dual gabor frames theory and computational aspects we consider a general method for constructing dual gabor elements different from the canonical dual our approach is based on combining two gabor frames such that the generated frame type operator s sub g spl gamma is nonsingular we provide necessary and sufficient conditions on the gabor window functions g and spl gamma such that s sub g spl gamma is nonsingular for rational oversampling considering both the continuous time and the discrete time settings in contrast to the frame operator the operator s sub g spl gamma is in general not positive therefore all results in gabor analysis that are based on the positivity of the frame operator cannot be applied directly the advantage of the proposed characterization is that the algebraic system for computing the gabor dual elements preserves the high structure of usual gabor frames leading to computationally efficient algorithms in particular we consider examples in which both the condition number and the computational complexity in computing the proposed dual gabor elements decrease in comparison to the canonical dual gabor elements

frame theory based analysis and design of oversampled filter banks direct computational method this paper studies the frames corresponding to oversampled filter banks fbs for this class of fb frames we present a state space parameterization of perfect reconstruction fb frames and explicit and numerically efficient formulas to compute the tightest frame bounds to obtain the dual fb frame and to construct a tight paraunitary fb frame from a given untight nonparaunitary fb frame the derivation uses well developed techniques from modern control theory which results in the unified formulas for generic infinite impulse response iir and finite impulse response fir fbs these formulas involve only algebraic manipulations of real matrices and can be computed efficiently reliably and exactly without the approximation required in the existing methods for generic fbs the results provide a unified framework for frame theory based analysis and systematic design of oversampled filter banks

the generalized lapped pseudo biorthogonal transform oversampled linear phase perfect reconstruction filterbanks with lattice structures we investigate a lattice structure for a special class of n channel oversampled linear phase perfect reconstruction filterbanks with a decimation factor m smaller than n we deal with systems in which all analysis and synthesis filters have the same finite impulse response fir length and share the same center of symmetry we provide the minimal lattice factorization of a polyphase matrix of a particular class of these oversampled filterbanks fbs all filter coefficients are parameterized by rotation angles and positive values the resulting lattice structure is able to provide fast implementation and allows us to determine the filter coefficients by solving an unconstrained optimization problem we consider next the case where we are given the generalized lapped pseudo biorthogonal transform glpbt lattice structure with specific parameters and we a priori know the correlation matrix of noise that is added in the transform domain in this case we provide an alternative lattice structure that suppress the noise we show that the proposed systems with the lattice structure cover a wide range of linear phase perfect reconstruction fbs we also introduce a new cost function for oversampled fb design that can be obtained by generalizing the conventional coding gain finally we exhibit several design examples and their properties

generalized fixed point algebras and square integrable group actions we analyze rieffel s construction of generalized fixed point algebras in the setting of group actions on hilbert modules let g be a locally compact group acting on a c algebra b we construct a hilbert module f over the reduced crossed product of g and b using a pair e r where e is an equivariant hilbert module over b and r is a dense subspace of e with certain properties the generalized fixed point algebra is the c algebra of compact operators on f any hilbert module over the reduced crossed product arises by this construction for a pair e r that is unique up to isomorphism a necessary condition for the existence of r is that e be square integrable the consideration of square integrable representations of abelian groups on hilbert space shows that this condition is not sufficient and that different choices for r may yield different generalized fixed point algebras if b is proper in kasparov s sense there is a unique r with the required properties thus the generalized fixed point algebra only depends on e

semidirect products of c quantum groups multiplicative unitaries approach c quantum groups with projection are the noncommutative analogues of semidirect products of groups radford s theorem about hopf algebras with projection suggests that any c quantum group with projection decomposes uniquely into an ordinary c quantum group and a braided c quantum group we establish this on the level of manageable multiplicative unitaries

linear phase perfect reconstruction filter bank lattice structure design and application in image coding a lattice structure for an m channel linear phase perfect reconstruction filter bank lpprfb based on the singular value decomposition svd is introduced the lattice can be proven to use a minimal number of delay elements and to completely span a large class of lpprfbs all analysis and synthesis filters have the same fir length sharing the same center of symmetry the lattice also structurally enforces both linear phase and perfect reconstruction properties is capable of providing fast and efficient implementation and avoids the costly matrix inversion problem in the optimization process from a block transform perspective the new lattice can be viewed as representing a family of generalized lapped biorthogonal transform glbt with an arbitrary number of channels m and arbitrarily large overlap the relaxation of the orthogonal constraint allows the glbt to have significantly different analysis and synthesis basis functions which can then be tailored appropriately to fit a particular application several design examples are presented along with a high performance glbt based progressive image coder to demonstrate the potential of the new transforms

frame theoretic analysis of oversampled filter banks we provide a frame theoretic analysis of oversampled finite impulse response fir and infinite impulse response fir uniform filter banks fbs our analysis is based on a new relationship between the fbs polyphase matrices and the frame operator corresponding to an fb for a given oversampled analysis fb we present a parameterization of all synthesis fbs providing perfect reconstruction we find necessary and sufficient conditions for an oversampled fb to provide a frame expansion a new frame theoretic procedure for the design of paraunitary fbs from given nonparaunitary fbs is formulated we show that the frame bounds of an fb can be obtained by an eigen analysis of the polyphase matrices the relevance of the frame bounds as a characterization of important numerical properties of an fb is assessed by means of a stochastic sensitivity analysis we consider special cases in which the calculation of the frame bounds and synthesis filters is simplified finally simulation results are presented

linear phase paraunitary filter banks theory factorizations and designs m channel maximally decimated filter banks have been used in the past to decompose signals into subbands the theory of perfect reconstruction filter banks has also been studied extensively nonparaunitary systems with linear phase filters have also been designed the authors study paraunitary systems in which each individual filter in the analysis synthesis banks has linear phase specific instances of this problem have been addressed by other authors and linear phase paraunitary systems have been shown to exist this property is often desirable for several applications particularly in image processing they begin by answering several theoretical questions pertaining to linear phase paraunitary systems next they develop a minimal factorization for a large class of such systems this factorization will be proved to be complete for even m further they structurally impose the additional condition that the filters satisfy pairwise mirror image symmetry in the frequency domain this significantly reduces the number of parameters to be optimized in the design process they then demonstrate the use of these filter banks in the generation of m band orthonormal wavelets several design examples are also given to validate the theory

from multiplicative unitaries to quantum groups ii abstract it is shown that all important features of a c algebraic quantum group a defined by a modular multiplicative w depend only on the pair a rather than the multiplicative unitary operator w the proof is based on thorough study of representations of quantum groups as an application we present a construction and study properties of the universal dual of a quantum group defined by a modular multiplicative unitary without assuming existence of haar weights

the analysis of partial differential operators i hormander springer 1985 vol 1 391 pp vol 2 390 pp vol 3 525 pp vol 4 351 pp

oversampled filter banks perfect reconstruction oversampled filter banks are equivalent to a particular class of frames in l sup 2 z these frames are the subject of this paper first the necessary and sufficient conditions of a filter bank for implementing a frame or a tight frame expansion are established as well as a necessary and sufficient condition for perfect reconstruction using fir filters after an fir analysis complete parameterizations of oversampled filter banks satisfying these conditions are given further we study the condition under which the frame dual to the frame associated with an fir filter bank is also fir and give a parameterization of a class of filter banks satisfying this property then we focus on non subsampled filter banks non subsampled filter banks implement transforms similar to continuous time transforms and allow for very flexible design we investigate the relations of these filter banks to continuous time filtering and illustrate the design flexibility by giving a procedure for designing maximally flat two channel filter banks that yield highly regular wavelets with a given number of vanishing moments

structures factorizations and design criteria for oversampled paraunitary filterbanks yielding linear phase filters we present here a special class of oversampled filterbanks fbs namely paraunitary fbs with linear phase filters we propose some necessary conditions for the existence of such banks based on the repartition between type i ii and type ii iv linear phase filters in the bank for a subset of these fbs we develop a factorization that leads to a minimal implementation as well as a direct parameterization of the fbs in terms of elementary rotation angles this factorization is applied to some design examples with two different optimization criteria coding gain and reconstructibility of lost coefficients

equivariant poincare duality for quantum group actions we extend the notion of poincare duality in kk theory to the setting of quantum group actions an important ingredient in our approach is the replacement of ordinary tensor products by braided tensor products along the way we discuss general properties of equivariant kk theory for locally compact quantum groups including the construction of exterior products as an example we prove that the standard podle sphere is equivariantly poincare dual to itself

tight weyl heisenberg frames in l sup 2 z tight weyl heisenberg frames in l sup 2 z are the tool for short time fourier analysis in discrete time they are closely related to paraunitary modulated filter banks and are studied here using techniques of the filter bank theory good resolution of short time fourier analysis in the joint time frequency plane is not attainable unless some redundancy is introduced that is the reason for considering overcomplete weyl heisenberg expansions the main result of this correspondence is a complete parameterization of finite length tight weyl heisenberg frames in l sup 2 z with arbitrary rational oversampling ratios this parameterization follows from a factorization of polyphase matrices of paraunitary modulated filter banks which is introduced first

operator equalities related to the quantum e 2 group the paper deals with normal operatorsr ands satisfying simple commutation relations that we encounter investigating the quantum deformation of thee 2 group we show thatr s admits a normal extension if and only ifr 1s satisfies a certain spectral condition a number of related formulae are derived in particular all the functionsf satisfying the character equationf r s f r f s are found

a new approach to induction and imprimitivity results given a closed quantum subgroup of a locally compact quantum group we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group more generally we study induction given a coaction of the quantum subgroup on a c algebra we prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups this means that we deflne quantum homogeneous spaces as c algebras and that we prove morita equivalence of crossed products and homogeneous spaces we essentially use von neumann algebraic techniques to prove these morita equivalences between c algebras the theory of induced representations of locally compact l c groups was introduced by mackey 21 who discovered the imprimitivity theorem characterizing induced representations through the presence of a covariant representation of a homogeneous space riefiel 26 provided a modern approach using the language of hilbert c modules after the work of riefiel several induction procedures and imprimitivity results have been obtained both for actions and for coactions of groups we shall brie y review them below the purpose of this paper is to develop such an induction and imprimitivity machinery in the setting of locally compact quantum groups 16 17 in this way we provide a unifled approach to several results on actions and coactions of groups on c algebras at the same time our proofs in the setting of l c quantum groups are simpler than the classical proofs dealing with coactions of groups another motivation comes from quantum group theory meyer and nest have undertaken a reformulation of the baum connes conjecture 24 in which induction and restriction play a crucial role the development of induction and imprimitivity in the current paper should play an equally important role in the formulation of a baum connes conjecture for quantum groups

role of anticausal inverses in multirate filter banks i system theoretic fundamentals in a maximally decimated filter bank with identical decimation ratios for all channels the perfect reconstructibility property and the nature of reconstruction filters causality stability fir property and so on depend on the properties of the polyphase matrix various properties and capabilities of the filter bank depend on the properties of the polyphase matrix as well as the nature of its inverse in this paper we undertake a study of the types of inverses and characterize them according to their system theoretic properties i e properties of state space descriptions mcmillan degree degree of determinant and so forth we find in particular that causal polyphase matrices with anticausal inverses have an important role in filter bank theory we study their properties both for the fir and iir cases techniques for implementing anticausal iir inverses based on state space descriptions are outlined it is found that causal fir matrices with anticausal fir inverses cafacafi have a key role in the characterization of fir filter banks

podleś spheres for the braided quantum su 2 abstract starting with the braided quantum group su q 2 for a complex deformation parameter q we perform the construction of the quotient su q 2 t which serves as a model of a quantum sphere then we follow the reasoning of podle who for real q classified quantum spaces with the action of su q 2 with appropriate spectral properties these properties can also be expressed in the context of the braided quantum su q 2 with complex q and we find that they lead to precisely the same family of quantum spaces as found by podle for the real parameter q

extensions of locally compact quantum groups and the bicrossed product construction abstract in the framework of locally compact quantum groups we study cocycle actions we develop the cocycle bicrossed product construction starting from a matched pair of locally compact quantum groups we define exact sequences and establish a one to one correspondence between cocycle bicrossed products and cleft extensions in this way we obtain new examples of locally compact quantum groups

oversampled linear phase perfect reconstruction filterbanks theory lattice structure and parameterization the paper presents the theory lattice structure and parameterization for a general class of p channel oversampled linear phase perfect reconstruction filterbanks olpprfbs systems with sampling factor m p spl ges m and filter length of l km k spl ges 1 each for these olpprfbs the necessary existence conditions on the number of symmetric filters n sub spl beta and antisymmetric filters n sub spl alpha i e symmetry polarity are first investigated vlsi friendly lattice structures are then developed for two types of olpprfbs type i system n sub spl beta n sub spl alpha and type ii system n sub spl beta spl ne n sub spl alpha the completeness and minimality of each type of lattice are also analyzed compared with existing work the proposed lattices are the most general and efficient ones for olpprfbs besides through the lattice structures the sufficiency of the existence conditions is also verified next lifting based structures are proposed to parameterize a left invertible matrix and all of its left inverses which leads to unconstrained optimization as well as robust implementation of olpprfbs finally several design examples are presented to confirm the validity of the theory and demonstrate the versatility of synthesis filterbanks in the oversampled system

design of multirate filter banks by spl hscr sub spl infin optimization a procedure is developed to design iir synthesis filters in a multirate filter bank the filters minimize the l sub 2 induced norm of the error system between the multirate filter bank and a desired pure time delay system this criterion is reduced to one of spl hscr sub spl infin optimization for which there is ready made software

the structure of hopf algebras with a projection the smash product algebra and the smash coproduct coalgebra are well known in the context of hopf algebras 1 5 and these notions can be viewed as being motivated by the semidirect product construction in the theory of groups and in the theory of afiine group schemes respectively let

closed quantum subgroups of locally compact quantum groups abstract we investigate the fundamental concept of a closed quantum subgroup of a locally compact quantum group two definitions one due to s vaes and one due to s l woronowicz are analyzed and relations between them discussed among many reformulations we prove that the former definition can be phrased in terms of quasi equivalence of representations of quantum groups while the latter can be related to an old definition of podle from the theory of compact quantum groups the cases of classical groups duals of classical groups compact and discrete quantum groups are singled out and equivalence of the two definitions is proved in the relevant context a deep relationship with the quantum group generalization of the herz restriction theorem from classical harmonic analysis is also established in particular in the course of our analysis we give a new proof of the herz restriction theorem

quantum groups with projection on von neumann algebra level abstract we introduce an axiomatization of the notion of a semidirect product of locally compact quantum groups and study their properties our approach is slightly different from the one introduced in the thesis of s roy 16 and unlike the investigations of roy we work within the von neumann algebraic picture this allows the use of powerful techniques related to crossed products by actions of locally compact quantum groups in particular we show existence of a braided comultiplication on the algebra spanned by slices of braided multiplicative unitary

a fast algorithm to computer the h norm of a transfer function matrix abstract a fast algorithm is presented to compute the h norm of a transfer function matrix based on the relation between the singular values of the transfer function matrix and the eigenvalues of a related hamiltonian matrix the norm is computed with guaranteed accuracy a very simple method to compute a rather close lower bound on the h norm is given

noise reduction in oversampled filter banks using predictive quantization we introduce two methods for quantization noise reduction in oversampled filter banks these methods are based on predictive quantization noise shaping or linear prediction it is demonstrated that oversampled noise shaping or linear predictive subband coders are well suited for subband coding applications where for technological or other reasons low resolution quantizers have to be used in this case oversampling combined with noise shaping or linear prediction improves the effective resolution of the subband coder at the expense of increased rate simulation results are provided to assess the achievable quantization noise reduction and resolution enhancement and to investigate the rate distortion properties of the proposed methods