the multiplier of the interval 1 1 for the dunkl transform on the real line abstract we study the boundedness of the multiplier of the interval 1 1 for the dunkl transform of order 1 2 on weighted l p spaces with 1 p in particular we get that it is bounded from l p r x 2 1 d x into itself if and only if 4 1 2 3 p 4 1 2 1

yang s system of particles and hecke algebras the graded hecke algebra has a simple realization as a certain algebra of operators acting on a space of smooth functions this operator algebra arises from the study of the root system analogue of yang s system of n particles on the real line with delta function potential it turns out that the spectral problem for this generalization of yang s system is related to the problem of finding the spherical tempered representations of the graded hecke algebra this observation turns out to be very useful for both these problems application of our technique to affine hecke algebras yields a simple formula for the formal degree of the generic iwahori spherical discrete series representations

bessel functions of two variables some power series and plots for a finite reflection group g there is a rich theory developed by dunkl heckman and opdam leading to the notion of a commuting set of bessel differential operators these systems play an important role in the study of calogero moser systems and other problems of physical interest when g acts on the line one recovers the usual bessel function with a well known power series expansion at the origin we obtain some such expansions in the case of g a2 acting in the plane and we use these to produce plots of some of these functions

two new families of sixth order methods for solving non linear equations in this paper we developed two new families of sixth order methods for solving simple roots of non linear equations per iteration these methods require two evaluations of the function and two evaluations of the first order derivatives which implies that the efficiency indexes of our methods are 1 565 these methods have more advantages than newton s method and other methods with the same convergence order as shown in the illustration examples finally using the developing methodology described in this paper two new families of improvements of jarratt method with sixth order convergence are derived in a straightforward manner notice that kou s method in jisheng kou yitian li an improvement of the jarratt method appl math comput 189 2007 1816 1821 and wang s method in xiuhua wang jisheng kou yitian li a variant of jarratt method with sixth order convergence appl math comput 204 2008 14 19 are the special cases of the new improvements

fixed points koebe uniformization and circle packings a domain in the riemann sphere hat mathbb c is called a circle domain if every connected component of its boundary is either a circle or a point in 1908 p koebe ko1 posed the following conjecture known as koebe s kreisnormierungsproblem any plane domain is conformally homeomorphic to a circle domain in hat mathbb c when the domain is simply connected this is the content of the riemann mapping theorem

a new integral formula for heckman opdam hypergeometric functions abstract we provide harish chandra type formulas for the multivariate bessel functions and heckman opdam hypergeometric functions as representation valued integrals over dressing orbits our expression is the quasi classical limit of the realization of macdonald polynomials as traces of intertwiners of quantum groups given by etingof and kirillov jr in 10 integration over the liouville tori of the gelfand tsetlin integrable system and adjunction for higher calogero moser hamiltonians recovers and gives a new proof of the integral realization over gelfand tsetlin polytopes which appeared in the recent work 4 of borodin and gorin on the jacobi corners ensemble

intertwining operators associated to the group for any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values containing only certain negative rational numbers this paper constructs an integral transform implementing the intertwining operator for the group s3 the symmetric group on three objects for parameter value the transform is realized as an absolutely continuous measure on a compact subset of m2 r which contains the group as a subset of its boundary the construction of the integral formula involves integration over the unitary group u 3 associated to any finite reflection group g on an euclidean space there is a parametrized commutative algebra of differential difference operators with as many parameters as there are conjugacy classes of reflections in g it has been shown that there exists a linear isomorphism on polynomials which intertwines this algebra with the algebra of partial differential operators with constant coefficients for all but a singular set of parameter values this singular set contains only negative values and is closely linked to the zero set of the poincare series of g this paper constructs an integral transform implementing the intertwining map for the group s3 the symmetric group on three objects for positive parameter values previously this had been done only for the group z2 acting by sign change on r where the transform is a classical fractional integral the transform in this paper has its origin in the adjoint action of the unitary group u 3 on the linear space of real diagonal 3 x 3 matrices the complexification of the maximal torus this will lead to a transform realized as an absolutely continuous measure on a certain compact subset of m2 r here is a concise statement of the main result rephrased from formulas 5 1 5 6 the operator v intertwines the differential difference operators received by the editors august 22 1994 1991 mathematics subject classification primary 22e30 33c80 secondary 33c50 20b30

one dimensional double hecke algebra and gaussian sums we introduce and describe one dimensional cyclotomic gauss selberg sums generalizing the classical gaussian sums they correspond to irreducible self dual unitary spherical representations of the one dimensional double affine hecke algebra

interpolation by weighted paley wiener spaces associated with the dunkl transform abstract given 1 2 0 and 1 p we study the interpolation problem in the space pw p of entire functions f c c of exponential type for which r f x p x 2 1 d x with nodes of interpolation at s j j z where s j j n is the increasing sequence of all positive roots of the bessel function j 1 z of order 1 and s j s j for all j z we prove that if 4 1 2 3 p 1 p p 2 4 1 2 1 the interpolation problem f pw p and f 1 s j c j for all j z has a unique solution for every sequence c j of complex numbers satisfying j z c j p 1 j 2 1 and that if p p 1 the corresponding interpolation problem may not have a solution and that the solution if exists is unique if and only if p p 2 finally we show that r f x p x 2 1 d x 2 2 j z f s j 1 p 1 j 2 1 with the constant of equivalence depending only on p and holds for all entire functions f of exponential type if and only if p 1 p p 2

paley wiener theorems for the dunkl transform we conjecture a geometrical form of the paley wiener theorem for the dunkl transform and prove three instances thereof by using a reduction to the one dimensional even case shift operators and a limit transition from opdam s results for the graded hecke algebra respectively these paley wiener theorems are used to extend dunkl s intertwining operator to arbitrary smooth functions furthermore the connection between dunkl operators and the cartan mo tion group is established it is shown how the algebra of radial parts of in variant differential operators can be described explicitly in terms of dunkl operators this description implies that the generalized bessel functions coin cide with the spherical functions in this context of the cartan motion group the restriction of dunkl s intertwining operator to the invariants can be inter preted in terms of the abel transform we also show that for certain values of the multiplicities of the restricted roots the abel transform is essentially inverted by a differential operator 1 introduction and overview in recent times the study of special functions associated with root systems has developed to a considerable degree starting with a number of conjectures by mac donald and the work of heckman and opdam on multivariable hypergeometric functions in the late 1980 s the development of the theory was greatly enhanced by the introduction of rational dunkl operators by dunkl 6 through various intermediate steps of generalization these operators can even be said to have ulti mately provided crucial building blocks for cherednik s work on double affine hecke algebras and the q macdonald conjectures originally before the introduction of dunkl operators the idea when studying special functions related to root systems was to consider root multiplicities in the theory of spherical functions on lie groups as parameters and then to develop a theory for weyl group invariant objects in this more general situation without the aid of the presence of the group it was this point of view which underlay the macdonald conjectures and which led heckman and opdam to the development of their theory of hypergeometric functions in higher dimension one of the main technical problems in this context is the description of the generalized radial parts of invariant differential operators apart from an explicit formula for the generalized radial part of the laplacian an expression which was in fact the starting point for heckman and opdam the other operators remain somewhat intangible

riesz distributions and laplace transform in the dunkl setting of type a abstract we study riesz distributions in the framework of rational dunkl theory associated with root systems of type a as an important tool we employ a laplace transform involving the associated dunkl kernel which essentially goes back to macdonald 18 but was so far only established at a formal level we give a rigorous treatment of this transform based on suitable estimates of the type a dunkl kernel our main result is a precise analogue in the dunkl setting of a well known result by gindikin stating that a riesz distribution on a symmetric cone is a positive measure if and only if its exponent is contained in the wallach set for riesz distributions in the dunkl setting we obtain an analogous characterization in terms of a generalized wallach set which depends on the multiplicity parameter on the root system

strichartz estimates for the dunkl wave equation and application in this paper we prove dispersive and strichartz estimates associated for the dunkl wave equation

segal bargmann transforms associated with finite coxeter groups using a polarization of a suitable restriction map and heat kernel analysis we construct a generalized segal bargmann transform associated with every finite coxeter group g on n we find the integral representation of this transform and we prove its unitarity to define the segal bargmann transform we introduce a hilbert space open image in new window of holomorphic functions on open image in new window with reproducing kernel equal to the dunkl kernel the definition and properties of open image in new window extend naturally those of the well known classical fock space the generalized segal bargmann transform allows to exhibit some relationships between the dunkl theory in the schrodinger model and in the fock model further we prove a branching decomposition of open image in new window as a unitary open image in new window module and a general version of hecke s formula for the dunkl transform

riesz transforms for dunkl hermite expansions abstract in the present paper we establish that riesz transforms for dunkl hermite expansions introduced by nowak and stempak are singular integral operators with hormander s type condition we prove that they are bounded on l p r d d for 1 p and from l 1 r d d into l 1 r d d

dunkl operators and a family of realizations of osp 1 2 in this paper a family of radial deformations of the realization of the lie superalgebra osp 1 vertical bar 2 in the theory of dunkl operators is obtained this leads to a dirac operator depending on 3 parameters several function theoretical aspects of this operator are studied such as the associated measure the related laguerre polynomials and the related fourier transform for special values of the parameters it is possible to construct the kernel of the fourier transform explicitly as well as the related intertwining operator

mixed norm estimates for the cesaro means associated with dunkl hermite expansions our main goal in this article is to study mixed norm estimates for the cesaro means associated with dunkl hermite expansions on r d these expansions arise when one considers the dunkl hermite operator or dunkl harmonic oscillator h kappa delta kappa vertical bar x vertical bar 2 where delta kappa stands for the dunkl laplacian it is shown that the desired mixed norm estimates are equivalent to vector valued inequalities for a sequence of cesaro means for laguerre expansions with shifted parameter in order to obtain such vector valued inequalities we develop an argument to extend these laguerre operators for complex values of the parameters involved and apply a version of the three lines lemma

besov type spaces for the dunkl operator on the real line in this paper we shall introduce functions spaces as subspaces of lp that we call besov dunkl spaces we provide characterizations of these spaces involving bochner riesz means associated to the dunkl operator and partial dunkl integrals

global well posedness and scattering for a class of nonlinear dunkl schrodinger equations in this paper we introduce a class of nonlinear schrodinger equations associated with the dunkl operators in this regard we study local and global well posedness and the scattering theory associated with these equations

on the integrability of a representation of sl 2 r abstract the dunkl operators involve a multiplicity function k as parameter c f dunkl differential difference operators associated to reflection groups trans amer math soc 311 1989 167 183 for positive real values of this function we consider on the schwartz space s r n a representation k of sl 2 r defined in terms of the dunkl laplacian operator by means of a beautiful theorem due to e nelson we prove that k exponentiates to a unique unitary representation of the universal covering group of sl 2 r the representation theory is used to derive an identity of bochner type for the dunkl transform

nonlinear generalized dunkl wave equations and applications abstract in this paper we introduce a class of nonlinear wave equations associated with the generalized dunkl laplace operator we study local and global well posedness next we establish the linearization of bounded energy solutions in the spirit of gerard 1996 9 the proof uses strichartz type inequalities and the energy estimate

lp fourier multipliers for the dunkl operator on the real line we consider fourier multipliers for lp associated with the dunkl operator on r and establish a version of hormander s multiplier theorem in applying this version we come up with some results regarding the oscillating multipliers partial sum operators and generalized bessel potentials

fefferman stein inequalities for the mathbb z _2 d dunkl maximal operator in this article we establish the fefferman stein inequalities for the dunkl maximal operator associated with a finite reflection group generated by the sign changes similar results are also given for a large class of operators related to dunkl s analysis

radial mollifiers mean value operators and harmonic functions in dunkl theory abstract in this paper we show how to use mollifiers to regularize functions relative to a set of dunkl operators in r d with coxeter weyl group w multiplicity function k and weight function k in particular for a w invariant open subset of r d for d r d a radial function and u l loc 1 k x d x we study the dunkl convolution product u k and the action of the dunkl laplacian and the volume mean operators on these functions the results are then applied to obtain an analog of the weyl lemma for dunkl harmonic functions and to characterize them by invariance properties relative to mean value and convolution operators

explicit formulas for the dunkl dihedral kernel and the κ a generalized fourier kernel abstract in this paper a new method is developed to obtain explicit and integral expressions for the kernel of the a generalized fourier transform for 0 in the case of dihedral groups this method is also applied to the dunkl kernel as well as the dunkl bessel function the method uses the introduction of an auxiliary variable in the series expansion of the kernel which is subsequently laplace transformed the kernel in the laplace domain takes on a much simpler form by making use of the poisson kernel the inverse laplace transform can then be computed using the generalized mittag leffler function to obtain integral expressions in case the parameters involved are integers explicit formulas are obtained using partial fraction decomposition new bounds for the kernel of the a generalized fourier transform are obtained as well

mean convergence of fourier dunkl series abstract in the context of the dunkl transform a complete orthogonal system arises in a very natural way this paper studies the weighted norm convergence of the fourier series expansion associated to this system we establish conditions on the weights in terms of the a p classes of muckenhoupt which ensure the convergence necessary conditions are also proved which for a wide class of weights coincide with the sufficient conditions

spherical dunkl monogenics and a factorization of the dunkl laplacian in this paper we consider and study a factorization of the dunkl laplacian in terms of spherical coordinates this allows for the construction of a direct sum decomposition of spherical dunkl harmonics by explicit representation in spherical coordinates of dunkl harmonics one obtains explicit projection operators from dunkl harmonics to inner resp outer dunkl monogenics concrete examples of spherical dunkl monogenics will be given at the end

riesz transform and riesz potentials for dunkl transform analogues of riesz potentials and riesz transforms are defined and studied for the dunkl transform associated with a family of weight functions that are invariant under a reflection group the lp boundedness of these operators is established in certain cases

hormander s multiplier theorem for the dunkl transform abstract for a normalized root system r in r n and a multiplicity function k 0 let n n r k denote by d w x r x k d x the associated measure in r n let f stand for the dunkl transform given a bounded function m on r n we prove that if there is s n such that m satisfies the classical hormander condition with the smoothness s then the multiplier operator t m f f 1 m f f is of weak type 1 1 strong type p p for 1 p and is bounded on a relevant hardy space h 1 to this end we study the dunkl translations and the dunkl convolution operators and prove that if f is sufficiently regular for example its certain schwartz class seminorm is finite then the dunkl convolution operator with the function f is bounded on l p d w for 1 p we also consider boundedness of maximal operators associated with the dunkl convolutions with schwartz class functions

multivariable continuous hahn and wilson polynomials related to integrable difference systems multivariable generalizations of the continuous hahn and wilson polynomials are introduced as eigenfunctions of rational ruijsenaars type difference systems with an external field

on fundamental harmonic analysis operators in certain dunkl and bessel settings we consider several harmonic analysis operators in the multi dimensional context of the dunkl laplacian with the underlying group of reflections isomorphic to z2n also negative values of the multiplicity function are admitted our investigations include maximal operators g functions lusin area integrals riesz transforms and multipliers of laplace and laplace stieltjes transform type using the general calderon zygmund theory we prove that these objects are bounded in weighted lp spaces 1 p and from l1 into weak l1

translation operator and maximal function for the k 1 generalized fourier transform abstract in this paper we study a translation operator associated with the n dimensional k 1 generalized fourier transform where k is a multiplicity function for the dunkl operators in particular we prove that the translation is a positivity preserving operator acting on a suitable space of radial functions on r n we then use it to define a hardy littlewood type maximal operator where weak type 1 1 and strong type p p estimates for the maximal operator are established with a precise behavior in n and k

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

riesz transforms and fractional integration for orthogonal expansions on spheres balls and simplices abstract this paper studies the hardy littlewood sobolev hls inequality and the riesz transforms for fractional integration associated to weighted orthogonal polynomial expansions on spheres balls and simplexes with weights being invariant under a general finite reflection group on r d the sharp index for the validity of the hls inequality is determined and the l p boundedness of the riesz transforms is established in particular our results extend a classical inequality of muckenhoupt and stein on conjugate ultraspherical polynomial expansions our idea is based on a new decomposition of the dunkl laplace beltrami operator on the sphere and some sharp asymptotic estimates of the weighted christoffel functions

generalized hermite polynomials and the heat equation for dunkl operators based on the theory of dunkl operators this paper presents a general concept of multivariable hermite polynomials and hermite functions which are associated with finite reflection groups on n the definition and properties of these generalized hermite systems extend naturally those of their classical counterparts partial derivatives and the usual exponential kernel are here replaced by dunkl operators and the generalized exponential kernel k of the dunkl transform in the case of the symmetric group s n our setting includes the polynomial eigenfunctions of certain calogero sutherland type operators the second part of this paper is devoted to the heat equation associated with dunkl s laplacian as in the classical case the corresponding cauchy problem is governed by a positive one parameter semigroup this is assured by a maximum principle for the generalized laplacian the explicit solution to the cauchy problem involves again the kernel k which is on the way proven to be nonnegative for real arguments

littlewood paley operators associated with the dunkl operator on r we consider a vector version of l p multipliers for the dunkl transform on r and we prove l p inequalities for the littlewood paley operators

on the dunkl intertwining operator abstract dunkl operators are differential difference operators parametrized by a finite reflection group and a weight function the commutative algebra generated by these operators generalizes the algebra of standard differential operators and intertwines with this latter by the so called intertwining operator in this paper we give an integral representation for the operator v k e 2 for an arbitrary weyl group and a large class of regular weights k containing those of nonnegative real parts our representing measures are absolutely continuous with respect the lebesgue measure in r d which allows us to derive out new results about the intertwining operator v k and the dunkl kernel e k we show in particular that the operator v k e 2 extends uniquely as a bounded operator to a large class of functions which are not necessarily differentiables in the case of nonnegative weights this operator is shown to be positivity preserving

fundamental solutions for partial differential equations with reflection group invariance the dunkl differential difference operator associated with a finite reflection group is used to extend the weyl heisenberg algebra the subalgebra of mastersymmetries graded by the characters of the coxeter group w is constructed it provides us with a new class of intertwining operators for appropriate w invariant differential equations these intertwining operators turn out to be suitable for constructing fundamental solutions the remarkable example is given by the iterated wave operator with a calogero type potential for which the fundamental solution can be derived through the relevant intertwining operator in an explicit form it is found that under appropriate conditions such an equation satisfies huygens s principle in the sense that its fundamental solution possesses a nontrivial inner lacuna

a positive radial product formula for the dunkl kernel it is an open conjecture that generalized bessel functions associated with root systems have a positive product formula for nonnegative multiplicity parameters of the associated dunkl operators in this paper a partial result towards this conjecture is proven namely a positive radial product formula for the non symmetric counterpart of the generalized bessel function the dunkl kernel radial here means that one of the factors in the product formula is replaced by its mean over a sphere the key to this product formula is a positivity result for the dunkl type spherical mean operator it can also be interpreted in the sense that the dunkl type generalized translation of radial functions is positivity preserving as an application we construct dunkl type homogeneous markov processes associated with radial probability distributions

a variant of cowling price s theorem for the dunkl transform on r abstract in this paper we prove a new version of cowling price s theorem for the dunkl transform on r

biorthogonal polynomials associated with reflection groups and a formula of macdonald abstract dunkl operators are differential difference operators on r n which generalize partial derivatives they lead to generalizations of laplace operators fourier transforms heat semigroups hermite polynomials and so on in this paper we introduce two systems of biorthogonal polynomials with respect to dunkl s gaussian distributions in a canonical way these systems called appell systems admit many properties known from classical hermite polynomials and turn out to be useful for the analysis of dunkl s gaussian distributions in particular these polynomials lead to a new proof of a generalized formula of macdonald due to dunkl the ideas for this paper are taken from recent works on non gaussian white noise analysis and from the umbral calculus

the heckman opdam markov processes we introduce and study the natural counterpart of the dunkl markov processes in a negatively curved setting we give a semimartingale decomposition of the radial part and some properties of the jumps we prove also a law of large numbers a central limit theorem and the convergence of the normalized process to the dunkl process eventually we describe the asymptotic behavior of the infinite loop as it was done by anker bougerol and jeulin in the symmetric spaces setting in iberoamericana 18 41 97 2002