differential systems of type 1 1 on hermitian symmetric spaces and their solutions abstract this paper concerns g invariant systems of second order differential operators on irreducible hermitian symmetric spaces g k the systems of type 1 1 are obtained from k invariant subspaces of p p we show that all such systems can be derived from a decomposition p p h l h c here l gives the laplace beltrami operator and h h l is the celebrated hua system which has been extensively studied elsewhere our main result asserts that for g k of rank at least two a bounded real valued function is annihilated by the system l h c if and only if it is the real part of a holomorphic function in view of previous work one obtains a complete characterization of the bounded functions that are solutions for any system of type 1 1 which contains the laplace beltrami operator

the limits of stochastic integrals of differential forms this paper is concerned with the integration of 1 forms against the markov stochastic process associated with a second order elliptic differential operator in divergence form it focuses on the limiting behavior of the integral as the process leaves a fixed point or goes to infinity this extends previous work in the area where advantage was usually taken of the fact that the operator was self adjoint and started with the associated invariant measure applications are given for example it is a trivial consequence that the diffusion associated to a uniformly elliptic operator on a negatively curved cartan hadamard manifold has an asymptotic direction recovering and strengthening the previous arguments of pratt sullivan and others the approach can also be used to construct a levy area for such processes to study the thinness of sets for the elliptic operator and probably has wider applications

harmonic analysis as spectral theory of laplacians by writing the fourier inversion formula on euclidean space in polar coordinates we obtain f x 0 fgl x dx where fgl x 2fgl x which is the spectral theory of the laplacian how do properties of f relate to properties of the family of eigenfunctions fgl answers are provided for the following spaces l2 s s d in odd dimensions analogous results are obtained for harmonic analysis on hyperbolic space constant curvature semi riemannian spaces the heisenberg group and for differential forms on hyperbolic space for f l2 there is a plancherel formula f 22 limt o 1t bt z f x 2dx d where bt z denotes the ball of radius t about an arbitrary point z which is independent of the dimension and the identical formula holds in hyperbolic space if the parametrization of the eigenvalues is shifted for certain semi riemannian symmetric spaces we obtain a paley wiener theorem that explains the role of discrete series in producing functions of compact support and also involve certain non unitary representations that contain the discrete series representations for the heisenberg group the plancherel formula is of a different nature requiring that the eigenfunctions be almost periodic in one variable

the principal series for a reductive symmetric space ii eisenstein integrals in this paper we develop a theory of eisenstein integrals related to the principal series for a reductive symmetric space g h here g is a real reductive group of harish chandra s class an involution of g and h an open subgroup of the group g of xed points for the group g itself is a symmetric space for the left right action of g g we refer to this setting as the group case up to a normalization our eisenstein integrals generalize those of harish chandra 18 associated with a minimal parabolic subgroup in the group case

asymptotic expansions and boundary values of eigenfunctions on riemannian symmetric spaces soit x g k un espace de riemann symetrique de type non compacte a bord b k m et soit d x l algebre de tous les operateurs differentiels g invariants sur x on obtient des developpements asymptotiques pour une grande classe de fonctions jointives f sur x

filtration de certains espaces de fonctions sur un espace symetrique reductif we introduce a filtration of a g k module of some space of functions on a reductive symmetric space g h and compute the associated grading as a direct sum of induced representations as an application of this result to the reductive groups viewed as symmetric spaces we are able to realize any harish chandra module as a subquotient of a direct sum of induced representations from parabolic subgroups the inducing representations being trivial on the unipotent radical

subband filters optimized for lost coefficient reconstruction transmission of digital subband coded images over lossy packet networks presents a reconstruction problem at the decoder this correspondence addresses the design of a family of quadrature mirror filters qmfs that minimize the mean squared error of mean reconstructed low frequency subband coefficients the resulting filters perform well as qmfs as well as provide the desired reconstruction properties in the event of loss

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

unicity of best one sidedl1 approximations this paper deals with the problem of uniqueness in one sidedl 1 approximation the chief purpose is to characterize finite dimensional subspacesg of the space of continuous or differentiable functions which have a unique best one sidedl 1 approximation in addition we study a related problem in moment theory these considerations have an important application to the uniqueness of quadrature formulae of highest possible degree of precision

fourier transforms on a semisimple symmetric space let g h be a semisimple symmetric space that is g is a connected semisimple real lie group with an involution and h is an open subgroup of the group of xed points for in g the main purpose of this paper is to study an explicit fourier transform on g h in terms of general representation theory the abstract fourier transform of a compactly supported smooth function f 2 c 1 c g h is given by see 5 1 f f zg h f x x dx for h a unitary irreducible representation of g and 2 h an h invariant distribution vector for here dx is the invariant measure on g h thus f isa smooth vector for h depending linearly on our goal is to obtain an explicit version of the restriction of this fourier transform to representations h in the minimal unitary principal series h for g h under the assumption that the center of g is nite in a sequel 9 to this paper it will be proved that a function f 2 c 1 c g h is uniquely determined by the restriction of f to this series a priori it is known that f is determined by f

hua operators on bounded homogeneous domains in cnand alternative reproducing kernels for holomorphic functions let d be a bounded homogeneous domain in cn note that d is not assumed to be hermitian symmetric in this work we are interested in studying various classes of harmonic functions on d and the possibility of representing them as poisson integrals over the bergman shilov boundary one such class of harmonic functions is the hua harmonic functions specifically by forming a contraction of with the holomorphic curvature tensor we define a canonical system of differential operators which generalizes the classical hua system this system is invariant under all bi holomorphisms of d the hua harmonic functions are by definition the nullspace of this system our main result concerning this system is that every bounded hua harmonic function is the poisson integral over the bergman shilov boundary of a uniquel function against the poisson kernel for the laplace beltrami operator we also consider spaces of harmonic functions defined as the kernel of a single real differential operator which is invariant under a particular solvable lie group which acts transitively on d we show that there exists such an operator which a annihilates holomorphic functions b satisfies the hormander condition and c has the bergman shilov boundary as its maximal boundary it follows that the corresponding bounded harmonic functions are in one to one correspondence with thel functions on the bergman shilov boundary under poisson integration

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative

a haar type theory of best approximation with constraints a general setting for constrained l 1 approximation is presented let u n be a finite dimensional subspace of c a b and l be a linear operator from u n to c r k r 0 1 where k is a finite union of disjoint closed bounded intervals for v u c r k with v u the approximating set is u n v u p u n v lp u on k and the norm is formule where w a positive continuous function on a b we obtain necessary and sufficient conditions for u n v u to admit unique best w approximations to all f c a b for all positive continuous w and all v u c r k r 0 1 satisfying a nonempty interior condition

fonctions d g h finies sur un espace symetrique reductif abstract it is well known that on r n every smooth function annihilated by a finite codimensional ideal in the algebra of constant coefficient differential operators is a linear combination of polynomial exponential functions p x e x hom r n c furthermore the polynomial functions are obtained by applying to the exponential functions e x some constant coefficient differential operator in the parameter we generalize this fact to the reductive symmetric spaces case the role of the exponential functions being taken by the eisenstein integrals

hua system and pluriharmonicity for symmetric irreducible siegel domains of type ii abstract we consider here a generalization of the hua system that was proved by johnson and koranyi to characterize poisson szego integrals for siegel domains of tube type we show that the situation is completely different when dealing with non tube type symmetric irreducible symmetric domains then all functions that are annihilated by this second order system and satisfy an h 2 type integrability condition are pluriharmonic functions

approximation with constraints in this paper we study problems of best l approximation to continuous functions from finite dimensional subspaces under a variety of constraints included are problems of bounded coefficient approximation approximation with interpolation restricted range approximation and restricted range and derivative approximation emphasis is placed on problems of uniqueness

uniqueness of positive solutions to the rotationally symmetric p harmonic map equations abstract let and be positive solutions on 0 to the rotationally symmetric p harmonic map equation on model manifolds m f and m g where f is assumed to be sufficiently large near infinity and g y 0 for y 0 we show that if and have the same limit at infinity then on 0

subband coded image reconstruction for lossy packet networks transmission of digital subband coded images over lossy packet networks presents a reconstruction problem at the decoder this paper presents two techniques for reconstruction of lost subband coefficients one for low frequency coefficients and one for high frequency coefficients the low frequency reconstruction algorithm is based on inherent properties of the hierarchical subband decomposition to maintain smoothness and exploit the high intraband correlation a cubic interpolative surface is fit to known coefficients to interpolate lost coefficients accurate edge placement crucial for visual quality is achieved by adapting the interpolation grid in both the horizontal and vertical directions as determined by the edges present an edge model is used to characterize the adaptation and a quantitative analysis of this model demonstrates that edges can be identified by simply examining the high frequency bands without requiring any additional processing of the low frequency band high frequency reconstruction is performed using linear interpolation which provides good visual performance as well as maintains properties required for edge placement in the low frequency reconstruction algorithm the complete algorithm performs well on loss of single coefficients vectors and small blocks and is therefore applicable to a variety of source coding techniques

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

poisson transforms on vector bundles let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g let r v be an irreducible unitary representation of k and g x k tv the associated vector bundle in the algebra of invariant differential operators on g xk v the center of the universal enveloping algebra of lie g induces a certain commutative subalgebra z we are able to determine the characters of z given such a character we define a poisson transform from certain principal series representations to the corresponding space of joint eigensections we prove that for most of the characters this map is a bijection generalizing a famous conjecture by helgason which corresponds to r the trivial representation introduction let g be a connected real semisimple lie group with finite center and k a maximal compact subgroup of g then g k is a riemannian symmetric space of noncompact type we fix an iwasawa decomposition g kan let m be the centralizer of a in k let g and a be the lie algebras of g and a respectiarely and s 0 a the root system for 0 a let s 0 a be the positiare roots in s a for the ordering giaren by n let d g k be the algebra of inarariant differential operators on g k it is well known that the characters of d g k are parametrized by a e a the complex dual space of a let a g k denote the space of joint eigenfunctions corresponding to a we write g k g exp h g n g for each g e g according to g kan for each x e c k m we define pxx e c g k by p x g t k e p h g lk dk k here p is the half sum of s 0 a including multiplicities it turns out that pxq e v g k one can easily extend the definition of px to the space d k m resp a k m of distributions resp analytic functionals on k m in this paragraph we fix a e a such that 2 a ct ct ct is not in n 0 for each ct c s 0 a it is proared by helgason helg2 that pa defines a bijection from c o k m k finite onto e g k k finite he also proares in the rank one case p is a bijection from a k m onto 6 g k he then conjectured this should be true for the higher rank case the conjecture was earentually proared by six japanese mathematicians in kkmoot it should be mentioned that a representation theoretic proof by schmid starting from the kfinite result is indicated in sch lewis then a student of helgason made the following obseraration let 6 g k be the subspace of e g k whereeach element increases at most exporlentially see 02 for definition received by the editors september 28 1994 and in revised form january 30 1995 1991 mathematics subject classification primary 22e46 secondary 43a85 1998 american mathematical society 857 this content downloaded from 157 55 39 198 on fri 17 jun 2016 05 05 10 utc all use subject to http about jstor org terms

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

bounded harmonic functions on riemannian manifolds of nonpositive curvature certain general conditions are put forth on a complete simply connected riemannian manifold of nonpositive curvature which guarantee that they support nontrivial bounded harmonic functions this result includes the cartan hadamard manifolds with curvature pinched between two negative constants and the bounded symmetric domains mathfrak r i n n and mathfrak r ii n n 2 as special cases

asymptotic expansions and hua harmonic functions on bounded homogeneous domains let d be a homogeneous siegel domain of type ii we prove that every bounded hua harmonic function f on d is pluriharmonic the proof is based on asymptotic expansion of f

lyapunov exponents for a stochastic analogue of the geodesic flow new invariants for a riemannian manifold are defined as lya punov exponents of a stochastic analogue of the geodesic flow a lower bound is given reminiscent of corresponding results for the geodesic flow and an upper bound is given for surfaces of positive curvature for surfaces of con stant negative curvature a direct method via the doob rt transform is used to determine the full lyapunov structure relating the stable manifolds to the horocycles

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

patterson sullivan distributions in higher rank for a compact locally symmetric space x of non positive curvature we consider sequences of normalized joint eigenfunctions which belong to the principal spectrum of the algebra of invariant differential operators using an h pseudo differential calculus on x we define and study lifted quantum limits as weak limit points of wigner distributions the helgason boundary values of the eigenfunctions allow us to construct patterson sullivan distributions on the space of weyl chambers these distributions are asymptotic to lifted quantum limits and satisfy additional invariance properties which makes them useful in the context of quantum ergodicity our results generalize results for compact hyperbolic surfaces obtained by anantharaman and zelditch

unbounded positive entire solutions of rotationally symmetric harmonic map equations abstract we study the existence of unbounded positive entire c 2 solutions of the rotationally symmetric harmonic map equations using the existence result we solve the dirichlet problem at infinity for any nonnegative boundary value at infinity

van den ban schlichtkrull wallach asymptotic expansions on nonsymmetric domains if it is annihilated by every element of dg x the algebra of all g invariant differential operators without constant term one of the most beautiful results in the harmonic analysis of symmetric spaces is the helgason conjecture which states that on a riemannian symmetric space of noncompact type a function is harmonic if and only if it is the poisson integral of a hyperfunction over the furstenberg boundary g po where po is a minimal parabolic subgroup see 14 17 one of the more remarkable aspects of this theorem is its generality one obtains a complete description of all solutions to the system of invariant differential operators on x without imposing any boundary or growth conditions if x is a hermitian symmetric space then one is typically interested in complex function theory in which case one is interested in functions whose boundary values are supported on the shilov boundary rather than the furstenberg boundary the shilov boundary is g p where p is a certain maximal parabolic containing po in this case it turns out that the algebra of g invariant differential operators is not necessarily the most appropriate one for

geometric realization of discrete series for semisimple symmetric spaces let g be a connected semisimple lie group a an involution of g and h the connected component of the fixed point set of a containing the identity then g h is a semisimple symmetric space we assume g is a real form of a complex lie group and rank g h rank k k c h where k is a a stable maximal compact subgroup of g flensted jensen using his duality principle and poisson transform constructed countably many discrete series in u g h f j i then using deep results of hyperfunction theory and microlocal analysis oshima and matsuki gave a detailed characterization of these discrete series including a proof of flensted jensen s conjecture c 0 om based on these results we give the discrete series a geometric interpretation in locally symmetric spaces f c where g k and f cg is a cocompact discrete group such that f c h c h is cocompact our geometric interpretation is the following first of all to a flensted jensen function which is the generating function of a discrete series v we associate a harmonic form cb z on g with values in a finite dimensional g module v right g translates of coefficients of 5z realizes the contragredient t of r and tba is left invariant under h and under right k action it transforms according to the minimal k type with highest weight of 3 is the pull back of a v valued from coz on and v determines a locally constant bundle e on f so that if convergence holds

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

the plancherel decomposition for a reductive symmetric space i spherical functions we prove the plancherel formula for spherical schwartz functions on a reductive symmetric space our starting point is an inversion formula for spherical smooth compactly supported functions the latter formula was earlier obtained from the most continuous part of the plancherel formula by means of a residue calculus in the course of the present paper we also obtain new proofs of the uniform tempered estimates for normalized eisenstein integrals and of the maass selberg relations satisfied by the associated c functions

local boundary data of eigenfunctions on a riemannian symmetric space let x g k be a riemannian symmetric space of non compact type and let d x be the algebra of invariant differential operators on x in a previous paper 1 we developed a theory of asymptotic expansions for joint eigenfunctions of d x of at most exponential growth in the present paper we show that local asymptotic data determine the eigenfunctions completely we also develop a theory of asymptotic expansions along walls both results are of importance for the theory of the discrete series of a semisimple symmetric space let a be a maximal abelian split subspace of the lie algebra 9 of g and denote by a its complexified dual let e be the restricted root system of a in g and w the associated weyl group then harish chandra s isomorphism d x s a w determines a bijection 2 z from a w to the set d x of algebra homomorphisms d x c given 2ca let ga x denote the joint eigenspace of functions f e c o x satisfying

existence of non trivial harmonic functions on cartan hadamard manifolds of unbounded curvature the liouville property of a complete riemannian manifold m i e the question whether there exist non trivial bounded harmonic functions on m attracted a lot of attention for cartan hadamard manifolds the role of lower curvature bounds is still an open problem we discuss examples of cartan hadamard manifolds of unbounded curvature where the limiting angle of brownian motion degenerates to a single point on the sphere at infinity but where nevertheless the space of bounded harmonic functions is as rich as in the non degenerate case to see the full boundary the point at infinity has to be blown up in a non trivial way such examples indicate that the situation concerning the famous conjecture of greene and wu about existence of non trivial bounded harmonic functions on cartan hadamard manifolds is much more complicated than one might have expected

the plancherel decomposition for a reductive symmetric space ii representation theory we obtain the plancherel decomposition for a reductive symmetric space in the sense of representation theory our starting point is the plancherel formula for spherical schwartz functions obtained in part i the formula for schwartz functions involves eisenstein integrals obtained by a residual calculus in the present paper we identify these integrals as matrix coefficients of the generalized principal series

holomorphic l p type for sub laplacians on connected lie groups we study the problem of determining all connected lie groups g which have the following property hlp every sub laplacian l on g is of holomorphic lp type for 1 p p 2 first we show that semi simple non compact lie groups with finite center have this property by using holomorphic families of representations in the class one principal series of g and the kunze stein phenomenon we then apply an lp transference principle essentially due to anker to show that every connected lie group g whose semi simple quotient by its radical is non compact has property hlp for the convenience of the reader we give a self contained proof of this transference principle which generalizes the well known coifman weiss principle one is thus reduced to studying those groups for which the semi simple quotient is compact i e to compact extensions of solvable lie groups in this article we consider semi direct extensions of exponential solvable lie groups by connected compact lie groups it had been proved in joint work by w hebisch and the two first named authors that every exponential solvable lie group s which has a non regular co adjoint orbit whose restriction to the nilradical is closed has property hlp and we show here that hlp remains valid for compact extensions of these groups

classical and quantum resonances for hyperbolic surfaces for compact and for convex co compact oriented hyperbolic surfaces we prove an explicit correspondence between classical ruelle resonant states and quantum resonant states except at negative integers where the correspondence involves holomorphic sections of line bundles

harmonic maps between 3 dimensional hyperbolic spaces we prove that a quasiconformal map of the sphere s 2 admits a harmonic quasi isometric extension to the hyperbolic space h 3 thus confirming the well known schoen conjecture in dimension 3

euler and navier stokes equations on the hyperbolic plane we show that nonuniqueness of the leray hopf solutions of the navier stokes equation on the hyperbolic plane 2 observed by chan and czubak is a consequence of the hodge decomposition we show that this phenomenon does not occur on n whenever n 3 we also describe the corresponding general hamiltonian framework of hydrodynamics on complete riemannian manifolds which includes the hyperbolic setting

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

a fatou theorem for f harmonic functions in this paper we study a class of functions that appear naturally in some equidistribution problems and that we call f harmonic these are functions of the universal cover of a closed and negatively curved manifold which possess an integral representation analogous to the poisson representation of harmonic functions where the role of the poisson kernel is played by a holder continuous kernel more precisely we prove a theorem a la fatou about the nontangential convergence of quotients of such functions from which we deduce some basic properties such as the uniqueness of the f harmonic function on a compact manifold and of the integral representation of f harmonic functions

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

plongements de series discretes pour un espace symetrique reductif resume using the realization of a discrete series a of a semi simple symmetric space g h as a space of boundary values in the dual space gd kd we construct an explicit embedding of a in a principal series ind 1 associated to an open subgroup q1 of a minimal stable parabolic subgoup q of g the construction of q1 and uses the properties of what we call a fundamental system of roots

the hodge cohomology of a conformally compact metric on etudie le laplacien de hogde agissant sur des k formes differentielles pour une classe de varietes de riemann completes a courbure sectionnelle negative proche de l infini ces varietes ont des compactifications c sur lesquelles la metrique est conforme a une metrique lisse jusqu au bord avec un facteur conforme 2 une fonction definissante pour le bord

brownian motion and dirichlet problems at infinity we discuss angular convergence of riemannian brownian motion on a cartan hadamard manifold and show that the dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions ce 2 ar x km x a2 vspace 0 5pt 0 and cr x 2 km x 1 r x 2 2 2 respectively

scattering theory for the laplacian on symmetric spaces of noncompact type and its application to a conjecture of strichartz abstract in this paper we develop the scattering theory for the laplacian on symmetric spaces of noncompact type we study the asymptotic properties of the resolvent in the framework of the agmon hormander space our approach is based on a detailed analysis of the helgason fourier transform and generalized spherical functions on symmetric spaces of noncompact type as an application of our scattering theory we prove a conjecture by strichartz concerning a characterization of a family of generalized eigenfunctions of the laplacian

support d asymptotiques et croissance des fonctions propres sur un espace symetrique semi simple abstract we associate several distribution boundary values to an eigenfunction with moderate growth on a riemannian symmetric space g k the associated character of the algebra d g k of invariant differential operators is allowed to be non regular we prove results on the support of these boundary values these allow us to recover the theorems of matsuki oshima and oshima on the equivalence between growth of an eigenfunction and limitations on the supports of its boundary values our approach is based on an asymptotic analysis that makes no use of hyperfunction theory

quadrature mirror filter design in the time domain a new technique for designing quadrature mirror filters is described the formulation carried out in the time domain is shown to result in an optimization problem requiring minimization of a quartic multinomial an iterative solution is suggested which involves computation of the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps our experiments show that convergence to the optimum tap weights is stable and the accuracy of the final solution is limited only by the accuracy of the eigenvalue eigenvector routine as in an earlier technique the user can specify the stop band frequency the relative weights of the passband ripple energy and the stopband residual energy and of course the number of filter taps

the limiting angle of certain riemannian brownian motions let m be a cartan hadamard manifold of dimension d 3 let p m and x exp r x x be geodesic polar coordinates with pole p and let x be the brownian motion on m let sectm x denote the sectional curvature of any plane section in mx we prove that for each c 2 there is a 0 1 such that if l2r x 2 sectm x cr x 2 for all x in the complement of a compact set then limt xt exists a s and defines a nontrivial invariant random variable the dirichlet problem at infinity and a conjecture of greene and wu are also discussed

the paley wiener theorem for the hua system abstract e damek a hulanicki and r penney j funct anal in press studied a canonical system of differential equations the hua system denoted hjk which is definable on any kahlerian manifold m functions annihilated by this system are called hua harmonic in the case where m is a bounded homogeneous domain in c n with its bergman metric it was shown that every bounded hua harmonic function has a boundary value on the bergman shilov boundary and that the function is reproducible from the shilov boundary by integration against the reduction of the poisson kernel for the laplace beltrami operator to the shilov boundary this then provided a partial generalization of the results of johnson and koranyi to the stated context significantly however no characterization of the resulting space of boundary functions so obtained was given the current work extends these results in several ways we show that for a tube domain i e a siegel domain of type i the cauchy szego poisson kernel also reproduces the hua harmonic functions since the two kernels agree only in the symmetric case it follows that the space of boundary functions is dense in l if and only if the domain is symmetric we also show that an l 2 function is the boundary function for a hua harmonic function if and only if its fourier transform is supported in a certain typically non convex cone this cone is characterized in terms of the fourier transformation of the cauchy szego poisson kernel

a characterization of the l2 range of the poisson transform related to strichartz conjecture on symmetric spaces of noncompact type abstract the main purpose of this paper is to solve strichartz conjecture concerning an image characterization of the poisson transform in the l 2 theory on symmetric spaces of noncompact type we prove that the poisson transform provides an isomorphism between the l 2 space on the boundary and a certain weighted l 2 space consisting of joint eigenfunctions on symmetric spaces of noncompact type our approach is based on the scattering theory for the schrodinger operator moreover we give a fourier restriction estimate and an asymptotic formula for the poisson transform

asymptotic dirichlet problems for harmonic functions on riemannian manifolds we define the asymptotic dirichlet problem and give a sufficient condition for solving it this proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete riemannian manifolds 0 introduction in this paper we will prove the existence of nonconstant bounded harmonic functions on certain classes of noncompact riemannian manifolds by defining and solving an asymptotically defined dirichlet problem for harmonic functions the motivation comes from the classical uniformization theorem of riemann surfaces which says that a simply connected riemann surface is biholo morphic to the riemann sphere s2 the complex plane c or the unit disk d this is a geometric theorem but its original proof due to koebe relies heavily on function theory the function theoretic interpretation of this theorem is the following among the noncompact simply connected surfaces c is characterized by the fact that it admits no nonconstant bounded harmonic functions and d by that it admits nonconstant bounded harmonic functions the geometric aspect of the uniformization theorem could be roughly stated as follows let m be a simply connected riemann surface equipped with a complete riemannian metric with gaussian curvature km if km c 0 then m is biholo morphic to s2 if km c 0 then m is biholomorphic to d if km is close to zero then m is biholomorphic to c the following precise version is due to greene and wu gw1 p 120