lattice tiling and density conditions for subspace gabor frames abstract the well known density theorem in time frequency analysis establishes the connection between the existence of a gabor frame g g a b e 2 i b m x g x a n m n z d for l 2 r d and the density of the time frequency lattice a z d b z d this is also tightly related to lattice tiling and packing in this paper we investigate the density theorem for gabor systems in l 2 s with s being an a z d periodic subset of r d we characterize the existence of a gabor frame for l 2 s in terms of a condition that involves the haar measure of the group generated by a z d and b t 1 z d this new characterization is used to recover the density theorem and several related known results in the literature additionally we apply this approach to obtain the density theorems for multi windowed and super gabor frames for l 2 s

convergence in measure with respect to a matrix valued measure and some matrix completion problems for a matrix valued measure m we introduce a notion of convergence in measure m which generalizes the notion of convergence in measure with respect to a scalar measure and takes into account the matrix structure of m let s be a subset of the set of matrices of given size it is easy to see that the set of s valued measurable functions is closed under convergence in measure with respect to a matrix valued measure if and only if s is a closed set i e if and only if sp is closed for any orthoprojector p we discuss the behaviour of closed sets under operations of linear algebra and the closedness of particular classes of matrices

the existence of subspace wavelet sets let h be a reducing subspace of l2 rd that is a closed subspace of l2 rd with the property that f amt l h for any f h m z and l zd where a is a d d expansive matrix it is known that h is a reducing subspace if and only if there exists a measurable subset m of rd such that atm m and f h l2 rd m under some given conditions of m it is known that there exist a dilation subspace wavelet sets iwith respect to h in this paper we prove that this holds in general

free monotone transport without a trace we adapt the free monotone transport results of guionnet and shlyakhtenko to the type iii case as a direct application we obtain that the q deformed araki woods algebras are isomorphic for sufficiently small q

l 2 homology for von neumann algebras the aim of this paper is to introduce a notion of l homology in the context of von neumann algebras finding a suitable co homology theory for von neumann algebras has been a dream for several generations see kr71a kr71b jkr72 ss95 and references therein one s hope is to have a powerful invariant to distinguish von neumann algebras unfortunately little positive is known about the kadison ringrose cohomology h b m m except that it vanishes in many cases furthermore there does not seem to be a good connection between the bounded cohomology theory of a group and of the bounded cohomology of its von neumann algebra our interest in developing an l cohomology theory was revived by the introduction of lcohomology invariants in the field of ergodic equivalence relations in the paper of gaboriau gab02 his results in particular imply that l betti numbers 2 i of a discrete group are the same for measure equivalent groups i e for groups that can generate isomorphic ergodic measure preserving equivalence relations parallels between the worlds of von neumann algebras and measurable equivalence relations have been noted for a long time starting with the parallel between the work of murray and von neumann mvn and that of h dye dy thus there is hope that an invariant of a group that survives measure equivalence will survive also von neumann algebra equivalence i e will be an invariant of the von neumann algebra of the group the original motivation for our construction comes from the well understood analogy between the theory of ii1 factors and that of discrete groups based on the theory of correspondences con con94 this analogy has been remarkably efficient to transpose analytic properties such as amenability or property t from the group context to the factor context con80 cj and more recently in the breakthrough work of popa popa con03 we use the theory of correspondences together with the algebraic description of l betti numbers given by luck his definition involves the computation of the algebraic group homology with coefficients in the group von neumann algebra following the guiding idea that the category of bimodules over a von neumann algebra is the analogue of the category of modules over a group we are led to the following algebraic definition of l homology of a von neumann algebra m

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

vector valued weak gabor dual frames on discrete periodic sets the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples the notion of weak dual frames is a generalization of that of dual frames gabor analysis on discrete periodic sets has potential applications in signal processing this paper addresses vector valued weak gabor dual frames on discrete periodic sets we introduce the notions of its weak oblique gabor dual weak gabor duals of types i and ii for a gabor system on a discrete periodic set using the zak transform matrix method we characterize these three kinds of weak duals and their uniqueness finally we give an explicit expression of a class of weak gabor duals and provide some examples

freely independent random variables with non atomic distributions 2015 american mathematical society we examine the distributions of non commutative polynomials of non atomic freely independent random variables in particular we obtain an analogue of the strong atiyah conjecture for free groups thus proving that the measure of each atom of any n n matricial polynomial of nonatomic freely independent random variables is an integer multiple of n 1 in addition we show that the cauchy transform of the distribution of any matricial polynomial of freely independent semicircular variables is algebraic and thus the polynomial has a distribution that is real analytic except at a finite number of points

the characterization of a class of multivariate mra and semi orthogonal parseval frame wavelets abstract let a be a d d expansive matrix with deta 2 this paper addresses parseval frame wavelets pfws in the setting of reducing subspaces of l 2 r d we prove that all semi orthogonal pfws semi orthogonal mra pfws are precisely the ones with their dimension functions being non negative integer valued 0 or 1 we also characterize all mra pfws some examples are provided

levels of subnormality abstract three progressively more stringent constraints are added to the requirement that a matrix be subnormal for the case that a is normal this results in the definition of four classes of subnormal matrices two of these classes are categorized and it is shown by way of examples and counterexamples that these four classes are nontrivial and distinct

wavelet frames for not necessarily reducing affine subspaces ii the structure of affine subspaces abstract this is a continuation of the investigation into the theory of wavelet frames for general affine subspaces the main focus of this paper is on the structural properties of affine subspaces we show that every affine subspace is the orthogonal direct sum of at most three purely non reducing subspaces while every reducing subspace with respect to the dilation and translation operators is the orthogonal direct sum of two purely non reducing ones this result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine motivated by the fundamental question as to whether every affine subspace is singly generated and by a recent result that every singly generated purely non reducing subspace admits a singly generated wavelet frame we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of small size as a consequence we establish a connection between the above mentioned two questions

gabor super frames with hermite functions we investigate vector valued gabor frames sometimes called gabor superframes based on hermite functions hn let h h0 h1 hn be the vector of the first n 1 hermite functions we give a complete characterization of all lattices lambda subseteq mathbb r 2 such that the gabor system rm e 2 pi i lambda 2 t bf h t lambda 1 lambda lambda 1 lambda 2 in lambda is a frame for l 2 mathbb r mathbb c n 1 as a corollary we obtain sufficient conditions for a single hermite function to generate a gabor frame and a new estimate for the lower frame bound the main tools are growth estimates for the weierstrass function a new type of interpolation problem for entire functions on the bargmann fock space and structural results about vector valued gabor frames

hyponormal 2 2 matrices are subnormal abstract given an n n matrix a and a k n matrix b both complex define h a b a a aa b b where a and b are the conjugate transposes of a and b respectively the matrix a b is said to be hyponormal if h a b is positive semidefinite we answer a question of p r halmos hyponormality implies subnormality when k n 2 but not when k n 2

construction and reconstruction of tight framelets and wavelets via matrix mask functions abstract the paper develops construction procedures for tight framelets and wavelets using matrix mask functions in the setting of a generalized multiresolution analysis gmra we show the existence of a scaling vector of a gmra such that its first component exhausts the spectrum of the core space near the origin the corresponding low pass matrix mask has an especially advantageous form enabling an effective reconstruction procedure of the original scaling vector we also prove a generalization of the unitary extension principle for an infinite number of generators this results in the construction scheme for tight framelets using low pass and high pass matrix masks generalizing the classical mra constructions we prove that our scheme is flexible enough to reconstruct all possible orthonormal wavelets as an illustration we exhibit a pathwise connected class of non msf non mra wavelets sharing the same wavelet dimension function

the structure of shift invariant subspaces of l2 rn using the range function approach to shift invariant spaces in l2 rn we give a simple characterization of frames and riesz families generated by shifts of a countable set of generators in terms of their behavior on subspaces of l2 zn this in turn gives a simplified approach to the analysis of frames and riesz families done by gramians and dual gramians we prove a decomposition of a shift invariant space into the orthogonal sum of spaces each of which is generated by a quasi orthogonal generator as an application of this fact we characterize shift preserving operators in terms of range operators and prove some facts about the dimension function

efficient algorithms for discrete gabor transforms on a nonseparable lattice the discrete gabor transform dgt is the most commonly used transform for signal analysis and synthesis using a linear frequency scale it turns out that the involved operators are rich in structure if one samples the discrete phase space on a subgroup most of the literature focuses on separable subgroups in this paper we will survey existing methods for a generalization to arbitrary groups as well as present an improvement on existing methods comparisons are made with respect to the computational complexity and the running time of optimized implementations in the c programming language the new algorithms have the lowest known computational complexity for nonseparable lattices and the implementations are freely available for download by summarizing general background information on the state of the art this article can also be seen as a research survey sharing with the readers experience in the numerical work in gabor analysis

dirichlet forms and markovian semigroups on standard forms of von neumann algebras we characterize w continuous markovian semigroups on a von neumann algebra m which are 0 symmetric w r t a faithful normal state 0in m in terms of quadratic forms on the hilbert space h of a standard form m h p j we characterize also symmetric strongly continuous contraction semigroups on a real hilbert space h which leave invariant a closed convex set in h in terms of a contraction property of the associated quadratic forms we apply the results to give criteria of essential selfadjointness for quadratic form sums and to give a characterization of w continuous markovian semigroups on m which commute with the modular automorphism group 0t

on finite free fisher information for eigenvectors of a modular operator abstract suppose m is a von neumann algebra equipped with a faithful normal state and generated by a finite set g g g 2 we show that if g consists of eigenvectors of the modular operator with finite free fisher information then the centralizer m is a ii 1 factor and m is either a type ii 1 factor or a type iii factor 0 1 depending on the eigenvalues of g furthermore m m c m does not have property and m is full provided it is type iii 0 1

rational time frequency super gabor frames and their duals abstract this paper addresses general super gabor systems with rational time frequency product lattices a zak transform matrix method is developed for such super gabor systems we characterize complete super gabor systems and super gabor frames given a super gabor frame we obtain an explicit expression of its canonical dual and a parametrization of all its super gabor duals and prove that the canonical dual is the norm minimal one among all super gabor duals we also prove that a super gabor frame has a unique super gabor dual if and only if it is a riesz basis

the structure of finitely generated shift invariant spaces in l2 rd a simple characterization is given of finitely generated subspaces of l2 rd which are invariant under translation by any multi integer and is used to give conditions under which such a space has a particularly nice generating set namely a basis and more than that a basis with desirable properties such as stability orthogonality or linear independence the last property makes sense only for local spaces i e shift invariant spaces generated by finitely many compactly supported functions and special attention is paid to such spaces as an application we prove that the approximation order provided by a given local space is already provided by the shift invariant space generated by just one function with this function constructible as a finite linear combination of the finite generating set for the whole space hence compactly supported this settles a question of some 20 years standing

true bargmann transforms for rank one automorphic functions associated with landau levels for a given discrete subgroup of c and given real number 0 we study the spectral properties of the magnetic laplacian operator acting on the hilbert space l 2 c of l2 automorphic functions see below for notations we show that its spectrum is reduced to the eigenvalues m m 0 1 we also give a concrete description of each eigenspace in terms of the hermite polynomials this description will be used to characterize the range of true bargmann transform of l2 periodic functions on the real line r

free fisher information with respect to a completely positive map and cost of equivalence relations given a family of isometries in a tracial von neumann algebra m a unital subalgebra b m and a completely positive map we define the free fisher information of relative to b and using this notion we define the free dimension of relative to b id

characterizations of subnormal matrices abstract suppose that is said to be subnormal if there exist matrices x and y such that is normal we develop characterizations of subnormal matrices as well as constructive techniques for normal extensions for example we show that geometrical properties of certain subsets of the spectrum of a determine if normal extensions exist our results add to as well as generalize those of an earlier paper

a note about proving non γ under a finite non microstates free fisher information assumption we prove that if x1 xn n 1 are self adjoints in a w probability space with finite non microstates free fisher information then the von neumann algebra w x1 xn they generate doesn t have property especially is not amenable this is an analog of a well known result of voiculescu for microstates free entropy we also prove factoriality under finite non microstates entropy

vector valued gabor frames associated with periodic subsets of the real line the notion of vector valued frame also called superframe was first introduced by balan in the context of multiplexing it has significant applications in mobile communication satellite communication and computer area network for vector valued gabor analysis existent literatures mostly focus on l 2 r c l instead of its subspace let a 0 and s be an a z periodic measurable set in r i e s a z s this paper addresses gabor frames in l 2 s c l with rational time frequency product they can model vector valued signals to appear periodically but intermittently and the projections of gabor frames in l 2 r c l onto l 2 s c l cannot cover all gabor frames in l 2 s c l if s r by introducing a suitable zak transform matrix we characterize completeness and frame condition of gabor systems obtain a necessary and sufficient condition on gabor duals of type i resp ii for a general gabor frame and establish a parametrization expression of gabor duals of type i resp ii all our conclusions are closely related to corresponding zak transform matrices this allows us to easily realize these conclusions by designing the corresponding matrix valued functions an example theorem is also presented to illustrate the efficiency of our method

the analogues of entropy and of fisher s information measure in free probability theory i analogues of the entropy and fisher information measure for random variables in the context of free probability theory are introduced monotonicity properties and an analogue of the cramer rao inequality are proved

derivations as square roots of dirichlet forms abstract the aim of this work is to analyze the structure of a tracially symmetric dirichlet form on a c algebra in terms of a killing weight and a closable derivation taking values in a hilbert space with a bimodule structure it is shown that the generator of the associate markovian semigroup always appears in a natural way as the divergence of a closable derivation applications are shown to the decomposition of dirichlet forms and to the construction of differential calculus on metric spaces

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

the duals of gabor frames on discrete periodic sets due to its good potential for digital signal processing analysis the discrete version of gabor analysis has interested many mathematicians in this paper we focus on gabor analysis on discrete periodic sets which can model a signal to appear periodically but intermittently for a gabor frame on a discrete periodic set three types of dual frames with gabor structure are introduced and investigated in terms of discrete zak transform given a gabor frame on a discrete periodic set a characterization of a gabor system being separately its dual of types i iii is obtained a method to construct these three types of duals is presented an explicit expression of its canonical dual is established the uniqueness of these three types of duals is characterized some examples are also provided to illustrate the generality of the theory

affine systems in l2 rd the analysis of the analysis operator abstract discrete affine systems are obtained by applying dilations to a given shift invariant system the complicated structure of the affine system is due first and foremost to the fact that it is not invariant under shifts affine frames carry the additional difficulty that they are global in nature it is the entire interaction between the various dilation levels that determines whether the system is a frame and not the behaviour of the system within one dilation level we completely unravel the structure of the affine system with the aid of two new notions the affine product and a quasi affine system this leads to a characterization of affine frames the induced characterization of tight affine frames is in terms of exact orthogonality relations that the wavelets should satisfy on the fourier domain several results such as a general oversampling theorem follow from these characterizations most importantly the affine product can be factored during a multiresolution analysis construction and this leads to a complete characterization of all tight frames that can be constructed by such methods moreover this characterization suggests very simple sufficient conditions for constructing tight frames from multiresolution of particular importance are the facts that the underlying scaling function does not need to satisfy any a priori conditions and that the freedom offered by redundancy can be fully exploited in these constructions an

free entropy dimension and regularity of non commutative polynomials abstract we show that the spectral measure of any non constant non commutative polynomial evaluated at a non commutative n tuple cannot have atoms if the free entropy dimension of that n tuple is n see also work of mai speicher and weber under stronger assumptions on the n tuple we prove that the spectral measure of any non constant non commutative polynomial function is not singular and measures of intervals surrounding any point may not decay slower than polynomially as a function of the interval s length

a linear cost algorithm to compute the discrete gabor transform in this paper we propose an alternative efficient method to calculate the gabor coefficients of a signal given a synthesis window with a support of size much lesser than the length of the signal the algorithm uses the canonical dual of the window which does not need to be calculated beforehand and achieves a computational cost that is linear with the signal length in both analysis and synthesis this is done by exploiting the block structure of the matrices and using an ad hoc cholesky decomposition of the gabor frame matrix

the separability problem and normal completions abstract in this paper we introduce new normal completion problems that are directly related to the separability problem in quantum information in fact we show that generically the n m separability problem may be reduced in dimension via a multi matrix normal completion problem specifying the result for the 2 m separability problem yields the equivalence to a normal completion problem that is a variation of the one introduced by p halmos in addition upper bounds on the number of states in a separable representation are given in terms of the normal completions

multiwindow real valued discrete gabor transform and its fast algorithms based on the biorthogonal analysis approach a multiwindow real valued discrete gabor transform m rdgt for periodic sequences is presented to efficiently analyze the dynamic time frequency content of a signal containing components with multiple and or time varying frequencies the m rdgt offers a computationally efficient implementation as well as a real valued formulation of the multiwindow complex valued discrete gabor transform m cdgt the completeness condition of the m rdgt is proved to be equivalent to its biorthogonality constraint between analysis windows and synthesis windows the m rdgt can utilize the fast discrete hartley transform algorithms for fast computation and has a simple relationship with the m cdgt such that its coefficients can be directly computed from the m rdgt coefficients therefore the m rdgt offers an efficient method to compute the m cdgt since the analyzed sequence analysis and synthesis windows in the existing m cdgt must have an equal period if the period of a sequence is very long solving its windows requires a huge amount of computation and memory and could lead to numerical instability to overcome this problem a modified m rdgt for long periodic or even infinite sequences is presented and its corresponding biorthogonality constraint between analysis windows and synthesis windows is modified in which the period of the analysis and synthesis windows is independent of the period of a analyzed sequence so that one can apply short windows to process any long periodic or even in finite sequence finally the multirate based parallel implementation of the m rdgt is presented which has shown to be effective and fast for time frequency analysis

convergence of an iterative time variant filtering based on discrete gabor transform an iterative time variant filtering based on the discrete gabor transform dgt has been previously proposed by the authors in this article we present a proof of the convergence of the iterative algorithm under a sufficient condition on the analysis and synthesis window functions of the dgt in the meantime we show that the iterative algorithm refines the least squares solution

discrete multi gabor expansions a discrete multi gabor expansion dmge is developed to meet the requirements of localized and refined time frequency tf representation of signals the dmge uses multiple windows and their translations and complex modulations as synthesis or analysis waveforms it includes and generalizes the metaplectic translation modulation and dilation representations which are useful in signal analysis uniform nonuniform and proportional time sampling schemes are analyzed the fundamental features and the importance of the dmge are discussed we focus on the construction of dmge and deriving fast algorithms for the computation of related multi analysis sequences with matrix algebra the algorithms derived apply to both multi gabor expansions and uni window gabor expansions another useful feature of the dmge lies in the fact that the multi gabor transform can be realized in a parallel fft based implementation structure examples of dmge and their applications to tf analysis are also discussed