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

families of lyapunov functions for nonlinear systems in critical cases lyapunov functions are constructed for nonlinear systems of ordinary differential equations whose linearized system at an equalized point possesses either a simple zero eigenvalue or a complex conjugate pair of simple pure imaginary eigenvalues the construction is explicit and yields parameterized families of lyapunov functions for such systems in the case of a zero eigenvalue the lyapunov functions contain quadratic and cubic terms in the state quartic terms appear as well for the case of a pair of pure imaginary eigenvalues predictions of local asymptotic stability using these lyapunov functions are shown to coincide with those of pertinent bifurcation theoretic calculations the development of the paper is carried out using elementary properties of multilinear functions the lyapunov function families thus obtained are amenable to symbolic computer coding

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

convergence of nonstationary cascade algorithms a nonstationary multiresolution of l 2 mathbb r s is generated by a sequence of scaling functions phi k in l 2 mathbb r s k in mathbb z we consider phi k that is the solution of the nonstationary refinement equations phi k m sum j h k 1 j phi k 1 m cdot j k in mathbb z where h k is finitely supported for each k and m is a dilation matrix we study various forms of convergence in l 2 mathbb r s of the corresponding nonstationary cascade algorithm phi k n m sum j h k 1 j phi k 1 n 1 m cdot j as k or n tends to infty it is assumed that there is a stationary refinement equation at infty with filter sequence h and that sum k h k j h j infty the results show that the convergence of the nonstationary cascade algorithm is determined by the spectral properties of the transition operator associated with h

on a class of generating vector fields for the extremum seeking problem lie bracket approximation and stability properties abstract in this paper we describe a broad class of control functions for extremum seeking problems we show that it unifies and generalizes existing extremum seeking strategies which are based on lie bracket approximations and allows to design new controls with favorable properties in extremum seeking and vibrational stabilization tasks the second result of this paper is a novel approach for studying the asymptotic behavior of extremum seeking systems it provides a constructive procedure for defining frequencies of control functions to ensure the practical asymptotic and exponential stability in contrast to many known results we also prove asymptotic and exponential stability in the sense of lyapunov for the proposed class of extremum seeking systems under appropriate assumptions on the vector fields

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

kohn rossi cohomology and its application to the complex plateau problem ii let x be a compact connected strongly pseudoconvex cr manifold of real dimension 2n 1 in c n 1 tanaka introduced a spectral sequence e p q r x with e p q 1 x being the kohn rossi cohomology group and e k 0 2 x being the holomorphic de rham cohomology denoted by h k h x we study the holomorphic de rham cohomology in terms of the s invariant of the isolated singularities of the variety v bounded by x we give a characterization of the singularities with vanishing s invariants for n 3 yau used the kohn rossi cohomology groups to solve the classical complex plateau problem in 1981 for n 2 the problem has remained unsolved for over a quarter of a century in this paper we make progress in this direction by putting some conditions on x so that v will have very mild singularities specifically we prove that if dim x 3 and h 2 h x 0 then x is a boundary of complex variety v with only isolated quasi homogeneous singularities such that the dual graphs of the exceptional sets in the resolution are star shaped and all curves are rational

on higher dimensional complex plateau problem let x be a compact connected strongly pseudoconvex cr manifold of real dimension 2n 1 in mathbb c n it has been an interesting question to find an intrinsic smoothness criteria for the complex plateau problem for n ge 3 and n n 1 yau found a necessary and sufficient condition for the interior regularity of the harvey lawson solution to the complex plateau problem by means of kohn rossi cohomology groups on x in 1981 for n 2 and n ge n 1 the first and third authors introduced a new cr invariant g 1 1 x of x the vanishing of this invariant will give the interior regularity of the harvey lawson solution up to normalization for n ge 3 and n n 1 the problem still remains open in this paper we generalize the invariant g 1 1 x to higher dimension as g lambda n 1 x and show that if g lambda n 1 x 0 then the interior has at most finite number of rational singularities in particular if x is calabi yau of real dimension 5 then the vanishing of this invariant is equivalent to give the interior regularity up to normalization

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

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

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

stabilization of nonlinear systems in compound critical cases in this paper we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique three degenerate cases are considered wherein the linearized model of the system has two zero eigenvalues one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues or two distinct pairs of nonzero pure imaginary eigenvalues while the remaining eigenvalues are stable using a local nonlinear mapping normal form reduction and liapunov stability criteria one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics the stabilizing control laws in linear and or nonlinear feedback forms are then designed for both linearly controllable and linearly uncontrollable cases the normal form transformations obtained in this paper have been verified by using code macsyma

analysis of asymptotically equivalent binary subdivision schemes non stationary uniform subdivision schemes are presented including some important examples as the up function and exponential splines the multivariate analysis of subdivision schemes is extended to a class of non stationary schemes which are asymptotically equivalent to converging stationary or non stationary schemes

variational analysis of non lipschitz spectral functions we consider spectral functions f where f is any permutation invariant mapping from cn to r and is the eigenvalue map from the set of n n complex matrices to cn ordering the eigenvalues lexicographically for example if f is the function maximum real part then f is the spectral abscissa while if f is maximum modulus then f is the spectral radius both these spectral functions are continuous but they are neither convex nor lipschitz for our analysis we use the notion of subgradient extensively analyzed in variational analysis r t rockafellar and r j b wets springer 1998 we show that a necessary condition for y to be a subgradient of an eigenvalue function f at x is that y commutes with x we also give a number of other necessary conditions for y based on the schur form and the jordan form of x in the case of the spectral abscissa we refine these conditions and we precisely identify the case where subdifferential regularity holds we conclude by introducing the notion of a semistable program maximize a linear function on the set of square matrices subject to linear equality constraints together with the constraint that the real parts of the eigenvalues of the solution matrix are non positive semistable programming is a nonconvex generalization of semidefinite programming using our analysis we derive a necessary condition for a local maximizer of a semistable program and we give a generalization of the complementarity condition familiar from semidefinite programming

bifurcation stabilization with local output feedback local output feedback stabilization with smooth nonlinear controllers is studied for parameterized nonlinear systems for which the linearized system possesses either a simple zero eigenvalue or a pair of imaginary eigenvalues and the bifurcated solution is unstable at the critical value of the parameter it is assumed that the unstable mode corresponding to the critical eigenvalue of the linearized system is not linearly controllable results are established for bifurcation stabilization using output feedback where the critical mode can be either linearly observable or linearly unobservable the stabilizability conditions are characterized in explicit forms that can be used to synthesize stabilizing controllers the results obtained in this paper are applied to rotating stall control for axial flow compressors as an application example

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

asymptotic behavior of solutions of a nonlinear system in the critical case of q pairs of purely imaginary eigenvalues abstract this paper is devoted to the stability problem of a nonlinear system in the critical case of q pairs of purely imaginary eigenvalues the center manifold theory and the normal form method are exploited in this study the main result of the paper is a power estimate for the solutions in the case when stability is ensured by third order forms a lyapunov function is constructed explicitly as a part of the proof of this estimate the result obtained is illustrated by an example of a double pendulum with partial dissipation

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

stabilizability conditions in terms of critical hamiltonians and symbols symmetric functions of critical hamiltonians called symbols were used in such problems of nonlinear control as the characterization of symmetries and feedback invariants we derive here a stabilizability condition in the class of almost continuous feedback controls based on symbols the methodology proposed consists of defining a selector of the multivalued covector field satisfying an extra degree condition on a sphere close to the origin then the above selector is extended to some neighborhood in order to get an exact differential form integration of that form gives us a control lyapunov function candidate theorem 3 1 the condition proposed is applied to the analysis of a planar control affine system with polynomial homogeneous vector fields unlike known results in the literature we consider the case when each vector field vanishes at the origin and the control is bounded we give stabilizability conditions explicitly in terms of the control system parameters theorem 4 1

optimal stabilization problem with minimax cost in a critical case this work addresses the optimal stabilization problem of a nonlinear control system by using a smooth output feedback the optimality criterion is the maximization of the decay rate of solutions in a neighborhood of the origin we formulate this criterion as a minimax problem with respect to non integral functional an explicit construction of a lyapunov function is proposed to evaluate the optimal cost this design methodology is justified for nonlinear systems in a critical case of stability with a pair of purely imaginary eigenvalues as an example a minimax optimal controller is obtained for a spring pendulum system with partial measurements of the state vector

local feedback stabilization and bifurcation control i hopf bifurcation abstract local bifurcation control problems are defined and employed in the study of the local feedback stabilization problem for nonlinear systems in critical cases sufficient conditions are obtained for the local stabilizability of general nonlinear systems whose linearizations have a pair of simple nonzero imaginary eigenvalues the conditions show in particular that generically these nonlinear critical systems can be stabilized locally even if the critical modes are uncontrollable the analysis also yields a direct method for computing stabilizing feedback controls use is made of bifurcation formulae which require only a series expansion of the vector field

various numerical invariants for isolated singularities definition 0 let v be a stein analytic space with x as its only singular point let 7r m v be a resolution of the singularity of v we shall denote dim h1 m 9 1 i n 1 by h i and dim hq m up for 1 p n 1 q n by hp q m so far as the classification problem is concerned h n i is one of the most important invariants in this paper we shall introduce a bunch of invariants cf definition 2 6 definition 4 1 and definition 5 1 which are naturally attached to isolated singularities these invariants are used to characterize the different notions of sheaves of germs of holomorphic differential forms on analytic spaces various formulae which relate all these invariants are proved we also show how to calculate these invariants explicitly our paper is organized as follows in section 2 we discuss the relationship between three different kinds of sheaves of germs of holomorphic forms introduced by grauert grothendieck noether and ferrari respectively and the dualizing sheaf on a complex analytic space which admits only isolated singularities we remark that the torsion sheaf of the sheaf of germs of holomorphic p forms in the sense of grauert grothendieck was studied by brieskorn 9 greuel 48 kantor 19 suzuki 38 vetter 39 and the author 43 in section 3 we relate the invariant g n l1 of a hypersurface isolated singularity with analytic invariants and topological invariants of any resolution of the singularity the noether s formula for the rank two bundle

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order