a multilevel domain decomposition algorithm for fast o n sup 2 logn reprojection of tomographic images a novel algorithm for fast computation of tomographic image projections is presented the method comprises a decomposition of an image into subimages followed by an aggregation of projections computed for the subimages the multilevel domain decomposition algorithm is formulated as a recursive procedure the computational cost of the proposed algorithm is comparable to that of fft based techniques while it appears to be more flexible than the latter numerical results demonstrate the effectiveness of the method

hierarchical construction of orientation and velocity selective filters as a step towards the early measurement of visual primitives the authors outline design criteria for the extraction of orientation and velocity information and present a variety of tools useful in the construction of simple linear filters a hierarchical parallel processing scheme is used in which nodes compute a weighted sum of inputs from within a small spatio temporal neighborhood the resulting scheme is easily analyzed and provides mechanisms sensitive to narrow ranges of both image velocity and orientation the hierarchical approach in combination with separability in the first levels yields an efficient implementation

spectral analysis for a three dimensional superradiance problem in this paper three dimensional superradiance problem is considered and a differential operator that commutes with the integral operator related to the problem is obtained all the eigenfunctions of the differential operator are found and a complete set of eigensolutions for the three dimensional superradiance problem is obtained

algebraic heun operator and band time limiting we introduce the algebraic heun operator associated to any bispectral pair of operators this operator can be presented as a generic bilinear combination of these bispectral operators we show that the resulting operators are natural generalizations of the ordinary heun operator this leads to a simple construction of the operators commuting with the projection operators in problems of band time limiting and it gives a way to adapt a construction first used by perline in a purely finite setup to general situations we also extend this algebraic construction to cover some purely finite cases which lie beyond this scheme in particular for the anti krawtchouk polynomials in the latter case the commuting operator is represented by a pentadiagonal matrix which can be constructed explicitly as a polynomial of degree 4 of the pair of bispectral operators

an active set newton algorithm for large scale nonlinear programs with box constraints a new algorithm for large scale nonlinear programs with box constraints is introduced the algorithm is based on an efficient identification technique of the active set at the solution and on a nonmonotone stabilization technique it possesses global and superlinear convergence properties under standard assumptions a new technique for generating test problems with known characteristics is also introduced the implementation of the method is described along with computational results for large scale problems

product formulas and convolutions for angular and radial spheroidal wave functions product formulas for angular spheroidal wave functions on 0 and for radial spheroidal wave functions on 0 are presented which generalize results for the ultraspherical polynomials and functions as well as for the mathieu functions although these functions cannot be given in closed form the kernels of the product formulas are represented in an explicit and surprisingly simple way in terms of bessel functions so that the exact range of positivity can easily be read off the formulas are used to introduce two families of convolution structures on 0 and 0 many of which provide new hypergroups we proceed from the fact that the spheroidal wave functions are eigenfunctions of sturm liouville equations of confluent heun type and employ a partial differential equation technique based on riemann s integration method

the discrete prolate spheroidal filter as a digital signal processing tool the discrete prolate spheroidal dps filter is one of the class of nonrecursive finite impulse response fir filters the dps filter first introduced by tufts and francis 1 is superior to other filters in this class in that it has maximum energy concentration in the frequency passband and minimum ringing in the time domain slepian 2 gives a complete discussion of dps function properties we give a mathematical development of the dps filter properties provide information required to easily and accurately construct even many coefficient filters and compare the properties of this filter with those of the more commonly used filters of the same class we note that use of the dps filter allows for particularly useful statements of data time frequency resolution cell values and that overall it forms an especially useful though little known tool for digital signal processing

mse estimates for multitaper spectral estimation and off grid compressive sensing we obtain estimates for the mean squared error mse for the multitaper spectral estimator and certain compressive acquisition methods for multi band signals we confirm a fact discovered by thomson spectrum estimation and harmonic analysis proc ieee 1982 assuming bandwidth w and n time domain observations the average of the square of the first k lfloor 2nw rfloor slepian functions approaches as k grows an ideal bandpass kernel for the interval w w we provide an analytic proof of this fact and measure the corresponding rate of convergence in l 1 norm this validates a heuristic approximation used to control the mse of the multitaper estimator the estimates have also consequences for the method of compressive acquisition of multi band signals introduced by davenport and wakin giving mse approximation bounds for the dictionary formed by modulation of the critical number of prolates

inverse methods for reconstruction of channel taps in ofdm systems we describe a novel pilot aided method for estimation of doubly selective wireless channels in ofdm systems we compute the first few fourier coefficients of each channel tap from the pilot information we then estimate the bem coefficients of the channel taps from their respective fourier coefficients using a recently developed inverse reconstruction method for a system with l channel taps the proposed method uses o l log l operations and o l memory per ofdm symbol we validate our method by simulating a system conforming to the ieee 802 16e standard

optimal energy transfer in band limited communication channels maximization of the energy transfer ratio for time limited signals over linear channels is considered the case of linear channels with rational transfer function is addressed and a general procedure for the analytic solution of the maximization problem is outlined the maximum energy transfer ratio over a fixed time interval is evaluated in some cases of interest two performance metrics the energy transfer ratio and the energy intersymbol interference isi ratio are evaluated for optimal signals and compared to those of commonly used rectangular and sinusoidal pulses in order to determine the achievable gain

an energy approach to space time galerkin bem for wave propagation problems in this paper we consider dirichlet or neumann wave propagation problems reformulated in terms of boundary integral equations with retarded potential starting from a natural energy identity a space time weak formulation for 1d integral problems is briefly introduced and continuity and coerciveness properties of the related bilinear form are proved then a theoretical analysis of an extension of the introduced formulation for 2d problems is proposed pointing out the novelty with respect to existing literature results at last various numerical simulations will be presented and discussed showing unconditional stability of the space time galerkin boundary element method applied to the energetic weak problem copyright 2009 john wiley sons ltd

on two stage nyquist pulse shaping filters this correspondence outlines a method for designing two stage nyquist filters the nyquist filter is split into two equal and linear phase finite length impulse response spectral factors the per time unit multiplicative complexity of the overall structure is included as the objective function examples are then provided where nyquist filters are designed so as to minimize the multiplicative complexity subject to the constraints on the overall nyquist filter in comparison to the single stage case the two stage realization reduces the multiplicative complexity by an average of 48 for two stage sampling rate conversion src the correspondence shows that it is better to have a larger src ratio in the first stage

a square root nyquist m filter design for digital communication systems designing matched transmit and receive filters whose combination satisfies the nyquist condition is a classical problem in digital communication systems in this correspondence we propose a novel method for designing such filters the proposed method is based on a cost function whose minimization leads to designs that can strike a balance between the stopband attenuation the residual intersymbol interference isi robust sensitivity to timing jitter and or reduced peak to average power ratio par an iterative algorithm for optimizing the proposed cost function is suggested and its excellent performance is shown by presenting a variety of design examples compared to the published works the proposed method offers the following unique advantages by introducing a symmetry in the filter coefficients filters with reduced computational complexity can be designed we also introduce a design parameter that allows one to strike a balance between the par and other features of the desired filter

box constrained quadratic programming with proportioning and projections two new closely related concepts are introduced that depend on a positive constant gamma an iteration is proportional if the norm of violation of the kuhn tucker conditions at active variables does not excessively exceed the norm of the part of the gradient that corresponds to free variables while a progressive direction determines a descent direction that enables the released variables to move far enough from the boundary in a step called proportioning an algorithm that uses the conjugate gradient method to explore the face of the region defined by the current iterate until a disproportional iteration is generated is proposed it then changes the face by means of the progressive direction it is proved that for strictly convex problems the proportioning is a spacer iteration so that the algorithm converges to the solution if the solution is nondegenerate then the algorithm finds the solution in a finite number of steps moreover a simple lower bound on gamma is given to ensure finite termination even for problems with degenerate solutions the theory covers a class of algorithms allowing many constraints to be added or dropped at a time and accepting approximate solutions of auxiliary problems preliminary numerical results are promising

stability estimates for an inverse scattering problem at high frequencies we consider an inverse scattering problem and its near field approximation at high frequencies we first prove for both problems lipschitz stability results for determining the low frequency component of the potential then we show that in the case of a radial potential supported sufficiently near the boundary infinite resolution can be achieved from measurements of the near field operator in the monotone case

a decomposition method for large scale box constrained optimization a decomposition method for solving large scale box constrained optimization is proposed the algorithm is motivated by the successful use of the decomposition method presented by joachims for training support vector machines in particular a new technique based on the new definition kkt violating index is introduced for working set identification finally the numerical experiments and implementation details show that this method is practical for large scale problems

adaptive radial basis function and hermite function pseudospectral methods for computing eigenvalues of the prolate spheroidal wave equation for very large bandwidth parameter asymptotic approximations show that the lowest modes of the prolate spheroidal wave equation are concentrated with an o 1 c length scale where c is the bandwidth parameter of the prolate differential equation accurate computation of the ground state eigenvalue by the long known legendre galerkin method requires roughly 3 8 c legendre polynomials some studies have therefore applied a grid with 20 000 points in conjunction with high order finite differences to achieve c 10 7 here we show that by adaptively applying either hermite functions or gaussian radial basis functions rbfs it is never necessary to use more than eighty degrees of freedom to calculate the lowest dozen eigenvalues of each symmetry class for small c the eigenmodes are not confined to a small portion of the domain 2 2 in latitude but are global we show that by periodizing the basis functions via imbricate series it is possible to apply hermite and rbfs even in the limit c 0 the legendre method is probably a little more efficient in this limit since the prolate functions tend to legendre polynomials in this limit a sideband truncation restricts the discretization to a small block taken from the large hermite galerkin matrix we show that sideband truncation with blocks as small as 5 5 is a very efficient way to compute high order modes in an appendix we prove a rigorous convergence theorem for the periodized hermite expansion

a survey on orthogonal matrix polynomials satisfying second order differential equations the subject of orthogonal polynomials cuts across a large piece of mathematics and its applications two notable examples are mathematical physics in the 19th and 20th centuries as well as the theory of spherical functions for symmetric spaces it is also clear that many areas of mathematics grew out of the consideration of problems like the moment problem that are intimately associated to the study of scalar valued orthogonal polynomials matrix orthogonality on the real line has been sporadically studied during the last half century since krein devoted some papers to the subject in 1949 see ams translations series 2 vol 97 providence rhode island 1971 pp 75 143 dokl akad nauk sssr 69 2 1949 125 in the last decade this study has been made more systematic with the consequence that many basic results of scalar orthogonality have been extended to the matrix case the most recent of these results is the discovery of important examples of orthogonal matrix polynomials many families of orthogonal matrix polynomials have been found that as the classical families of hermite laguerre and jacobi in the scalar case satisfy second order differential equations with coefficients independent of n the aim of this paper is to give an overview of the techniques that have led to these examples a small sample of the examples themselves and a small step in the challenging direction of finding applications of these new examples

design of optimum nyquist signals based on generalized sampling theory for data communications a new method is given for the optimal design of bandlimited nyquist type signal shapes for data communications which maximizes its energy in a given time interval the method is based on the periodically nonuniform sampling pns theory making use of the linear splines the computation is straightforward and the constraint for intersymbol interference is shown to be easy to include in the problem a numerical example is given and performance of the optimal signal shapes is compared with that resulting from the use of previously obtained signal shapes in the literature it is also concluded that the optimal signal shapes thus obtained are almost immune to small offsets at the sampling instants

global convergence of a class of trust region algorithms for optimization with simple bounds this paper extends the known excellent global convergence properties of trust region algorithms for unconstrained optimization to the case where bounds on the variables are present weak conditions on the accuracy of the hessian approximations are considered it is also shown that when the strict complementarily condition holds the proposed algorithms reduce to an unconstrained calculation after finitely many iterations allowing a fast asymptotic rate of convergence

a sampling theorem for wavelet subspaces the classical shannon sampling theorem is extended to the subspaces used in the multiresolution analysis in wavelet theory under weak hypotheses these subspaces are first shown to have a riesz basis formed from the reproducing kernels these in turn are used to construct the sampling sequences examples are given

mean convergence of prolate spheroidal series and their extensions abstract the aim of this paper is to establish the range of p s for which the expansion of a function f l p in a generalized prolate spheroidal wave function pswfs basis converges to f in l p two generalizations of pswfs are considered here the circular pswfs introduced by d slepian and the weighted pswfs introduced by wang and zhang both cases cover the classical pswfs for which the corresponding results have been previously established by barcelo and cordoba to establish those results we prove a general result that allows to extend mean convergence in a given basis e g jacobi polynomials or bessel basis to mean convergence in a second basis here the generalized pswfs

stochastic phase lockings in a relaxation oscillator forced by a periodic input with additive noise a first passage time approach noise effects on phase lockings in a system consisting of a piecewise linear van der pol relaxation oscillator driven by a periodic input are studied the problem of finding the period of the oscillator is reduced to the first passage time problem of the ornstein uhlenbeck process with time varying boundary the probability density functions of the first passage time are used to define the operator which governs a transition of an input phase density after one cycle of the oscillator phase lockings in a stochastic sense are investigated on the basis of the density evolution by the operator

spectral analysis of stochastic phase lockings and stochastic bifurcations in the sinusoidally forced van der pol oscillator with additive noise noise effects on the phase lockings and bifurcations in the sinusoidally forced van der pol relaxation oscillator are investigated deterministic noise free one dimensional poincare mapping is extended to the iteration of the operator defined by a stochastic kernel function stochastic phase lockings and bifurcations are analyzed in terms of the density evolution by the operator in particular a new method which uses spectra eigenvalues and eigenfunctions of the operator to analyze stochastic bifurcations intensively is proposed

the gridding method for image reconstruction by fourier transformation the authors explore a computational method for reconstructing an n dimensional signal f from a sampled version of its fourier transform f spl circ the method involves a window function w spl circ and proceeds in three steps first the convolution g spl circ w spl circ f spl circ is computed numerically on a cartesian grid using the available samples of f spl circ then g wf is computed via the inverse discrete fourier transform and finally f is obtained as g w due to the smoothing effect of the convolution evaluating w spl circ f spl circ is much less error prone than merely interpolating f spl circ the method was originally devised for image reconstruction in radio astronomy but is actually applicable to a broad range of reconstructive imaging methods including magnetic resonance imaging and computed tomography in particular it provides a fast and accurate alternative to the filtered backprojection the basic method has several variants with other applications such as the equidistant resampling of arbitrarily sampled signals or the fast computation of the radon hough transform

the darboux process and time and band limiting for matrix orthogonal polynomials abstract we extend to a situation involving matrix valued orthogonal polynomials a scalar result that originates in work of claude shannon in laying the mathematical foundations of information theory and a remarkable series of papers by d slepian h landau and h pollak at bell labs in the 1960 s we show that in this case an algebraic miracle that plays a very important role in the classical case survives an application of the so called darboux process in the matrix valued context

prolate spheroidal wavefunctions as an alternative to chebyshev and legendre polynomials for spectral element and pseudospectral algorithms prolate spheroidal functions of order zero are generalizations of legendre polynomials which when the bandwidth parameter c 0 oscillate more uniformly on x 1 1 than either chebyshev or legendre polynomials this suggests that compared to these polynomials prolate functions give more uniform spatial resolution further when used as the spatial discretization for time dependent partial differential equations in combination with explicit time marching prolate functions allow a longer stable timestep than legendre polynomials we show that these advantages are real and further that it is almost trivial to modify existing pseudospectral and spectral element codes to use the prolate basis however improvements in spatial resolution are at most a factor of 2 approached slowly as n the timestep can be lengthened by several times but not by a factor that grows rapidly with n the prolate basis is not likely to radically expand the range of problems that can be done on a workstation however for production runs on the bleeding edge edge of arithmurgy such as numerical weather prediction the rewards for switching to a prolate basis may be considerable

toeplitz matrices commuting with tridiagonal matrices abstract we prove that if r is a nonscalar toeplitz matrix ri j r i j which commutes with a tridiagonal matrix with simple spectrum then r k r 1 u k 1 r 2 r 1 cos p u k 1 cos p k 4 5 with uk the chebychev polynomial of the second kind where p is determined from cos p 1 2 r 2 1 r 1 r 3 r 2 2 r 1 r 3

efficient computation of slepian functions for arbitrary regions on the sphere in this paper we develop a new method for the fast and memory efficient computation of slepian functions on the sphere slepian functions which arise as the solution of the slepian concentration problem on the sphere have desirable properties for applications where measurements are only available within a spatially limited region on the sphere and or a function is required to be analyzed over the spatially limited region slepian functions are currently not easily computed for large band limits for an arbitrary spatial region due to high computational and large memory storage requirements for the special case of a polar cap the symmetry of the region enables the decomposition of the slepian concentration problem into smaller subproblems and consequently the efficient computation of slepian functions for large band limits by exploiting the efficient computation of slepian functions for the polar cap region on the sphere we develop a formulation supported by a fast algorithm for the approximate computation of slepian functions for an arbitrary spatial region to enable the analysis of modern datasets that support large band limits for the proposed algorithm we carry out accuracy analysis of the approximation computational complexity analysis and review of memory storage requirements we illustrate through numerical experiments that the proposed method enables faster computation and has smaller storage requirements while allowing for sufficiently accurate computation of the slepian functions

on landau s eigenvalue theorem and information cut sets a variation of landau s eigenvalue theorem describing the phase transition of the eigenvalues of a time frequency limiting self adjoint operator is presented the total number of degrees of freedom of square integrable multidimensional bandlimited functions is defined in terms of kolmogorov s n width and computed in some limiting regimes where the original theorem cannot be directly applied results are used to characterize up to order the total amount of information that can be transported in time and space by multiple scattered electromagnetic waves rigorously addressing a question originally posed in the early works of toraldo di francia and gabor applications in the context of wireless communication and electromagnetic sensing are discussed

design and hybrid realization of fir nyquist filters with quantized coefficients and low sensitivity to timing jitter a design of finite impulse response fir nyquist filters with zero intersymbol interference isi and low sensitivity to timing jitter is presented using an affine scaling linear programming algorithm a near optimum quantized coefficient set can be obtained in a feasible computational time by varying a parameter the design provides a tradeoff between the tail energy of the impulse response in the time domain and the stopband of the magnitude response in the frequency domain we also present a pipelined multiplier free fir filter realization with periodically time varying ptv coefficients based on a hybrid form the realizations exploit the coefficient symmetry to reduce the hardware by about one half by placing most of the shifts followed by addition toward the back end of the structure hardware is reduced due to the shorter wordlength of the adders the proposed structure has a provision to increase the speed by adjusting a design parameter but at the expense of more hardware

on hp convergence of prolate spheroidal wave functions and a new well conditioned prolate collocation scheme the first purpose of this paper is to provide further illustrations from both theoretical and numerical perspectives for the nonconvergence of h refinement in hp approximation by the prolate spheroidal wave functions pswfs a surprising convergence property that was first discovered by boyd et al 2013 3 the second purpose is to offer a new basis that leads to prolate collocation systems with condition numbers independent of c n the intrinsic bandwidth parameter and the number of collocation points we highlight that the collocation scheme together with a very practical rule for pairing up c n significantly outperforms the legendre polynomial based method and likewise other jacobi polynomial based methods in approximating highly oscillatory bandlimited functions

the heun racah and heun bannai ito algebras the heun racah and heun bannai ito algebras are introduced specializations of these algebras are seen to be realized by the operators obtained by applying the algebraic heun construct to the bispectral operators of the racah and bannai ito polynomials the study supplements the results on the heun askey wilson algebra and completes the description of the heun algebras associated with the polynomial families at the top of the askey scheme its q analog and the bannai ito one

some comments on fourier analysis uncertainty and modeling investigation of the problem of simultaneously concentrating a function and its fourier transform has led to some interesting special functions that have widespread applications in engineering they provide a means of proving a rigorous version of an engineering folk theorem called the 2wt theorem many generalizations of these ideas seem to possess a similar elegant mathematical structure a brief descriptive review is given of these developments

a donoho stark criterion for stable signal recovery in discrete wavelet subspaces abstract we derive a sufficient condition by means of which one can recover a scale limited signal from the knowledge of a truncated version of it in a stable manner following the canvas introduced by donoho and stark 1989 4 the proof follows from simple computations involving the zak transform well known in solid state physics geometric harmonics in the terminology of coifman and lafon 2006 22 for scale limited subspaces of l 2 r are also displayed for several test cases finally some algorithms are studied for the treatment of zero angle problems

the degrees of freedom of wireless networksvia cut set integrals the problem of determining the number of spatial degrees of freedom d o f of the signals carrying information in a wireless network is reduced to the computation of the geometric variation of the environment with respect to the cut through which the information must flow physically this has an appealing interpretation in terms of the diversity induced on the cut by the possible richness of the scattering environment mathematically this variation is expressed as an integral along the cut which we call cut set integral and whose scaling order is evaluated exactly in the case of planar networks embedded in arbitrary three dimensional 3 d environments presented results shed some new light on the problem of computing the capacity of wireless networks showing a fundamental limitation imposed by the size of the cut through which the information must flow in an attempt to remove what may appear as apparent inconsistencies with previous literature we also discuss how our upper bounds relate to corresponding lower bounds obtained using the techniques of multihop hierarchical cooperation and interference alignment

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

compact representations of kohn sham invariant subspaces we present a method to compute a hierarchical approximate representation of the solutions of the kohn sham equations the method is based on a recursive bisection algorithm and yields one particle wave functions localized on subdomains of varying sizes the accuracy of the representation is set a priori by specifying the highest acceptable error in 2 norm for any solution applications to large systems are used to illustrate the achievable reduction in data size implications for the development of linear scaling methods and for the acceleration of conventional iterative methods are discussed

band limited functions lp convergence we consider the set bp q functions of lp tl whose fourier spectrum lies in i2 i2 we prove that the prolate spheroidal wave func tions constitute a basis of this space if and only if 4 3 p 4 the result is obtained as a consequence of the analogous problem for the spherical bessel functions the proof rely on a weighted inequality for the hubert transform

analytical approach to an integrate and fire model with spike triggered adaptation the calculation of the steady state probability density for multidimensional stochastic systems that do not obey detailed balance is a difficult problem here we present the analytical derivation of the stationary joint and various marginal probability densities for a stochastic neuron model with adaptation current our approach assumes weak noise but is valid for arbitrary adaptation strength and time scale the theory predicts several effects of adaptation on the statistics of the membrane potential of a tonically firing neuron i a membrane potential distribution with a convex shape ii a strongly increased probability of hyperpolarized membrane potentials induced by strong and fast adaptation and iii a maximized variability associated with the adaptation current at a finite adaptation time scale

transient signal detection using prior information in the likelihood ratio test a uniform approach to the problem of detecting a deterministic transient signal of unknown waveshape in gaussian noise that is applicable for a general class of signals is provided it is assumed that the vector representation consists of a group of zero coefficients plus a group of unknown nonzero coefficients starting at a certain index performance indices for a likelihood ratio test detector of such a generalized transient signal are derived for three cases a both the starting index and the number of unknown nonzero signal coefficients are known b only the number of unknown nonzero signal coefficients is known and c neither of these is known it is shown that the performance of detector a is always better than that of b and c and conditions under which the performance of detector b is better than that of c are derived the theoretical results are demonstrated by considering signals of different assumed structures

a new active set algorithm for box constrained optimization an active set algorithm asa for box constrained optimization is developed the algorithm consists of a nonmonotone gradient projection step an unconstrained optimization step and a set of rules for branching between the two steps global convergence to a stationary point is established for a nondegenerate stationary point the algorithm eventually reduces to unconstrained optimization without restarts similarly for a degenerate stationary point where the strong second order sufficient optimality condition holds the algorithm eventually reduces to unconstrained optimization without restarts a specific implementation of the asa is given which exploits the recently developed cyclic barzilai borwein cbb algorithm for the gradient projection step and the recently developed conjugate gradient algorithm cg descent for unconstrained optimization numerical experiments are presented using box constrained problems in the cuter and minpack 2 test problem libraries

critical dynamics of the bonhoeffer van der pol equation and its chaotic response to periodic stimulation abstract we consider the bonhoeffer van der pol equation as a prototype model for excitable systems for this equation we investigate the effect of application of a periodic stimulus in a critical range of values for stimulus strength and period we find depending on the choice of those values many different response types which include quasi periodic and chaotic behavior we present phase portraits and bifurcation diagrams to illustrate these responses the study of an associated local equation shows that the same types of behavior may be expected in a whole class of related equations we discuss the relevance of our results for a number of biological and physiological applications

spatio spectral concentration of convolutions differential equations may possess coefficients that vary on a spectrum of scales because coefficients are typically multiplicative in real space they turn into convolution operators in spectral space mixing all wavenumbers however in many applications only the largest scales of the solution are of interest and so the question turns to whether it is possible to build effective coarse scale models of the coefficients in such a manner that the large scales of the solution are left intact here we apply the method of numerical homogenisation to deterministic linear equations to generate sub grid scale models of coefficients at desired frequency cutoffs we use the fourier basis to project filter and compute correctors for the coefficients the method is tested in 1d and 2d scenarios and found to reproduce the coarse scales of the solution to varying degrees of accuracy depending on the cutoff we relate this method to mode elimination renormalisation group rg and discuss the connection between accuracy and the cutoff wavenumber the tradeoff is governed by a form of the uncertainty principle for convolutions which states that as the convolution operator is squeezed in the spectral domain it broadens in real space as a consequence basis sparsity is a high virtue and the choice of the basis can be critical

noise induced order ii a curious noise effect in certain maps reported earlier is investigated further a striking feature of these maps is obtained in the symbolic dynamical approach the decrease of entropy is attributed to a simple mechanism which deletes certain states in the symbolic dynamics and the value of the modified entropy is calculated based on this picture

optimum fir transmitter and receiver filters for data transmission over band limited channels the paper deals with the design of digital transmitter and receiver filters with finite impulse response fir for data transmission over band limited channels the filters are matched and satisfy a zero intersymbol interference constraint when cascaded for baseband transmission the filters achieve optimum spectral concentration in the frequency range 1 beta 2t 1 beta 2t mathematically the filter design leads to a generalized eigenvalue problem which is solved numerically by a projected gradient procedure for transmission over bandpass channels by combined amplitude and phase modulation the design technique is modified so that filters with complex valued impulse response and optimum spectral concentration in the range of positive bandpass frequencies f c 1 beta 2t f c 1 beta 2t are obtained in addition the complex formulation allows the design of impulse responses with enhanced spectral attenuation in the corresponding range of negative frequencies in order to minimize imageband interference results are shown in terms of filter coefficients signal spectra and spectral concentrations obtained for example filters designed for a voiceband data modem operating at a symbol rate of 2400 baud achieve a spectral concentration of 98 5 percent with 24 coefficients and beta 0 1 and with only 0 001 percent of the total energy in the imageband region

a subspace limited memory quasi newton algorithm for large scale nonlinear bound constrained optimization in this paper we propose a subspace limited memory quasi newton method for solving large scale optimization with simple bounds on the variables the limited memory quasi newton method is used to update the variables with indices outside of the active set while the projected gradient method is used to update the active variables the search direction consists of three parts a subspace quasi newton direction and two subspace gradient and modified gradient directions our algorithm can be applied to large scale problems as there is no need to solve any subproblems the global convergence of the method is proved and some numerical results are also given

approximate factorizations of fourier matrices with nonequispaced knots abstract we propose a new algorithm for the multiplication of vectors with fourier matrices of the form a tf e 2 i x k v j n n 2 1 j k n 2 where both x k and v j are arbitrary knots in n 2 n 2 the algorithm is based on an approximate factorization of the transform matrix a tf into sparse matrices which entries are chosen to minimize the frobenius norm of certain error matrices numerical experiments demonstrate that the approximation error introduced by our new algorithm is about 10 2 times smaller than the approximation error of previously reported algorithms with the gaussian or b splines as window functions and as good as a previous algorithm with the kaiser bessel function as window function

recovering band limited signals under noise we consider the problem of recovering a band limited signal f t from noisy data yk f k spl tau spl gt epsilon sub k where spl tau is the sampling rate starting from the truncated whittaker shannon cardinal expansion with or without sampling windows both cases yield inconsistent estimates of f t we propose estimators that are convergent to f t in the pointwise and uniform sense the basic idea is to cut down high frequencies in the data and to use suitable oversampling spl tau spl les spl pi spl omega spl omega being the bandwidth maximum frequency of f t the simplest estimator we propose is given by f spl circ sub n t spl tau spl sigma t k spl tau spl les n spl tau yksin spl omega t k spl tau spl pi t k spl tau t spl les n spl tau generalizations of f spl circ sub n including sampling windows are also examined the main aim is to examine the mean squared error mse properties of such estimators in order to determine the optimal choice of the sampling rate spl tau yielding the fastest possible rate of convergence the best rate for the mse we obtain is o in n n

rapid prolate pseudospectral differentiation and interpolation with the fast multipole method pseudospectral methods utilizing prolate spheroidal wave functions as basis functions have been shown to possess advantages over the conventional pseudospectral methods based on trigonometric and orthogonal polynomials however the spectral differentiation and interpolation steps of the prolate pseudospectral method involve matrix vector products which if evaluated directly entail o n2 memory requirement and computational complexity where n is the number of unknowns utilized for discretization and interpolation in this work we show that the fast multipole method fmm can be used to reduce the memory requirement and computational complexity of the prolate pseudospectral method to o n example simulation results demonstrate the speed and accuracy of the resulting fast prolate pseudospectral solver

characterization of stochastic bifurcations in a simple biological oscillator this study of the effect of noise on bifurcations in a simple biological oscillator with a periodically modulated threshold uses the first passage time problem of the ornstein uhlenbeck process with a periodic boundary to define the operator governing the transition of a threshold phase density stochastic phase locking is analyzed numerically by evaluating the evolution of the probability density function of the threshold phase a firing phase map in a noisy environment is extended to a stochastic kernel so that stochastic bifurcations can be investigated by spectral analysis of the kernel

the chirplet transform physical considerations we consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis we propose the use of quadratic chirp functions which we will call q chirps for short giving rise to a parameter space that includes both the time frequency plane and the time scale plane as 2 d subspaces the parameter space contains a time frequency scale volume and thus encompasses both the short time fourier transform as a slice along the time and frequency axes and the wavelet transform as a slice along the time and scale axes in addition to time frequency and scale there are two other coordinate axes within this transform space shear in time obtained through convolution with a q chirp and shear in frequency obtained through multiplication by a q chirp signals in this multidimensional space can be obtained by a new transform which we call the q chirplet transform or simply the chirplet transform the proposed chirplets are generalizations of wavelets related to each other by 2 d affine coordinate transformations translations dilations rotations and shears in the time frequency plane as opposed to wavelets which are related to each other by 1 d affine coordinate transformations translations and dilations in the time domain only

the use of the wavelet transform in the detection of an unknown transient signal for the detection of a not perfectly known signal in noise usually no uniformly most powerful test exists and thus a detector performance depends on the signal representation the use of the wavelet representation of signals to perform a new detection scheme is discussed the advantage of using this particular representation is shown it is shown that prior information regarding the relative bandwidth and the time bandwidth product of the signal to be detected is efficiently incorporated into the detection problem formulation thus the proposed detection scheme is most suitable for detection of unknown transient signals when prior information about the signal time bandwidth product and relative bandwidth exists in these cases the wavelet representation based detector performs better than any other the structure of the proposed detectors is discussed and its performance is evaluated using monte carlo simulations

computation of component image velocity from local phase information we present a technique for the computation of 2d component velocity from image sequences initially the image sequence is represented by a family of spatiotemporal velocity tuned linear filters component velocity computed from spatiotemporal responses of identically tuned filters is expressed in terms of the local first order behavior of surfaces of constant phase justification for this definition is discussed from the perspectives of both 2d image translation and deviations from translation that are typical in perspective projections of 3d scenes the resulting technique is predominantly linear efficient and suitable for parallel processing moreover it is local in space time robust with respect to noise and permits multiple estimates within a single neighborhood promising quantiative results are reported from experiments with realistic image sequences including cases with sizeable perspective deformation

generalized sampling theorems in multiresolution subspaces it is well known that under very mild conditions on the scaling function multiresolution subspaces are reproducing kernel hilbert spaces rkhss this allows for the development of a sampling theory in this paper we extend the existing sampling theory for wavelet subspaces in several directions we consider periodically nonuniform sampling sampling of a function and its derivatives oversampling multiband sampling and reconstruction from local averages all these problems are treated in a unified way using the perfect reconstruction pr filter bank theory we give conditions for stable reconstructions in each of these cases sampling theorems developed in the past do not allow the scaling function and the synthesizing function to be both compactly supported except in trivial cases this restriction no longer applies for the generalizations we study here due to the existence of fir pr banks in fact with nonuniform sampling oversampling and reconstruction from local averages we can guarantee compactly supported synthesizing functions moreover local averaging schemes have additional nice properties robustness to the input noise and compression capabilities we also show that some of the proposed methods can be used for efficient computation of inner products in multiresolution analysis after this we extend the sampling theory to random processes we require autocorrelation functions to belong to some subspace related to wavelet subspaces it turns out that we cannot recover random processes themselves unless they are bandlimited but only their power spectral density functions we consider both uniform and nonuniform sampling

a study on smo type decomposition methods for support vector machines decomposition methods are currently one of the major methods for training support vector machines they vary mainly according to different working set selections existing implementations and analysis usually consider some specific selection rules this paper studies sequential minimal optimization type decomposition methods under a general and flexible way of choosing the two element working set the main results include 1 a simple asymptotic convergence proof 2 a general explanation of the shrinking and caching techniques and 3 the linear convergence of the methods extensions to some support vector machine variants are also discussed

spectral analysis of the finite hankel transform and circular prolate spheroidal wave functions in this paper we develop two practical methods for the computation of the eigenvalues as well as the eigenfunctions of the finite hankel transform operator these different eigenfunctions are called circular prolate spheroidal wave functions cpswfs this work is motivated by the potential applications of the cpswfs as well as the development of practical methods for computing their values also in this work we should prove that the cpswfs form an orthonormal basis of the space of hankel band limited functions an orthogonal basis of l 2 0 1 and an orthonormal system of l 2 0 our computation of the cpswfs and their associated eigenvalues is done by the use of two different methods the first method is based on a suitable matrix representation of the finite hankel transform operator the second method is based on the use of an efficient quadrature method based on a special family of orthogonal polynomials also we give two maple programs that implement the previous two methods finally we present some numerical results that illustrate the results of this work

spatiospectral concentration on a sphere we pose and solve the analogue of slepian s time frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere or alternatively of strictly spacelimited functions that are optimally concentrated in the spherical harmonic domain such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications as well as in medical imaging computer science cosmology and numerical analysis the spherical slepian functions can be found by solving either an algebraic eigenvalue problem in the spectral domain or a fredholm integral equation in the spatial domain the associated eigenvalues are a measure of the spatiospectral concentration when the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a sturm liouville operator this enables the eigenfunctions to be computed extremely accurately and efficiently even when their area bandwidth product or shannon number is large in the asymptotic limit of a small spatial region and a large spherical harmonic bandwidth the spherical concentration problem reduces to its planar equivalent which exhibits self similarity when the shannon number is kept invariant

a new conjugate gradient method based on quasi newton equation for unconstrained optimization abstract the spectral conjugate gradient method is an effective method for large scale unconstrained optimization problems in this paper based on quasi newton direction and quasi newton equation a new spectral conjugate gradient method is proposed this method is motivated by the three term modified polak ribiyre polyak prp method and spectral parameters the global convergence of algorithm is proved for general functions under a strong wolfe line search numerical results show that the new algorithm is superior to the three term modified prp method

the zak transform and sampling theorems for wavelet subspaces the zak transform is used for generalizing a sampling theorem of g waiter see ieee trans informat theory vol 38 p 881 884 1992 for wavelet subspaces cardinal series based on signal samples f a n n in z with a possibly unequal to 0 waiter s case are considered the condition number of the sampling operator and worst case aliasing errors are expressed in terms of zak transforms of scaling function and wavelet this shows that the stability of the resulting interpolation formula depends critically on a

time frequency localization operators a geometric phase space approach the author defines a set of operators which localize in both time and frequency these operators are similar to but different from the low pass time limiting operator the singular functions of which are the prolate spheroidal wave functions the author s construction differs from the usual approach in that she treats the time frequency plane as one geometric whole phase space rather than as two separate spaces for disk shaped or ellipse shaped domains in time frequency plane the associated localization operators are remarkably simple their eigenfunctions are hermite functions and the corresponding eigenvalues are given by simple explicit formulas involving the incomplete gamma functions

eigenvalue estimates and mutual information for the linear time varying channel we consider linear time varying channels with additive white gaussian noise for a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time varying transfer function and thus provide a theoretical justification for a relationship that has been frequently observed in the literature we then use this eigenvalue estimate to derive an estimate of the mutual information of the channel our approach is constructive and is based on a careful balance of the tradeoff between approximate operator diagonalization signal dimension loss and accuracy of eigenvalue estimates

q prolate spheroidal wave functions and q sampling theorem in this paper we introduce a q version of the prolate spheroidal wave functions which is the eigenfunction of the concentration problem related to sampling and interpolation of q hankel bandlimited functions as application we give a q sampling theorem which extrapolates functions in l 2 space with q n email protected z as sampling points

ode solvers using band limited approximations we use generalized gaussian quadratures for exponentials to develop a new ode solver nodes and weights of these quadratures are computed for a given bandlimit c and user selected accuracy e so that they integrate functions e i b x for all b

estimation of the optical constants and the thickness of thin films using unconstrained optimization the problem of estimating the thickness and the optical constants of thin films using transmission data only is very challenging from the mathematical point of view and has a technological and an economic importance in many cases it represents a very ill conditioned inverse problem with many local nonglobal solutions in a recent publication we proposed nonlinear programming models for solving this problem well known software for linearly constrained optimization was used with success for this purpose in this paper we introduce an unconstrained formulation of the nonlinear programming model and we solve the estimation problem using a method based on repeated calls to a recently introduced unconstrained minimization algorithm numerical experiments on computer generated films show that the new procedure is reliable

orthogonal sets of data windows constructed from trigonometric polynomials suboptimal easily computable substitutes for the discrete prolate spheroidal windows used by d j thomson proc ieee vol 70 p 1055 1096 1982 for spectral estimation are given trigonometric coefficients and energy leakages of the window polynomials are tabulated

energy optimization of band limited nyquist signals in the time domain this paper presents a theoretical consideration of the optimal design of band limited nyquist type signal shapes for data transmission which maximizes its energy in a given time interval and which generates no intersymbol interference at the periodic sampling instants a method based on a completely analytical approach is given for design of such signals the optimal signal is a solution of an inhomogeneous linear integral equation of fredholm type a technique for solving this equation is given the computation is straightforward and involves the determination of eigenvalues and eigenfunctions of a positive definite and symmetric kernel in terms of prolate spheroidal wave functions the constraint for intersymbol interference is shown to be easily included into the problem finally some numerical examples are given and the performance of the optimal signal shapes is compared to that resulting from the use of the raised cosine type of signals it is also concluded that especially for small values of rolloff factor the optimal signals thus obtained are almost maximally immune to small timing offsets at the sampling instants

a new property of reproducing kernels for classical orthogonal polynomials abstract we exhibit a second order differential operator commuting with the reproducing kernel n 0 t n n h n each time that n is one of the classical orthogonal polynomials jacobi laguerre hermite and bessel this is the analog of a known property in the study of time and band limited signals

optimized signaling waveforms to reduce the effects of intersymbol interference and timing jitter the mainlobe energy of a member from the class of symbol waveforms with null derivatives at zero crossings is maximized thus a bandlimited pulse shape without intersymbol interference robust against jitter and timing errors is obtained comparisons to known pulses show advantages for these signals at the expense of bandwidth which linearly increases with the number of null derivatives

stochastic and chaotic relaxation oscillations for relaxation oscillators stochastic and chaotic dynamics are investigated the effect of random perturbations upon the period is computed for an extended system with additional state variables chaotic behavior can be expected as an example the van der pol oscillator is changed into a third order system admitting period doubling and chaos in a certain parameter range the distinction between chaotic oscillation and oscillation with noise is explored return maps power spectra and lyapunov exponents are analyzed for that purpose

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

eigenvector matrices of symmetric tridiagonals a simple test is given for determining whether a given matrix is the eigenvector matrix of an unknown unreduced symmetric tridiagonal matrix a list of known necessary conditions is also provided a lower bound on the separation between eigenvalues of tridiagonals follows from our theorem 3

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

superoscillations faster than the nyquist rate it is commonly assumed that a signal bandlimited to mu 2 hz cannot oscillate at frequencies higher than mu hz in fact however for any fixed bandwidth there exist finite energy signals that oscillate arbitrarily fast over arbitrarily long time intervals these localized fast transients called superoscillations can only occur in signals that possess amplitudes of widely different scales this paper investigates the required dynamical range and energy squared l2 norm as a function of the superoscillation s frequency number and maximum derivative it briefly discusses some of the implications of superoscillating signals in reference to information theory and time frequency analysis for example it also shows among other things that the required energy grows exponentially with the number of superoscillations and polynomially with the reciprocal of the bandwidth or the reciprocal of the superoscillations period

the eps method a new method for constructing pseudospectral derivative operators we develop a new type of derivative matrix for pseudospectral methods the norm of these matrices grows at the optimal rate o n 2 for n by n matrices in contrast to standard pseudospectral constructions that result in o n 4 growth of the norm the smaller norm has a big advantage when using the derivative matrix for solving time dependent problems such as wave propagation the construction is based on representing the derivative operator as an integral kernel and does not rely on the interpolating polynomials in particular we construct second derivative matrices that incorporate dirichlet or neumann boundary conditions on an interval and on the disk but the method can be used to construct a wide variety of commonly used operators for solving pdes and integral equations the construction can be used with any quadrature including traditional gauss legendre quadratures but we have found that by using quadratures based on prolate spheroidal wave functions we can achieve a near optimal sampling rate close to two points per wavelength even for non periodic problems we provide numerical results for the new construction and demonstrate that the construction achieves similar or better accuracy than traditional pseudospectral derivative matrices while resulting in a norm that is orders of magnitude smaller than the standard construction to demonstrate the advantage of the new construction we apply the method for solving the wave equation in constant and discontinuous media and for solving pdes on the unit disk we also present two compression algorithms for applying the derivative matrices in o nlogn operations

tridiagonalization and the heun equation it is shown that the tridiagonalization of the hypergeometric operator l yields the generic heun operator m the algebra generated by the operators l m and z l m is quadratic and a one parameter generalization of the racah algebra the racah heun orthogonal polynomials are introduced as overlap coefficients between the eigenfunctions of the operators l and m an interpretation in terms of the racah problem for su 1 1 and separation of variables in a superintegrable system are discussed

superlinear and quadratic convergence of affine scaling interior point newton methods for problems with simple bounds without strict complementarity assumption a class of affine scaling interior point methods for bound constrained optimization problems is introduced which are locally q superlinear or q quadratic convergent it is assumed that the strong second order sufficient optimality conditions at the solution are satisfied but strict complementarity is not required the methods are modifications of the affine scaling interior point newton methods introduced by t f coleman and y li math programming 67 189 224 1994 there are two modifications one is a modification of the scaling matrix the other one is the use of a projection of the step to maintain strict feasibility rather than a simple scaling of the step a comprehensive local convergence analysis is given a simple example is presented to illustrate the pitfalls of the original approach by coleman and li in the degenerate case and to demonstrate the performance of the fast converging modifications developed in this paper

maximizing the output energy of a linear channel with a time and amplitude limited input the problem of maximizing the output energy of a linear time invariant channel given that the input signal is time and amplitude limited is considered it is shown that a necessary condition for an input mu to be optimal assuming a unity amplitude constraint is that it satisfy the fixed point equation sgn f mu where the functional f is the convolution of mu with the autocorrelation function of the channel impulse response it is also shown that all solutions to this equation for which mod mu mod 1 almost everywhere correspond to local maxima of the output energy iteratively recomputing mu from the fixed point equation leads to an algorithm for finding local optima numerical results are given for the cases where the transfer function is ideal low pass and has two poles these results support the conjecture that in the ideal low pass case the optimal input signal is a single square pulse a generalization of the preceding fixed point condition is also derived for the problem of maximally separating n outputs of a discrete time linear time invariant channel