unbounded subnormal weighted shifts on directed trees ii abstract a new method of verifying the subnormality of unbounded hilbert space operators based on an approximation technique is proposed diverse sufficient conditions for subnormality of unbounded weighted shifts on directed trees are established an approach to this issue via consistent systems of probability measures is invented the role played by determinate stieltjes moment sequences is elucidated lambert s characterization of subnormality of bounded operators is shown to be valid for unbounded weighted shifts on directed trees that have sufficiently many quasi analytic vectors which is a new phenomenon in this area

derivations vanishing on s lets be the group of finite permutations on countably many symbols we exhibit an embedding ofs into a uhf algebra mathfrak a of glimm typen such that if is a derivation vanishing ons and satisfying 0 where is the unique trace on mathfrak a then admits an extension which is the generator of ac dynamics

essential self adjointness of the graph laplacian we study the operator theory associated with such infinite graphs g as occur in electrical networks in fractals in statistical mechanics and even in internet search engines our emphasis is on the determination of spectral data for a natural laplace operator associated with the graph in question this operator will depend not only on g but also on a prescribed positive real valued function c defined on the edges in g in electrical network models this function c will determine a conductance number for each edge we show that the corresponding laplace operator is automatically essential self adjoint by this we mean that is defined on the dense subspace d of all the real valued functions on the set of vertices g0 with finite support in the hilbert space l2 g0 the conclusion is that the closure of the operator is self adjoint in l2 g0 and so in particular that it has a unique spectral resolution determined by a projection valued measure on the borel subsets of the infinite half line we

subnormality of unbounded composition operators over one circuit directed graphs exotic examples abstract a recent example of a non hyponormal injective composition operator in an l 2 space generating stieltjes moment sequences invented by three of the present authors was built over a non locally finite directed tree the main goal of this paper is to solve the problem of whether there exists such an operator over a locally finite directed graph and in the affirmative case to find the simplest possible graph with these properties simplicity refers to local valency the problem is solved affirmatively for the locally finite directed graph g 2 0 which consists of two branches and one loop the only simpler directed graph for which the problem remains unsolved consists of one branch and one loop the consistency condition the only efficient tool for verifying subnormality of unbounded composition operators is intensively studied in the context of g 2 0 which leads to a constructive method of solving the problem the method itself is partly based on transforming the krein and the friedrichs measures coming either from shifted al salam carlitz q polynomials or from a quartic birth and death process

visual error criteria for qualitative smoothing abstract an important gap between the classical mathematical theory and the practice and implementation of nonparametric curve estimation is due to the fact that the usual norms on function spaces measure something different from what the eye can see visually in a graphical presentation mathematical error criteria that more closely follow visual impression are developed and analyzed from both graphical and mathematical viewpoints examples from wavelet regression and kernel density estimation are considered

favard separation method for almost periodic stochastic differential equations abstract favard separation method is an important means to study almost periodic solutions to linear differential equations later amerio applied favard s idea to nonlinear differential equations in this paper by appropriate choosing separation and almost periodicity in distribution sense we obtain the favard and amerio type theorems for stochastic differential equations

unbounded derivations commuting with compact group actions ii abstract let be a closed derivation in a c algebra 21 which commuteswith an ergodic action of a compact group on 5 then generates a c dynamics of 91 similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators 1 introduction in 9 sakai showed that a non zero closed derivation in c t commuting withtranslations is a constant multiple of the derivative following this it wasshown in 4 7 that if g is a locally compact group and is a closed derivationin c 0 g commuting with left translations by elements of g then is the generatorof a c dynamics i e strongly continuous one parameter group of automor phisms of c 0 g a like result holds for c 0 g h h a closed sub group of g provided that g is either separable or the projective limit of lie groups in thisnote we assume g is compact and we obtain similar results for an ergodic action ofg on an arbitrary c algebra theorem 2 1 and for an arbitrary action of g on anabelian c algebra theorem 3 2 or on the algebra of compact operators theorem 4 1 we refer to 1 3 9 for background on unbounded derivations in c algebras

non commutative moment problems we are concerned with two problems called the generalized moment problem gmp and the quantum moment problem qmp both problems can be considered as non commutative versions of the classical moment problem et g be a connected lie group and 7b e g the universal enveloping algebra of the lie algebra of g suppose f is a given linear functional on 7b e g and is a fixed unitary representation of g we then ask gmp when do there exist a unitary representation u of g and a cyclic c vector for u such that f x x 7b e g qmp when is f of the form f x trdv x x 7b e g where is an operator such that dv x is of trace class for every x 7b e g

asymptotically dichotomic almost periodic differential equations abstract consider a non linear differential equation in r n which asymptotically behaves as a linear equation admitting an exponential dichotomy we wonder if almost periodic solutions exist when we add to the equation an almost periodic forcing term large enough and not vanishing too much a positive answer has been given in 3 for the scalar case n 1 and our aim is to extend that result to higher dimensions we discover that the extension seems to be driven by a new ingredient namely the type of the exponential dichotomy besides the pure stable types the mixed hyperbolic type is now possible and leads to a weaker than expected extension an example shows that a stronger extension cannot be obtained by the same method the approach is blended and mixes methods of differential equations and functional analysis especially when estimating norm and spectral radius of some crucial positive but non compact linear integral operators

on estimation of monotone and concave frontier functions abstract when analyzing the productivity of firms one may want to compare how the firms transform a set of inputs x typically labor energy or capital into an output y typically a quantity of goods produced the economic efficiency of a firm is then defined in terms of its ability to operate close to or on the production frontier the boundary of the production set the frontier function gives the maximal level of output attainable by a firm for a given combination of its inputs the efficiency of a firm may then be estimated via the distance between the attained production level and the optimal level given by the frontier function from a statistical viewpoint the frontier function may be viewed as the upper boundary of the support of the population of firms density in the input and output space it is often reasonable to assume that the production frontier is a concave monotone function then a famous estimator in the univariate input and output case is the data envelopment analysis dea estimato

unbounded derivations tangential to compact groups of automorphisms ii let g be a compact abelian group and an action of g on a c algebra u such that u u u 0 u for all g where u is the spectral subspace of u corresponding to the character on g derivations which are defined on the algebra uf of g finite elements are considered in the special case u 0 these derivations are characterized by a cocycle on g with values in the relative commutant of u in the multiplier algebra of u and these derivations are inner if and only if the cocycles are coboundaries and bounded if and only if the cocycles are bounded under various restrictions on g and properties of the cocycle are deduced which again give characterizations of in terms of decompositions into generators of one parameter subgroups of g and approximately inner derivations finally a perturbation technique is devised to reduce the case uf uf to the case uf uf and u 0 this is used to show that any derivation with d uf is wellbehaved and if furthermore g t1 and uf uf the closure of generates a one parameter group of automorphisms of u in the case g td d 2 3 finite and uf uf it is shown that extends to a generator of a group of automorphisms of the weak closure of u in any g covariant representation

discontinuous regression surfaces fitting we suggest a three stage procedure to recover discontinuous regression surfaces when noisy data are present in the first stage jump candidate points are detected using a jump detection criterion a local principal component line is then fitted through these points in a neighborhood of a design point this line provides a first order approximation to the true jump location curve in that neighborhood in the third stage observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point if there are no jump candidate points in the neighborhood then all observations in that neighborhood are used in the surface fitting if however the center of the neighborhood is on a jump location curve only those observations on one side of the line are used thus blurring is automatically avoided around the jump locations this methodology requires o n k 2 computation where n is the sample size and k is the window width its assumptions on the model are flexible some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

asymptotical minimax recovery of sets with smooth boundaries in this paper optimal rates of convergence are derived for estimates of sets in n dimensional black and white pictures under smoothness conditions it is assumed that the boundaries of the black regions have a smooth parameterisation that is that the boundaries are given by smooth functions from the sphere s n 1 into r n furthermore classes of convex regions are considered two models are studied edge estimation models motivated by image segmentation problems and density support estimation

commuting self adjoint extensions of symmetric operators defined from the partial derivatives we consider the problem of finding commuting self adjoint extensions of the partial derivatives 1 i xj j 1 d with domain cc where the self adjointness is defined relative to l2 and is a given open subset of rd the measure on is lebesgue measure on rd restricted to the problem originates with segal and fuglede and is difficult in general in this paper we provide a representation theoretic answer in the special case when i 2 and i is an open interval we then apply the results to the case when is a d cube id and we describe possible subsets rd such that e id is an orthonormal basis in l2 id

on the ϰth root of a stieltjes moment sequence abstract stieltjes moment sequences a n n 0 whose th roots a n n 0 are stieltjes moment sequences are studied is a fixed integer greater than or equal to 2 a formula connecting the closed supports of representing measures of a n n 0 and a n n 0 is established the relationship between the holes of the supports of these measures is investigated the set of all pairs m n of positive integers for which there exists a stieltjes moment sequence whose square root is a stieltjes moment sequence and both of them have representing measures supported on subsets of 0 of cardinality m and n respectively is described

analytic continuation of local representations of symmetric spaces we prove the following analytic continuation theorem which applies to any virtual representation of any symmetric space g k the problem of passing from the euclidean group to the poincare group appears first to have been addressed and solved this way by klein and landau let g be a lie group k a closed subgroup and an involutive automorphism with k as fixed point subgroup if g k m is the corresponding symmetric lie algebra we form g k im and let g denote the simply connected lie group with g as lie algebra we consider virtual representations of g on a fixed complex hilbert space h adopting the definitions due to j frohlich k osterwalder and e seiler in particular g 1 g possibly unbounded operators for g in a neighborhood of e in g we prove that every such continues analytically to a strongly continuous unitary representation of g on h our theorem extends results due to klein landau frohlich et al and others earlier for special cases previous results were known only for special g k and then only for certain

on favard s theorems abstract in this paper we improve and extend the classical favard s theorems i e favard s theorem of the module containment favard s theorem of linear differential equations we study favard s theory of linear differential equations with piecewise constant argument an example shows that the new module containment is necessary in the study of differential equations with piecewise constant argument the equivalence between almost automorphic functions and n almost periodic ones is studied

fields over unsharp coordinates it has been shown that space time coordinates can exhibit only very few types of short distance structures if described by linear operators they can be continuous discrete or unsharp in one of two ways in the literature various quantum gravity models of space time at short distances point towards one of these two types of unsharpness here we investigate the properties of fields over such unsharp coordinates we find that these fields are continuous but possess only a finite density of degrees of freedom similar to fields on lattices as a special case we recover the shannon sampling theorem of information theory

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

local likelihood tracking of fault lines and boundaries summary we suggest locally parametric methods for estimating curves such as boundaries of density supports or fault lines in response surfaces in a variety of spatial problems the methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane as a function of that point the local likelihood might be a regular likelihood computed locally with kernel weights e g in the case of support boundary estimation or a local version of a likelihood ratio statistic e g in fault line estimation in either case the local likelihood surface represents a function which is relatively large near the target curve and relatively small elsewhere therefore the curve may be estimated as a ridge line of the surface we require only a numerical algorithm for tracking the projection of a ridge into the plane this approach offers several potential advantages over alternative methods first the local log likelihood surface can be graphed and the degree of ridginess assessed visually to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately secondly the local likelihood surface does not need to be computed in anything like its entirety once we have a reasonable approximation to a point on the curve we may track it by numerically walking along the ridge line thirdly the method is appropriate without change for many different types of spatial explanatory vanables gridded stochastic or otherwise three examples are explored in detail fault lines in response surfaces and in intensity or density surfaces and boundaries of supports of probability densities

spectral theory for borel sets in rn of finite measure abstract let be a pair of subsets in r n such that has finite positive lebesgue measure the pair is called a spectral pair if the functions e i x for form an orthogonal total family in l 2 i e when restricted to our present work continues earlier papers on simultaneous diagonalization of vector fields for the special case when is also assumed open however the setting here is more general and the results are stated in the measure theoretic category in considering fundamental domains for lattices for example we allow distinct lattice point translates of the domain to overlap on sets of measure zero rather than the traditional condition of empty overlap two classes of examples of spectral pairs are known fundamental domains and simple factors the latter may be identified in terms of a certain finite group action we show that every spectral pair can be factored in the sense of cartesian products as a product of two pairs of lower dimension where the first factor is a fundamental domain and the second a simple factor the dimensions add a simple factor yields in a natural way a finite covering of an associated compact torus but not every such covering corresponds to a spectral pair we give a complete spectral characterization of the finite coverings and identify those which correspond to simple factors the latter identification is phrased in terms of generators and relations for certain c algebras which are analogous to but different from algebras studied earlier by cuntz and arveson

on the notion of derivo periodicity a de blasi like differentiable multivalued function is shown to have a periodic derivative i e to be derivo periodic if and only if it is a sum of a function of a continuous single valued periodic function linear function and a bounded interval a multivalued constant at the same time the single valued part is derivo periodic a e in the usual sense in the single valued case a characterization of a more general class of derivo periodic acg functions is given derivo periodicity in terms of the clarke subdifferentials and an impossibility of an almost periodic analogy are also discussed the obtained results are finally applied to differential equations and inclusions 2004 published by elsevier inc

nonmonotone equations with large almost periodic forcing terms abstract we consider the scalar differential equation u f u c h t where f u is a jumping nonlinearity and h t is an almost periodic function while c is a real parameter deciding the size of the forcing term the main result is that if h t does not vanish too much in some suitable sense then the equation admits a unique almost periodic solution for large values of the parameter c the class of the h t s to which the result applies is studied in detail it includes all the nontrivial trigonometric polynomials and is generic in the baire sense

commutative algebras of unbounded operators abstract we consider algebras u of unbounded operators defined on a common invariant dense domain d in a hilbert space h we assume that u is generated by a finite number of operators s1 sd satisfying the inequality x s x x s x x 0 for all finite sets of vectors x indexed by 1 d i 0 1 1 i d where s s 1 1 s d d and denotes the inner product in h we show that every such algebra dilates to a commutative algebra in a generally bigger hilbert space the latter algebra is generated by commuting formally normal extension operators n1 nd such that sjx njx x d 1 j d moreover the adjoint operators n j are represented as multiplication operators on a quotient of the space of d valued polynomials in complex variables z1 zd c d n j z j z d z j u z j z d if it is further assumed that d contains a dense space of vectors analytic for each of the operators sj then we prove that the nj s are essentially normal and there is a spectral measure e on the product of the spectrum of the closed normal operators n j such that n i f de and sjx jde x for x d as an application we consider representations of the canonical commutation relations in d degrees of freedom if p1 pd q1 qd denote the momentum resp position operators and sj pj iqj we show that s1 sd admits a representation as above we also show using our analytic vector theorem that the corresponding extension operators n 1 n d are commuting and normal for the schrodinger representation finally we calculate the spectral representation for the operators n j and prove absolute continuity and uniform multiplicity equal infinity the exposition of the paper begins with the case d 1 this is the unbounded version of the subnormal operators which have been studied extensively in the past in the case of bounded operators we stress the differences between the theory for the bounded and the unbounded case even in the case d 1 the applications of the unbounded theory are surprising

almost periodic linear differential equations with non separated solutions a celebrated result by favard states that for certain almost periodic linear differential systems the existence of a bounded solution implies the existence of an almost periodic solution a key assumption in this result is the separation among bounded solutions here we prove a theorem of anti favard type if there are bounded solutions which are non separated in a strong sense sometimes almost periodic solutions do not exist strongly non separated solutions appear when the associated homogeneous system has homoclinic solutions this point of view unifies two fascinating examples by zhikov levitan and johnson for the scalar case our construction uses the ideas of zhikov levitan together with the theory of characters in topological groups

a flexible and fast method for automatic smoothing abstract the choice of smoothing parameter or bandwidth is crucial when applying nonparametric regression estimators such as kernel estimators the optimal choice depends on the data at hand a data driven bandwidth selection close to the optimal one would make these curve estimators objective more reliable and easier to use minimizing residual mean squared error criteria such as cross validation have been frequently proposed to estimate the optimal smoothing parameter empirical and theoretical evidence indicates that cross validation rules and related methods lead to rather variable estimated optimal bandwidths the method presented here builds on estimating the asymptotically optimal bandwidth from the data since estimators for the residual variance and for an asymptotic expression for the bias are plugged into the asymptotic formula such selection rules are called plug in estimators the functional that quantifies bias is approximated by the integrated squared second derivative of the regre

on bounded solutions of almost periodic linear differential systems has an almost periodic solution with frequency module contained in the joint frequency module of a and f whenever it has a bounded solution the method of proof was his so called minimax method favard s result has been extended to equations in banach space and partial differential equations amerio and prouse 11 sibuya 171 sell 151 zhikov and levitan 191 however apart from some results of bochner 3 and cameron 4 cf also fink 10 p 1081 no conditions have been given which ensure that bounded solutions of the homogeneous system 1 itself are almost periodic as shown by an example of lillo 13 p 4061 cf also bochner 3 p 285 it can happen that all the solutions of 1 are bounded that all systems in the hull have the favard property and still no nontrivial solution is almost periodic it is the main purpose of this paper to show nevertheless that favard s minimax method can be used to show that under certain conditions some bounded solutions of 1 are almost periodic in particular if a t a t is real with u f 0 for i j and all systems in the hull have the favard property then 1 has a nontrivial almost periodic solution if it has a nontrivial bounded solution whose components are nonnegative this

tiling the line with translates of one tile a region t is a closed subset of the real line of positive finite lebesgue measure which has a boundary of measure zero call a region t a tile if bbb r can be tiled by measure disjoint translates of t for a bounded tile all tilings of bbb r with its translates are periodic and there are finitely many translation equivalence classes of such tilings the main result of the paper is that for any tiling of bbb r by a bounded tile any two tiles in the tiling differ by a rational multiple of the minimal period of the tiling from it we deduce a structure theorem characterizing such tiles in terms of complementing sets for finite cyclic groups

likelihood based confidence bands for fault lines in response surfaces we consider the problem of constructing asymptotic confidence bands both pointwise and simultaneous for a smooth fault line in a response surface when the design is represented by a point process either deterministic or stochastic with intensity n diverging to infinity the estimator of the fault line is defined as the ridge line on the likelihood surface which arises from locally fitting a model that employs a linear approximation to the fault line to a kernel smooth of the data the construction is based on analysis of the limiting behaviour of perpendicular distance from a point on the true fault line to the nearest point on the ridge we derive asymptotic properties of bias and the limiting distribution of stochastic error this distribution is given by the location of the maximum of a gaussian process with quadratic drift although the majority of attention is focused on the regression problem the limiting distribution is shown to have wider relevance to local likelihood approaches to fault line estimation for density or intensity surfaces

lie algebras of unbounded derivations abstract in this paper some new results on analytic domination of operators and on integrability of lie algebras of operators are proved and then our methods are applied to the study of lie algebras of unbounded derivations in c algebras

anticommuting selfadjoint operators abstract several equivalent definitions of anticommutativity for selfadjoint operators are presented it is shown that the anticommutativity of two selfadjoint operators a and b satisfying af bg bf ag 0 for all and g in the intersection of the domains of a and b is equivalent to the selfadjointness of their sum and difference in studying the structure of anticommuting selfadjoint operators a and b attention may be restricted to the case where both a and b are injective in that case a is unitarily equivalent to a moreover a and b are uniquely determined by a triplet b p q where b is unitary and p and q are commuting injective positive and selfadjoint operators this is a natural correspondence

on global performance of approximations to smooth curves using gridded data approximating boundaries using data recorded on a regular grid induces discrete rounding errors in both vertical and horizontal directions in cases where grid points exhibit at least some degree of randomness an extensive theory has been developed for local polynomial boundary estimators it is inapplicable to regular grids however in this paper we impose strict regularity of the grid and describe the performance of local linear estimators in this context unlike the case of classical curve estimation problems pointwise convergence rates vary erratically along the boundary depending on number theoretic properties of the boundary s slope however average convergence rates expressed in the l 1 metric are much less susceptible to fluctuation we derive theoretical bounds to performance coming within no more than a logarithmic factor of the optimal convergence rate

square mean pseudo almost automorphic process and its application to stochastic evolution equations abstract this paper introduces the concept of the square mean pseudo almost automorphy for a stochastic process also it introduces the properties on the completeness and the composition of the space that consists of such processes with appropriate settings and by virtue of the theory of the semigroups of the operators to an evolution family the banach fixed point theorem and the stochastic analysis techniques this paper investigates the existence the uniqueness and the global stability of the square mean pseudo almost automorphic solutions for a general class of stochastic evolution equations finally this paper provides an illustrative example to justify the practical usefulness of the established theoretical results

comparison of commuting one parameter groups of isometries abstract let a s be two commuting strongly continuous one parametergroups of isometries on a banach space j with generators 5q and 5 andanalytic elements srff srff respectively then it is easy to show that if oais relatively bounded by 5 then srff c saf and in this paper we establishthe inverse implication for unitary one parameter groups on hubert spaces andfor one parameter groups of automorphisms of abelian c algebras it is notknown in general whether the inverse implication holds or not but it does nothold for one parameter semigroups of contractions 1 introduction our intention in this paper is to examine the extent to which the analyticelements determine a strongly continuous one parameter group of isometries ona banach space for example if if v are two one parameter unitary groups on the hubert space sv with analytic elements a jfia v and generators h k and if u commutes with v then we prove that s w u c s jv if

derivations commuting with abelian gauge actions on lattice systems let be an action of a compact abelian groupg on ac algebraa and assume that the fixed point subalgebraa is an af algebra we show that if is a closed derivation ona commuting with and the restriction of toa generates a one parameter group of automorphisms then itself is a generator in particular the result applies if is an infinite product action ofg on a uhf algebra furthermore if in this situation 1 and 2 are two derivations both satisfying the hypotheses on and 1 and 2 have the same restriction toa then there exists a one parameter subgroup of the action with generator 0 such thatd 1 d 2 d 0 is a joint core for the three derivations and 2 1 0 on this core

spectral sets whose spectrum is a lattice with a base abstract let l be a finite subset of euclidean d dimensional space r d and let s be a lattice in r d such that l l l l l s 0 a classification of all lebesgue measurable subsets of r d for which the functions e x exp i 2 x x l s l s l l s s form an orthogonal basis for l 2 is given in terms of geometric and combinatorial data the data is derived from l and s

on diffraction by aperiodic structures this paper gives a rigorous treatment of some aspects of diffraction by aperiodic structures such as quasicrystals it analyses diffraction in the limit of the infinite system through an appropriately defined autocorrelation the main results are a justification of the standard way of calculating the diffraction spectrum of tilings obtained by the projection method and a proof of a variation on a conjecture by bombieri and taylor

almost automorphic linear dynamics by favard theory abstract we prove that linear almost periodic systems always carry almost automorphic dynamics thus confirming in this case a more general conjecture by shen and yi the result is based on the accurate description of the breakdown of the favard separation condition

unitary representations of lie groups with reflection symmetry we consider the following class of unitary representations of some real lie groupgwhich has a matched pair of symmetries described as follows i supposeghas a period 2 automorphism and that the hilbert spaceh carries a unitary operatorjsuch thatj j i e selfsimilarity ii an added symmetry is implied ifh further contains a closed subspacek0having a certainorder covarianceproperty and satisfying thek0 restricted positivity v jv 0 v k0 where is the inner product inh from i ii we get an induced dual representation of an associated dual groupgc all three properties selfsimilarity order covariance and positivity are satisfied in a natural context whengis semisimple and hermitean but whengis the ax b group or the heisenberg group positivity is incompatible with the other two axioms for the infinite dimensional irreducible representations we describe a class ofg containing the latter two which admits a classification of the possible spacesk0 h satisfying the axioms of selfsimilarity and order covariance

favard theory for the adjoint equation and fredholm alternative abstract fredholm alternative is a classical tool of periodic linear equations allowing to describe the existence of periodic solutions of an inhomogeneous equation in terms of the adjoint equation a few partial extensions have been proposed in the literature for recurrent equations our aim is to point out that they have a common root and discuss whether such a root gives rise to a general fredholm type alternative sacker sell spectral theory and favard theory are main ingredients in this discussion a considerable effort is devoted to understand how favard theory is affected by adjunction at least for planar equations

unbounded derivations commuting with compact group actions let be a closed derivation in ac algebra mathfrak a which commutes with an ergodic action of a compact group on mathfrak a then generates ac dynamics of mathfrak a similar results are obtained for non ergodic actions on abelianc algebras and on the algebra of compact operators

spectral theory of finite volume domains in rn abstract let be an arbitrary open subset of r n of finite positive measure and assume the existence of a subset r n such that the exponential functions e exp i 1x1 nxn 1 n form an orthonormal basis for l 2 with normalized measure assume 0 and define subgroups k and a of r n by k 0 r n 2 z a a r n ua m u a m where ut is the unitary representation of r n on l 2 given by ute eit e t r n and where m is the multiplication algebra of l on l2 assume that a is discrete then there is a discrete subgroup d a of dimension n a fundamental domain d for d and finite sets of representers r r r each containing 0 r for a k in k0 and r for a k in a such that is disjoint union of translates of d a r a d neglecting null sets and r d0 if r is a set of representers for d a in d then r k is a translation set for i e r n direct sum neglecting null sets the case a r n corresponds to d d0 and k this last case corresponds in turn to a function theoretic assumption of forelli

harmonic analysis and fractal limit measures induced by representations of a certain c algebra we describe a class of measurable subsets in rd such that l2 has an orthogonal basis of frequencies e x ei2 x x indexed by rd we show that such spectral pairs have a self similarity which may be used to generate associated fractal measures typically with cantor set support the hilbert space l2 does not have a total set of orthogonal frequencies but a harmonic analysis of may be built instead from a natural representation of the cuntz c algebra which is constructed from a pair of lattices supporting the given spectral pair we show conversely that such a pair may be reconstructed from a certain cuntz representation given to act on l2

spectral theory of commuting self adjoint partial differential operators let denote a connected and open subset of rn the existence of n commuting self adjoint operators h1 hn on l2 such that each hj is an extension of i xj acting on cc is shown to be equivalent to the existence of a measure on rn such that f tf the fourier transform of f is unitary from l2 onto it is shown that the support of can be chosen as a subgroup of rn iff h1 hn can be chosen such that the unitary groups generated by h1 hn act multiplicatively on l2 this happens iff after correction by a null set forms a system of representatives for the quotient of rn by some subgroup i e iff is essentially a fundamental domain

the classical moment problem as a self adjoint finite difference operator abstract this is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators among the advantages of this approach is that the nevanlinna functions appear as elements of a transfer matrix and convergence of pade approximants appears as the strong resolvent convergence of finite matrix approximations to a jacobi matrix as a bonus of this we obtain new results on the convergence of certain pade approximants for series of hamburger

analytic vectors and integrability of lie algebra representations abstract we prove that a representation of a finite dimensional lie algebra in a banach space is integrable if it admits a suitable invariant set of analytic vectors

the heat semigroup and integrability of lie algebras abstract let du x 1 s t c t 1 2 0 denote a continuous representation u of a lie group g acting on a banach space b and s the semigroup generated by the closed laplacian associated with a basis of the lie algebra g in the derived representation du we prove there is a c 0 such that du x i s t c t 1 2 0 for all xi in the basis of g and we also prove that this property is characteristic for integrability of a lie algebra of operators satisfying the usual dissipativity requirements for generators of one parameter groups the basic estimate is established from bounds on the heat kernel associated with s

jump surface estimation edge detection and image restoration surface estimation is important in many applications when conventional smoothing procedures e g running averages local polynomial kernel smoothing procedures smoothing spline procedures are used for estimating jump surfaces from noisy data jumps are blurred at the same time when noise is removed in recent years new smoothing methodologies have been proposed in the statistical literature for detecting jumps in surfaces and for estimating jump surfaces with jumps preserved we provide a review of these methodologies because a monochrome image can be considered a jump surface of the image intensity function with jumps at the outlines of objects edge detection and image restoration problems in image processing are closely related to the jump surface estimation problem in statistics we also review major methodologies on edge detection and image restoration and discuss connections and differences among these methods and related methods in the statistical literature

cube root asymptotics we establish a new functional central limit theorem for empirical processes indexed by classes of functions in a neighborhood of a fixed parameter point an n 1 3 rescaling of the parameter is compensated for by an n 2 3 rescaling of the empirical measure resulting in a limiting gaussian process by means of a modified continuous mapping theorem for the location of the maximizing value we deduce limit theorems for several statistics defined by maximization or constrained minimization of a process derived from the empirical measure these statistics include the short rousseeuw s least median of squares estimator manski s maximum score estimator and the maximum likelihood estimator for a monotone density the limit theory depends on a simple new sufficient condition for a gaussian process to achieve its maximum almost surely at a unique point

the complex moment problem and subnormality a polar decomposition approach abstract it has been known that positive definiteness does not guarantee a bisequence to be a complex moment however it turns out that positive definite extendibility does theorems 1 and 22 and this is the main theme of this paper the main tool is generally understood polar decomposition to strengthen applicability of our approach we work out a criterion for positive definite extendibility in a fairly wide context theorems 9 and 29 all this enables us to prove characterizations of subnormality of unbounded operators having invariant domain theorems 37 and 39 and their further applications theorems 41 and 43 and a description of the complex moment problem on real algebraic curves theorems 52 and 56 the latter question is completed in the appendix in which we relate the complex moment problem to the two dimensional real one with emphasis on real algebraic sets