wavelet based integral representation for solutions of the wave equation an integral representation of solutions of the wave equation as a superposition of other solutions of this equation is built the solutions from a wide class can be used as building blocks for the representation considerations are based on mathematical techniques of continuous wavelet analysis the formulae obtained are justified from the point of view of distribution theory a comparison of the results with those by g kaiser is carried out methods of obtaining physical wavelets are discussed

bounds on variable length compound jumps in euclidean space there is a trivial upper bound on the maximum length of a compound walk built up of variable length jumps and a considerably less trivial lower bound on its minimum length the existence of this non trivial lower bound is intimately connected to the triangle inequalities and the more general polygon inequalities moving beyond euclidean space when a modified version of these bounds is applied in rapidity space they provide upper and lower bounds on the relativistic composition of velocities similarly when applied to transfer matrices these bounds place constraints either in a scattering context on transmission and reflection coefficients or in a parametric excitation context on particle production physically these are very different contexts but mathematically there are intimate relations between these superficially very distinct systems

joint estimation of real squeezing and displacement we study the problem of joint estimation of real squeezing and amplitude of the radiation field deriving the measurement that maximizes the probability density of detecting the true value of the unknown parameters more generally we provide a solution for the problem of estimating the unknown unitary action of a nonunimodular group in the maximum likelihood approach remarkably in this case the optimal measurements do not coincide with the so called square root measurements in the case of squeezing and displacement we analyse in detail the sensitivity of estimation for coherent states and displaced squeezed states deriving the asymptotic relation between the uncertainties in the joint estimation and the corresponding uncertainties in the optimal separate measurements of squeezing and displacement a two mode set up is also analysed showing how entanglement between optical modes can be used to approximate perfect estimation

ridge regression biased estimation for nonorthogonal problems in multiple regression it is shown that parameter estimates based on minimum residual sum of squares have a high probability of being unsatisfactory if not incorrect if the prediction vectors are not orthogonal proposed is an estimation procedure based on adding small positive quantities to the diagonal of x x introduced is the ridge trace a method for showing in two dimensions the effects of nonorthogonality it is then shown how to augment x x to obtain biased estimates with smaller mean square error

functions analytic on the half plane as quantum mechanical states a transform between the state space of one dimensional quantum mechanical systems and a direct sum of two spaces of square integrable functions analytic on the open upper half plane is constructed it gives a representation of usual quantum mechanics on which the free evolution is trivial and the representation of canonical transformation very simple generalizations to higher dimensions are also discussed

a calculation with a biorthogonal wavelet transformation the use of a biorthogonal basis for continuous wavelet transformations is explored thus relaxing the so called admissibility condition on the analyzing wavelet as an application the eigenvalues and corresponding radial eigenfunctions of the hamiltonian of relativistic hydrogen like atoms are determined

star products a group theoretical point of view adopting a purely group theoretical point of view we consider the star product of functions which is associated in a natural way with a square integrable in general projective representation of a locally compact group next we show that for this implicitly defined star product explicit formulas can be provided two significant examples are studied in detail the group of translations on phase space and the one dimensional affine group the study of the first example leads to the groenewold moyal star product in the second example the link with wavelet analysis is clarified

generalized phase space representation of operators introducing asymmetry into the weyl representation of operators leads to a variety of phase space representations and new symbols specific generalizations of the husimi and the glauber sudarshan symbols are explicitly derived

nonnegative matrix factorization for spectral data analysis data analysis is pervasive throughout business engineering and science very often the data to be analyzed is nonnegative and it is often preferable to take this constraint into account in the analysis process here we are concerned with the application of analyzing data obtained using astronomical spectrometers which provide spectral data which is inherently nonnegative the identification and classification of space objects that cannot be imaged in the normal way with telescopes is an important but difficult problem for tracking thousands of objects including satellites rocket bodies debris and asteroids in orbit around the earth in this paper we develop an effective nonnegative matrix factorization algorithm with novel smoothness constraints for unmixing spectral reflectance data for space object identification and classification purposes promising numerical results are presented using laboratory and simulated datasets

dimensionality reduction for data of unknown cluster structure dimensionality reduction that preserves certain characteristics of data is needed for numerous reasons in this work we focus on data coming from a mixture of gaussian distributions and we propose a method that preserves the distinctness of the clustering structure although this structure is assumed to be yet unknown the rationale behind the method is the following i had one known the clusters classes within the data one could facilitate further analysis and reduce space dimensionality by projecting the data to the fisher s linear subspace which by definition best preserves the structure of the given classes ii under some reasonable assumptions this can be done albeit approximately without prior knowledge of the clusters classes in this paper we show how this approach works we present a method of preliminary data transformation that brings the directions of largest overall variability close to the directions of the best between class separation hence for the transformed data simple pca provides an approximation to the fisher s subspace we show that the transformation preserves the distinctness of the unknown structure in the data to a great extent

localization properties of wavelet transforms estimates are given of the localization of wavelet transforms on the half plane in terms of the localization of the analyzed function in time and frequency then a precise formulation of the wavelet analysis of tempered distributions is given the results in this paper are well known to the wavelet community however they are nowhere explicitly written up though they are quite useful this paper therefore serves as reference for these kinds of results

wavelet analysis on the circle the construction of a wavelet analysis over the circle is presented the spaces of infinitely times differentiable functions tempered distributions and square integrable functions over the circle are analyzed by means of the wavelet transform

an overview of wavelet based multiresolution analyses in this paper an overview of wavelet based multiresolution analyses is presented first the continuous wavelet transform in its simplest form is discussed then the definition of a multiresolution analysis is given and how wavelets fit into it is shown the authors take a closer look at orthogonal biorthogonal and semiorthogonal wavelets the fast wavelet transform wavelets on an interval multidimensional wavelets and wavelet packets are discussed several examples of wavelet families are introduced and compared finally the essentials of two major applications are outlined data compression and compression of linear operators

quantization of a classical system on a coadjoint orbit of the poincare group in 1 1 dimensions the connection is investigated between quantization by means of coherent states and geometric quantization when the base manifold is a coadjoint orbit and hence a homogeneous space of the 1 1 dimensional poincare group coherent states of the poincare group stem from a representation that is square integrable modulo closed subgroup and so they depend on a measurable section on the given homogeneous space for each section that leads to a tight frame a geometric prequantization is constructed i e a hermitian line bundle with metric connection conditions are given under which the two forms associated to the connection and to the coadjoint orbit structure of the base manifold coincide

frame transforms star products and quantum mechanics on phase space using the notions of frame transform and of square integrable projective representation of a locally compact group g we introduce a class of isometries tight frame transforms from the space of hilbert schmidt operators in the carrier hilbert space of the representation into the space of square integrable functions on the direct product group g g these transforms have remarkable properties in particular their ranges are reproducing kernel hilbert spaces endowed with a suitable star product which mimics at the level of functions the original product of operators a phase space formulation of quantum mechanics relying on the frame transforms introduced in the present paper and the link of these maps with both the wigner transform and the wavelet transform are discussed

transfer matrices for one dimensional potentials the one dimensional schrodinger equation can be written as a first order multicomponent equation by considering and d dx or combinations thereof as independent variables a potential barrier is then represented by a matrix belonging to one of the homomorphic groups su 1 1 so 2 1 sp 2 r or sl 2 r the relationship between these groups is clarified in various applications one of them may turn out more convenient than others in particular so 2 1 which is obtained by using as a basis some bilinear combinations of and d dx leads to remarkable results the schrodinger wavefunction is represented by a trajectory on a unit hyperboloid a periodic potential corresponds to a pseudorotation around a fixed axis a random potential gives a random walk on the hyperboloid this method can also be used to calculate bound states in potential wells and may have many other interesting applications

a note on irregular discrete wavelet transforms estimates of frame bounds for wavelet frames generated by discrete sets in phase space satisfying only a certain density condition are deduced numerical examples show that these estimates which are the sharpest results of this kind known to the authors are rather poor compared to daubechies estimates see ibid vol 36 no 5 p 961 1005 1990 for the usual discrete wavelet transforms

phase space analysis of the identity decompositions the connections between the decompositions of the identity operator on l2 rn and the decompositions of the function 1 on the phase space r2n are considered the phase space properties of various transformations such as the fourier transform gabor transform and wavelet transform etc are investigated

wavelet coorbit spaces viewed as decomposition spaces abstract in this paper we show that the fourier transform induces an isomorphism between the coorbit spaces defined by feichtinger and grochenig of the mixed weighted lebesgue spaces l v p q with respect to the quasi regular representation of a semi direct product r d h with suitably chosen dilation group h and certain decomposition spaces d q l p l u q essentially as introduced by feichtinger and grobner where the localized parts of a function are measured in the f l p norm this equivalence is useful in several ways it provides access to a fourier analytic understanding of wavelet coorbit spaces and it allows to discuss coorbit spaces associated to different dilation groups in a common framework as an illustration of these points we include a short discussion of dilation invariance properties of coorbit spaces associated to different types of dilation groups

continuous wavelet transform on a special homogeneous space we consider a semidirect product of two locally compact groups s and t with s abelian denoted by s t an action of s t on s is introduced to make s a homogeneous space of s t then we define a unitary representation from s t into the unitary group of l2 s which is our main tool for defining the continuous wavelet transform on l2 s also the main properties of the transform are discussed we prove the plancherel and inversion formulas and reproducing kernel s formula for this transform this is finally specialized to the case of the continuous wavelet transform on l2 rd

singular value decomposition in additive multiplicative and logistic forms singular value decomposition svd is widely used in data processing reduction and visualization applied to a positive matrix the regular additive svd by the first several dual vectors can yield irrelevant negative elements of the approximated matrix we consider a multiplicative svd modification that corresponds to minimizing the relative errors and produces always positive matrices at any approximation step another logistic svd modification can be used for decomposition of the matrices of proportions when a regular svd can yield the elements beyond the zero one range while the modified svd decomposition produces all the elements within the correct range at any step of approximation several additional modifications of matrix approximation are also considered

modular algebras in geometric quantization presented here is a discussion on the connection between geometric quantization and algebraic quantization the former procedure relies on a construction of unitary irreducible representations that starts from co adjoint orbits and uses polarizations while the latter depends on the purely algebraic characterization of unitary irreducible representations which is based on central decompositions of von neumann algebras in involutive duality and their decompositions in terms of maximal abelian subalgebras intermediate stages of these two quantization methods turn out to be complementary leading thus to a new characterization of the so called discrete series representations

banach spaces related to integrable group representations and their atomic decompositions i abstract we present a general theory of banach spaces which are invariant under the action of an integrable group representation and give their atomic decompositions with respect to coherent states i e the atoms arise from a single element under the group action several well known decomposition theories are contained as special examples and are unified under the aspect of group theory

alternative commutation relations star products and tomography invertible maps from operators of quantum observables onto functions of c number arguments and their associative products are first assessed different types of maps such as the weyl wigner stratonovich map and s ordered quasi distribution are discussed the recently introduced symplectic tomography map of observables tomograms related to the heisenberg weyl group is shown to belong to the standard framework of the maps from quantum observables onto the c number functions the star product for symbols of the quantum observable for each one of the maps including the tomographic map and explicit relations among different star products are obtained deformations of the moyal star product and alternative commutation relations are also considered

more on the analysis of local regularity through wavelets in this paper we want to extend the results of pointwise analysis through wavelet transforms to the class of functions where the local fluctuation is bounded by any submultiplicative function this generalizez the results obtained before in the well known case of holder regularity

the weyl correspondence and time frequency analysis describes the weyl correspondence and its properties showing how it gives a window independent definition of time frequency concentration for use in models in signal detection the definition of concentration is justified by showing that it gives reasonable answers in certain intuitive cases the weyl correspondence expresses a linear transformation as a weighted superposition of time frequency shifts of the signal and then authors explain why this is not the same as transforming a signal into the time frequency domain multiplying by a weight in the transform domain and taking the inverse the investigation into time frequency concentration and the weyl correspondence is justified by a new result the authors show that convolving the wigner distribution with a general smoothing function is equivalent to evaluating a weighted sum of spectrograms this is a new interpretation of the process of smoothing the wigner distribution to reduce cross terms it relates smoothing of the wigner distribution to the multiple window technique pioneered by thomson 1982

transforms associated to square integrable group representations i general results let g be a locally compact group which need not be unimodular let x u x x g be an irreducible unitary representation of g in a hilbert space h u assume that u is square integrable i e that there exists in h u at least one nonzero vector g such that u x g g 2 dx we give here a reasonably self contained analysis of the correspondence associating to every vector f h u the function u x g f on g discussing its isometry characterization of the range inversion and simplest interpolation properties this correspondence underlies many properties of generalized coherent states

functional calculus using wavelet transforms it is shown how the wavelet transform may be used to compute for a function s the symbol s a for any not necessarily self adjoint operator a whose spectrum is contained in the upper half plane for self adjoint operators it is shown that this functional calculus coincides with the usual one in particular it is shown how the exponential eita can be written in terms of the resolvent rz a z 1 of a as follows eita 1 c 0 da an 2 dbg at eitbrb ian a with c 2i 0 d i n 1g e and n n and the integral is understood as the cesaro limit this shows explicitly how the behavior for large t is determined by the behavior of rz at iz 1 t

wavelet frames and admissibility in higher dimensions this paper is concerned with the relations between discrete and continuous wavelet transforms on k dimensional euclidean space we start with the construction of continuous wavelet transforms with the help of square integrable representations of certain semidirect products thereby generalizing results of bernier and taylor we then turn to frames of l2 rk and to the question when the functions occurring in a given frame are admissible for a given continuous wavelet transform for certain frames we give a characterization which generalizes a result of daubechies to higher dimensions

a composite fourier wavelet transform and square integrable group representations a formula for a composite fourier wavelet transform is suggested from which the noncommutative fourier and wavelet transforms can be obtained as particular examples as well as several other fourier type transforms it is shown how this composite transform can be used to extend certain results originally obtained for square integrable group representations to more general measures arising in quantum probability theory

finding cluster centers and sizes via multinomial parameterization the clustering problem consists in dividing a data set into groups of observations that are similar within but different across this paper presents a method for assessment the clusters centers and sizes in a non linear least squares optimization with multinomial parameterization the method is especially useful for large data sets as it operates on the summary statistics only this approach also works for the problem of finding clusters centers and sizes by the covariance matrix when the original data is not available estimation of the clusters centers and sizes can be followed by actual clustering example of application to marketing research problem is discussed

tunneling in one dimensional ideal barriers general properties of the transmission coefficient of an ideal one dimensional potential barrier of arbitrary shape are studied it is proved that an arbitrary symmetric barrier is perfectly transparent for at least one energy in each energy band of the related band problem where the barrier potential is periodically continued on the whole real axis recursion relations are obtained for transmission coefficients of barriers consisting of 2k structural units they are used in a simple proof showing that transmission coefficients of finite barriers composed of m identical arbitrary structural units have chaotic behavior for almost all energies for m in each energy band there exists however becoming more dense with m a countable set of energies in each energy band where finite repeated barriers are perfectly transparent the results are illustrated by a numerical example

wavelets associated with representations of the affine weyl heisenberg group the affine weyl heisenberg group generated by time and frequency translations and time dilations is considered and some associated resolutions of the identity are derived as a result it will be shown that they can all be obtained from those associated with the affine group and the weyl heisenberg group by frequency translations of the analyzed function and the analyzing and reconstructing wavelets

necessary and sufficient conditions for constructing orthonormal wavelet bases this paper proves a previous conjecture of the author characterizing sequences h l2 z that yield orthonormal wavelet bases of l2 r in terms of the multiplicity of the eigenvalue 1 of an operator associated to h the proof utilizes a result of cohen characterizing these sequences in terms of the real zeros of their fourier transforms the mapping from sequences to wavelets is shown to define a continuous mapping from a subset of l2 z into l2 r related conjectures are discussed

wavelet transforms and discrete frames associated to semidirect products we consider a semidirect product g rn h and its unitary representations u of the form indg0g p0m where ind is the unitary induction p0 is in the dual group of rn g0 is the stability group of p0 and m is a unitary representation of g0 h we give sufficient conditions such that u defines a wavelet transform and a discrete frame

a generalized wigner function on the space of irreducible representations of the weyl heisenberg group and its transformation properties a natural extension of the wigner function to the space of irreducible unitary representations of the weyl heisenberg group is discussed the action of the automorphisms group of the weyl heisenberg group onto wigner functions and their generalizations and onto symplectic tomograms is elucidated some examples of physical systems are considered to illustrate some aspects of the characterization of the wigner functions as solutions of differential equations

wigner based formulation of the chirplet transform using the wigner distribution we derive and analyze a matrix formulation for the chirplet transform a signal analysis tool that generalizes the wavelet and short time fourier transforms the formulation expresses the translations scalings and shears of the chirplet transform in terms of affine matrix transformations on the time frequency plane our approach leads naturally to several new signal transforms which we derive analyze and extend

an algebraic approach to multiresolution analysis the notion of a weak multiresolution analysis is defined over an arbitrary field in terms of cyclic modules for a certain affine group ring in this setting the basic properties of weak multiresolution analyses are established including characterizations of their submodules and quotient modules the existence and uniqueness of reduced scaling equations and the existence of wavelet bases these results yield some sta ndard facts on classical multiresolution analyses over the rea ls as special cases but provide a different perspective by not relying on orthogonality or topology connections with other areas of algebra and possible further directions are mentioned

coorbit spaces and wavelet coefficient decay over general dilation groups we study continuous wavelet transforms associated to matrix dilation groups giving rise to an irreducible square integrable quasi regular representation on l 2 r d it turns out that these representations are integrable as well with respect to a wide variety of weights thus allowing to consistently quantify wavelet coefficient decay via coorbitspace norms we then show that these spaces always admit an atomic decomposition in terms of bandlimited schwartz wavelets we exhibit spaces of schwartz functions contained in all coorbit spaces and dense in most of them we also present an example showing that for a consistent definition of coorbit spaces the irreducibility requirement cannot be easily dispensed with we then address the question how to predict wavelet coefficient decay from vanishing moment assumptions to this end we introduce a new condition on the open dual orbit associated to a dilation group if the orbit is temperately embedded it is possible to derive rather general weighted l p q estimates for the wavelet coefficients from vanishing moment conditions on the wavelet and the analyzed function these estimates have various applications they provide very explicit admissibility conditions for wavelets and integrable vectors as well as sufficient criteria for membership in coorbit spaces as a further consequence one obtains a transparent way of identifying elements of coorbit spaces with certain cosets of tempered distributions we then show that for every dilation group in dimension two the associated dual orbit is temperately embedded in particular the general results derived in this paper apply to the shearlet group and its associated family of coorbit spaces where they complement and generalize the known results

a statistical view of some chemometrics regression tools chemometrics is a field of chemistry that studies the application of statistical methods to chemical data analysis in addition to borrowing many techniques from the statistics and engineering literatures chemometrics itself has given rise to several new data analytical methods this article examines two methods commonly used in chemometrics for predictive modeling partial least squares and principal components regression from a statistical perspective the goal is to try to understand their apparent successes and in what situations they can be expected to work well and to compare them with other statistical methods intended for those situations these methods include ordinary least squares variable subset selection and ridge regression

complete signal processing bases and the jacobi group the continuous windowed fourier and wavelet transforms are created from the actions of the heisenberg and affine groups respectively both wavelet and windowed fourier bases are known to be complete that is the only signal which is orthogonal to every element of each basis is the zero signal the jacobi group is a group which contains both the heisenberg and affine groups and it can also be used to produce bases for signal processing this paper investigates completeness for bases of one and two real variables which are produced by the jacobi group

the wavelet transform on sobolev spaces and its approximation properties we extend the continuous wavelet transform to sobolev spacesh s for arbitrary reals and show that the transformed distribution lies in the fiber spaces l 2 left left mathbb r 0 frac da a 2 right h s left mathbb r right right cong h 0 s left mathbb r 2 frac dadb a 2 right this generalisation of the wavelet transform naturally leads to a unitary operator between these spaces further the asymptotic behaviour of the transforms ofl 2 functions for small scaling parameters is examined in special cases the wevelet transform converges to a generalized derivative of its argument we also discuss the consequences for the discrete wavelet transform arising from this property numerical examples illustrate the main result

a new transform for time frequency analysis the psi decomposition of a signal in which the signal is written as a weighted sum of certain elementary synthesizing functions is described the set s of synthesizing functions consists of dilated and translated copies of two parent functions which are concentrated in both the time and the frequency domains the weighting constants in the psi decomposition define a transform called the phi transform the phi transform of a signal captures both the frequency content and the temporal evolution of a nonstationary signal the phi transform is linear continuous and continuously invertible the set s of synthesizing functions used in the psi decomposition is nonorthogonal hence considerable flexibility is permitted in its construction it is shown with the help of two examples that the set s is easy to construct

non negative matrix factorization with sparseness constraints non negative matrix factorization nmf is a recently developed technique for finding parts based linear representations of non negative data although it has successfully been applied in several applications it does not always result in parts based representations in this paper we show how explicitly incorporating the notion of sparseness improves the found decompositions additionally we provide complete matlab code both for standard nmf and for our extension our hope is that this will further the application of these methods to solving novel data analysis problems

wiener measures for path integrals with affine kinematic variables the results obtained earlier have been generalized to show that the path integral for the affine coherent state matrix element of a unitary evolution operator exp ith can be written as a well defined wiener integral involving wiener measure on the lobachevsky half plane in the limit that the diffusion constant diverges this approach works for a wide class of hamiltonians including e g d2 dx2 v x on l2 r with v sufficiently singular at x 0

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the euclidean motion group inspired by the early visual system of many mammalians we consider the construction of and reconstruction from an orientation score it u f mathbb r 2 times s 1 to mathbb c as a local orientation representation of an image f mathbb r 2 to mathbb r the mapping f mapsto it u f is a wavelet transform mathcal w psi corresponding to a reducible representation of the euclidean motion group onto mathbb l 2 mathbb r 2 and oriented wavelet psi in mathbb l 2 mathbb r 2 this wavelet transform is a special case of a recently developed generalization of the standard wavelet theory and has the practical advantage over the usual wavelet approaches in image analysis constructed by irreducible representations of the similitude group that it allows a stable reconstruction from one single scale orientation score since our wavelet transform is a unitary mapping with stable inverse we directly relate operations on orientation scores to operations on images in a robust manner furthermore by geometrical examination of the euclidean motion group g mathbb r 2 mathbb r times mathbb t which is the domain of our orientation scores we deduce that an operator on orientation scores must be left invariant to ensure that the corresponding operator mathcal w psi 1 phi mathcal w psi on images is euclidean invariant as an example we consider all linear second order left invariant evolutions on orientation scores corresponding to stochastic processes on g as an application we detect elongated structures in medical images and automatically close the gaps between them finally we consider robust orientation estimates by means of channel representations where we combine robust orientation estimation and learning of wavelets resulting in an auto associative processing of orientation features here linear averaging of the channel representation is equivalent to robust orientation estimation and an adaptation of the wavelet to the statistics of the considered image class leads to an auto associative behavior of the system

continuous wavelet transforms with abelian dilation groups this paper is devoted to the construction of wavelet or coherent state systems arising from the action of certain semidirect products g rk h on l2 rk for this purpose previous results which guarantee the existence of inversion formulas are applied to the special case where h is abelian the questions of systematic construction and conjugacy of such groups are completely resolved by setting up a correspondence to unit groups of commutative associative algebras as an application the numbers of conjugacy classes of possible abelian groups are computed for k 2 3 4 for k 7 there are uncountably many conjugacy classes we then compute the admissibility conditions belonging to abelian groups the final section contains a characterization of abelian matrix groups acting ergodically on some subset of rk this result ensures that the approach via associative algebras yields all possible groups

remarks on resolutions of hilbert schmidt operators a generalization due to ali antoine and gazeau of the notion of square integrability of representations is used to construct operators as integrals over compact homogeneous spaces x g h it is shown that such operators are hilbert schmidt and a formula for the hilbert schmidt norm is given finally two examples are given demonstrating the dependence of the hilbert schmidt norm on the section x g used to construct it one example involving the three dimensional euclidean group makes interesting use of bessel coefficients

asymptotic wavelet and gabor analysis extraction of instantaneous frequencies the behavior of the continuous wavelet and gabor coefficients in the asymptotic limit using stationary phase approximations are investigated in particular it is shown how under some additional assumptions these coefficients allow the extraction of some characteristics of the analyzed signal such as frequency and amplitude modulation laws applications to spectral line estimations and matched filtering are briefly discussed

using principal component analysis and correspondence analysis for estimation in latent variable models abstract correspondence analysis ca and principal component analysis pca are often used to describe multivariate data in certain applications they have been used for estimation in latent variable models the theoretical basis for such inference is assessed in generalized linear models where the linear predictor equals j xi j or aj bj xi xi j 2 i 1 n j 1 m and xi is treated as a latent fixed effect the pca and ca eigenvectors column scores are evaluated as estimators of j and uj and as estimators of uj with m fixed and n consistent estimators cannot be obtained due to the incidental parameters problem unless sufficient moment conditions are imposed on xi pca is equivalent to maximum likelihood estimation for the linear gaussian model and gives a consistent estimator of j up to a scale change when the second sample moment of xi is positive and finite in the limit it is inconsistent for poisson and bernoulli distributions but when bj is constant its first and or sec

tomograms and other transforms a unified view a general framework is presented which unifies the treatment of wavelet like quasidistribution and tomographic transforms explicit formulae relating the three types of transforms are obtained the case of transforms associated with the symplectic and affine groups is treated in some detail special emphasis is given to the properties of the scale time and scale frequency tomograms tomograms are interpreted as a tool to sample the signal space by a family of curves or as the matrix element of a projector

efficient integration of the one dimensional time independent wave equation for bound states and for wave propagation the wave equation is solved with the phase integral approximation in the asymptotic regions and integrated numerically elsewhere in this application the phase integral approximation is relatively simple in higher orders and its accuracy can be precisely controlled with the derived error estimates for the bound states the analytical formulas defining the termination error in the wave function are derived for the wave propagation an efficient formalism is given for calculating the reflection transmission and absorption coefficients

square integrable imprimitivity systems we give a definition of square integrability for imprimitivity systems and we prove that the square integrable ones share most of the properties of square integrable representations of groups

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

subspace learning for unsupervised feature selection via matrix factorization dimensionality reduction is an important and challenging task in machine learning and data mining feature selection and feature extraction are two commonly used techniques for decreasing dimensionality of the data and increasing efficiency of learning algorithms specifically feature selection realized in the absence of class labels namely unsupervised feature selection is challenging and interesting in this paper we propose a new unsupervised feature selection criterion developed from the viewpoint of subspace learning which is treated as a matrix factorization problem the advantages of this work are four fold first dwelling on the technique of matrix factorization a unified framework is established for feature selection feature extraction and clustering second an iterative update algorithm is provided via matrix factorization which is an efficient technique to deal with high dimensional data third an effective method for feature selection with numeric data is put forward instead of drawing support from the discretization process fourth this new criterion provides a sound foundation for embedding kernel tricks into feature selection with this regard an algorithm based on kernel methods is also proposed the algorithms are compared with four state of the art feature selection methods using six publicly available datasets experimental results demonstrate that in terms of clustering results the proposed two algorithms come with better performance than the others for almost all datasets we experimented with here highlightspropose a new feature selection based on matrix factorization present a fast convergent algorithm for matrix factorization on certain constraints incorporate kernel tricks into feature selection problems construct a unified framework for feature extraction feature selection and clustering

a general theorem on square integrability vector coherent states we derive a generalization of the well known theorem for the square integrability of a unitary irreducible representation of a locally compact group the generalization covers the case of representations admitting vector coherent states the result is illustrated by an example drawn from the isochronous galilei group the construction yields a wide variety of coherent states labeled by phase space points which satisfy a resolution of the identity condition and incorporate spin degrees of freedom

tight frames of compactly supported affine wavelets this paper extends the class of orthonormal bases of compactly supported wavelets recently constructed by daubechies commun pure appl math 41 909 1988 for each integer n 1 a family of wavelet functions having support 0 2n 1 is constructed such that jk x 2j 2 2jx k kj k z is a tight frame of l2 r i e for every f l2 r f c jk jk f jk for some c 0 this family is parametrized by an algebraic subset vn of r4n furthermore for n 2 a proper algebraic subset wn of vn is specified such that all points in vn outside of wn yield orthonormal bases the relationship between these tight frames and the theory of group representations and coherent states is discussed

square integrability of metaplectic wave packet representations on l2 r abstract this paper presents a systematic study for harmonic analysis of metaplectic wave packets in l 2 r via group representation theory the abstract notions of metaplectic wave packet groups and metaplectic wave packet representations will be presented and as the main result we prove an admissibility condition on closed subgroups of the special linear group sl 2 r which guarantees square integrability of the associated metaplectic wave packet representation on l 2 r finally we present an analytic study of metaplectic wave packet representations over admissible subgroups of sl 2 r

pca and svd with nonnegative loadings principal component analysis pca and singular value decomposition svd are widely used in multivariate statistical analysis for data reduction the work considers exponential logit and multinomial parameterization of the eigenvectors elements that always yields nonnegative loadings of shares for variable aggregation in contrast to regular pca and svd matrix decomposition by the positive shares shows explicitly which variables and with which percent are composed into each group so what is each variable contribution to data approximation the least squares objective of matrix fit is reduced to rayleigh quotient for variational description of the eigenvalues eigenvectors with the nonlinear parameterization can be found in newton raphson optimizing procedure numerical examples compare the classical and nonnegative loadings results with interpretation by the perron frobenius theory for each subset of variables identified by sparse loading vectors

on informational completeness of covariant localization observables and wigner coefficients an orthogonality theorem is obtained for square integrable representations on homogeneous spaces of a locally compact group from this a series of lemmas is derived showing the informational completeness of natural covariant localization operators as well as of the generalized wigner distributions matrix elements of the group some of these results give explicit reconstruction formulas for the quantum state from its expectation values against these families of operators the results are applied to special phase space representations of the heisenberg affine and galilei groups