a fast algorithm for apes and capon spectral estimation in this paper we derive a novel implementation of some very computationally demanding matched filter bank based spectral estimators namely the amplitude and phase estimator apes the amplitude spectrum capon asc estimator and the power spectrum capon psc estimator filter bank based spectral estimation methods that adopt data dependent filter banks can provide spectra characterized by a significantly improved resolution compared to classical approaches however the computational complexity of the currently available implementation algorithms is extremely high a novel technique is introduced that provides efficient algorithms for the computation of the apes asc and psc spectra the proposed method is based on suitable displacement representations of all pertinent data matrices that are subsequently utilized for the computation of the associated complex valued polynomials the computational complexity of the proposed algorithms is lower than all relevant existing methods

iaa spectral estimation fast implementation using the gohberg semencul factorization we consider fast implementations of the weighted least squares based iterative adaptive approach iaa for one dimensional 1 d and two dimensional 2 d spectral estimation of uniformly sampled data iaa is a robust user parameter free and nonparametric adaptive algorithm that can work with a single data sequence or snapshot compared to the conventional periodogram iaa can be used to significantly increase the resolution and suppress the sidelobe levels however due to its high computational complexity iaa can only be used in applications involving small sized data we present herein novel fast implementations of iaa using the gohberg semencul g s type factorization of the iaa covariance matrices by exploiting the toeplitz structure of the said matrices we are able to reduce the computational cost by at least two orders of magnitudes even for moderate data sizes

on transformation group algebras with continuous trace in this paper we answer some questions posed by dana williams in 19 concerning the problem under which conditions the transformation group c algebra c g q of a locally compact transformation group g li has continuous trace one consequence will be for compact g that c g q2 has continuous trace if and only if the stabilizer map is continuous we also give a complete solution to the problem if g is discrete

pull backs of algebras and crossed products by certain diagonal actions let g be a locally compact group and p q t a principal g bundle if a is a c algebra with primitive ideal space t the pull back p a of a along p is the balanced tensor product co q c t a if 3 g aut a consists of c t module automorphisms and y g aut co q is the natural action then the automorphism group y 0x 3 of co q 0 a respects the balancing and induces the diagonal action p 3 of g on p a we discuss some examples of such actions and study the crossed product p a x p g we suggest a substitute d for the fixed point algebra prove p a x g is strongly morita equivalent to d and investigate the structure of d in various cases in particular we ask when d is strongly morita equivalent to a sometimes but by no means always and investigate the case where a has continuous trace let b be a c algebra and g a locally compact group acting on b as a strongly continuous automorphism group a our goal here is to study the crossed product c algebra b x g for two classes of diagonal actions for which the induced action of g on b is free the first class includes actions of the form y 0x3 on b co q 0 a where p q t is a principal g bundle y is the dual action of g on co q and 3 g aut a is an action of g on another c algebra a we also consider diagonal actions on algebras which are the pull backs of another algebra a along a principal bundle p q t if a is a c algebra with primitive ideal space t then the pull back p a is the balanced tensor product co q cb t a when 3 g aut a consists of c t module automorphisms the product action ty 0 3 preserves the balancing and the diagonal action p 3 is by definition the induced action on p a in general if f x y and q prima y are continuous then cb y acts on co x by composition with f and on a by composition with q and the daunshofmann theorem we can therefore define the pull back f a of a along f as the c algebraic tensor product co x cb y a the reason for the name is that when a is the algebra of sections of some c bundle e over y there is a natural isomorphism of f a onto the algebra of sections of the pull back f e in 1 we discuss this and other basic properties of pull backs and give some evidence to show they are likely to be of interest in particular we show that if g is abelian and a g aut a is locally unitary in the sense of 18 then the crossed product received by the editors october 20 1983 and in revised form february 15 1984 1980 mathematics subject classification primary 46l40 46l55

the pseudospectral method accurate representation of interfaces in elastic wave calculations when finite difference methods are used to solve the elastic wave equation in a discontinuous medium the error has two dominant components dispersive errors lead to artificial wave trains errors from interfaces lead to circular wavefronts emanating from each location where the interface appears jagged to the rectangular grid the pseudospectral method can be viewed as the limit of finite differences with infinite order of accuracy with this method dispersive errors are essentially eliminated the mappings introduced in this paper also eliminate the other dominant error source test calculations confirm that these mappings significantly enhance the already highly competitive pseudospectral method with only a very small additional cost although the mapping method is described here in connection with the pseudospectral method it can also be used with high order finite difference approximations

induced representations of crossed products by coactions abstract let a m a c r g be a coaction of a locally compact group g on a c algebra a then for any closed normal amenable subgroup h of g we define a coaction a m a c r g h of g h on a we present dense subalgebras of the crossed products a g and a g h and use these to induce representations of a g h to representations of a g we then formulate an imprimitivity theorem for the induction process that is we classify the induced representations we also investigate the continuity of the induction process and are able to prove rieffel s conjecture that the dual action to a coaction is proper and saturated

satisfaction of boundary conditions for chebyshev collocation methods in cuboidal domains abstract a collocation strategy for the satisfaction of boundary conditions in the application of chebyshev spectral methods to poisson and biharmonic type problems in cuboidal domains is presented this strategy leads in both cases to nonsingular linear systems for the determination of the unknown coefficients

a chebyshev spectral collocation method for the computation of buoyancy induced flows abstract a spectral powell hybrid algorithm is presented for the solution of the boussinesq equations the method is applied to a streamfunction formulation of the governing equations and used to solve the double glazing test problem excellent qualitative agreement with benchmark solutions is obtained given the relatively few degrees of freedom used

an equivariant brauer semigroup and the symmetric imprimitivity theorem suppose that x g is a second countable locally compact transformation group we let sg x denote the set of morita equivalence classes of separable dynamical systems a g where a is a c0 x algebra and is compatible with the given g action on x we prove that sg x is a commutative semigroup with identity with respect to the binary operation a g b g a x b g x for an appropriately defined balanced tensor product on c0 x algebras if g and h act freely and properly on the left and right of a space x then we prove that sg x h and sh g x are isomorphic as semigroups if the isomorphism maps the class of a g to the class of b h then a o g is morita equivalent to b o h

numerical solution of the 3 d buoyancy driven cavity problem by a pseudospectral method the pseudospectral continuation technique introduced in karageorghis and sivaloganathan for 2 d recirculating flows is extended to three dimensions the pseudospectral continuation strategy is applied to the vorticity vector potential formulation of the boussinesq equations and used to successfully solve the three dimensional air filled buoyancy driven cavity problem at various rayleigh numbers

superfast approximative implementation of the iaa spectral estimate in this correspondence we develop superfast approximative one dimensional algorithms for the computationally efficient implementation of the recent iterative adaptive approach iaa spectral estimate the proposed methods are based on rewriting the iaa algorithm with suitable gohberg semencul representations solving the resulting linear systems of equations using the preconditioned conjugate gradient method where a novel preconditioning is applied using an incomplete factorization of the toeplitz matrix numerical simulations illustrate the efficiency of both the proposed preconditioning as well as the overall algorithm offering a computational reduction of up to two orders of magnitude as compared to our recently proposed efficient and exact iaa implementation

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

induction in stages for crossed products of c algebras by maximal coactions let be a maximal coaction of a locally compact group g on a c algebra b and let n and h be closed normal subgroups of g with n h we show that the process indg hg which uses mansfield s bimodule to induce representations of b g from those of b g h is equivalent to the two stage induction process indg ng indg hg n the proof involves a calculus of symmetric imprimitivity bimodules which relates the bimodule tensor product to the fibred product of the underlying spaces

crossed products byc0 x actions abstract suppose thatghas a representation grouph thatgab g g g is compactly generated and thatais ac algebra for which the complete regularization of prim a is a locally compact hausdorff spacex in a previous article we showed that there is a bijection z f between the collection of exterior equivalence classes of locally inner actions g aut a and the collection of principalgab bundlesz together with continuous functionsf x h2 g t in this paper we compute the crossed productsa gin terms of the dataz f andc h

fast newton transversal filters a new class of adaptive estimation algorithms a class of adaptive algorithms for the estimation of fir finite impulse response transversal filters is presented the main characteristic of this class is the fast computation of the gain vector needed for the adaptation of the transversal filters the method for deriving these algorithms is based on the assumption that the input signal is autoregressive of order m where m can be much smaller than the order of the filter to be estimated under this assumption the covariance matrix of the input signal is estimated by extending in a min max way the m order sample covariance matrix this estimate can be regarded as a generalization of the diagonal covariance matrix used in lms and leads to an efficient computation of the gain needed for the adaptation the new class of algorithms contains the lms and the fast versions of ls as special cases the complexity changes linearly with m starting from the complexity of the lms for m 0 and ending at the complexity of the fast versions of ls

computation of natural convection in two dimensional cavities with chebyshev polynomials abstract a semi implicit spectral method for the computation of buoyancy induced two dimensional flow of a viscous incompressible fluid is presented the incompressible navier stokes equations expressed in terms of the primitive variables velocity and pressure and an additional equation for the temperature are considered all the variables are expanded in double truncated series of chebyshev polynomials the pressure distribution at the boundary of the computational domain is determined by means of an influence matrix technique in order to satisfy the incompressibility condition everywhere in the field the algorithm is applied to the classical problem of natural convection of a boussinesq fluid within a differentially heated vertical square cavity results show that for a prandtl number of 0 71 corresponding to air and up to rayleigh numbers of 106 very accurate solutions are obtained with a spatial resolution of 33 polynomials in each direction new interesting phenomena are shown for higher values of the rayleigh number

proper actions on imprimitivity bimodules and decompositions of morita equivalences we consider a class of proper actions of locally compact groups on imprimitivity bimodules over c algebras which behave like the proper actions on c algebras introduced by rieffel in 1988 we prove that every such action gives rise to a morita equivalence between a crossed product and a generalized fixed point algebra and in doing so make several innovations which improve the applicability of rieffel s theory we then show how our construction can be used to obtain canonical tensor product decompositions of important morita equivalences our results show for example that the different proofs of the symmetric imprimitivity theorem for actions on graph algebras yield isomorphic equivalences and this gives new information about the amenability of actions on graph algebras

transformation group c algebras with continuous trace we obtain several results characterizing when transformation group c algebras have continuous trace these results can be stated most succinctly when g l is second countable and the stability groups are contained in a fixed abelian subgroup in this case c g q has continuous trace if and only if the stability groups vary continuously on r and compact subsets of q are wandering in an appropriate sense in general we must assume that the stability groups vary continuously and if g j is not second countable that the natural maps of g s onto g x are homeomorphisms for each x then c g 0 has continuous trace if and only if compact subsets of 0 are wandering and an additionai c algebra constructed from the stability groups and 0 has continuous trace

efficient implementation of iterative adaptive approach spectral estimation techniques this paper presents computationally efficient implementations for several recent algorithms based on the iterative adaptive approach iaa for uniformly sampled one and two dimensional data sets considering both the complete data case and the cases when the data sets are missing samples either lacking arbitrary locations or having gaps or periodically reoccurring gaps by exploiting the method s inherent low displacement rank together with the development of suitable gohberg semencul representations and the use of data dependent trigonometric polynomials the proposed implementations are shown to offer a reduction of the necessary computational complexity by at least one order of magnitude numerical simulations together with theoretical complexity measures illustrate the achieved performance gain

a fully conforming spectral collocation scheme for second and fourth order problems abstract a spectral element scheme is presented for the solution of second and fourth order problems in two and three space dimensions this scheme is based on a collocation formulation and for a certain class of problems results in spectral approximations which are c1 pointwise continuous across the element interfaces for second order problems and c3 pointwise continuous across the element interfaces for fourth order problems

an equivariant brauer group and actions of groups onc algebras suppose that g t is a second countable locally compact trans formation group given by a homomorphism l g homeo t and that a is a separable continuous trace c algebra with spectrum t an action g aut a is said to cover l if the induced action of g on t coincides with the original one we prove that the set brg t of morita equivalence classes of such systems forms a group with multiplication given by the balanced tensor product a b a c0 t b and we refer to brg t as the equivariant brauer group we give a detailed analysis of the structure of brg t in terms of the moore cohomology of the group g and the integral cohomology of the space t using this we can characterize the stable continuous trace c algebras with spectrum t which admit actions covering l in particular we prove that if g r then every stable continuous trace c algebra admits an essentially unique action covering l thereby substantially improving results of raeburn and rosenberg

fast lcmv based methods for fundamental frequency estimation recently optimal linearly constrained minimum variance lcmv filtering methods have been applied to fundamental frequency estimation such estimators often yield preferable performance but suffer from being computationally cumbersome as the resulting cost functions are multimodal with narrow peaks and require matrix inversions for each point in the search grid in this paper we therefore consider fast implementations of lcmv based fundamental frequency estimators exploiting the estimators inherently low displacement rank of the used toeplitz like data covariance matrices using as such either the classic time domain averaging covariance matrix estimator or if aiming for an increased spectral resolution the covariance matrix resulting from the application of the recent iterative adaptive approach iaa the proposed exact implementations reduce the required computational complexity with several orders of magnitude but as we show further computational savings can be obtained by the adoption of an approximative iaa based data covariance matrix estimator reminiscent of the recently proposed quasi newton iaa technique furthermore it is shown how the considered pitch estimators can be efficiently updated when new observations become available the resulting time recursive updating can reduce the computational complexity even further the experimental results show that the performances of the proposed methods are comparable or better than that of other competing methods in terms of spectral resolution finally it is shown that the time recursive implementations are able to track pitch fluctuations of synthetic as well as real life signals

complexity of multiplication with vectors for structured matrices abstract fast algorithms for computing the product with a vector are presented for a number of classes of matrices whose properties relate to the properties of toeplitz vandermonde or cauchy matrices these matrices are defined using the concept of displacement of a matrix and also for their inverses all the actions which are not dependent upon the coordinates of the input vector are singled out in a separate preprocessing stage the proposed algorithms are based on new representations of these matrices involving factor circulants

mansfield s imprimitivity theorem for full crossed products for any maximal coaction a g s and any closed normal subgroup n of g there exists an imprimitivity bimodule y g g n a between the full crossed product a g n and a g n together with inf dec compatible coaction sy of g the assignment a y g g n a y implements a natural equivalence between the crossed product functors x g x n and x g n in the category whose objects are maximal coactions of g and whose morphisms are isomorphism classes of right hilbert bimodule coactions of g

spectral analysis of nonuniformly sampled data a new approach versus the periodogram we begin by revisiting the periodogram to explain why arguably the plain least squares periodogram lsp is preferable to the ldquoclassicalrdquo fourier periodogram from a data fitting viewpoint as well as to the frequently used form of lsp due to lomb and scargle from a computational standpoint then we go on to introduce a new enhanced method for spectral analysis of nonuniformly sampled data sequences the new method can be interpreted as an iteratively weighted lsp that makes use of a data dependent weighting matrix built from the most recent spectral estimate because this method is derived for the case of real valued data which is typically more complicated to deal with in spectral analysis than the complex valued data case it is iterative and it makes use of an adaptive i e data dependent weighting we refer to it as the real valued iterative adaptive approach riaa lsp and riaa are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra however they are most suitable for data sequences with discrete spectra i e sinusoidal data which is the case we emphasize in this paper

displacement structure theory and applications in this survey paper we describe how strands of work that are important in two different fields matrix theory and complex function theory have come together in some work on fast computational algorithms for matrices with what we call displacement structure in particular a fast triangularization procedure can be developed for such matrices generalizing in a striking way an algorithm presented by schur 1917 j reine angew math 147 1917 pp 205 232 in a paper on checking when a power series is bounded in the unit disc this factorization algorithm has a surprisingly wide range of significant applications going far beyond numerical linear algebra we mention among others inverse scattering analytic and unconstrained rational interpolation theory digital filter design adaptive filtering and state space least squares estimation

a spectral collocation method based on integrated chebyshev polynomials for two dimensional biharmonic boundary value problems this paper reports a new spectral collocation method for numerically solving two dimensional biharmonic boundary value problems the construction of the chebyshev approximations is based on integration rather than conventional differentiation this use of integration allows i the imposition of the governing equation at the whole set of grid points including the boundary points and ii the straightforward implementation of multiple boundary conditions the performance of the proposed method is investigated by considering several biharmonic problems of first and second kinds more accurate results and higher convergence rates are achieved than with conventional differential methods

product rules for the displacement of near toeplitz matrices abstract we consider displacements which are linear operations mapping a near toeplitz matrix into a low rank matrix the objective is to use this low rank representation in solving matrix equations in place of using the full matrix two formulas for the displacement of the product of two matrices are presented and applied applications include multiplication of toeplitz matrices inversion of near toeplitz matrices and finding the eigenvectors of a special class of matrices

efficient algorithms for adaptive capon and apes spectral estimation in this paper fast algorithms for adaptive capon and amplitude and phase estimation apes methods for spectral analysis of time varying signals are derived fast stable nonrecursive formulae are derived based on time shifting properties of the pertinent variables as a consequence efficient frequency domain recursive least squares rls based as well as fast rls based algorithms for the adaptive estimation of the power spectra are developed stability issues of the frequency domain estimators are considered and stabilization procedures are proposed the computational complexity of the proposed algorithms is lower than relevant existing methods the performance of the proposed algorithms is demonstrated through extensive simulations

duality of restriction and induction for c coactions consider a coaction of a locally compact group g on a c algebra a and a closed normal subgroup n of g we prove following results of echterhoff for abelian g that mansfield s imprimitivity between a g n and a g r n implements equivalences between mansfield induction of representations from a g n to a g and restriction of representations froma g rn to a g and between restriction of representations from a g to a g n and green induction of representations from a g to a g r n this allows us to deduce properties of mansfield induction from the known theory of ordinary crossed products