wavelets and filter banks theory and design the wavelet transform is compared with the more classical short time fourier transform approach to signal analysis then the relations between wavelets filter banks and multiresolution signal processing are explored a brief review is given of perfect reconstruction filter banks which can be used both for computing the discrete wavelet transform and for deriving continuous wavelet bases provided that the filters meet a constraint known as regularity given a low pass filter necessary and sufficient conditions for the existence of a complementary high pass filter that will permit perfect reconstruction are derived the perfect reconstruction condition is posed as a bezout identity and it is shown how it is possible to find all higher degree complementary filters based on an analogy with the theory of diophantine equations an alternative approach based on the theory of continued fractions is also given these results are used to design highly regular filter banks which generate biorthogonal continuous wavelet bases with symmetries

sampling theory for not necessarily band limited functions a historical overview shannon s sampling theorem is one of the most powerful results in signal analysis the aim of this overview is to show that one of its roots is a basic paper of de la vallee poussin of 1908 the historical development of sampling theory from 1908 to the present especially the matter dealing with not necessarily band limited functions which includes the duration limited case actually studied in 1908 is sketched emphasis is put on the study of error estimates as well as on the delicate point wise behavior of sampling sums at discontinuity points of the signal to be reconstructed

wavelets and dilation equations a brief introduction wavelets are new families of basis functions that yield the representation f x sum b jk w 2 j x k their construction begins with the solution phi x to a dilation equation with coefficients c k then w comes from phi and the basis comes by translation and dilation of w it is shown in part 1 how conditions on the c k lead to approximation properties and orthogonality properties of the wavelets part 2 describes the recursive algorithms also based on the c k that decompose and reconstruct f the object of wavelets is to localize as far as possible in both time and frequency with efficient algorithms

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

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

stability estimate for the aligned magnetic field in a periodic quantum waveguide from dirichlet to neumann map in this article we study the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic schrodinger equation in a periodic quantum cylindrical waveguide by knowledge of the dirichlet to neumann map we prove a holder stability estimate with respect to the dirichlet to neumann map by means of the geometrical optics solutions of the magnetic schrodinger equation

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

the wavelet transform time frequency localization and signal analysis two different procedures for effecting a frequency analysis of a time dependent signal locally in time are studied the first procedure is the short time or windowed fourier transform the second is the wavelet transform in which high frequency components are studied with sharper time resolution than low frequency components the similarities and the differences between these two methods are discussed for both schemes a detailed study is made of the reconstruction method and its stability as a function of the chosen time frequency density finally the notion of time frequency localization is made precise within this framework by two localization theorems

optimal stability estimate in the inverse boundary value problem for periodic potentials with partial data abstract we consider the inverse boundary value problem for operators of the form q in an infinite domain r r 1 n n 3 with a periodic potential q for dirichlet to neumann data localized on a portion of the boundary of the form 1 r 1 with 1 being the complement either of a flat or spherical portion of we prove that a log type stability estimate holds

on sampling theorem wavelets and wavelet transforms the classical shannon sampling theorem has resulted in many applications and generalizations from a multiresolution point of view it provides the sine scaling function in this case for a band limited signal its wavelet series transform wst coefficients below a certain resolution level can be exactly obtained from the samples with a sampling rate higher than the nyquist rate the authors study the properties of cardinal orthogonal scaling functions cosf which provide the standard sampling theorem in multiresolution spaces with scaling functions as interpolants they show that cosf with compact support have and only have one possibility which is the haar pulse they present a family of cosf with exponential decay which are generalizations of the haar function with these cosf an application is the computation of wst coefficients of a signal by the mallat 1989 algorithm they present some numerical comparisons for different scaling functions to illustrate the advantage of cosf for signals which are not in multiresolution spaces they estimate the aliasing error in the sampling theorem by using uniform samples

inverse problems for the schrodinger equations with time dependent electromagnetic potentials and the aharonov bohm effect we consider the inverse boundary value problem for the schrodinger operator with time dependent electromagnetic potentials in domains with obstacles we extend the resuls of the author s works inverse probl 19 49 2003 19 985 2003 20 1497 2004 to the case of time dependent potentials we relate our results to the aharonov bohm effect caused by magnetic and electric fluxes

sharp sufficient conditions for the observation control and stabilization of waves from the boundary for the observation or control of solutions of second order hyperbolic equation in mathbb r t times omega ralston s construction of localized states comm pure appl math 22 1969 pp

fast algorithms for discrete and continuous wavelet transforms several algorithms are reviewed for computing various types of wavelet transforms the mallat algorithm 1989 the a trous algorithm and their generalizations by shensa the goal of this work is to develop guidelines for implementing discrete and continuous wavelet transforms efficiently and to compare the various algorithms obtained and give an idea of possible gains by providing operation counts most wavelet transform algorithms compute sampled coefficients of the continuous wavelet transform using the filter bank structure of the discrete wavelet transform although this general method is already efficient it is shown that noticeable computational savings can be obtained by applying known fast convolution techniques such as the fft fast fourier transform in a suitable manner the modified algorithms are termed fast because of their ability to reduce the computational complexity per computed coefficient from l to log l within a small constant factor for large filter lengths l for short filters smaller gains are obtained fast running fir finite impulse response filtering techniques allow one to achieve typically 30 savings in computations

stability estimate for an inverse problem for the magnetic schrodinger equation from the dirichlet to neumann map abstract we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic schrodinger equation in a bounded smooth domain of r n with input dirichlet data from measured neumann boundary observations this information is enclosed in the dynamical dirichlet to neumann map associated to the solutions of the magnetic schrodinger equation we prove in dimension n 2 that the knowledge of the dirichlet to neumann map for the magnetic schrodinger equation measured on the boundary determines uniquely the magnetic field and we prove a holder type stability in determining the magnetic field induced by the magnetic potential

holder stable determination of a quantum scalar potential in unbounded cylindrical domains we consider the inverse problem of determining the time independent scalar potential of the dynamic schrodinger equation in an infinite cylindrical domain from one boundary neumann observation of the solution we prove holder stability by choosing the dirichlet boundary condition suitably

the discrete wavelet transform wedding the a trous and mallat algorithms two separately motivated implementations of the wavelet transform are brought together it is observed that these algorithms are both special cases of a single filter bank structure the discrete wavelet transform the behavior of which is governed by the choice of filters in fact the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact moreover it is shown that the commonly used lagrange a trous filters are in one to one correspondence with the convolutional squares of the daubechies filters for orthonormal wavelets of compact support a systematic framework for the discrete wavelet transform is provided and conditions are derived under which it computes the continuous wavelet transform exactly suitable filter constraints for finite energy and boundedness of the discrete transform are also derived relevant signal processing parameters are examined and it is observed that orthonormality is balanced by restrictions on resolution

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