frame expansions in separable banach spaces abstract banach frames are defined by straightforward generalization of hilbert space frames we characterize banach frames and x d frames in separable banach spaces and relate them to series expansions in banach spaces in particular our results show that we can not expect banach frames to share all the nice properties of frames in hilbert spaces

interpolation sets and the bohr topology of locally compact groups abstract rosenthal s theorem describing those banach spaces containing no copy of l 1 is extended to topological groups replacing l 1 basis by interpolation sets in the sense of hartman and ryll nardzewsky colloq math 12 1964 23 39 this extension provides a characterization of those locally compact groups containing no interpolation sets and of those locally compact groups which respect compactness i e such that every bohr compact subset is compact the approach followed in this paper sheds some light on other questions related to the duality theory of non abelian locally compact groups

reproducing pairs and the continuous nonstationary gabor transform on lca groups in this paper we introduce and investigate the concept of reproducing pairs as a generalization of continuous frames reproducing pairs yield a bounded analysis and synthesis process while the frame condition can be omitted at both stages moreover we will investigate certain continuous frames resp reproducing pairs on lca groups which can be described as a continuous version of nonstationary gabor systems and state sufficient conditions for these systems to form a continuous frame resp reproducing pair as a byproduct we identify the structure of the frame operator resp resolution operator we will apply our results to systems generated by a unitary action of a subset of the affine weyl heisenberg group in this setup will also serve as a nontrivial example of a system for which whereas continuous frames exist no dual system with the same structure exists even if we drop the frame property

existence and uniqueness of monotone measure preserving maps introduction given a pair of borel probability measures and v on rd it is natural to ask whether v can be obtained from by redistributing its mass in a canonical way in the case of the line d 1 the answer is clear as long as both measures are free from atoms x v x 0mthere is a map y x of the line to itself for which uniquely determined almost everywhere this map may be taken to be nondecreasing by a suitable choice of y x e rw o at the remaining points interpreting and v as the initial and final distribution of a one dimensional fluid the transformation y x gives a rearrangement of fluid particles yielding final state v from the initial state this rearrangement is characterized by the fact that it preserves particle ordering obviating any need for two particles to cross although the generalization of this construction to higher dimensions is the focus of this paper the one dimensional case will be pursued slightly further when the measures are absolutely continuous with respect to lebesgue d x f x dx and dr y g y dymthen formally at least neglecting regularity issues the fundamental theorem of calculus yields

representation of the inverse of a frame multiplier certain mathematical objects appear in a lot of scientific disciplines like physics signal processing and naturally mathematics in a general setting they can be described as frame multipliers consisting of analysis multiplication by a fixed sequence called the symbol and synthesis in this paper we show a surprising result about the inverse of such operators if any as well as new results about a core concept of frame theory dual frames we show that for semi normalized symbols the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones furthermore one of those dual frames is uniquely determined and the other one can be arbitrarily chosen we investigate sufficient conditions for the special case when both dual frames can be chosen to be the canonical duals in connection to the above we show that the set of dual frames determines a frame uniquely furthermore for a given frame the union of all coefficients of its dual frames is dense in l2 we also introduce a class of frames called pseudo coherent frames which includes gabor frames and coherent frames and investigate invertible pseudo coherent frame multipliers allowing a classification for frame type operators for these frames finally we give a numerical example for the invertibility of multipliers in the gabor case

the unconditional constants for hilbert space frame expansions abstract the most fundamental notion in frame theory is the frame expansion of a vector although it is well known that these expansions are unconditionally convergent series no characterizations of the unconditional constant were known this has made it impossible to get accurate quantitative estimates for problems which require using subsequences of a frame we will prove some new results in frame theory by showing that the unconditional constants of the frame expansion of a vector in a hilbert space are bounded by b a where a b are the frame bounds of the frame tight frames thus have unconditional constant one which we then generalize by showing that bessel sequences have frame expansions with unconditional constant one if and only if the sequence is an orthogonal sum of tight frames we give further results concerning frame expansions in which we examine when b a is actually attained or not we end by discussing the connections of this work to frame multipliers these results hold in both real and complex hilbert spaces

hamiltonians defined by biorthogonal sets in some recent papers studies on biorthogonal riesz bases have found renewed motivation because of their connection with pseudo hermitian quantum mechanics which deals with physical systems described by hamiltonians that are not self adjoint but may still have real point spectra also their eigenvectors may form riesz not necessarily orthonormal bases for the hilbert space in which the model is defined those riesz bases allow a decomposition of the hamiltonian as already discussed in some previous papers however in many physical models one has to deal not with orthonormal bases or with riesz bases but just with biorthogonal sets here we consider the more general concept of quasi basis and we show a series of conditions under which a definition of non self adjoint hamiltonian with purely point real spectra is still possible

canonical forms of unconditionally convergent multipliers multipliers are operators that combine frame like analysis a multiplication with a fixed sequence called the symbol and synthesis they are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing it is known that bounded symbols and bessel sequences guarantee unconditional convergence in this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers in particular we show that under mild conditions unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence into multipliers with symbol 1 and bessel sequences called multipliers in canonical form

remarks on fourier multipliers and applications to the wave equation exploiting continuity properties of fourier multipliers on modulation spaces and wiener amalgam spaces we study the cauchy problem for the nlw equation local wellposedness for rough data in modulation spaces and wiener amalgam spaces is shown the results formulated in the framework of modulation spaces refine those in a benyi k a okoudjou local well posedness of nonlinear dispersive equations on modulation spaces preprint april 2007 available at arxiv 0704 0833v1 the same arguments may apply to obtain local wellposedness for the nlkg equation

multivariate quantiles and multiple output regression quantiles from l1 optimization to halfspace depth a new multivariate concept of quantile based on a directional version of koenker and bassett s traditional regression quantiles is introduced for multivariate location and multiple output regression problems in their empirical version those quantiles can be computed efficiently via linear programming techniques consistency bahadur representation and asymptotic normality results are established most importantly the contours generated by those quantiles are shown to coincide with the classical halfspace depth contours associated with the name of tukey this relation does not only allow for efficient depth contour computations by means of parametric linear programming but also for transferring from the quantile to the depth universe such asymptotic results as bahadur representations finally linear programming duality opens the way to promising developments in depth related multivariate rank based inference

a dichotomy property for locally compact groups abstract we extend to metrizable locally compact groups rosenthal s theorem describing those banach spaces containing no copy of l 1 for that purpose we transfer to general locally compact groups the notion of interpolation i 0 set which was defined by hartman and ryll nardzewsky 24 for locally compact abelian groups thus we prove that for every sequence g n n in a locally compact group g then either g n n has a weak cauchy subsequence or contains a subsequence that is an i 0 set this result is subsequently applied to obtain sufficient conditions for the existence of sidon sets in a locally compact group g an old question that remains open since 1974 see 31 and 19 finally we show that every locally compact group strongly respects compactness extending thereby a result by comfort trigos arrieta and wu 13 who established this property for abelian locally compact groups

an operator theory approach to the approximate duality of hilbert space frames abstract the purpose of this paper is to state some properties and applications of approximate duals constructed by bounded operators it is shown that there are close relationships between this kind of approximate duals and some concepts in frame theory such as the closeness of two frames nearly parseval and scalable frames we get some equivalent conditions for a frame to be close to another frame in terms of approximate duals also using approximate duals some conditions for a frame to be nearly parseval or scalable are derived moreover some other properties of approximate duals are obtained in particular some characterizations of them are stated and some of their applications for gabor frames and erasures are given especially the excesses of them are considered

u cross gram matrices and their invertibility abstract the gram matrix is defined for bessel sequences by combining synthesis with subsequent analysis operators if different sequences are used and an operator u is inserted we reach so called u cross gram matrices this can be seen as reinterpretation of the matrix representation of operators using frames in this paper we investigate some necessary or sufficient conditions for schatten p class properties and the invertibility of u cross gram matrices in particular we show that under mild conditions the pseudo inverse of a u cross gram matrix can always be represented as a u cross gram matrix with dual frames of the given ones we link some properties of u cross gram matrices to approximate duals finally we state several stability results more precisely it is shown that the invertibility of u cross gram matrices is preserved under small perturbations

bessel fusion multipliers in this paper we study properties of a bessel multiplier when the symbol involved belongs to lp furthermore we introduce the concept of bessel fusion multiplier which generalizes a bessel multiplier for bessel fusion sequences we study their behavior when the symbol belongs to lp and some continuity properties

interpolation sets in spaces of continuous metric valued functions abstract let x and k be a cech complete topological group and a compact group respectively we prove that if g is a non equicontinuous subset of c h o m x k the set of all continuous homomorphisms of x into k then there is a countably infinite subset l g such that l k x is canonically homeomorphic to the stone cech compactifcation of the natural numbers as a consequence if g is an infinite subset of c h o m x k such that for every countable subset l g and compact separable subset y x it holds that either l k y has countable tightness or l k y c then g is equicontinuous given a topological group g denote by g the algebraic group g equipped with the bohr topology it is said that g respects a topological property p when g and g have the same subsets satisfying p as an application of our main result we prove that if g is an abelian locally quasiconvex locally k group then the following holds i g respects any compact like property p stronger than or equal to functional boundedness ii g strongly respects compactness

an approach to multivariate covariate dependent quantile contours with application to bivariate conditional growth charts multivariate quantile contours are useful in numerous applications and have been studied in different contexts however no easy solutions exist when dynamic and conditional quantile contours are needed without strong distributional assumptions in this article we propose a new form of bivariate quantile contours and a two stage estimation procedure to take time effect into account the proposed procedure relies on quantile regression for longitudinal data and is flexible to include potentially important covariates as necessary in addition we propose a visual model assessment tool and discuss a practical guideline for model selection the performance of the proposed methodology is demonstrated by a simulation study as well as an application to joint height weight screening of young children in the united states we construct bivariate growth charts by a nested sequence of age dependent and covariate varying quantile contours of height and weight and use it to locate an individual subject s percentile

compact subsets of the first baire class perhaps the earliest results about pointwise compact sets of baire class 1 functions are the two selection theorems of e helly found in most of the standard texts on real variable see e g lo n these two theorems are really theorems about a particular example of a compact set of baire class 1 functions known today as helly space the space of all nondecreasing functions from the unit interval i 0 1 into itself more recently the notion of baire class 1 function turned out to also be important in some areas of functional analysis see r3 for example odell and rosenthal or showed that the double dual of a separable banach space e with the weak topology consists only of baire class 1 functions defined on the unit ball of e if and only if the space e contains no subspace isomorphic to 1 this resulted in a renewed interest in this class of spaces for example building on the work of rosenthal r2 bourgain fyemlin and talagrand bft proved analogues of the two theorems of helly for the whole first baire class using their results godefroy go showed that this class of spaces enjoys some interesting permanence properties for example if a compact space k is representable as a compact set of baire class 1 functions then so is p k the space of all radon probability measures on k with the weak topology some further permanence properties of this class of spaces were obtained by marciszewski ml m2 and an excellent survey of the early results is given by r pol po2 our paper is an attempt towards a fine structure theory of compact subsets of first baire class the first result that we give is a positive answer to a natural question one usually asks in such a context

basic definition and properties of bessel multipliers this paper introduces the concept of bessel multipliers these operators are defined by a fixed multiplication pattern which is inserted between the analysis and synthesis operators the proposed concept unifies the approach used for gabor multipliers for arbitrary analysis synthesis systems which form bessel sequences like wavelet or irregular gabor frames the basic properties of this class of operators are investigated in particular the implications of summability properties of the symbol for the membership of the corresponding operators in certain operator classes are specified as a special case the multipliers for riesz bases are examined and it is shown that multipliers in this case can be easily composed and inverted finally the continuous dependence of a bessel multiplier on the parameters i e the involved sequences and the symbol in use is verified using a special measure of similarity of sequences

unconditionally convergent multipliers and bessel sequences abstract we prove that every unconditionally summable sequence in a hilbert space can be factorized as the product of a square summable scalar sequence and a bessel sequence some consequences on the representation of unconditionally convergent multipliers are obtained thus providing positive answers to a conjecture by balazs and stoeva in some particular cases

nonparametric estimation of nonadditive random functions we present estimators for nonparametric functions that are nonadditive in unobservable random terms the distributions of the unobservable random terms are assumed to be unknown we show that when a nonadditive nonparametric function is strictly monotone in an unobservable random term and it satisfies some other properties that may be implied by economic theory such as homogeneity of degree one or separability the function and the distribution of the unobservable random term are identified we also present convenient normalizations to use when the properties of the function other than strict monotonicity in the unobservable random term are unknown the estimators for the nonparametric function and for the distribution of the unobservable random term are shown to be consistent and asymptotically normal we extend the results to functions that depend on a multivariate random term the results of a limited simulation study are presented

adaptive methods for boundary integral equations complexity and convergence estimates this paper is concerned with developing numerical techniques for the adaptive application of global operators of potential type in wavelet coordinates this is a core ingredient for a new type of adaptive solvers that has so far been explored primarily for pdes we shall show how to realize asymptotically optimal complexity in the present context of global operators asymptotically optimal means here that any target accuracy can be achieved at a computational expense that stays proportional to the number of degrees of freedom within the setting determined by an underlying wavelet basis that would ideally be necessary for realizing that target accuracy if full knowledge about the unknown solution were given the theoretical findings are supported and quantified by the first numerical experiments

m estimation convexity and quantiles the paper develops a class of extensions of the univariate quantile function to the multivariate case m quantiles related in a certain way to m parameters of a probability distribution and their m estimators the spatial geometric quantiles recently introduced by koltchinskii and dudley and by chaudhuri as well as the regression quantiles of koenker and basset are the examples of the m quantile function discussed in the paper we study the main properties of m quantiles and develop the asymptotic theory of empirical m quantiles we use m quantiles to extend l parameters and l estimators to the multivariate case to introduce a bootstrap test for spherical symmetry of a multivariate distribution and to extend the notion of regression quantiles to multiresponse linear regression models

vector quantile regression an optimal transport approach we propose a notion of conditional vector quantile function and a vector quantile regression a conditional vector quantile function cvqf of a random vector y taking values in d given covariates z z taking values in k is a map u qy z u z which is monotone in the sense of being a gradient of a convex function and such that given that vector u follows a reference non atomic distribution fu for instance uniform distribution on a unit cube in d the random vector qy z u z has the distribution of y conditional on z z moreover we have a strong representation y qy z u z almost surely for some version of u the vector quantile regression vqr is a linear model for cvqf of y given z under correct specification the notion produces strong representation y u f z for f z denoting a known set of transformations of z where u u f z is a monotone map the gradient of a convex function and the quantile regression coefficients u u have the interpretations analogous to that of the standard scalar quantile regression as f z becomes a richer class of transformations of z the model becomes nonparametric as in series modelling a key property of vqr is the embedding of the classical monge kantorovich s optimal transportation problem at its core as a special case in the classical case where y is scalar vqr reduces to a version of the classical qr and cvqf reduces to the scalar conditional quantile function several applications to diverse problems such as multiple engel curve estimation and measurement of financial risk are considered

invariance principles for sums of banach space valued random elements and empirical processes almost sure and probability invariance principles are established for sums of independent not necessarily measurable random elements with values in a not necessarily separable banach space it is then shown that empirical processes readily fit into this general framework thus we bypass the problems of measurability and topology characteristic for the previous theory of weak convergence of empirical processes