algebraic structure of multiparameter quantum groups let g be a connected semi simple complex lie group we define and study the multi parameter quantum group cq p g in the case where q is a complex parameter that is not a root of unity using a method of twisting bigraded hopf algebras by a cocycle 2 we develop a unified approach to the construction of cq p g and of the multi parameter drinfeld double dq p using a general method of deforming bigraded pairs of hopf algebras we construct a hopf pairing between these algebras from which we deduce a peterweyl type theorem for cq p g we then describe the prime and primitive spectra of cq p g generalizing a result of joseph in the one parameter case this description was conjectured and established in the sl n case by the first and second authors in 15 16 it was proved in the general case by joseph in 18 19 in particular the orbits in prim cq p g under the natural action of the maximal torus h are indexed as in the one parameter case by the elements of the double weyl group w w unlike the one parameter case there is not in general a bijection between symp g and prim cq p g however in the case when

complex quantum groups and their quantum enveloping algebras we construct complexified versions of the quantum groups associated with the lie algebras of typean 1 bn cn anddn following the ideas of faddeev reshetikhin and takhtajan we obtain the hopf algebras of regular functionals u on these complexified quantum groups in the special examplea1 we derive theq deformed enveloping algebrauq sl 2 in the limitq 1 the undeformeduq sl 2 is recovered

holomorphic mappings of bounded type abstract for a banach space e we prove that the frechet space h b e is the strong dual of an lb space b b e which leads to a linearization of the holomorphic mappings of bounded type it is also shown that the holomorphic functions defined on dfc spaces are of uniformly bounded type

quantum deformation of lorentz group a one parameter quantum deformations l 2 ofsl 2 is introduced and investigated an analog of the iwasawa decomposition is proved the compact part of this decomposition coincides withs u 2 whereas the solvable part is identified as a pontryagin dual ofs u 2 it shows thats l 2 is the result of the dual version of drinfeld s double group construction applied tos u 2 the same construction applied to any compact quantum groupgc is discussed in detail in particular the explicit formulae for the haar measures on the pontryagin dualgd ofgc and on the double groupg are given we show that there exists remarkable 1 1 correspondence between representations ofg and bicovariant bimodules tensor bundles overgc the theory of smooth representations ofs l 2 is the same as that ofsl 2 clebsh gordon coefficients are however modified the corresponding tame bicovariant bimodules ons u 2 are classified an application to 4d differential calculus is presented the nonsmooth case is also discussed

a karhunen loeve like expansion for 1 f processes via wavelets while so called 1 f or scaling processes emerge regularly in modeling a wide range of natural phenomena as yet no entirely satisfactory framework has been described for the analysis of such processes orthonormal wavelet bases are used to provide a new construction for nearly 1 f processes from a set of uncorrelated random variables

low rank variance approximation in gmrf models single and multiscale approaches we present a versatile framework for tractable computation of approximate variances in large scale gaussian markov random field estimation problems in addition to its efficiency and simplicity it also provides accuracy guarantees our approach relies on the construction of a certain low rank aliasing matrix with respect to the markov graph of the model we first construct this matrix for single scale models with short range correlations and then introduce spliced wavelets and propose a construction for the long range correlation case and also for multiscale models we describe the accuracy guarantees that the approach provides and apply the method to a large interpolation problem from oceanography with sparse irregular and noisy measurements and to a gravity inversion problem

the wavelet transform of stochastic processes with stationary increments and its application to fractional brownian motion the wavelet transform of random processes with wide sense stationary increments is shown to be a wide sense stationary process whose correlation function and spectral distribution are determined the second order properties of the coefficients in the wavelet orthonormal series expansion of such processes is obtained applications to the spectral analysis and to the synthesis of fractional brownian motion are given

on the wavelet transform of fractional brownian motion a theorem characterizing fractional brownian motion by the covariance structure of its wavelet transform is established the authors examine whether there are alternate gaussian processes whose wavelet transforms have a natural covariance structure in addition the authors examine if there are any gaussian processes whose wavelet transform is stationary with respect to the affine group i e the statistics of the wavelet transform do not depend on translations and dilations of the process

finite dimensional representations of the quantum lorentz group all finite dimensional irreducible representations of the quantum lorentz group sl q 2 are described explicitly and it is proved all finite dimensional representations of sl q 2 are completely reducible the conjecture of podle and woronowicz will be answered affirmatively

linearization of bounded holomorphic mappings on banach spaces the main result in this paper is the following linearization theorem for each open set u in a complex banach space e there is a complex banach space goo u and a bounded holomorphic mapping gu u g u with the following universal property for each complex banach space f and each bounded holomorphic mapping f u f there is a unique continuous linear operator tf goo u f such that tf gu f the correspondence f tf is an isometric isomorphism between the space ho u f of all bounded holomorphic mappings from u into f and the space l g u f of all continuous linear operators from g u into f these properties characterize g u uniquely up to an isometric isomorphism the rest of the paper is devoted to the study of some aspects of the interplay between the spaces h o u f and l g u f this paper consists of five sections in 1 we establish our notation and terminology in 2 we prove the aforementioned linearization theorem in 3 we translate certain properties of a mapping f e h u f into properties of the corresponding operator tf e l g u f we show for instance that f has a relatively compact range if and only if tf is a compact operator in 4 we give a seminorm characterization of the unique locally convex topology t y on h u f such that the correspondence f tf is a topological isomorphism between the spaces h u f ty and l g u f t where tc denotes the compact open topology finally in 5 we use the preceding results to establish necessary and sufficient conditions for the spaces g u and h u to have the approximation property these are holomorphic analogues of classical results of a grothendieck 8 and complement results of r aron and m schottenloher 2 we show in particular that if u is a balanced bounded open set in a complex banach space e then g u has the approximation property if and only if e has the approximation property we also show that if u is an arbitrary open set in a complex banach space e then h u has the approximation property if and only if for each complex banach space f each mapping in h u f with a relatively compact range can be uniformly approximated on u by mappings in h u f with finite dimensional range since it is still unknown whether received by the editors april 10 1989 1980 mathematics subject classification 1985 revision primary 46g20 46e15 secondary 46e10 d 1991 american mathematical society 0002 9947 91 1 00 25 per page

correlation structure of the discrete wavelet coefficients of fractional brownian motion it is shown that the discrete wavelet coefficients of fractional brownian motion at different scales are correlated and that their auto and cross correlation functions decay hyperbolically fast at a rate much faster than that of the autocorrelation of the fractional brownian motion itself the rate of decay of the correlation function in the wavelet domain is primarily determined by the number of vanishing moments of the analyzing wavelet

double quantum groups and iwasawa decomposition abstract the double quantum groups c q d g c q g c q g are the hopf algebras underlying the complex quantum groups of which the simplest example is the quantum lorentz group they are nonstandard quantizations of the double groupg g we construct a corresponding quantized universal enveloping algebrauq d g and prove that the pairing between c q d g anduq d g is nondegenerate we analyze the representation theory of these c q d g give a detailed version of the iwasawa decomposition proved by podles and woronowicz for the quantum lorentz group and show that c q d g is noetherian finally we outline how to construct more general nonstandard quantum groups using quantum double groups and their generalizations

mra of processes synthesized by fractional integration in this correspondence we show that the multiresolution analysis mra of the reproducing kernel hilbert space associated with processes synthesized by fractional integration of white processes leads to an orthogonal multiresolution decomposition of the process we show that the decomposition converges almost surely and uniformly on finite intervals we also show that the random coefficients in the decomposition can be evaluated almost surely

multiresolution sequence detection in rapidly fading channels based on focused wavelet decompositions the problem of detecting a sequence using the maximum likelihood strategy when the time varying multipath channel is unknown can be in principle solved using the generalized likelihood detector gld the gld metric involves finding the orthogonal projection of the received signal onto the subspace of the transmitted signal distorted by the multipath channel the faded signal corresponding to one of the transmitted sequences is in fact known to lie in a subspace but its exact location is not known because the channel parameters are unknown this detector which in the case of a static channel collapses to the per survivor processing method is also called a matched subspace detector because its statistic is matched to the a priori known signal subspace unfortunately the computation of the perfectly matched orthogonal projection of the received signal onto the multipath faded signal subspace is in the general time varying case impossible we introduce in this work the idea of using new wavelet based subspaces that approximate the original signal subspace the nested sequence of linear vector spaces defined by a wavelet based multiresolution decomposition of the fading channel time variations provides a set of subspaces that at an increasingly high level of detail are efficient representations of the original signal subspace for each of these representations sequence detection at different levels of resolution can be performed

braided momentum in the q poincare group the q poincare group of m schlieker et al z phys c 53 79 1992 is shown to have the structure of a semidirect product and coproduct b soq 1 3 where b is a braided quantum group structure on the q minkowski space of four momentum with braided coproduct p p 1 1 p here the necessary b is not a usual kind of quantum group but one with braid statistics similar braided vectors and covectors v r v r exist for a general r matrix the abstract structure of the q lorentz group is also studied

twisting 2 cocycles for the construction of new non standard quantum groups we introduce a new class of 2 cocycles defined explicitly on the generators of certain multiparameter standard quantum groups these allow us through the process of twisting the familiar standard quantum groups to generate new as well as previously known examples of non standard quantum groups in particular we are able to construct generalisations of both the cremmer gervais deformation of sl 3 and the so called esoteric quantum groups of fronsdal and galindo in an explicit and straightforward manner

on the spectrum of fractional brownian motions fractional brownian motions fbms provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order however due to the nonstationary nature of these processes the precise meaning of such spectra remains generally unclear two complementary approaches are proposed which are intended to clarify this point the first one based on a time frequency analysis takes into account the nonstationary nature of fbm and puts emphasis on time averaged measurements the second one based on a time scale analysis is matched to self similarity properties of fbm and reveals an underlying stationary structure relative to each time scaling

linearization of functions given a space u of functions f u which are continuous we construct another space u and a map e u u linearizing all functions f u i e there are lf u such that lf e f such linearizations are stronger than mere preduals for u for example for u l1 linearizations correspond to preduals of l1 which are isomorphic to c0 we also address the vector valued case a number of such linearizing constructions are to be found in the literarture mostly for certain spaces of holomorphic functions the procedure presented here generalizes all these special cases

efficient multiscale regularization with applications to the computation of optical flow a new approach to regularization methods for image processing is introduced and developed using as a vehicle the problem of computing dense optical flow fields in an image sequence the solution of the new problem formulation is computed with an efficient multiscale algorithm experiments on several image sequences demonstrate the substantial computational savings that can be achieved due to the fact that the algorithm is noniterative and in fact has a per pixel computational complexity that is independent of image size the new approach also has a number of other important advantages specifically multiresolution flow field estimates are available allowing great flexibility in dealing with the tradeoff between resolution and accuracy multiscale error covariance information is also available which is of considerable use in assessing the accuracy of the estimates in particular these error statistics can be used as the basis for a rational procedure for determining the spatially varying optimal reconstruction resolution furthermore if there are compelling reasons to insist upon a standard smoothness constraint the new algorithm provides an excellent initialization for the iterative algorithms associated with the smoothness constraint problem formulation finally the usefulness of the approach should extend to a wide variety of ill posed inverse problems in which variational techniques seeking a smooth solution are generally used

inference for a random wavelet packet model of single channel event related potentials event related potentials erps are brain electrical potentials associated with sensory and cognitive processing erp researchers typically wish to separate a recorded time series into functionally distinct component waveforms and to estimate the effects of experimental conditions on each component we present an integrated statistical approach to the decomposition of single channel erps and to inference concerning the component waveforms and the effects of experimental conditions on the amplitude and latency lag from stimulus presentation of each component a wavelet packet model of a single individual s data defines a unique decomposition based on prior time frequency information and variation among experimental conditions a particular orthogonal wavelet packet basis is selected using the best basis algorithm with a special cost function that incorporates prior information our statistical model allows individual specific parameters to vary randomly among individuals because the number of observati

matching wavelet packets to gaussian random processes we consider the problem of approximating a set of arbitrary discrete time gaussian random processes by a single representative wavelet based gaussian process we measure the similarity between the original processes and the wavelet based process with the bhattacharyya 1943 coefficient by manipulating the bhattacharyya coefficient we reduce the task of defining the representative process to finding an optimal unitary matrix of wavelet based eigenvectors an associated diagonal matrix of eigenvalues and a mean vector the matching algorithm we derive maximizes the nonadditive bhattacharyya coefficient in three steps a migration algorithm that determines the best basis by searching through a wavelet packet tree for the optimal unitary matrix of wavelet based eigenvectors and two separate fixed point algorithms that derive an appropriate set of eigenvalues and a mean vector we illustrate the method with two different classes of processes first order markov and bandlimited the technique is also applied to the problem of robust terrain classification in polarimetric sar images

wavelet estimation by bayesian thresholding and model selection many wavelet based algorithms have been proposed in recent years to solve the problem of function estimation from noisy samples in particular it has been shown that threshold approaches lead to asymptotically optimal estimation and are extremely effective when dealing with real data working under a bayesian perspective in this paper we first study optimality of the hard and soft thresholding rules when the function is modelled as a stochastic process with known covariance function next we consider the case where the covariance function is unknown and propose a novel approach that models the covariance as a certain wavelet combination estimated from data by bayesian model selection simulated data are used to show that the new method outperforms traditional threshold approaches as well as other wavelet based bayesian techniques proposed in the literature

modeling and estimation of multiresolution stochastic processes an overview is provided of the several components of a research effort aimed at the development of a theory of multiresolution stochastic modeling and associated techniques for optimal multiscale statistical signal and image processing a natural framework for developing such a theory is the study of stochastic processes indexed by nodes on lattices or trees in which different depths in the tree or lattice correspond to different spatial scales in representing a signal or image in particular it is shown how the wavelet transform directly suggests such a modeling paradigm this perspective then leads directly to the investigation of several classes of dynamic models and related notions of multiscale stationarity in which scale plays the role of a time like variable the investigation of models on homogeneous trees is emphasized the framework examined here allows for consideration in a very natural way of the fusion of data from sensors with differing resolutions also thanks to the fact that wavelet transforms do an excellent job of compressing large classes of covariance kernels it is seen that these modeling paradigms appear to have promise in a far broader context than one might expect

orthonormal bases of compactly supported wavelets we construct orthonormal bases of compactly supported wavelets with arbitrarily high regularity the order of regularity increases linearly with the support width we start by reviewing the concept of multiresolution analysis as well as several algorithms in vision decomposition and reconstruction the construction then follows from a synthesis of these different approaches

quantum groups and quantum determinants abstract we construct function coordinate algebras of multi parameter quantum deformations of g sl n gl n so n o n and sp n when g is of rank l the obtained function algebra has 1 l l 1 2 continuous parameters and some discrete parameters whose values are roots of unity to do this we introduce a notion of co quasitriangular bialgebras and study their group like elements quantum determinants

wavelet based model for stochastic analysis of beam structures a wavelet based model for stochastic analysis of beam structures is presented in this model the random processes representing the stochastic material and geometric properties are treated as stationary gaussian processes with specified mean and correlation functions using the karhunen loeve expansion the process is represented as a linear sum of orthonormal eigenfunctions with uncorrelated random coefficients the correlation and the eigenfunctions are approximated as truncated linear sums of compactly supported orthogonal wavelets and the integral eigenvalue problem is converted to a finite dimensional eigenvalue problem the energy principle based finite element approach is used to obtain the equilibrium and boundary conditions neumann expansion of the stiffness matrix is used to write the nodal displacement vector in terms of random coefficients the expectation operator is applied to the nodal displacements and their squares to obtain the mean and standard deviation of the displacements studies show that the results obtained using this method compare well with monte carlo and semianalytical techniques

wavelet estimation of fractional brownian motion embedded in a noisy environment this correspondence proposes a wavelet based fractional brownian motion fbm signal estimation scheme despite the fact that wavelet transform approximately whitens the fbm processes it is observed that statistical dependencies still exist across adjacent wavelet scales and between neighboring wavelet coefficients these dependencies can be exploited to improve the estimation of fbm signals embedded into noise the idea is to reorganize the wavelet coefficients into a scale time mixture model and then carry out the minimum mean square error estimation mmse using the model experiments show that the proposed scheme obtains better estimates than wornell and oppenheim s algorithm in which the wavelet dependencies are not utilized

complex quantum group dual algebra and bicovariant differential calculus the method used to construct the bicovariant bimodule in ref csww is applied to examine the structure of the dual algebra and the bicovariant differential calculus of the complex quantum group the complex quantum group fun q sl n c is defined by requiring that it contains fun q su n as a subalgebra analogously to the quantum lorentz group analyzing the properties of the fundamental bimodule we show that the dual algebra has the structure of the twisted product fun q su n fun q su n reg then the bicovariant differential calculi on the complex quantum group are constructed

on the correlation structure of the wavelet coefficients of fractional brownian motion shows that the interdependence of the discrete wavelet coefficients of fractional brownian motion defined by normalized correlation decays exponentially fast across scales and hyperbolically fast along time

primitive ideals of c q sl 3 the primitive ideals of the hopf algebrac q sl 3 are classified in particular it is shown that the orbits in primc q sl 3 under the action of the representation grouph c c are parameterized naturally byw w wherew is the associated weyl group it is shown that there is a natural one to one correspondence between primitive ideals ofc q sl 3 and symplectic leaves of the associated poisson algebraic groupsl 3 c

wavelet analysis and synthesis of fractional brownian motion fractional brownian motion fbm offers a convenient modeling for nonstationary stochastic processes with long term dependencies and 1 f type spectral behavior over wide ranges of frequencies statistical self similarity is an essential feature of fbm and makes natural the use of wavelets for both its analysis and its synthesis a detailed second order analysis is carried out for wavelet coefficients of fbm it reveals a stationary structure at each scale and a power law behavior of the coefficients variance from which the fractal dimension of fbm can be estimated conditions for using orthonormal wavelet decompositions as approximate whitening filters are discussed consequences of discretization are considered and some connections between the wavelet point of view and previous approaches based on length measurements analysis or dyadic interpolation synthesis are briefly pointed out

fast positive definite linear system solvers the authors show that the m band wavelet transforms of a wide class of covariance matrices consist of subblocks that are essentially banded furthermore they prove that the cholesky factors of the transformed covariance matrices also consist of subblocks that are essentially banded they combine these two observations to construct a fast o n sup 2 algorithm for solving the n spl times n linear positive definite systems of equations that arise in statistical signal processing finally they provide an error analysis of the proposed linear positive definite system solver

reciprocal processes on a tree modeling and estimation issues motivated by multiresolution decomposition methods such as the discrete wavelet transformation the authors introduce reciprocal processes on truncated n ary trees the authors discuss the relationship between such processes and nearest neighbor models the authors show that they can derive a recursive description of the process and that all reciprocal processes on n ary trees reduce to autoregressive processes in the case of zero valued boundary values at the bottom of the tree corresponding to truncation of the tree the authors then study the smoothing equations associated with such models

wavelet representations of stochastic processes and multiresolution stochastic models deterministic signal analysis in a multiresolution framework through the use of wavelets has been extensively studied very successfully in recent years in the context of stochastic processes the use of wavelet bases has not yet been fully investigated we use compactly supported wavelets to obtain multiresolution representations of stochastic processes with paths in l sup 2 defined in the time domain we derive the correlation structure of the discrete wavelet coefficients of a stochastic process and give new results on how and when to obtain strong decay in correlation along time as well as across scales we study the relation between the wavelet representation of a stochastic process and multiresolution stochastic models on trees proposed by basseville et al see ieee trans inform theory vol 38 p 766 784 mar 1992 we propose multiresolution stochastic models of the discrete wavelet coefficients as approximations to the original time process these models are simple due to the strong decorrelation of the wavelet transform experiments show that these models significantly improve the approximation in comparison with the often used assumption that the wavelet coefficients are completely uncorrelated

frt duals as quantum enveloping algebras the hopf algebra generated by the l functionals on the quantum double cq g cq g is considered where cq g is the coordinate algebra of a standard quantum group and q is not a root of unity it is shown to be isomorphic to cq g op uq g this proves a conjecture by t hodges as an algebra it can be embedded into uq g uq g here it is proven that there is no bialgebra structure on uq g uq g for which this embedding becomes a homomorphism of bialgebras in particular it is not an isomorphism as a preliminary a lemma of hodges concerning the structure of l functionals on cq g is generalized for the classical groups a certain choice of root vectors is expressed in terms of l functionals a formula for their coproduct is derived

a theory for multiresolution signal decomposition the wavelet representation multiresolution representations are effective for analyzing the information content of images the properties of the operator which approximates a signal at a given resolution were studied it is shown that the difference of information between the approximation of a signal at the resolutions 2 sup j 1 and 2 sup j where j is an integer can be extracted by decomposing this signal on a wavelet orthonormal basis of l sup 2 r sup n the vector space of measurable square integrable n dimensional functions in l sup 2 r a wavelet orthonormal basis is a family of functions which is built by dilating and translating a unique function psi x this decomposition defines an orthogonal multiresolution representation called a wavelet representation it is computed with a pyramidal algorithm based on convolutions with quadrature mirror filters wavelet representation lies between the spatial and fourier domains for images the wavelet representation differentiates several spatial orientations the application of this representation to data compression in image coding texture discrimination and fractal analysis is discussed