interpreting proper orthogonal modes of randomly excited vibration systems proper orthogonal modes poms of displacements are interpreted for linear vibration systems under random excitation excitations are considered for which the fourier transform is convergent meaning that the input must have zero mean and no sustained sinusoidal component in such a case the poms in undamped discrete linear symmetric systems can represent linear natural modes if the mass distribution is known poms in one dimensional distributed parameter self adjoint systems can approximately represent the linear normal modes if the mass distribution is known simulation examples are presented simulations show that these ideas are also applicable under light modal damping

primes in short intervals contrary to what would be predicted on the basis of cramer s model con cerning the distribution of prime numbers we develop evidence that the distribution of x h x for 0 x n is approximately normal with mean h and variance h log n h when n h n 1

on the physical interpretation of proper orthogonal modes in vibrations abstract the authors relate the proper orthogonal modes as applied in discrete vibration systems to normal modes of vibration in systems with a known mass matrix in the case of undamped free vibration the proper orthogonal modes converge to the linear normal modes as the amount of data increases this interpretation is also practical for lightly modally damped systems forced resonances lead to proper orthogonal modes which approximate vibration modes a particular case of non linear normal modes are looked at in which the motion of a single mode follows a curve in the co ordinate space in this case the proper orthogonal modes represent the principal axes of inertia formed by the distribution of data on the modal co ordinate curve more generally the proper orthogonal modes represent the principal axes of inertia of the data

on the exploitation of chaos to build reduced order models the present study focuses on the model reduction of non linear systems the proper orthogonal decomposition is exploited to compute eigenmodes from time series of displacement these eigenmodes called the proper orthogonal modes are optimal with respect to energy content and are used to build a low dimensional model of the non linear system for this purpose the proper orthogonal modes obtained from a chaotic orbit are considered indeed such an orbit is assumed to cover a portion of the phase space of higher dimension and hence of greater measure this higher dimensional data is further assumed to contain more information about the system dynamics than data of a lower dimensional periodic orbit in an example it is shown that the modes for this particular behaviour are more representative of the system dynamics than any other set of modes extracted from a non chaotic response this is applied to a buckled beam with two permanent magnets and the reduced order model is validated using both qualitative and quantitative comparisons

b series analysis of stochastic runge kutta methods that use an iterative scheme to compute their internal stage values in recent years implicit stochastic runge kutta srk methods have been developed both for strong and weak approximations for these methods the stage values are only given implicitly however in practice these implicit equations are solved by iterative schemes such as simple iteration modified newton iteration or full newton iteration we employ a unifying approach for the construction of stochastic b series which is valid both for it o and stratonovich stochastic differential equations sdes and applicable both for weak and strong convergence to analyze the order of the iterated runge kutta method moreover the analytical techniques applied in this paper can be of use in many other similar contexts

complex architecture of primes and natural numbers prime numbers and their distribution in the sequence of natural numbers are of interest for fundamental as well as practical reasons the authors introduce a stochastic model based on a network representation that closely matches some statistical properties of the prime numbers and that can be useful as a tool in the study of the distribution of primes

partitioned runge kutta methods for separable hamiltonian problems separable hamiltonian systems of differential equations have the form dp dt dh dq dq dt dh dp with a hamiltonian function h that satisfies h t p k q t and v are respectively the kinetic and potential energies we study the integration of these systems by means of partitioned runge kutta methods i e by means of methods where different runge kutta tableaux are used for the p and q equations we derive a suffi cient and almost necessary condition for a partitioned runge kutta method to be canonical i e to conserve the symplectic structure of phase space thereby reproducing the qualitative properties of the hamiltonian dynamics we show that the requirement of canonicity operates as a simplifying assumption for the study of the order conditions of the method

proper orthogonal decomposition pod of a class of vibroimpact oscillations abstract this study employs the method of proper orthogonal decomposition pod also known as the karhunen loeve k l method to extract dominant coherent structures modes that approximate the system behavior from time series data these mode shapes can be used in a galerkin reconstruction process to obtain lower dimensional models for the structural systems under consideration the mode shapes and the energies obtained by this process can also be used to predict certain kinds of periodic or non periodic non linear motions the k l method has been applied successfully to fluid dynamical thermal processes and signal processing however only a handful of works exist in the area of vibration analysis and structural mechanics an extensive k l analysis is performed numerically for two vibroimpacting systems a beam and a rotor lower dimensional models are created and used to study non linear energy transmission from low to higher k l modes extensive reconstructions are also performed to prove the efficacy of the k l method to provide accurate low dimensional dynamical models in addition experimental investigations are presented for the case of the overhung impacting rotor and qualitative comparisons with the theoretical analysis are presented

downstream evolution of proper orthogonal decomposition eigenfunctions in a lobed mixer a two dimensional one space and time scalar adaptation of the proper orthogonal decomposition was applied to streamwise velocity data obtained in a lobed mixer flowfield using a rake of 15 single component hot wires through the application of the proper orthogonal decomposition the amount of streamwise turbulent kinetic energy contained in the various proper orthogonal modes was examined for two different downstream locations z h 2 6 and 3 9 the large eddy or dominant mode was shown to have a measurable decrease in the relative streamwise component of the kinetic energy between these two downstream locations this indicates that the large eddy as defined by the proper orthogonal decomposition breaks down and the flow becomes more homogeneous a pseudoflow visualization technique was then employed to help visualize this process

a nonsymmetric state variable decomposition for modal analysis a modal decomposition strategy based on state variable ensembles is formulated a nonsymmetric generalized eigenvalue problem is constructed the data based eigenvalue problem is related to the generalized eigenvalue problem associated with free vibration solutions of the state variable formulation of linear multi degree of freedom systems for linear free response data the inverse transpose of the eigenvector matrix converges to the state variable modal eigenvectors and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state variable model as such the eigenvalues lead to estimates of frequencies and modal damping the interpretation holds for linear systems with multi modal free responses whether damping is small or somewhat large modal or nonmodal and without the need of input data

a determinant representation for a correlation function of the scaling lee yang model we consider the scaling lee yang model it corresponds to the unique perturbation of the minimal cft model this model is not unitary we are using an expression for form factors in terms of symmetric polynomials in order to obtain a closed expression for the correlation function of the trace of the energy momentum tensor this expression is a determinant of an integral operator similar determinant representations were proven to be useful not only for quantum correlation functions but also in matrix models

an optimal modal reduction of a system with frictional excitation abstract the proper orthogonal decomposition theorem is used in an optimal modal reduction of a frictionally excited system the chaotic dynamics are exploited in obtaining a basis that broadly represents the system s dynamics this basis is used in a galerkin s approximation to reduce the system order the reduced model is validated using qualitative and quantitative means and also from the occurrence of bifurcations the results confirm the validity of the method and expose the benefits of using proper orthogonal modes rather than the linear natural modes of the system

finite size corrections in random matrix theory and odlyzko s dataset for the riemann zeros odlyzko has computed a dataset listing more than 109 successive riemann zeros starting from a zero number to beyond 1023 this dataset relates to random matrix theory as according to the montgomery odlyzko law the statistical properties of the large riemann zeros agree with the statistical properties of the eigenvalues of large random hermitian matrices moreover keating and snaith and then bogomolny and co workers have used n n random unitary matrices to analyse deviations from this law we contribute to this line of study in two ways first we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability and we proceed to compute empirical two point correlation functions and nearest neighbour spacings in this setting second we show how to characterize the order 1 n 2 correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are painleve transcendents and where the thinning parameter appears only in the boundary condition this equation can be solved numerically using a power series method in comparison to the riemann zero data accurate agreement is exhibited

on using the hilbert transform for blind identification of complex modes a practical approach abstract the modal identification of dynamical systems under operational conditions when subjected to wide band unmeasured excitations is today a viable alternative to more traditional modal identification approaches based on processing sets of measured frfs or impulse responses among current techniques for performing operational modal identification the so called blind identification methods are the subject of considerable investigation in particular the sobi second order blind identification method was found to be quite efficient sobi was originally developed for systems with normal modes to address systems with complex modes various extension approaches have been proposed in particular a using a first order state space formulation for the system dynamics b building complex analytic signals from the measured responses using the hilbert transform in this paper we further explore the latter option which is conceptually interesting while preserving the model order and size focus is on applicability of the sobi technique for extracting the modal responses from analytic signals built from a set of vibratory responses the novelty of this work is to propose a straightforward computational procedure for obtaining the complex cross correlation response matrix to be used for the modal identification procedure after clarifying subtle aspects of the general theoretical framework we demonstrate that the correlation matrix of the analytic responses can be computed through a hilbert transform of the real correlation matrix so that the actual time domain responses are no longer required for modal identification purposes the numerical validation of the proposed technique is presented based on time domain simulations of a conceptual physical multi modal system designed to display modes ranging from normal to highly complex while keeping modal damping low and nearly independent of the modal complexity and which can prove very interesting in test bench applications numerical results for complex modal identifications are presented and the quality of the identified modal matrix and modal responses extracted using the complex sobi technique and implementing the proposed formulation is assessed

spatiotemporal analysis of complex signals theory and applications we present a space time description of regular and complex phenomena which consists of a decomposition of a spatiotemporal signal into orthogonal temporal modes that we call chronos and orthogonal spatial modes that we call topos this permits the introduction of several characteristics of the signal three characteristic energies and entropies one temporal one spatial and one global and a characteristic dimension although the technique is general we concentrate on its applications to hydrodynamic problems specifically the transition to turbulence we consider two cases of application a coupled map lattice as a dynamical system model for spatiotemporal complexity and the open flow instability on a rotating disk in the latter we show a direct relation between the global entropy and the different instabilities that the flow undergoes as reynolds number increases

mean square convergence of one step methods for neutral stochastic differential delay equations this paper deals with strong approximations of the solutions of neutral stochastic differential delay equations nsddes in ito sense a general framework for the strong convergence of a class of drift implicit one step schemes to the solutions of nsddes is established two examples to illustrate the applicability of our results are provided

damping identification in multi degree of freedom systems via a wavelet logarithmic decrement part 1 theory a wavelet based formula similar to the logarithmic decrement formula is introduced to estimate damping in multi degree of freedom systems from time domain responses both analytical and numerical approaches are investigated

gaussian unitary ensemble eigenvalues and riemann ζ function zeros a nonlinear equation for a new statistic for infinite gaussian unitary ensemble random matrices the probability density function s mathrm nn t for the nearest neighbor eignenvalue spacing as distinct from the spacing between consecutive eigenvalues is computed in terms of the solution of a certain nonlinear equation which generalizes the ensuremath sigma form of the painlev e v equation comparison is made with the empirical value of s mathrm nn t for the zeros of the riemann ensuremath zeta function on the critical line including data from 10 6 consecutive zeros near zero number 10 20

t stability of the semi implicit euler method for delay differential equations with multiplicative noise the paper deals with the t stability of the semi implicit euler method for delay differential equations with multiplicative noise a difference equation is obtained by applying the numerical method to a linear test equation in which the wiener increment is approximated by a discrete random variable with two point distribution the conditions under which the method is t stable are considered and the numerical experiments are given

numerical methods for stochastic systems preserving symplectic structure stochastic hamiltonian systems with multiplicative noise phase flows of which preserve symplectic structure are considered to construct symplectic methods for such systems sufficiently general fully implicit schemes i e schemes with implicitness both in deterministic and stochastic terms are needed a new class of fully implicit methods for stochastic systems is proposed increments of wiener processes in these fully implicit schemes are substituted by some truncated random variables a number of symplectic integrators is constructed special attention is paid to systems with separable hamiltonians some results of numerical experiments are presented they demonstrate superiority of the proposed symplectic methods over very long times in comparison with nonsymplectic ones

stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with multiplicative noise a new class of sprk methods and their order conditions are given symplectic conditions of the sprk methods are obtained stochastic generating functions of the ssprk methods are obtained it is proved that the ssprk methods can preserve the quadratic invariants some low stage ssprk methods are constructed some new stochastic partitioned runge kutta sprk methods are proposed for the strong approximation of partitioned stochastic differential equations sdes the order conditions up to strong global order 1 0 are calculated the sprk methods are applied to solve stochastic hamiltonian systems with multiplicative noise some conditions are captured to guarantee that a given sprk method is symplectic it is shown that stochastic symplectic partitioned runge kutta ssprk methods can be written in terms of stochastic generating functions in addition this paper also proves that the ssprk methods can conserve the quadratic invariants of original stochastic systems based on the order and symplectic conditions some low stage ssprk methods with strong global order 1 0 are constructed finally some numerical results are presented to demonstrate the efficiency of the ssprk methods

karhunen loeve decomposition of the transient dynamics of a multibay truss we compute the transient dynamics of an impulsively loaded multibay truss in computing the transient responses we avoid numerical instabilities due to exponential dichotomy of the bay transfer matrices by employing the direct global matrix approach this method eliminates the need of transfer matrix multiplications that are prone to numerical instabilities and instead requires a numerically stable inversion of a global transfer matrix of large dimension the transient responses at multiple points of the truss are used to perform system identification of the dynamics using karhunen loeve k l decomposition this analysis computes dominant coherent structures proper orthogonal modes of the truss under specific transient excitations together with the energy of each identified coherent structure the identified k l modes can be used to develop accurate low order models of the truss dynamics for control or substructure synthesis purposes

two level discretizations of nonlinear closure models for proper orthogonal decomposition proper orthogonal decomposition has been successfully used in the reduced order modeling of complex systems its original promise of computationally efficient yet accurate approximation of coherent structures in high reynolds number turbulent flows however still remains to be fulfilled to balance the low computational cost required by reduced order modeling and the complexity of the targeted flows appropriate closure modeling strategies need to be employed since modern closure models for turbulent flows are generally nonlinear their efficient numerical discretization within a proper orthogonal decomposition framework is challenging this paper proposes a two level method for an efficient and accurate numerical discretization of general nonlinear closure models for proper orthogonal decomposition reduced order models the two level method computes the nonlinear terms of the reduced order model on a coarse mesh compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step the two level method attains the same level of accuracy while dramatically reducing the computational cost we numerically illustrate these improvements in the two level method by using it in three settings the one dimensional burgers equation with a small diffusion parameter n 10 3 the two dimensional flow past a cylinder at reynolds number re 200 and the three dimensional flow past a cylinder at reynolds number re 1000

nearest neighbour spacing distributions of the β hermite ensemble of random matrices the evolution with of the distributions of the spacing s between nearest neighbour levels of unfolded spectra of random matrices from the hermite ensemble he is investigated by monte carlo simulations the random matrices from the he are real symmetric and tridiagonal where which can take any positive value is the reciprocal of the temperature in the classical electrostatic interpretation of eigenvalues the distribution of eigenvalues coincide with those of the three classical gaussian ensembles for 1 2 4 the use of the he ensemble results in an incomparable speed up and efficiency of numerical simulations of all spectral characteristics of large random matrices generalized gamma distributions are shown to be excellent approximations of the nearest neighbor spacing nns distributions for any while being still simple they account both for the level repulsion in s when s 0 and for the whole shape of the nns distributions in the range of s which is accessible to experiment or to most numerical simulations the exact nns distribution of the goe 1 is in particular significantly better described by a generalized gamma distribution than it is by the wigner surmise while the best generalized gamma approximation coincides essentially with the wigner surmise for 2 they describe too the evolution of the level repulsion between that of a poisson distribution and that of a goe distribution when increases from 0 to 1 the distribution of ln s related to the electrostatic interaction energy between neighbouring charges is accordingly well approximated by a generalized gumbel distribution for any 0 the distributions of the minimum nn spacing between eigenvalues of matrices from the he obtained both from as calculated eigenvalues and from unfolded eigenvalues are brody distributions which are classically used to characterize the spectral fluctuations of various physical systems

on the application of karhunen loeve transform to transient dynamic systems abstract the karhunen loeve transform klt has become a popular method in various fields of engineering science due to its ability to identify the most prominent features in the underlying system dynamics the klt is a favorable method for such tasks as process monitoring model order reduction or optimum control however it is a well known fact that the klt is case sensitive that is that changes in the dynamic system behavior can decisively affect the klt results as much as this property is desired for monitoring problems it limits the performance of klt in model order reduction or optimum control problems if systems are subject to structural changes recent research interest focuses on extending applications of klt to systems with transient dynamic behavior or changing boundary conditions approaches have been published that circumvent the limitations of klt by either assuming reasonable comparability of system dynamics or by measuring the representative performance of klt bases a posteriori however such methods require additional simulations of the full size system and thus jeopardize the idea of model order reduction in this paper we introduce a novel a priori measure to evaluate the performance of the current klt basis this procedure can be of great help in either monitoring or adaptive control of systems that show intermittent transient and quasi stationary dynamic behavior this a priori measure prepares the path for adaptive model order reduction schemes moreover it can be used to measure the stationarity of multidimensional dynamic processes

non commutative q painleve vi equation by applying suitable centrality conditions to non commutative non isospectral lattice modified gel fand dikii type systems we obtain the corresponding non autonomous equations then we derive the non commutative q discrete painleve vi equation with full range of parameters as the 2 2 similarity reduction of the non commutative non isospectral and non autonomous lattice modified korteweg de vries equation we also comment on the fact that in making the analogous reduction starting from the schwarzian korteweg de vries equation no such non isospectral generalization is needed

correlations for the cauchy and generalized circular ensembles with orthogonal and symplectic symmetry the generalized circular ensemble which specifies a spectrum singularity in random matrix theory is equivalent to the cauchy ensemble via a stereographic projection the cauchy weight function is classical and as such the n point distribution function in the cases of orthogonal and symplectic symmetry have expressions in terms of quaternion determinants with elements given in an explicit form suitable for asymptotic analysis the asymptotic analysis is undertaken in the neighbourhood of the spectrum singularity in both cases and it is shown that each quaternion determinant is specified by a single function involving bessel functions

a complex orthogonal decomposition for wave motion analysis a method is presented for decomposing wave motion into its principle components the basic idea is a complex generalization of proper orthogonal decomposition the method involves the representation of real oscillatory signals as complex analytic signals the relationship between complex modes and wave motion is explored from an ensemble of complex signals a complex correlation matrix is formed and its complex eigensolution is the basis of the decomposition like a complex singular value decomposition the complex eigenvectors contain standing and traveling characteristics a traveling index is proposed to quantify the relative degree of traveling and standing in a waveform a method of dissecting a wave mode into its traveling and standing parts is also proposed from the complex modes and modal coordinates frequencies wavelengths and characteristic wave speeds can be obtained the method is applied to traveling and standing wave examples

on proper orthogonal co ordinates as indicators of modal activity proper orthogonal decomposition pod is applied for examining modal activity the extraction of proper orthogonal modal co ordinates pocs is outlined the proper orthogonal values povs are the mean squared values of the pocs the number of povs above the noise floor provides a bound on the number of significant modes based on povs above the noise floor the ideas are illustrated on a linear cantilevered beam experiment the displacement ensembles are obtained by processing six strain measurements coherent proper orthogonal modes poms and povs above the noise floor together confirm that six active modes are detected in the system the distribution of modal components in pocs is discussed the characteristics of the poms povs and pocs are then examined in the presence of added noise

the determinant representation for quantum correlation functions of the sinh gordon model we consider the quantum sinh gordon model in this paper form factors in this model were calculated by mussardo and colleagues we sum up all contributions of form factors and obtain a closed expression for a correlation function this expression is a determinant of an integral operator similar determinant representations have been proven to be useful not only in the theory of correlation functions but also in matrix models

proper orthogonal decomposition and its applications part i theory abstract in view of the increasing popularity of the application of proper orthogonal decomposition pod methods in engineering fields and the loose description of connections among the pod methods the purpose of this paper is to give a summary of the pod methods and to show the connections among these methods firstly the derivation and the performance of the three pod methods karhunen loeve decomposition kld principal component analysis pca and singular value decomposition svd are summarized then the equivalence problem is discussed via a theoretical comparison among the three methods the equivalence of the matrices for processing the objective functions the optimal basis vectors the mean square errors and the asymptotic connections of the three methods are demonstrated and proved when the methods are used to handle the pod of discrete random vectors

multifractal analysis of a lobed mixer flowfield utilizing the proper orthogonal decomposition we use the theory of multifractal measures to throw a fresh perspective on the data analysis of velocity time traces in a lobed mixer flow a lobed mixer is a device that enhances mixing through secondary flows and streamwise vorticity a specially designed rake of hot wires with high spatial resolution was used to collect the streamwise velocity data through the use of the proper orthogonal decomposition a basic knowledge of the coherent structures on an energy weighted basis was studied the instantaneous velocity traces that are necessary for calculating the generalized fractal dimensions are reproduced from the ensemble average measurements taken from the proper orthogonal decomposition we compute the generalized fractal dimensions dq and the a multifractal spectrum of several proper eigenmodes for data samples from the velocity time trace of several hot wire probes at different energy levels this is a marked departure from previous multifractal theory applications to self similar cascades that mainly dealt with the dissipation fields of turbulent kinetic energy in the course of this exposition it will be pointed out that in certain cases a single dimension d may suffice to capture the entire spectrum of scaling exponents for the velocity time trace we verify that in such cases the location of the hot wire probes imply a lack of intermittency

sliding window proper orthogonal decomposition application to linear and nonlinear modal identification abstract the proper orthogonal decomposition pod method is a tool well adapted to analyze vector signals whereas continuous gabor transform cgt is suitable for scalar signal with multi frequency components in this paper a method named sliding window proper orthogonal decomposition swpod combining pod and cgt to analyze multi degrees of freedom mdof vibration system responses is presented swpod gives access to the evolution of the coherent spacial structures and their frequency components versus time the method is of principal interest in the case of swept sine excitation of linear or nonlinear systems to access the resonance frequencies mode shapes and modal damping ratios a procedure is proposed to extract the linear and nonlinear normal modes of weakly damped mdof mechanical systems and illustrated using numerical examples

smooth orthogonal decomposition for modal analysis of randomly excited systems modal parameter estimation in terms of natural frequencies and mode shapes is studied using smooth orthogonal decomposition for randomly excited vibration systems this work shows that under certain conditions the smooth orthogonal decomposition eigenvalue problem formulated from white noise induced response data can be tied to the unforced structural eigenvalue problem and thus can be used for modal parameter estimation using output response ensembles only the generalized eigenvalue problem is formed to estimate eigenfrequencies and modal vectors for an eight degree of freedom lightly damped vibratory system the estimated frequencies are compared against system frequencies obtained from structural eigenvalue problem and estimated modal vectors are checked using the modal assurance criterion for lightly damped simulations satisfactory results are obtained for estimating both system frequencies and modal vectors

physical interpretation of the proper orthogonal modes using the singular value decomposition abstract proper orthogonal decomposition is a statistical pattern analysis technique for finding the dominant structures called the proper orthogonal modes in an ensemble of spatially distributed data while the proper orthogonal modes are obtained through a statistical formulation they can be physically interpreted in the field of structural dynamics the purpose of this paper is thus to provide some insights into the physical interpretation of the proper orthogonal modes using the singular value decomposition

smooth orthogonal decomposition based vibration mode identification a new multivariate data analysis method called smooth orthogonal decomposition sod is proposed to extract linear normal modes and natural frequencies of multi degree of freedom and distributed parameter vibration systems it is demonstrated that for an undamped free vibration of a multi degree of freedom system the computed smooth orthogonal modes are in direct correspondence with the actual normal vibration modes and the smooth orthogonal values are related to the corresponding natural frequencies the same is also shown to be true for lightly damped free vibrations of both lumped and distributed parameter systems in contrast to the intrinsic limitations of the proper orthogonal decomposition pod analysis which requires the knowledge of system s mass matrix to extract normal modes and cannot uniquely identify modal subspaces that have similar proper orthogonal values the sod is shown to overcome both of these deficiencies numerical examples are provided to compare the performances of the pod and sod based modal identification in various types of vibration environment

high order stochastic symplectic partitioned runge kutta methods for stochastic hamiltonian systems with additive noise abstract in this paper a simple class of stochastic partitioned runge kutta sprk methods is proposed for solving stochastic hamiltonian systems with additive noise firstly the order conditions and symplectic condictions are analysised by using colored rooted tree theory then a family of mean square order 1 5 diagonally implicit stochastic symplectic partitioned runge kutta ssprk methods is presented moreover several explicit ssprk methods are constructed for systems with a separable hamiltonian h 0 v p u t q furthermore these methods are proved to converge with mean square order 2 0 to the solution when they are applied to second order stochastic hamiltonian systems with a separable hamiltonian and additive noise finally several numerical examples are performed to demonstrate efficiency of those ssprk methods

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