Some Problems On Jacobi Series And Laguerre Series

The function theory related to special functions is attached importance to by mathematicians more and more, and becomes a very active research field, in which the classical problems are Jacobi series, Laguerre series, Hermite series, Hankel transforms, and Dunkl transforms associated with reflection-invariant measures on Rⁿ. These problems are not only the generalization of the classical function theory, but also have close relations with the analysis problems on Lie groups and symmetric spaces. But, since the properties of special functions are more complicated then trigonometric polynomials, there would be many difficulties in study of these problems, such as the complexity of the associated translations and the associated convolutions, and in the meantime, some valuable new problems are produced.Conjugacy (Riesz transform or Hilbert transform) is one of the important concepts and tools in analysis. After B. Muckenhoupt and E. Stein studied in 1965 the conjugate ultraspherical series, the problems about congugacy related to special functions, such asconjugate Laguerre series, conjugate Hermite series and conjugate Jacobi series, become important topics in this field, and many results about them were obtained, for examples, Muckenhoupt [1969], [1970], Thangavelu [1990], [1993], Gosselin-Stempak [1994], Li [1996], Nowak [2004], Graczyk [2005].In this thesis, several problems associated to conjugate Jacobi series and conjugate Laguerre series are studied and some important results are achieved, and meanwhile, the Sobolev orthogonality of Laguerre matrix polynomials is studied. The results in the thesis include the following four aspects:? The expansions of functions in terms of the base system conjugate to Jacobi polynomials are studied. In this part, the associated generalized translations (?)_t2 and the generalized difference (?)_t2 are introduced. The Lipschitz functions in L_α,β^p((-1,1)) defined by the generalized translations (?)_t2 are characterized by the action to the associated Poisson integrals of the eigen-differential operator D_α,β of the conjugate base functions, and in the meantime, the Lipschitz functions in L_α,β^p ((—1,1)) defined by the generalized differences (?)_t2 are characterized by the first derivatives of the associated Poisson integrals.? The saturation problems of the (C, 5) means and the Poisson means of Jacobi series are studied, and the saturation classes are characterized by some weighted Lipschitz properties of the conjugate Jacobi series. This is an analogous result of Alexits and Zamansky, and generalizes that about Legendre series, but is different from that due to Bavinck.? By introducing the Laguerre generalized differences _?z and the kernel function G(y), a representation in principle value of conjugate Laguerre expansions is given. In this part, it is proved that the generalized differences (?)_z are bounded in some normed spaces and can be use to characterize the smoothness of functions. By estimating the kernel G(y) in various cases, it is verified that the (Laguerre-) conjugate function of a function f defined in principle value are consistent with that defined by its conjugate Laguerre-Poisson integrals f(x,y), in sense of some norms or pointwisely.· In the last part, by introducing a suitable matrix moment functional with derivatives, the Sobolev orthogonality in terms of this functional of the Laguerre matrix polynomials associated to a parameter matrix with negative and non-integer eigen-values is proved; and, the Laguerre matrix polynomials associated to a diagonalized parameter matrix with some negative integers are defined in an appropriate way, and it is proved that this generalized Laguerre matrix polynomials also have the Sobolev orthogonality in terms of the above functional.