marcinkiewicz zygmund inequalities and interpolation by spherical harmonics abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical harmonics in s d with respect to the l p norm moreover we prove that there are no complete interpolation families for p 2

splitting comodules over hopf algebras and application to representation theory of quantum groups of type a0 0 abstract in this work we study some properties of comodules over hopf algebras possessing integrals co frobenius hopf algebras in particular we give a necessary and sufficient condition for a simple comodule to be injective we apply the result obtained to the classification of representations of quantum groups of type a 0 0

decomposition of besov and triebel lizorkin spaces on the sphere abstract a discrete system of almost exponentially localized elements needlets on the n dimensional unit sphere s n is constructed it shown that the needlet system can be used for decomposition of besov and triebel lizorkin spaces on the sphere as an application of besov spaces on s n a jackson estimate for nonlinear m term approximation from the needlet system is obtained

solving the two dimensional constant quantum yang baxter equation a detailed analysis of the constant quantum yang baxter equation rk1k2j1j2 rl1k3k1j3rl2l3k2k3 rk2k3j2j3 rk1l3j1k3rl1l2k1k2 in two dimensions is presented leading to an exhaustive list of its solutions the set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the grobner basis methods each solution was then transformed into a canonical form based on the various trace matrices of r for final elimination of duplicates and subcases if we use homogeneous parametrization the solutions can be combined into 23 distinct cases modulo the well known c p and t reflections and rotations and scalings r q q r q q 1

co frobenius coalgebras abstract we investigate left and right co frobenius coalgebras and give equivalent characterizations which prove statements dual to the characterizations of frobenius algebras we prove that a coalgebra is left and right co frobenius if and only if c rat c c as right c modules and also that this is equivalent to the fact that the functors hom k k and hom c c from m c to m c are isomorphic this allows a definition of a left right symmetric concept of co frobenius coalgebras that is perfectly dual to the one of frobenius algebras and coincides to the existing notion left and right co frobenius coalgebra

characterizations of function spaces on the sphere using frames in this paper we introduce a polynomial frame on the unit sphere s d 1 of r d for which every distribution has a wavelet type decomposition more importantly we prove that many function spaces on the sphere s d 1 such as l p h p and besov spaces can be characterized in terms of the coefficients in the wavelet decompositions as in the usual euclidean case r d we also study a related nonlinear m term approximation problem on s d 1 in particular we prove both a jackson type inequality and a bernstein type inequality associated to wavelet decompositions which extend the corresponding results obtained by r a devore b jawerth and v popov compression of wavelet decompositions amer j math 114 1992 no 4 737 785

on the structure of quantum super groups glq m n abstract we show that a quantum super matrix in standard format is invertible if and only if its block matrices of even entries are invertible we prove the q analog of the well known formula for the berezinian

error estimates for scattered data interpolation on spheres we study sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the n sphere the interpolation knots are scattered our approach partly follows the general theory of golomb and weinberger and related estimates these error estimates are then based on series expansions of smooth functions in terms of spherical harmonics the markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

the kowalewski and henon heiles motions as manakov geodesic flows on so 4 a two dimensional family of lax pairs the invariant surfaces for the kowalewski top the henon heiles system and the manakov geodesic flow onso 4 complete into abelian surfacesa by adjoining in each case a divisord of arithmetic genus 9 these divisors belong to the same linear system ona and they each define a polarization 2 4 therefore there are rational maps transforming the kowalewski top and the henon heiles system into manakov s geodesic flow onso 4 this paper deals with the precise geometric relationship between these three problems it is based on the splitting of the 8 dimensional space of sections ofd theta functions into an even and an odd part and also on a normal form for the six quadrics describinga as embedded in 7 as a byproduct we get a 2 dimensional family of lax pairs for both the kowalewski top and the henon heiles system

on the dimension of the space of integrals on coalgebras abstract we study the injective envelopes of the simple right c comodules and their duals where c is a coalgebra this is used to give a short proof and to extend a result of iovanov on the dimension of the space of integrals on coalgebras we show that if c is right co frobenius then the dimension of the space of left m integrals on c is dim m for any left c comodule m of finite support and the dimension of the space of right n integrals on c is dim n for any right c comodule n of finite support some examples of integrals are computed for incidence coalgebras

maximal estimates for the cesaro means of weighted orthogonal polynomial expansions on the unit sphere abstract estimates of the maximal cesaro means at the critical index are established for the orthogonal polynomial expansions opes with respect to the weight function j 1 d x j 2 j on the unit sphere of r d these estimates allow us to improve several known results in this area including the almost everywhere a e convergence of the cesaro means at the critical index the sufficient conditions for the marcinkiewitcz multiplier theorem and a fefferman stein type inequality for the cesaro operators in addition several similar results for the weighted opes on the unit ball and on the simplex are deduced from the corresponding weighted results on the sphere

spherical marcinkiewicz zygmund inequalities and positive quadrature geodetic and meteorological data collected via satellites for example are genuinely scattered and not confined to any special set of points even so known quadrature formulas used in numerically computing integrals involving such data have had restrictions either on the sites points used or more significantly on the number of sites required here for the unit sphere embedded in r q we obtain quadrature formulas that are exact for spherical harmonics of a fixed order have nonnegative weights and are based on function values at scattered sites to be exact these formulas require only a number of sites comparable to the dimension of the space as a part of the proof we derive l 1 marcinkiewicz zygmund inequalities for such sites

cubature formulae for spheres simplices and balls we obtain in explicit form the unique gaussian cubature for balls spheres in rn based on integrals over balls spheres centered at the origin that integrates exactly all m harmonic functions in particular this formula is exact for all polynomials in n variables of degree 2m 1 a gaussian cubature for simplices is also constructed upper bounds for the errors for certain smoothness classes are derived

irreducible representations of quantum linear groups of type a1 0 a hecke symmetry is an invertible operator defined on the tensor square of a finite dimensional vector space which satisfied the quantized yang baxter equation and the hecke equation x 1 x q 0 see 4 6 for more details given such a matrix one can construct a hopf algebra which is the function algebra over an algebraic matrix quantum group one is interested in the representations of this quantum group that is the comodules over the hopf algebra it turns out that the dimension of the vector space on which the hecke symmetry is defined does not play a significant role but rather a certain inner characteristics of the hecke symmetry the birank a pair of non negative integers a hecke symmetry is called even if its birank is of the form r 0 and odd if its birank is of the form 0 r in these cases the representation category of the corresponding quantum group is like the representation category of the matrix group gl r it is semisimple and generally does not depend on the hecke symmetry but only on the birank if the hecke symmetry is neither even nor odd the representation category of the corresponding quantum group seems to be similar to the representation category of the super group gl r s this problem has not been settled yet the aim of this paper is to classify irreducible representation of the matrix quantum group corresponding to a hecke symmetry of birank 2 1 such a quantum group is usually called matrix quantum group of type 1 0 in this work we show that irreducible representations of a quantum group of type 1 0 can be indexed by tuples m n p of integers with m n we exhibit basic decomposition rules of the tensor product of these representations and compute their dimensions recall that representations of a quantum group are by definition comodules over the corresponding hopf algebra of functions using the koszul komplex we construct for each tuple m n p m n a comodule and prove that they are simple the difficulty here is that the comodule category is not semisimple our main technique is based on the theory of hopf algebras with integral co frobenius hopf algebras for such hopf algebras there is a special class of simple comodules that split in any comodule we show that our hopf algebra is co frobenius and use a criterion for a comodule to be splitting in any comodule which is equivalent to being injective and projective in kac s theory of representation of lie superalgebras irreducible representations that split in any representations are called typical 9

the kowalevski top its r matrix interpretation and bihamiltonian formulation the lax pair representation for the kowalevski top and for the kowalevski gyrostat are obtained directly via the r matrix method the construction of reyman and semenov tian shansky is then used to give a bihamiltonian description of the kowalevski gyrostat

finiteness conditions co frobenius hopf algebras and quantum groups abstract some finiteness conditions for infinite dimensional coalgebras particularly right or left semiperfect coalgebras or co frobenius hopf algebras are studied as well examples of co frobenius hopf algebras are constructed via a hopf algebra structure on an ore extension of a group algebra and it is shown that bicrossproducts of co frobenius hopf algebras are themselves co frobenius

greedy algorithm with regard to the needlet system on the sphere abstract order estimates of the best m term approximation for the besov classes and the sobolev classes with regard to the dictionary on the sphere are established where consists of needlets which are highly localized radial polynomials on the sphere with centers at the nodes of a suitable cubature rule moreover it is shown that for these classes the orders of the best m term approximation can be achieved by simple greedy type algorithm

lax representation and poisson geometry of the kowalevski top we discuss the poisson structure underlying the two field kowalevski gyrostat and the kowalevski top we start from their lax structure and construct a suitable pencil of poisson brackets which endows these systems with the structure of bi hamiltonian completely integrable systems we study the casimir functions of such pencils and show how it is possible to frame the kowalevski systems within the so called gel fand zakharevich bi hamiltonian setting for integrable systems

fast algorithms for discrete polynomial transforms consider the vandermonde like matrix p p k cos j n j k 0 n where the polynomials p k satisfy a three term recurrence relation if p k are the chebyshev polynomials t k then p coincides with c n 1 cos jk n j k 0 n this paper presents a new fast algorithm for the computation of the matrix vector product pa in o n log 2 n arithmetical operations the algorithm divides into a fast transform which replaces pa with c n 1 and a subsequent fast cosine transform the first and central part of the algorithm is realized by a straightforward cascade summation based on properties of associated polynomials and by fast polynomial multiplications numerical tests demonstrate that our fast polynomial transform realizes pa with almost the same precision as the clenshaw algorithm but is much faster for n 128

algebraic aspects of the quantum yang baxter equation in this paper we examine a variety of algebraic contexts in which the quantum yang baxter equation arises and derive methods for generating new solutions from given ones the solutions we describe are encoded in objects which have a module and a comodule structure over a bialgebra our work here is based in part on the ideas of dr1 dr2 suppose that r m m m m is a solution to the quantum yang baxter equation where m is a finite dimensional vector space over a field k in dr2 it was shown that r can be derived from a left h module structure and a right h comodule structure on m for some bialgebra h over k also see yet maj1 maj2 frt2 the module and comodule structure satisfy a natural compatibility condition m together with these structures is called a left quantum yang baxter h module every left quantum yang baxter module gives rise to a solution to the quantum yang baxter equation in a natural way we study the category hyb of left quantum yang baxter h modules and show that many of the familiar constructions in the module category hm can be made in hyb through these constructions solutions to the quantum yang baxter equation can be combined in various ways to generate new ones we place particular emphasis on the tensor product of objects of hyb and ways in which the tensor algebra t m of an object m of hyb can be viewed as an object of hyb using the tensor algebra we are able to extend certain solutions when dim m 2 to the quantum plane we turn the tensor algebra t m of an object m of hyb into an object of hyb by giving it a module algebra and a comodule algebra structure module algebra and comodule algebra structures play a very important role in this paper in section 9 we develop a calculus for uq sl2 starting with a right k slq 2 comodule algebra structure on the quantum plane k x y q where k slq 2 is the coordinate ring of quantum slq 2 over k this calculus has been described in other ways m s sud the right comodule algebra structure on k x y q accounts for the left uq sl2 module algebra structure on k x y q described in m s we show how the basic irreducible representations of tak follow by the calculus a module algebra action or comodule algebra action is a natural vehicle for extending structure on a submodule or subcomodule respectively to the subalgebra it generates the calculus and the solutions to the quantum yang baxter equation we describe in this paper are determined in this manner on small structures two dimensional in this case we believe that many more results will follow from this circle of ideas in section 2 we discuss basic aspects of coalgebra and bialgebras used in the sequel a good general reference for this is sweedler s book sb on hopf algebras in section 3 we review the quantum yang baxter equation in connection with quantum yang baxter modules the a r construction of faddeev reshetikhin and takhtajan frt is considered in this light and examined in detail in when r arises from two commuting operators even though this particular r is rather simple the algebra structure of a r is surprisingly complex using the scratchpad computer algebra system we determine the algebra structure of a r in an interesting specific case showing that our particular example has a finite non commutative groeb

polynomial operators and local approximation of solutions of pseudo differential equations on the sphere we study the solutions of an equation of the form lu f where l is a pseudo differential operator defined for functions on the unit sphere embedded in a euclidean space f is a given function and u is the desired solution we give conditions under which the solution exists and deduce local smoothness properties of u given corresponding local smoothness properties of f measured by local besov spaces we study the global and local approximation properties of the spectral solutions describe a method to obtain approximate solutions using values of f at points on the sphere and polynomial operators and describe the global and local rates of approximation provided by our polynomial operators

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

boundedness of projection operators and cesaro means in weighted l p space on the unit sphere for the weight function d 1 i 1 x i 2 i on the unit sphere sharp local estimates of the orthogonal projection operators are obtained and used to prove the convergence of the cesaro c means in the weighted l p space for above the critical index similar results are also proved for corresponding weight functions on the unit ball and on the simplex

on the convergence of a general class of finite volume methods in this paper we investigate numerical methods for solving hyperbolic conservation laws based on finite volumes and optimal recovery these methods can for example be applied in certain eno schemes their approximation properties depend in particular on the reconstruction from cell averages hence this paper is devoted to prove convergence results for such reconstruction processes from cell averages

moduli of smoothness and approximation on the unit sphere and the unit ball a new modulus of smoothness based on the euler angles is introduced on the unit sphere and is shown to satisfy all the usual characteristic properties of moduli of smoothness including direct and inverse theorem for the best approximation by polynomials and its equivalence to a k functional defined via partial derivatives in euler angles the set of results on the moduli on the sphere serves as a basis for defining new moduli of smoothness and their corresponding k functionals on the unit ball which are used to characterize the best approximation by polynomials on the ball

localized linear polynomial operators and quadrature formulas on the sphere the purpose of this paper is to construct universal auto adaptive localized linear polynomial valued operators based on scattered data on the hyper sphere mathbb s q q ge2 the approximation and localization properties of our operators are studied theoretically in deterministic as well as probabilistic settings numerical experiments are presented to demonstrate their superiority over traditional least squares and discrete fourier projection polynomial approximations an essential ingredient in our construction is the construction of quadrature formulas based on scattered data exact for integrating spherical polynomials of moderately high degree our formulas are based on scattered sites i e in contrast to such well known formulas as driscoll healy formulas we need not choose the location of the sites in any particular manner while the previous attempts to construct such formulas have yielded formulas exact for spherical polynomials of degree at most 18 we are able to construct formulas exact for spherical polynomials of degree 178

multivariate polynomial inequalities with respect to doubling weights and a weights abstract in one dimensional case various important weighted polynomial inequalities such as bernstein marcinkiewicz zygmund nikolskii schur remez etc have been proved under the doubling condition or the slightly stronger a condition on the weights by mastroianni and totik in a recent paper g mastroianni v totik weighted polynomial inequalities with doubling and a weights constr approx 16 1 2000 37 71 the main purpose of this paper is to prove multivariate analogues of these results we establish analogous weighted polynomial inequalities on some multivariate domains such as the unit sphere s d 1 the unit ball b d and the general compact symmetric spaces of rank one moreover positive cubature formulae based on function values at scattered sites are established with respect to the doubling weights on these multivariate domains some of these multi dimensional results are new even in the unweighted case our proofs are based on the investigation of a new maximal function for spherical polynomials

remez type inequalities and their applications abstract the remez inequality gives a sharp uniform bound on 1 1 for real algebraic polynomials p of degree at most n if the lebesgue measure of the subset of 1 1 where p is at most 1 is known remez type inequalities give bounds for classes of functions on a line segment on a curve or on a region of the complex plane given that the modulus of the functions is bounded by 1 on some subset of prescribed measure this paper offers a survey of the extensive recent research on remez type inequalities for polynomials generalized nonnegative polynomials exponentials of logarithmic potentials and muntz polynomials remez type inequalities play a central role in proving other important inequalities for the above classes the paper illustrates the power of remez type inequalities by giving a number of applications

marcinkiewicz zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights abstract we find necessary density conditions for marcinkiewicz zygmund inequalities and interpolation for spaces of spherical polynomials in the weighted l p 0 p space with respect to a doubling weight moreover we show that these conditions are sharp

approximation of smooth functions on compact two point homogeneous spaces estimates of kolmogorov n widths dn bpr lq and linear n widths n bpr lq 1 q of sobolev s classes bpr r 0 1 p on compact two point homogeneous spaces ctphs are established for part of p q 1 1 sharp orders of dn bpr lq or n bpr lq were obtained by bordin et al j funct anal 202 2 2003 307 in this paper we obtain the sharp orders of dn bpr lq and n bpr lq for all the remaining p q our proof is based on positive cubature formulas and marcinkiewicz zygmund type inequalities on ctphs

maximal function and multiplier theorem for weighted space on the unit sphere for a family of weight functions invariant under a finite reflection group the boundedness of a maximal function on the unit sphere is established and used to prove a multiplier theorem for the orthogonal expansions with respect to the weight function on the unit sphere similar results are also established for the weighted space on the unit ball and on the standard simplex

fast evaluation of quadrature formulae on the sphere recently a fast approximate algorithm for the evaluation of expansions in terms of standard l 2 s 2 orthonormal spherical harmonics at arbitrary nodes on the sphere s 2 has been proposed in s kunis and d potts fast spherical fourier algorithms j comput appl math 161 75 98 2003 the aim of this paper is to develop a new fast algorithm for the adjoint problem which can be used to compute expansion coefficients from sampled data by means of quadrature rules we give a formulation in matrix vector notation and an explicit factorisation of the spherical fourier matrix based on the former algorithm starting from this we obtain the corresponding factorisation of the adjoint spherical fourier matrix and are able to describe the associated algorithm for the adjoint transformation which can be employed to evaluate quadrature rules for arbitrary weights and nodes on the sphere we provide results of numerical tests showing the stability of the obtained algorithm using as examples classical gaus legendre and clenshaw curtis quadrature rules as well as the healpix pixelation scheme and an equidistribution

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

constructing pointed hopf algebras by ore extensions abstract we present a general construction producing pointed co frobenius hopf algebras and give some classification results for the examples obtained

a lax pair for kowasevski s top abstract we show that on each level surface of the invariants the equations of the kowalevski top are equivalent to a neumann system describing the motion of a mass point on the sphere s2 p 1 under the influence of force qp this allows us to write a global lax pair for the kowalevski system and to show that kowalevski s original reduction of the integration of the equations of motion to jacobi s inversion theorem is identical to the introduction of elliptical spherical coordinates for integrating c neumann s system

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

the kowalewski top 99 years later a lax pair generalizations and explicit solutions a natural lax pair for the kowalewski top is derived by using a general group theoretic approach this gives a new insight into the algebraic geometry of the top and leads to its complete solution via finite band integration theory