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

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

stability of error bounds for semi infinite convex constraint systems in this paper we are concerned with the stability of the error bounds for semi infinite convex constraint systems roughly speaking the error bound of a system of inequalities is said to be stable if all its small perturbations admit a local or global error bound we first establish subdifferential characterizations of the stability of error bounds for semi infinite systems of convex inequalities by applying these characterizations we extend some results established by aze and corvellec siam j optim 12 2002 pp 913 927 on the sensitivity analysis of hoffman constants to semi infinite linear constraint systems

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

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

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

h sub spl infin differential riccati equations convergence properties and finite escape phenomena the paper studies the transient and asymptotic behavior of the solution of the sign indefinite differential riccati equation dre arising in finite horizon h sub spl infin filtering and control problems differently from the sign definite h sub spl infin dre the solution of the h sub spl infin dre can have finite escape times even for nonnegative initial conditions sufficient and necessary conditions for boundedness and convergence are derived in correspondence to a fixed value of the h sub spl infin norm attenuation level spl gamma finally a stepwise spl gamma switching strategy is devised to guarantee boundedness as well as asymptotic performance

optimal input design for autoregressive model discrimination with constrained output variance the optimal input design problem is considered for discriminating autoregressive models when the output variance is constrained it is shown that a ds optimal input design can be achieved with an input comprised of a finite number of sinusoidal components with frequencies and amplitudes determined by solving a system of nonlinear equations digital simulation studies illustrate the usefulness of the ds optimal input for autoregressive model discrimination

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

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

stability of error bounds for convex constraint systems in banach spaces this paper studies stability of error bounds for convex constraint systems in banach spaces we show that certain known sufficient conditions for local and global error bounds actually ensure error bounds for the family of functions being in a sense small perturbations of the given one a single inequality as well as semi infinite constraint systems are considered

global error bounds for convex inequality systems in banach spaces we study conditions under which a global error bound in terms of a natural residual exists for a convex inequality system specifically we obtain an error bound result which unifies many existing results assuming a slater condition we also derive two characterizations for a convex inequality system to possess a global error bound one is in terms of metric regularity and the other is in terms of an associated convex inequality system as a consequence we show that in font bbbabstract msbm10 at 8pt mbox bbbabstract r n a global error bound holds for such a system under the assumption of the zero vector in the relative interior of the domain of an associated conjugate function along with metric regularity at every point of the feasible set defined by the system finally we discuss some applications of these results to convex programs

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

regularities and their relations to error bounds in this paper we mainly study various notions of regularity for a finite collection c1 cm of closed convex subsets of a banach space x and their relations with other fundamental concepts we show that a proper lower semicontinuous function f on x has a lipschitz error bound resp error bound if and only if the pair epi f x 0 of sets in the product space x is linearly regular resp regular similar results for multifunctions are also established next we prove that c1 cm is linearly regular if and only if it has the strong chip and the collection nc1 z ncm z of normal cones at z has property g for each z c i 1mci provided that c1 is a closed convex cone and that c2 y is a closed vector subspace of x we show that c1 y is linearly regular if and only if there exists 0 such that each positive relative to the order induced by c1 linear functional on y of norm one can be extended to a positive linear functional on x with norm bounded by similar characterization is given in terms of normal cones

dunkl kernel associated with dihedral groups abstract in this paper we pursue the investigations started in 18 where the authors provide a construction of the dunkl intertwining operator for a large subset of the set of regular multiplicity values more precisely we make concrete the action of this operator on homogeneous polynomials when the root system is of dihedral type and under a mild assumption on the multiplicity function in particular we obtain a formula for the corresponding dunkl kernel and another representation of the generalized bessel function already derived in 7 when the multiplicity function is everywhere constant our computations give a solution to the problem of counting the number of factorizations of an element from a dihedral group into a fixed number of non necessarily simple reflections in the remainder of the paper we supply another method to derive the dunkl kernel associated with dihedral systems from the corresponding generalized bessel function this time we use the shift principle together with multiple combinations of dunkl operators in the directions of the vectors of the canonical basis of r 2 when the dihedral system is of order six and only in this case a single combination suffices to get the dunkl kernel and agrees up to an isomorphism with the formula recently obtained by amri 2 lemma 1 in the case of a root system of type a 2 we finally derive an integral representation for the dunkl kernel associated with the dihedral system of order eight

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

auxiliary signal design for robust multimodel identification under the assumption that one of two given linear models is the real underlying model of the system a proper auxiliary signal is defined as an input signal that allows the selection of the correct model under the assumption that model noises have bounded energy the separability index is defined as the inverse of the energy of the proper auxiliary signal of least energy a method for the computation of this index and the construction of the auxiliary signal is presented finally an efficient procedure for online identification called the hyperplane test is developed

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

on the sensitivity analysis of hoffman constants for systems of linear inequalities relying on a general variational method developed by the authors and lucchetti nonlinear anal to appear the origin of which goes back to ioffe trans amer math soc 251 1979 pp 61 69 we give a formula for the best hoffman constant sigma inf x notin p a b frac ax b infty d x p a b where p a b x ax le b is a nonempty polyhedron in mathbb r n we also sharpen some results of luo and tseng siam j matrix anal appl 15 1994 pp 636 659 by characterizing the continuity set of some hoffman constants and by pointing out their locally lipschitzian character we apply these results to the study of the behavior of the solution set of a linear program inf ax le b u scriptscriptstyle t x with respect to a b u

calmness modulus of fully perturbed linear programs this paper provides operative point based formulas only involving the nominal data and not data in a neighborhood for computing or estimating the calmness modulus of the optimal set argmin mapping in linear optimization under uniqueness of nominal optimal solutions our analysis is developed in two different parametric settings first in the framework of canonical perturbations i e perturbations of the objective function and the right hand side of the constraints the paper provides a computationally tractable formula for the calmness modulus which goes beyond some preliminary results of the literature second in the framework of full perturbations perturbations of all coefficients after characterizing the calmness property for the optimal set mapping the paper provides an operative upper bound for the corresponding calmness modulus as well as some illustrative examples we provide two applications related to algorithms traced out from the literature the first one to a descent method in lp and the second to a regularization method for linear programs with complementarity constraints

a product formula and a convolution structure for a k hankel transform on r abstract the product b k x b k y of two dunkl type kernels is expressed as an integral in terms of b k z with explicit non positive kernel as a consequence a non positivity preserving convolution structure is established

volume of the set of unistochastic matrices of order 3 and the mean jarlskog invariant a bistochastic matrix b of size n is called unistochastic if there exists a unitary u such that bij uij 2 for i j 1 n the set u3 of all unistochastic matrices of order n 3 forms a proper subset of the birkhoff polytope which contains all bistochastic doubly stochastic matrices we compute the volume of the set u3 with respect to the flat lebesgue measure and analytically evaluate the mean entropy of an unistochastic matrix of this order we also analyze the jarlskog invariant j defined for any unitary matrix of order three and derive its probability distribution for the ensemble of matrices distributed with respect to the haar measure on u 3 and for the ensemble which generates the flat measure on the set of unistochastic matrices for both measures the probability of finding j smaller than the value observed for the cabbibo kobayashi maskawa matrix which describes the violation of the cp parity is shown to be small similar statistical reasoning may also be applied to the maki nakagawa sakata matrix which plays role in describing the neutrino oscillations some conjectures are made concerning analogous probability measures in the space of unitary matrices in higher dimensions

hilbert space for quantum mechanics on superspace in superspace a realization of sl2 is generated by the super laplace operator and the generalized norm squared in this paper an inner product on superspace for which this representation is skew symmetric is considered this inner product was already defined for spaces of weighted polynomials see k coulembier h de bie and f sommen orthogonality of hermite polynomials in superspace and mehler type formulae proc london math soc accepted arxiv 1002 1118 in this article it is proven that this inner product can be extended to the super schwartz space but not to the space of square integrable functions subsequently the correct hilbert space corresponding to this inner product is defined and studied a complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this hilbert space then the integrability of the sl2 representation is proven finally the heisenberg uncertainty principle for the super fourier transform is constructed

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

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

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

guaranteed active failure detection and isolation for linear dynamical systems a method for active failure detection and isolation in linear systems subject to inequality bounded perturbations is presented this method under a detectability condition can guarantee detection and isolation of failures to make the failures detectable the injection of a test signal is proposed a method for testing whether the test signal is proposed a method for testing whether the test signal does make the failures detectable is presented under certain conditions a method fro designing test signals which achieve detectability is also given

notes on the structure of the function interacting gas intertwining operator in the degenerate affine hecke algebra the wavefunction of the function interacting bose gas on the infinite interval is studied by introducing the intertwining operator of the degenerate affine hecke algebra the non symmetric eigenfunction of the dunkl operator is constructed

error bounds for vector valued functions necessary and sufficient conditions abstract in this paper we attempt to extend the definition and existing local error bound criteria to vector valued functions or more generally to functions taking values in a normed linear space some new derivative like objects slopes and subdifferentials are introduced and a general classification scheme of error bound criteria is presented

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

singular polynomials for finite reflection groups the dunkl operators involve a multiplicity function as parameter for generic values of this function the simultaneous kernel of these operators acting on polynomials is equal to the constants for special values however this kernel is larger we determine these singular values completely and give partial results on the representations of g that occur in this kernel

directional metric regularity of multifunctions in this paper we study relative metric regularity of set valued mappings with emphasis on directional metric regularity we establish characterizations of relative metric regularity without assuming the completeness of the image spaces by using the relative lower semicontinuous envelopes of the distance functions to set valued mappings we then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity

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

sufficient conditions for error bounds for a lower semicontinuous l s c inequality system on a banach space it is shown that error bounds hold provided every element in an abstract subdifferential of the constraint function at each point outside the solution set is norm bounded away from zero a sufficient condition for a global error bound to exist is also given for an l s c inequality system on a real normed linear space it turns out that a global error bound closely relates to metric regularity which is useful for presenting sufficient conditions for an l s c system to be regular at sets under the generalized slater condition a continuous convex system on rn is proved to be metrically regular at bounded sets

error bounds for lower semicontinuous functions in normed spaces without the convexity or analyticity assumption we study error bounds for an inequality system defined by a general lower semicontinuous function and establish sufficient necessary conditions on the existence of error bounds in infinite dimensional normed spaces some characterizations for a convex inequality system to possess an error bound in a reflexive banach space are also given as applications in dealing with the hoffman error bound result in normed spaces we give a computable lipschitz bound constant which is better than previous lipschitz bound constants in some examples we also consider error bounds for quadratic functions on rn

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

input design for detection of abrupt changes in dynamical systems the detection and diagnosis of changes in stationary dynamical systems via statistical methods and using input design to improve detection performance are discussed a cumulative sum test to detect a change towards one of several hypotheses is obtained by exploiting connections with the sequential probability ratio test for input design the objectives are taken to be to decrease the detection time and at the same time to ensure a tolerable false alarm rate both off line auxiliary inputs and on line generation of the input signal by a linear output feedback are considered the problem is first introduced for the two hypotheses case and then the design techniques are extended to the general multiple hypotheses case

metric subregularity of piecewise linear multifunctions and applications to piecewise linear multiobjective optimization in this paper we consider the metric subregularity property for a piecewise linear multifunction with respect to a piecewise linear constraint as well as the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem of these properties we pay special attention to the global ones we first provide a result on a certain relationship between two nonnegative piecewise linear numerical functions for which the kernel of one of them is contained in the kernel of the other using this result we establish the bounded global metric subregularity results for a piecewise linear multifunction with respect to a piecewise linear set as applications we study the weak sharp minimum property for a piecewise linear constrained multiobjective optimization problem

orthogonal polynomials and cubature formulae on balls simplices and spheres we report on recent developments on orthogonal polynomials and cubature formulae on the unit ball bd the standard simplex td and the unit sphere sd the main result shows that orthogonal structures and cubature formulae for these three regions are closely related this provides a way to study the structure of orthogonal polynomials for example it allows us to use the theory of h harmonics to study orthogonal polynomials on bd and on td it also provides a way to construct new cubature formulae on these regions

perturbation of error bounds our aim in the current article is to extend the developments in kruger et al siam j optim 20 6 3280 3296 2010 doi 10 1137 100782206 and more precisely to characterize in the banach space setting the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations we propose new concepts of arbitrary convex and linear perturbations of the given function defining the system under consideration which turn out to be a useful tool in our analysis the characterizations of error bounds for families of perturbations can be interpreted as estimates of the radius of error bounds the definitions and characterizations are illustrated by examples