hypergroups on R, in "Probability measures on groups and related structures Ⅺ", Edited by H. Heyer and A. Mukherjea, World Scientific, Singapore, 1995, pp. 292-304.
[Rs2] M. Rsler, Generalized Hermite polynomials and the heat equation for Dunkl operators, Commun. Math. Phys. 192(1998), 519-542.
[Rs3] M. Rsler, Positivity of Dunkl's intertwining operator, Duke Math. J. 98(1999), 445- 463.
[Rs4] M. Rsler, A positive radial product formula for the Dunkl kernel, Trans. Amer. Math. Soc. 355(2003), 2413-2438.
[Rs5] M. Rsler, Dunkl operators: theory and applications, in "Orthogonal Polynomials and Special Functions," Lecture Notes in Math. 1817, Edited by E. Koelink and W. Van Assche, Springer-Verlag, Berlin, 2003, pp. 93-135.
[RV1] M. Rsler and M. Voit, An uncertainty principle for Ultraspherical expansions, J. Math. Anal. Appl. 209(1997), 624-634.
[RV2] M. Rsler and M. Voit, An uncertainty principle for Hankel transforms, Proc. Amer. Math. Soc. 127(1999), 183-194.
[SW] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press,1971.
[Str] R.S. Strichartz, Uncertainty principles in harmonic analysis, J. Funct. Anal. 84(1989), 97-114.
[Sz] G.Szeg, Orthogonal Polynomials, Amer.Math.Soc. Colloq. Publ. 23,4th ed., Amer. Math. Soc..Providence. RI. ,1975.
[Th1] S. Thangavelu, Hermite expansions on R~n for radial function, Proc. Amer. Math. Soc, Vol. 118, No. 4, 1993.
[Th2] S. Thangavelu, Lectures on Hermite and Laguerre expansions, Princeton, New Jersey, 1993.
[Tr1] K. Trimèche, The Dunkl intertwining operator on spaces of functions and distributions and integral representation of its dual, Integral Transforms and Special Functions, 12(2001), 349-374.
[Tr2] K. Trimèche, Paley-Wiener theorems for the Dunkl transform and Dunkl translation operators, Integral Transforms and Special Functions 13(2002), 17-38.
[Xu1] Y.Xu, Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups,Proc. Amer. Math. Soc. Vol.125(1997), 2963-2973.
[Xu2] Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Can. J. Math. 49(1997), 175-192.
[Xu3] Y.Xu, Intertwining operator and h-harmonic associated with reflection groups, Can. J. Math. Vol.50(1),(1998),193-209.
[Xu4] Y. Xu, Generalized classical orthogonal polynomials on the ball and on the simplex, Constr. Approx. 17(2001), 383-412.
[Xu5] Y. Xu, Almost everywhere convergence of orthogonal expansions of several variables, preprint.
[Xu6] Y. Xu, Weighted approximation of functions on the unit sphere, preprint.
[We] G.V.Welland, Norm convergence of Riesz-Bochner means for radial function, Can. J. Math., 27 (1975) No.1,176-185.