wigner function for discrete phase space exorcising ghost images abstract we construct using simple geometrical arguments a wigner function defined on a discrete phase space of arbitrary integer hilbert space dimension that is free of redundancies ghost images plaguing other wigner functions for discrete phase spaces are thus revealed as artifacts it allows to devise a corresponding phase space propagator in an unambiguous manner we apply our definitions to eigenstates and propagator of the quantum baker map scars on unstable periodic points of the corresponding classical map become visible with unprecedented resolution

on semiclassical dispersion relations of harper like operators we describe some semiclassical spectral properties of harper like operators i e of one dimensional quantum hamiltonians periodic in both momentum and position the spectral region corresponding to the separatrices of the classical hamiltonian is studied for the case of integer flux we derive asymptotic formulae for the dispersion relations the width of bands and gaps and show how geometric characteristics and the absence of symmetries of the hamiltonian influence the form of the energy bands

factorization properties of finite spaces in 1960 schwinger j schwinger 1960 proc natl acad sci 46 570 9 proposed the algorithm for factorization of unitary operators in the finite m dimensional hilbert space according to a coprime decomposition of m using a special permutation operator a we generalize the schwinger factorization to every decomposition of m we obtain the factorized pairs of unitary operators and show that they obey the same commutation relations as schwinger s we apply the new factorization to two problems first we show how to generate two kq like mutually unbiased bases for any composite dimension then using a harper like hamiltonian model in the finite dimension m m1m2 we show how to design a physical system with m1 energy levels each having degeneracy m2

embedded image coding using zerotrees of wavelet coefficients the embedded zerotree wavelet algorithm ezw is a simple yet remarkably effective image compression algorithm having the property that the bits in the bit stream are generated in order of importance yielding a fully embedded code the embedded code represents a sequence of binary decisions that distinguish an image from the null image using an embedded coding algorithm an encoder can terminate the encoding at any point thereby allowing a target rate or target distortion metric to be met exactly also given a bit stream the decoder can cease decoding at any point in the bit stream and still produce exactly the same image that would have been encoded at the bit rate corresponding to the truncated bit stream in addition to producing a fully embedded bit stream the ezw consistently produces compression results that are competitive with virtually all known compression algorithms on standard test images yet this performance is achieved with a technique that requires absolutely no training no pre stored tables or codebooks and requires no prior knowledge of the image source the ezw algorithm is based on four key concepts 1 a discrete wavelet transform or hierarchical subband decomposition 2 prediction of the absence of significant information across scales by exploiting the self similarity inherent in images 3 entropy coded successive approximation quantization and 4 universal lossless data compression which is achieved via adaptive arithmetic coding

galois quantum systems irreducible polynomials and riemann surfaces finite quantum systems in which the position and momentum take values in the galois field gf pl are studied ideas from the subject of field extension are transferred in the context of quantum mechanics the frobenius automorphisms in galois fields lead naturally to the frobenius formalism in a quantum context the hilbert space splits into frobenius subspaces which are labeled with the irreducible polynomials associated with the ypl y the frobenius maps transform unitarily the states of a galois quantum system and leave fixed all states in some of its galois subsystems where the position and momentum take values in subfields of gf pl an analytic representation of these systems in the l sheeted complex plane shows deeper links between galois theory and riemann surfaces

generalized sampling stability and performance analysis generalized sampling provides a general mechanism for recovering an unknown input function f x spl epsiv spl hscr from the samples of the responses of m linear shift invariant systems sampled at 1 mth the reconstruction rate the system can be designed to perform a projection of f x onto the reconstruction subspace v spl phi span spl phi x k sub k spl epsiv z for example the family of bandlimited signals with spl phi x sinc x this implies that the reconstruction will be perfect when the input signal is included in v spl phi the traditional framework of papoulis 1977 generalized sampling theory otherwise one recovers a signal approximation f x spl epsiv v spl phi that is consistent with f x in the sense that it produces the same measurements to characterize the stability of the algorithm we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a riesz basis of v spl phi and we use the corresponding riesz bounds to define the condition number of the system we then use these results to analyze the stability of various instances of interlaced and derivative sampling next we consider the issue of performance which becomes pertinent once we have extended the applicability of the method to arbitrary input functions that is when spl hscr is considerably larger than v spl phi and the reconstruction is no longer exact we show that the generalized sampling solution is essentially equivalent to the optimal minimum error approximation we then perform a detailed analysis for the case in which the analysis filters are in l sub 2 and determine all relevant bound constants explicitly finally we use an interlaced sampling example to illustrate these various calculations

semiclassical wigner functions for quantum maps on a torus a semiclassical formula is derived for the wigner representation of the quasi energy states of a quantum map on a torus it is expressed as a finite sum over the classical fixed points of the map a criterion for the appearance of scars is presented semiclassical wigner functions of the cat map are calculated and shown to reproduce the exact quantum mechanical results

adaptive algorithms for optimal control of time dependent partial differential algebraic equation systems summary this paper describes an adaptive algorithm for optimal control of time dependent partial dierential algebraic equation pdae systems a direct method based on a modied multiple shooting type technique and sequential quadratic programming sqp is used for solving the optimal control problem while an adaptive mesh renement amr algorithm is employed to dynamically adapt the spatial integration mesh issues of coupling the amr solver to the optimization algorithm are addressed for time dependent pdaes which can benet from the use of an adaptive mesh the resulting method is shown to be highly ecient copyright 2003 john wiley sons ltd

biorthogonal bases of compactly supported wavelets orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide we show here that under fairly general conditions exact reconstruction schemes with synthesis filters different from the analysis filters give rise to two dual riesz bases of compactly supported wavelets we give necessary and sufficient conditions for biorthogonality of the corresponding scaling functions and we present a sufficient condition for the decay of their fourier transforms we study the regularity of these biorthogonal bases we provide several families of examples all symmetric corresponding to linear phase filters in particular we can construct symmetric biorthogonal wavelet bases with arbitrarily high preassigned regularity we also show how to construct symmetric biorthogonal wavelet bases close to a nonsymmetric orthonormal basis

image coding using wavelet transform a scheme for image compression that takes into account psychovisual features both in the space and frequency domains is proposed this method involves two steps first a wavelet transform used in order to obtain a set of biorthogonal subclasses of images the original image is decomposed at different scales using a pyramidal algorithm architecture the decomposition is along the vertical and horizontal directions and maintains constant the number of pixels required to describe the image second according to shannon s rate distortion theory the wavelet coefficients are vector quantized using a multiresolution codebook to encode the wavelet coefficients a noise shaping bit allocation procedure which assumes that details at high resolution are less visible to the human eye is proposed in order to allow the receiver to recognize a picture as quickly as possible at minimum cost a progressive transmission scheme is presented it is shown that the wavelet transform is particularly well adapted to progressive transmission

optimization of integer wavelet transforms based on difference correlation structures in this paper a novel lifting integer wavelet transform based on difference correlation structure dccs liwt is proposed first we establish a relationship between the performance of a linear predictor and the difference correlations of an image the obtained results provide a theoretical foundation for the following construction of the optimal lifting filters then the optimal prediction lifting coefficients in the sense of least square prediction error are derived dccs liwt puts heavy emphasis on image inherent dependence a distinct feature of this method is the use of the variance normalized autocorrelation function of the difference image to construct a linear predictor and adapt the predictor to varying image sources the proposed scheme also allows respective calculations of the lifting filters for the horizontal and vertical orientations experimental evaluation shows that the proposed method produces better results than the other well known integer transforms for the lossless image compression

a pde sensitivity equation method for optimal aerodynamic design the use of gradient based optimization algorithms in inverse design is well established as a practical approach to aerodynamic design a typical procedure uses a simulation scheme to evaluate the objective function from the approximate states and its gradient then passes this information to an optimization algorithm once the simulation scheme cfd flow solver has been selected and used to provide approximate function evaluations there are several possible approaches to the problem of computing gradients one popular method is to differentiate the simulation scheme and compute design sensitivities that are then used to obtain gradients although this black box approach has many advantages in shape optimization problems one must compute mesh sensitivities in order to compute the design sensitivity in this paper we present an alternative approach using the pde sensitivity equation to develop algorithms for computing gradients this approach has the advantage that mesh sensitivities need not be computed moreover when it is possible to use the cfd scheme for both the forward problem and the sensitivity equation then there are computational advantages an apparent disadvantage of this approach is that it does not always produce consistent derivatives however for a proper combination of discretization schemes one can showasymptotic consistencyunder mesh refinement which is often sufficient to guarantee convergence of the optimal design algorithm in particular we show that when asymptotically consistent schemes are combined with a trust region optimization algorithm the resulting optimal design method converges we denote this approach as thesensitivity equation method the sensitivity equation method is presented convergence results are given and the approach is illustrated on two optimal design problems involving shocks

there is no generalization of known formulas for mutually unbiased bases in a quantum system having a finite number n of orthogonal states two orthonormal bases ai and bj are called mutually unbiased if all inner products ai bj have the same modulus 1 n this concept appears in several quantum information problems the number of pairwise mutually unbiased bases is at most n 1 and various constructions of such n 1 bases have been found when n is a power of a prime number we study families of formulas that generalize these constructions to arbitrary dimensions using finite rings we then prove that there exists a set of n 1 mutually unbiased bases described by such formulas if and only if n is a power of a prime number

an angular momentum approach to quadratic fourier transform hadamard matrices gauss sums mutually unbiased bases the unitary group and the pauli group the construction of unitary operator bases in a finite dimensional hilbert space is reviewed through a nonstandard approach combining angular momentum theory and representation theory of su 2 a single formula for the bases is obtained from a polar decomposition of su 2 and is analyzed in terms of cyclic groups quadratic fourier transforms hadamard matrices and generalized gauss sums weyl pairs generalized pauli operators and their application to the unitary group and the pauli group naturally arise in this approach

adjoint sensitivity analysis for time dependent partial differential equations with adaptive mesh refinement a new adjoint sensitivity analysis approach is presented for time dependent partial differential equations with adaptive mesh refinement the new approach called adda combines the best features of both the adjoint of the discretization ad and discretization of the adjoint da approaches it removes the obstacles of applying ad to adaptive methods and in contrast to da requires for its use only a minimal amount of knowledge about the formulation of adjoint pdes and their boundary conditions the effectiveness and efficiency of adda are demonstrated for several numerical examples

oriented wavelet transform for image compression and denoising in this paper we introduce a new transform for image processing based on wavelets and the lifting paradigm the lifting steps of a unidimensional wavelet are applied along a local orientation defined on a quincunx sampling grid to maximize energy compaction the orientation minimizing the prediction error is chosen adaptively a fine grained multiscale analysis is provided by iterating the decomposition on the low frequency band in the context of image compression the multiresolution orientation map is coded using a quad tree the rate allocation between the orientation map and wavelet coefficients is jointly optimized in a rate distortion sense for image denoising a markov model is used to extract the orientations from the noisy image as long as the map is sufficiently homogeneous interesting properties of the original wavelet are preserved such as regularity and orthogonality perfect reconstruction is ensured by the reversibility of the lifting scheme the mutual information between the wavelet coefficients is studied and compared to the one observed with a separable wavelet transform the rate distortion performance of this new transform is evaluated for image coding using state of the art subband coders its performance in a denoising application is also assessed against the performance obtained with other transforms or denoising methods

a new algorithm for the embedding of a prediction mechanism into the jpeg2000 coding chain in this paper we propose the embedding of a prediction mechanism into a part of the coding structure of jpeg2000 image compression standard in order to reduce the amount of bits sent to the arithmetic coder without any significant changes into the standard architecture and without loosing performance the prediction is based upon an innovative processing of the data structures used by the standard jpeg2000 in progressive coding and the addition of a prediction matrix whose computation does not add any overhead at the decoder side experiments are performed to test the efficacy of the prediction mechanism and results are compared to the standard jpeg2000 and other similar approaches tests are documented over a set of well known images from literature also against different kinds of added noise performance in terms of saved bits are reported and a new figure of merit is defined to test the efficiency of prediction the results prove that the new proposal overcomes the standard and other related approaches for the entire set of referenced images with significant gain in synthetic images also in presence of noise

construction of biorthogonal wavelets starting from any two multiresolutions starting from any two given multiresolution analyses of l sub 2 v sub j sup 1 sub j spl isin z and v sub j sup 2 sub j spl isin z we construct biorthogonal wavelet bases that are associated with this chosen pair of multiresolutions thus our construction method takes a point of view opposite to the one of cohen daubechies feauveau 1992 which starts from a well choosen pair of biorthogonal discrete filters in our construction the necessary and sufficient condition is the nonperpendicularity of the multiresolutions

optimal control of two and three dimensional incompressible navier stokes flows the focus of this work is on the development of large scale numerical optimization methods for optimal control of steady incompressible navier stokes flows the control is affected by the suction or injection of fluid on portions of the boundary and the objective function represents the rate at which energy is dissipated in the fluid we develop reduced hessian sequential quadratic programming methods that avoid converging the flow equations at each iteration both quasi newton and newton variants are developed and compared to the approach of eliminating the flow equations and variables which is effectively the generalized reduced gradient method optimal control problems are solved for two dimensional flow around a cylinder and three dimensional flow around a sphere the examples demonstrate at least an order of magnitude reduction in time taken allowing the optimal solution of flow control problems in as little as half an hour on a desktop workstation

admitting the inadmissible adjoint formulation for incomplete cost functionals in aerodynamic optimization we derive the adjoint equations for problems in aerodynamic optimization which are improperly considered as inadmissible for example a cost functional which depends on the density rather than on the pressure is considered inadmissible for an optimization problem governed by the euler equations we show that for such problems additional terms should be included in the lagrangian functional when deriving the adjoint equations these terms are obtained from the restriction of the interior pde to the control surface demonstrations of the explicit derivation of the adjoint equations for inadmissible cost functionals are given for the potential euler and navier stokes equations

approximation power of biorthogonal wavelet expansions this paper looks at the effect of the number of vanishing moments on the approximation power of wavelet expansions the strang fix conditions imply that the error for an orthogonal wavelet approximation at scale a 2 sup i globally decays as a sup n where n is the order of the transform this is why for a given number of scales higher order wavelet transforms usually result in better signal approximations we prove that this result carries over for the general biorthogonal case and that the rate of decay of the error is determined by the order properties of the synthesis scaling function alone we also derive asymptotic error formulas and show that biorthogonal wavelet transforms are equivalent to their corresponding orthogonal projector as the scale goes to zero these results strengthen sweldens earlier analysis and confirm that the approximation power of biorthogonal and semi orthogonal wavelet expansions is essentially the same finally we compare the asymptotic performance of various wavelet transforms and briefly discuss the advantages of splines we also indicate how the smoothness of the basis functions is beneficial in reducing the approximation error

about mutually unbiased bases in even and odd prime power dimensions mutually unbiased bases generalize the x y and z qubit bases they possess numerous applications in quantum information science it is well known that in prime power dimensions n pm with p prime and m a positive integer there exists a maximal set of n 1 mutually unbiased bases in the present paper we derive an explicit expression for those bases in terms of the operations of the associated finite field galois division ring of n elements this expression is shown to be equivalent to the expressions previously obtained by ivanovic 1981 j phys a math gen 14 3241 in odd prime dimensions and wootters and fields 1989 ann phys 191 363 in odd prime power dimensions in even prime power dimensions we derive a new explicit expression for the mutually unbiased bases the new ingredients of our approach are basically the following we provide a simple expression of the generalized pauli group in terms of the additive characters of the field and we derive an exact groupal composition law between the elements of the commuting subsets of the generalized pauli group renormalized by a well chosen phase factor

symplectic transformations and quantum tomography in finite quantum systems quantum systems where the position and momentum are in the ring zd d is an odd integer are considered symplectic transformations are studied and the order of sp 2 zd is calculated quantum tomography is also discussed it is shown that measurements used in the inverse radon transform need to be made on j2 d lines where j2 d is the jordan totient function

hjelmslev geometry of mutually unbiased bases the basic combinatorial properties of a complete set of mutually unbiased bases mubs of a q dimensional hilbert space with p being a prime and r a positive integer are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective hjelmslev plane defined over a galois ring of characteristic p2 and rank r the q vectors of a basis of correspond to the q points of a so called neighbour class and the q 1 mubs answer to the total number of pairwise disjoint neighbour classes on the conic

mutually unbiased bases generalized spin matrices and separability abstract a collection of orthonormal bases for a complex d dimensional hilbert space is called mutually unbiased mub if for any two vectors v and w from different bases the square of the inner product equals 1 d v w 2 1 d the mub problem is to prove or disprove the existence of a maximal set of d 1 bases it has been shown in ann phys 191 1989 363 that such a collection exists if d is a power of a prime number p we revisit this problem and use d d generalizations of the pauli spin matrices to give a constructive proof of this result specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the mub problem additionally we give formulas from which the orthogonal bases can be readily computed we show how the techniques developed here provide a natural way to analyze the separability of the bases the techniques used require properties of algebraic field extensions and the relevant part of that theory is included in the appendix

periodic and discrete zak bases weyl s unitary operators for displacement in position and momentum commute with one another if the product of the elementary displacements equals planck s constant then their common eigenstates constitute the zak basis with each state specified by two phase parameters accordingly the transformation function from the position basis to the zak basis maps the hilbert space on the line onto the hilbert space on the torus this mapping is one to one provided that the zak basis states are periodic functions of their phase parameters but then the mapping cannot be continuous on the whole torus with the periodicity of the zak basis enforced the basis has a double fourier series the fourier coefficients identify a discrete basis which complements the periodic zak basis to form a pair of mutually unbiased bases the discrete basis states are the common eigenstates of the two complementary partners to the two unitary displacement operators these partner operators are of angular momentum type with integer eigenvalues and generate the fundamental rotations of the torus conversely the displacement operators are the ladder operators for their partners for each consistent phase convention for the periodic zak basis and thus for the line onto torus mapping there is a corresponding discrete zak basis and a corresponding pair of partner operators examples of particular interest are the conventions that give a continuous mapping in one phase parameter or are symmetric in both phase parameters the latter emphasizes the heisenberg weyl symmetry between position and momentum we discuss briefly the relation between the zak bases and aharonov s modular operators finally as an application of the zak operators for the torus we mention how they can be used to associate with the single degree of freedom of the line a pair of genuine qubits that are potentially entangled

a wavelet approach to adjoint state sensitivity computation for steady state differential equations 1 the computation of the sensitivity matrix is the most time consuming part of any parameter estimation algorithm that requires sensitivity coefficients an efficient wavelet approach to adjoint sensitivity analysis is proposed to reduce the computational cost of obtaining sensitivity coefficients the method exploits a wavelet reduction of the data space to reduce the size of the linear system encountered in steady state adjoint equations in this regard wavelet transform is used as a data compression tool numerical examples applied to spatial data are used to verify and show the effectiveness of the method

generation of bases with definite factorization for an n qubit system and mutually unbiased sets construction we propose a systematic procedure to construct all the possible bases with a definite factorization structure eigenstates of n commuting monomials constructed as products of pauli operators for an n qubit system as well as the possibility of collecting them into mutually unbiased sets we also discuss an algorithm for the determination of basis separability and propose a criteria for complementarity between such bases the results are applied to generate non isomorphic complete sets of mutually unbiased bases

geometrical approach to the discrete wigner function in prime power dimensions we analyse the wigner function in prime power dimensions constructed on the basis of the discrete rotation and displacement operators labelled with elements of the underlying finite field we separately discuss the case of odd and even characteristics and analyse the algebraic origin of the non uniqueness of the representation of the wigner function explicit expressions for the wigner kernel are given in both cases

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

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

a multiscale relaxation algorithm for snr maximization in nonorthogonal subband coding develops a technique for improving the applicability of complete nonorthogonal multiresolution transforms to image coding as is well known the l sup 2 norm of the quantization errors is not preserved by nonorthogonal transforms so the l sup 2 reconstruction error may be unacceptably large however given the quantizers and synthesis filters the authors show that this artifact can be eliminated by formulating the coding problem as that of minimizing the l sup 2 reconstruction error over the set of possible encoded images with this new formulation the coding problem becomes a high dimensional discrete optimization problem and features a coupling between the redundancy removing and quantization operations a practical solution to the optimization problem is presented in the form of a multiscale relaxation algorithm using inter and intrascale quantization noise feedback filters bounds on the coding gain over the standard coding technique are derived a simple extension of the algorithm allows for the use of a weighted l sup 2 error criterion and deadband non mmse quantizers experiments using biorthogonal spline filter banks demonstrate appreciable snr gains over the standard coding technique and comparable visual improvements

error estimation and adaptation for functional outputs in time dependent flow problems the paper presents an adjoint based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations the algorithm is derived for the unsteady euler equations that are discretized for second order accuracy in both space and time and takes into account the effect of dynamic meshes in addition to error due to temporal resolution the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time step the resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time step the method is demonstrated in the context of both time integrated and instantaneous functionals and the results are compared against traditional adaptation methods

analysis of low bit rate image transform coding calculations based on high resolution quantizations prove that the distortion rate d r of an image transform coding is proportional to 2 sup 2r when r is large enough in wavelet and block cosine bases we show that if r 1 bit pixel then d r varies like r sup 1 2 spl gamma where spl gamma remains of the order of 1 for most natural images the improved performance of embedded codings in wavelet bases is analyzed at low bit rates we show that the compression performance of an orthonormal basis depends mostly on its ability to approximate images with a few nonzero vectors

quantum mechanics on p adic numbers a quantum system with positions in and momenta in is studied displacement operators in the phase space of this system and the corresponding heisenberg weyl group are studied it is shown that such a system can be constructed from a semi infinite chain of spins which are coupled in a particular way

discrete adjoint based approach for optimization problems on three dimensional unstructured meshes a comprehensive strategy for developing and implementing discrete adjoint methods for aerodynamic shape optimization problems is presented by linearizing each procedure in the entire optimization problem transposing each linearization and reversing the sequential order of operations the adjoint of the complete optimization problem including flow equations and mesh motion equations is constructed in a modular and verifiable fashion this construction is also shown to produce minimal memory overheads and retain the same convergence characteristics of the original analysis problem in the sensitivity analysis these techniques are implemented in a three dimensional unstructured multigrid navierstokes solver and demonstrated on a transonic drag reduction problem for a wing body configuration

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

wigner functions and separability for finite systems a discussion of discrete wigner functions in phase space related to mutually unbiased bases is presented this approach requires mathematical assumptions which limits it to systems with density matrices defined on complex hilbert spaces of dimension pn where p is a prime number with this limitation it is possible to define a phase space and wigner functions in close analogy to the continuous case that is we use a phase space that is a direct sum of n two dimensional vector spaces each containing p2 points this is in contrast to the more usual choice of a two dimensional phase space containing p2n points a useful aspect of this approach is that we can relate complete separability of density matrices and their wigner functions in a natural way we discuss this in detail for bipartite systems and present the generalization to arbitrary numbers of subsystems when p is odd special attention is required for two qubits p 2 and our technique fails to establish the separability property for more than two qubits finally we give a brief discussion of hamiltonian dynamics in the language developed in the paper

tomographically complete sets of orthonormal bases in finite systems quantum systems where the position and momentum are in the ring d is an odd integer are considered a tomographically complete set of bases i e a set of bases such that probabilities from tomography experiments corresponding to these bases can be used for the calculation of an arbitrary density matrix is considered such a set of bases is constructed by considering a factorization of a system with dimension d p1p2 where p1 p2 are prime numbers in terms of two subsystems with dimensions p1 and p2 appropriate combination of the mutually unbiased bases in the subsystems leads to a tomographically complete set of bases in the full system this set leads to probabilities along all the lines through the origin 0 0 in the phase space with exactly d points it is shown that the number of these lines is d the dedekind function the general theory is exemplified with examples

feynman s path integral and mutually unbiased bases our previous work on quantum mechanics in hilbert spaces of finite dimension n is applied to elucidate the deep meaning of feynman s path integral pointed out by g svetlichny he speculated that the secret of the feynman path integral may lie in the property of mutual unbiasedness of temporally proximal bases we confirm the corresponding property of the short time propagator by using a specially devised n n approximation of quantum mechanics in applied to our finite dimensional analogue of a free quantum particle

linear transformations and aberrations in continuous and finite systems in geometric optics there is a natural distinction between the paraxial and aberration regimes which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space in the lie theoretical presentation linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates in a way parallel but distinct from the schrodinger quantization of continuous classical systems we quantize the geometric optical model into discrete finite dimensional systems based on the lie algebra whose wavefunctions are n point signals phase space is a sphere and transformations are represented by the n n unitary matrices that form the group we factor this group into linear and nonlinear unitary transformations of phase space and classify all its n2 4 aberrations this offers a new parametrization of based on a chosen subgroup

design of almost symmetric orthogonal wavelet filter bank via direct optimization it is a well known fact that compact support dyadic wavelets based on the two channel filter banks fbs cannot be simultaneously orthogonal and symmetric although orthogonal wavelets have the energy preservation property biorthogonal wavelets are preferred in image processing applications because of their symmetric property in this paper a novel method is presented for the design of almost symmetric orthogonal wavelet fb orthogonality is structurally imposed by using the unnormalized lattice structure and this leads to an objective function which is relatively simple to optimize the designed filters have good frequency response flat group delay almost symmetric filter coefficients and symmetric wavelet function

quantum systems with finite hilbert space galois fields in quantum mechanics a galois quantum system in which the position and momentum take values in the galois field gf pl is considered it is comprised of l component systems which are coupled in a particular way and is described by a certain class of hamiltonians displacements in the gf pl gf pl phase space and the corresponding heisenberg weyl group are studied symplectic transformations are shown to form the sp 2 gf pl group wigner and weyl functions are defined and their properties are studied frobenius symmetries which are based on frobenius automorphisms in the theory of galois fields are a unique feature of these systems for l 2 if they commute with the hamiltonian there are constants of motion which are discussed an analytic representation in the l sheeted complex plane provides an elegant formalism that embodies the properties of frobenius transformations the difference between a galois quantum system and other finite quantum systems where the position and momentum take values in the ring is discussed

von neumann lattices and magnetic orbitals in finite phase space it is proven that in the finite phase space of dimensionality m the number of independent von neumann states is m r where r is the number of distinct zeros of the kq function when applied to magnetic orbitals this leads to a linear dependence between them for each zero of the corresponding kq function different interesting identities follow from the linear dependence of von neumann states

from differential equations to the construction of new wavelet like bases in this paper an approach is introduced based on differential operators to construct wavelet like basis functions given a differential operator l with rational transfer function elementary building blocks are obtained that are shifted replicates of the green s function of l it is shown that these can be used to specify a sequence of embedded spline spaces that admit a hierarchical exponential b spline representation the corresponding b splines are entirely specified by their poles and zeros they are compactly supported have an explicit analytical form and generate multiresolution riesz bases moreover they satisfy generalized refinement equations with a scale dependent filter and lead to a representation that is dense in l sub 2 this allows us to specify a corresponding family of semi orthogonal exponential spline wavelets which provides a major extension of earlier polynomial spline constructions these wavelets are completely characterized and it is proven that they satisfy the following remarkable properties 1 they are orthogonal across scales and generate riesz bases at each resolution level 2 they yield unconditional bases of l sub 2 either compactly supported b spline type or with exponential decay orthogonal or dual type 3 they have n vanishing exponential moments where n is the order of the differential operator 4 they behave like multiresolution versions of the operator l from which they are derived and 5 their order of approximation is n m where n and m give the number of poles and zeros respectively last but not least the new wavelet like decompositions are as computationally efficient as the classical ones they are computed using an adapted version of mallat s filter bank algorithm where the filters depend on the decomposition level

symmetries of finite heisenberg groups for multipartite systems a composite quantum system comprising a finite number k of subsystems which are described with position and momentum variables in i 1 k is considered its hilbert space is given by a k fold tensor product of hilbert spaces of dimensions n1 nk the symmetry group of the respective finite heisenberg group is given by the quotient group of certain normalizer this paper extends our previous investigation of bipartite quantum systems to arbitrary multipartite systems of the above type it provides detailed description of the normalizers and the corresponding symmetry groups the new class of symmetry groups represents a very specific generalization of symplectic groups over modular rings as an application a new proof of existence of the maximal set of mutually unbiased bases in hilbert spaces of prime power dimensions is provided

wavelet filter evaluation for image compression choice of filter bank in wavelet compression is a critical issue that affects image quality as well as system design although regularity is sometimes used in filter evaluation its success at predicting compression performance is only partial a more reliable evaluation can be obtained by considering an l level synthesis analysis system as a single input single output linear shift variant system with a response that varies according to the input location module 2 sup l 2 sup l by characterizing a filter bank according to its impulse response and step response in addition to regularity we obtain reliable and relevant for image coding filter evaluation metrics using this approach we have evaluated all possible reasonably short less than 36 taps in the synthesis analysis pair minimum order biorthogonal wavelet filter banks of this group of over 4300 candidate filter banks we have selected and present here the filters best suited to image compression while some of these filters have been published previously others are new and have properties that make them attractive in system design

quantitative fourier analysis of approximation techniques i interpolators and projectors we present a general fourier based method that provides an accurate prediction of the approximation error as a function of the sampling step t our formalism applies to an extended class of convolution based signal approximation techniques which includes interpolation generalized sampling with prefiltering and the projectors encountered in wavelet theory we claim that we can predict the l sup 2 approximation error by integrating the spectrum of the function to approximate not necessarily bandlimited against a frequency kernel e spl omega that characterizes the approximation operator this prediction is easier yet more precise than was previously available our approach has the remarkable property of providing a global error estimate that is the average of the true approximation error over all possible shifts of the input function our error prediction is exact for stationary processes as well as for bandlimited signals we apply this method to the comparison of standard interpolation and approximation techniques our method has interesting implications for approximation theory in particular we use our results to obtain some new asymptotic expansions of the error as t spl rarr 0 as well as to derive improved upper bounds of the kind found in the strang fix 1971 theory we finally show how we can design quasi interpolators that are near optimal in the least squares sense

mathematical properties of the jpeg2000 wavelet filters the legall 5 3 and daubechies 9 7 filters have risen to special prominence because they were selected for inclusion in the jpeg2000 standard we determine their key mathematical features riesz bounds order of approximation and regularity holder and sobolev we give approximation theoretic quantities such as the asymptotic constant for the l sup 2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection we also derive new asymptotic error formulae that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet the daubechies 9 7 stands out because it is very close to orthonormal but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments

prime decomposition and correlation measure of finite quantum systems under the name prime decomposition pd a unique decomposition of an arbitrary n dimensional density matrix into a sum of separable density matrices with dimensions determined by the coprime factors of n is introduced for a class of density matrices a complete tensor product factorization is achieved the construction is based on the chinese remainder theorem and the projective unitary representation of by the discrete heisenberg group the pd isomorphism is unitarily implemented and it is shown to be co associative and to act on as comultiplication density matrices with complete pd are interpreted as group like elements of to quantify the distance of from its pd a trace norm correlation index is introduced and its invariance groups are determined

multicomplementary operators via finite fourier transform a complete set of d 1 mutually unbiased bases exists in a hilbert space of dimension d whenever d is a power of a prime we discuss a simple construction of d 1 disjoint classes each one having d 1 commuting operators such that the corresponding eigenstates form sets of unbiased bases such a construction works properly for prime dimension we investigate an alternative construction in which the real numbers that label the classes are replaced by a finite field having d elements one of these classes is diagonal and can be mapped to cyclic operators by means of the finite fourier transform which allows one to understand complementarity in a similar way as for the position momentum pair in standard quantum mechanics the relevant examples of two and three qubits and two qutrits are discussed in detail

non orthogonal wigner like mapping we have developed a general method for construction of wigner like mapping to the phase space associated with a non orthogonal set of mutually unbiased equally separated bases explicit expressions of the wigner functions in terms of measurements in non orthogonal bases are given and the limit cases are analyzed

the polyphase with advance representation and linear phase lifting factorizations a matrix theory is developed for the noncausal polyphase representation that underlies the theory of lifted filter banks and wavelet transforms the theory presented here develops an extensive matrix algebra framework for analyzing and implementing linear phase two channel filter banks via lifting cascade schemes whole sample symmetric and half sample symmetric linear phase filter banks are characterized completely in terms of the polyphase with advance representation and new proofs are given of linear phase lifting factorization theorems for these two principal classes of linear phase filter banks the theory benefits significantly from a number of group theoretic structures arising in the polyphase with advance representation and in the lifting factorization of linear phase filter banks these results form the foundations of the lifting methodology employed in part 2 of the iso iec jpeg 2000 still image coding standard

automatic differentiation of numerical integration algorithms automatic differentiation ad is a technique for automatically augmenting computer programs with statements for the computation of derivatives this article discusses the application of automatic differentiation to numerical integration algorithms for ordinary differential equations odes in particular the ramifications of the fact that ad is applied not only to the solution of such an algorithm but to the solution procedure itself this subtle issue can lead to surprising results when ad tools are applied to variable stepsize variable order ode integrators the computation of the final time step plays a special role in determining the computed derivatives we investigate these issues using various integrators and suggest constructive approaches for obtaining the desired derivatives

bi orthogonal mutually unbiased bases for n qubit systems we have developed a general method for constructing a set of non orthogonal bases with equal separations between all different basis states for n qubit systems using these bases we derive an explicit expression for the optimal tomography in non orthogonal bases and discuss the amount of non classical resources required for the bases preparation and the precision of the reconstructed state the special two qubit case is analysed separately

discrete wigner function for finite dimensional systems a phase space approach to finite dimensional systems is developed from basic principles for a system describable by a hilbert space of dimension d we define a one to one correspondence between operators and functions on a discrete and finite phase space with points valid for any dimension d the properties fulfilled by this correspondence and its uniqueness are examined this formalism is applied to the number difference and phase difference of a two mode field this case is compared with the marginal distribution for these variables arising from a two mode wigner function for number and phase

space time adaptive solution of inverse problems with the discrete adjoint method this paper develops a framework for the construction and analysis of discrete adjoint sensitivities in the context of time dependent adaptive grid adaptive step models discrete adjoints are attractive in practice since they can be generated with low effort using automatic differentiation however this approach brings several important challenges the space time adjoint of the forward numerical scheme may be inconsistent with the continuous adjoint equations a reduction in accuracy of the discrete adjoint sensitivities may appear due to the inter grid transfer operators moreover the optimization algorithm may need to accommodate state and gradient vectors whose dimensions change between iterations this work shows that several of these potential issues can be avoided through a multi level optimization strategy using discontinuous galerkin dg hp adaptive discretizations paired with runge kutta rk time integration we extend the concept of dual adjoint consistency to space time rk dg discretizations which are then shown to be well suited for the adaptive solution of time dependent inverse problems furthermore we prove that dg mesh transfer operators on general meshes are also dual consistent this allows the simultaneous derivation of the discrete adjoint for both the numerical solver and the mesh transfer logic with an automatic code generation mechanism such as algorithmic differentiation ad potentially speeding up development of large scale simulation codes the theoretical analysis is supported by numerical results reported for a two dimensional non stationary inverse problem

conjugate variables in finite phase plane we construct two pairs of quasicoordinates and quasimomenta in a finite phase plane which form sets of conjugate variables in such a plane the coordinate x is quantized with a step c and the momentum p with a step where mc is the size of the phase plane in the x direction the construction depends crucially on the possibility of writing m m1m2 with m1 and m2 relatively prime the conjugate variables are applied to harper like hamiltonians it is shown how to design physical systems with energy spectra containing any desired number of discrete energy levels say m1 each of them having a prescribed degeneracy m2

factorizations and physical representations a hilbert space in m dimensions is shown explicitly to accommodate representations that reflect the decomposition of m into prime numbers representations that exhibit the factorization of m into two relatively prime numbers the kq representation zak j 1970 phys today 23 51 and related representations termed q1q2 representations together with their conjugates are analysed as well as a representation that exhibits the complete factorization of m in this latter representation each quantum number varies in a subspace that is associated with one of the prime numbers that make up m

dynamics in discrete phase spaces and time interval operators abstract the von neumann liouville time evolution equation is represented in a discrete quantum phase space the mapped liouville operator and the corresponding wigner function are explicitly written for the problem of a magnetic moment interacting with a magnetic field and the precessing solution is found the propagator is also discussed and a time interval operator associated to a unitary operator which shifts the energy levels in the zeeman spectrum is introduced this operator is associated to the particular dynamical process and is not the continuous parameter describing the time evolution the pair of unitary operators which shifts the time and energy is shown to obey the weyl schwinger algebra

weak mutually unbiased bases quantum systems with variables in are considered the properties of lines in the phase space of these systems are studied weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases is equal to d 1 2 or alternatively to one of the d 1 2i 0 where di is a divisor of d apart from d 1 they are designed for the geometry of the phase space in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin in the special case of prime d there are no divisors of d apart from 1 d and the weak mutually unbiased bases are mutually unbiased bases

eigenvalues of spl darr 2 h and convergence of the cascade algorithm this paper is about the eigenvalues and eigenvectors of spl darr 2 h the ordinary fir filter h is a convolution with a vector h h o h n which is the impulse response the operator spl darr 2 downsamples the output y h x keeping the even numbered components y 2n where h is represented by a constant diagonal matrix this is a toeplitz matrix with h k on its kth diagonal the odd numbered rows are removed in spl darr 2 h the result is a double shift between rows yielding a block toeplitz matrix with 1 spl times 2 blocks iteration of the filter is governed by the eigenvalues if the transfer function h z spl sigma h k z sup k has a zero of order p at z 1 corresponding to spl omega spl pi then spl darr 2 h has p special eigenvalues 1 2 1 4 1 2 sup p we show how each additional zero at spl pi divides all eigenvalues by 2 and creates a new eigenvector for spl lambda 1 2 this eigenvector solves the dilation equation spl phi t 2 spl sigma h k spl phi 2t k at the integers t n the left eigenvectors show how 1 t t sup p 1 can be produced as combinations of spl phi t k the dilation equation is solved by the cascade algorithm which is an infinite iteration of m spl darr 2 2h convergence in l sup 2 is governed by the eigenvalues of t spl darr 2 2hh sup t corresponding to the response 2h z h z sup 1 we find a simple proof of the necessary and sufficient condition for convergence

factorization in finite quantum systems unitary transformations in an angular momentum hilbert space h 2j 1 are considered they are expressed as a finite sum of the displacement operators which play the role of su 2j 1 generators with the weyl function as coefficients the chinese remainder theorem is used to factorize large qudits in the hilbert space h 2j 1 in terms of smaller qudits in hilbert spaces h 2ji 1 all unitary transformations on large qudits can be performed through appropriate unitary transformations on the smaller qudits

data compression and harmonic analysis in this paper we review some recent interactions between harmonic analysis and data compression the story goes back of course to shannon s r d theory in the case of gaussian stationary processes which says that transforming into a fourier basis followed by block coding gives an optimal lossy compression technique practical developments like transform based image compression have been inspired by this result in this paper we also discuss connections perhaps less familiar to the information theory community growing out of the field of harmonic analysis recent harmonic analysis constructions such as wavelet transforms and gabor transforms are essentially optimal transforms for transform coding in certain settings some of these transforms are under consideration for future compression standards we discuss some of the lessons of harmonic analysis in this century typically the problems and achievements of this field have involved goals that were not obviously related to practical data compression and have used a language not immediately accessible to outsiders nevertheless through an extensive generalization of what shannon called the sampling theorem harmonic analysis has succeeded in developing new forms of functional representation which turn out to have significant data compression interpretations we explain why harmonic analysis has interacted with data compression and we describe some interesting recent ideas in the field that may affect data compression in the future

holomorphic quantization on the torus and finite quantum mechanics we explicitly construct the quantization of classical linear maps of on toroidal phase space of arbitrary modulus using the holomorphic chiral version of the metaplectic representation we show that finite quantum mechanics fqm on tori of arbitrary integer discretization is a consistent restriction of the holomorphic quantization of to the subgroup being the principal congruent subgroup on a finite dimensional hilbert space the generators of the rotation group for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates

finite dimensional schwinger basis deformed symmetries wigner function and an algebraic approach to quantum phase schwinger s finite d dimensional periodic hilbert space representations are studied on the toroidal lattice with specific emphasis on the deformed oscillator subalgebras and the generalized representations of the wigner function these subalgebras are shown to be admissible endowed with the non negative norm of hilbert space vectors hence they provide the desired canonical basis for the algebraic formulation of the quantum phase problem certain equivalence classes in the space of labels are identified within each subalgebra and connections with area preserving canonical transformations are examined the generalized representations of the wigner function are examined in the finite dimensional cyclic schwinger basis these representations are shown to conform to all fundamental conditions of the generalized phase space wigner distribution as a specific application of the schwinger basis the number phase unitary operator pair in is studied and based on the admissibility of the underlying q oscillator subalgebra an algebraic approach to the unitary quantum phase operator is established this being the focus of this work connections with the susskind glogower carruthers nieto phase operator formalism as well as standard action angle wigner function formalisms are examined in the infinite period limit the concept of continuously shifted fock basis is introduced to facilitate the fock space representations of the wigner function

galois quantum systems a finite quantum system in which the position and momentum take values in the galois field gf p is constructed from a smaller quantum system in which the position and momentum take values in using field extension the galois trace is used in the definition of the fourier transform the heisenberg weyl group of displacements and the sp 2 gf p group of symplectic transformations are studied a class of transformations inspired by the frobenius maps in galois fields is introduced the relationship of this galois quantum system with its subsystems in which the position and momentum take values in subfields of gf p is discussed

an extended weyl wigner transformation for special finite spaces abstract we extend the weyl wigner transformation to those particular degrees of freedom described by a finite number of states using a technique of constructing operator bases developed by schwinger discrete transformation kernels are presented instead of continuous coordinate momentum pair system and systems such as the one dimensional canonical continuous coordinate momentum pair system and the two dimensional rotation system are described by special limits expressions are explicitly given for the spin one half case

adjoint sensitivity analysis for differential algebraic equations the adjoint dae system and its numerical solution an adjoint sensitivity method is presented for parameter dependent differential algebraic equation systems daes the adjoint system is derived along with conditions for its consistent initialization for daes of index up to two hessenberg for stable linear daes stability of the adjoint system for semi explicit daes or of an augmented adjoint system for fully implicit daes is shown in addition it is shown for these systems that numerical stability is maintained for the adjoint system or for the augmented adjoint system

a differentiation index for partial differential algebraic equations a differentiation index for nonlinear partial differential algebraic equations is presented determination of the differentiation index with respect to a direction in the space of independent variables uncovers all equations that must be satisfied by cauchy data on the hyperplane orthogonal to that direction this index is thus a generalization of the differentiation index of differential algebraic equations the motivation for this work is consistent initialization of partial differential algebraic systems the analysis is demonstrated through several examples including some taken from fluid dynamics and chemical engineering

generalized daubechies wavelet families we present a generalization of the orthonormal daubechies wavelets and of their related biorthogonal flavors cohen daubechies feauveau 9 7 our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials this allows one to tune the corresponding wavelet transform to a specific class of signals thereby ensuring good approximation and sparsity properties the main difference with the classical construction of daubechies is that the multiresolution spaces are derived from scale dependent generating functions however from an algorithmic standpoint mallat s fast wavelet transform algorithm can still be applied the only adaptation consists in using scale dependent filter banks finite support ensures the same computational efficiency as in the classical case we characterize the scaling and wavelet filters construct them and show several examples of the associated functions we prove that these functions are square integrable and that they converge to their classical counterparts of the corresponding order

block circulant matrices with circulant blocks weil sums and mutually unbiased bases ii the prime power case in our previous paper combescure m circulant matrices gauss sums and the mutually unbiased bases i the prime number case cubo a mathematical journal unpublished we have shown that the theory of circulant matrices allows to recover the result that there exists p 1 mutually unbiased bases in dimension p p being an arbitrary prime number two orthonormal bases b b of cd are said mutually unbiased if b b b b one has that b b 1 d b b hermitian scalar product in cd in this paper we show that the theory of block circulant matrices with circulant blocks allows to show very simply the known result that if d pn p a prime number and n any integer there exists d 1 mutually unbiased bases in cd our result relies heavily on an idea of klimov et al geometrical approach to the discrete wigner function j phys a 39 14471 2006 as a subproduct we recover properties of quadratic weil sums for p 3 which generalizes the fact that in the prime case the quadratic gauss sum properties fo

partial order and a t0 topology in a set of finite quantum systems a whole part theory is developed for a set of finite quantum systems n with variables in z n the partial order subsystem is defined by embedding various attributes of the system m quantum states density matrices etc into their counterparts in the supersystem n for m n the compatibility of these embeddings is studied the concept of ubiquity is introduced for quantities which fit with this structure it is shown that various entropic quantities are ubiquitous the sets of various quantities become t0 topological spaces with the divisor topology which encapsulates fundamental physical properties these sets can be converted into directed complete partial orders by adding top elements the continuity of various maps among these sets is studied

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

the angle angular momentum quantum phase space the angle angular momentum quantum phase space is considered and the corresponding heisenberg weyl group is studied when 2j 1 is a power of a prime z 2j 1 is a galois field and stronger results can be proved sl 2 z 2j 1 transformations are explicitly constructed and various properties of the displacement operators are studied central extensions of the abelian group by z 2j 1 are studied and they provide all the ways of constructing the heisenberg weyl group from g and z 2j 1

quantitative fourier analysis of approximation techniques ii wavelets for pt i see ibid vol 47 no 10 p 2783 95 1999 in a previous paper we proposed a general fourier method that provides an accurate prediction of the approximation error irrespective of the scaling properties of the approximating functions here we apply our results when these functions satisfy the usual two scale relation encountered in dyadic multiresolution analysis as a consequence of this additional constraint the quantities introduced in our previous paper can be computed explicitly as a function of the refinement filter this is in particular true for the asymptotic expansion of the approximation error for biorthonormal wavelets as the scale tends to zero one of the contributions of this paper is the computation of sharp asymptotically optimal upper bounds for the least squares approximation error another contribution is the application of these results to b splines and daubechies 1988 1992 scaling functions which yields explicit asymptotic developments and upper bounds thanks to these explicit expressions we can quantify the improvement that can be obtained by using b splines instead of daubechies wavelets in other words we can use a coarser spline sampling and achieve the same reconstruction accuracy as daubechies specifically we show that this sampling gain converges to spl pi as the order tends to infinity

group theoretical construction of mutually unbiased bases in hilbert spaces of prime dimensions mutually unbiased bases in hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity an alternative proof of existence of a maximal collection of n 1 mutually unbiased bases in hilbert spaces of prime dimension n is given by exploiting the finite heisenberg group also called the pauli group and the action of on finite phase space implemented by unitary operators in the hilbert space crucial for the proof is that for prime is also a finite field

spectra of phase point operators in odd prime dimensions and the extended clifford group we analyze the role of the extended clifford group in classifying the spectra of phase point operators within the framework laid out by gibbons et al phys rev a 70 062101 2004 for setting up wigner distributions on discrete phase spaces based on finite fields to do so we regard the set of all the discrete phase spaces as a symplectic vector space over the finite field auxiliary results include a derivation of the conjugacy classes of esl 2 fn

geometrical approach to mutually unbiased bases we propose a unifying phase space approach to the construction of mutually unbiased bases for a two qubit system it is based on an explicit classification of the geometrical structures compatible with the notion of unbiasedness these consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties we also consider the feasible transformations between different kinds of curves and show that they correspond to local rotations around the bloch sphere principal axes we suggest how to generalize the method to systems in dimensions that are powers of a prime