fast multiresolution algorithms for solving linear equations a comparative study in g beylkin r coifman and v rokhlin comm pure appl math xliv 1991 pp 141 183 and b engquist s osher and s zhong siam j sci comput 15 1994 pp 755 775 orthonormal wavelet basis and the multiscale analysis they define are used as building blocks in the design of algorithms for the rapid numerical application of a number of linear operators to arbitrary vectors the algorithms may be viewed as a method for converting whenever possible dense matrices to sparse form we use the discrete multiresolution analysis described by harten in j appl numer math 12 1993 pp 153 193 as the building block in the algorithms and compare their performance with the wavelet based ones computationally both techniques give comparable results

general synthesis procedures for fir lossless transfer matrices for perfect reconstruction multirate filter bank applications a recently reported procedure see ibid vol assp 35 p 476 492 apr 1987 and proc ieee int conf acoust speech sig processing apr 1987 p 2169 2172 for the design of m channel perfect reconstruction quadrature mirror filter qmf bands uses the concept of lossless alias component matrices the synthesis of such qmf banks centers around the generation of a lossless m m finite impulse response fir transfer matrix e z recent results for such generation have been somewhat ad hoc i e not sufficiently general a general procedure is outlined for the generation of such transfer matrices the procedure is based on a cascaded lattice structure derived from a state space viewpoint the structure is such that it generates only m m lossless fir transfer matrices regardless of parameter values and conversely any m m lossless fir transfer matrix can be obtained by a suitable choice of parameters these parameters turn out to be angles theta sub k and the structure is such that the number of angles is minimal a design example is presented to demonstrate the main results

multiresolution based on weighted averages of the hat function i linear reconstruction techniques in this paper we analyze a particular example of the general framework developed in a harten it siam j numer anal 33 1996 pp 1205 1256 the case in which the discretization operator is obtained by taking local averages with respect to the hat function we consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution in part i we use data independent linear reconstruction techniques as our approximation tool we show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations

perfect reconstruction fir filter banks some properties and factorizations perfect reconstruction finite impulse response fir filter banks are analyzed both in the z transform and time domains showing the alternatives between designs in the two domains various classes of perfect reconstruction schemes are indicated and relations between previously known systems are given windowed modulated filter banks with low computational complexity and perfect reconstruction are shown new factorizations of polyphase filter matrices leading in particular to linear phase filters are given the computational complexity and the architecture of the new structures are indicated

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

fast multiresolution algorithms for matrix vector multiplication in this paper the authors present a class of multiresolution algorithms for fast application of structured dense matrices to arbitrary vectors which includes the fast wavelet transform of beylkin coifman and rokhlin and the multilevel matrix multiplication of brandt and lubrecht in designing these algorithms the authors first apply data compression techniques to the matrix and then show how to compute the desired matrix vector multiplication from the compressed form of the matrix in describing this class special attention is paid to an algorithm that is based on discretization by cell averages as it seems to be suitable for discretization of integral transforms with integrably singular kernels

on m channel linear phase fir filter banks and application in image compression this paper investigates the theory and structure of a large subclass of m channel linear phase perfect reconstruction fir filter banks systems with analysis and synthesis filters of length l sub i k sub i m spl beta where spl beta is an arbitrary integer 0 spl les spl beta m and k sub i is any positive integer for this subclass of systems we first investigate the necessary conditions for the existence of linear phase perfect reconstruction filter banks lpprfbs next we develop a complete and minimal factorization for all even channel linear phase paraunitary systems the most general lapped orthogonal transforms to date finally several design examples as well as comparisons with previous generalized lapped orthogonal transforms genlots in image compression are presented to confirm the validity of the theory

filters for distortion free two band multirate filter banks in this paper conditions are given for a two band multi rate filter bank to be alias free and to have a unity frequency response it is shown that the class of quadrature mirror filters qmf s that satisfies these conditions is quite limited a class of filters which does satisfy these conditions is given and a simple procedure for designing filters from this class is presented with an example

theory of regular m band wavelet bases orthonormal m band wavelet bases have been constructed and applied by several authors this paper makes three main contributions first it generalizes the minimal length k regular 2 band wavelets of daubechies 1988 to the m band case by deriving explicit formulas for k regular m band scaling filters several equivalent characterizations of k regularity are given and their significance explained second two approaches to the construction of the m 1 wavelet filters and associated wavelet bases are described one relies on a state space characterization with a novel technique to obtain the unitary wavelet filters the other uses a factorization approach third this paper gives a set of necessary and sufficient condition on the m band scaling filter for it to generate an orthonormal wavelet basis the conditions are very similar to those obtained by cohen 1990 and lawton 1990 for 2 band wavelets

linc a common theory of transform and subband coding a common theory of lapped orthogonal transforms lots and critically sampled filter banks called l into n coding linc is presented the theory includes a unified analysis of both coding methods and identity relations between the transform inverse transform analysis filter bank and synthesis filter bank a design procedure for linc analysis synthesis systems which satisfy the conditions for perfect reconstruction is developed the common linc theory is used to define an ideal linc system which is used together with the power spectral density of the input signal to calculate theoretical bounds for the coding gain a generalized overlapping block transform obt with time domain aliasing cancellation tdac is used to approximate the ideal linc a generalization of the obt includes multiple block overlap and additional windowing a recursive design procedure for windows of arbitrary lengths is presented the coding gain of the generalized obt is higher than that of the karhunen loeve transform klt and close to the theoretical bounds for linc in the case of image coding the generalized obt reduces the blocking effects when compared with the dct

subband coding of images using asymmetrical filter banks we address the problem of the choice of subband filters in the context of image coding the ringing effects that occur in subband based compression schemes are the major unpleasant distortions a new set of two band filter banks suitable for image coding applications is presented the basic properties of these filters are linear phase perfect reconstruction asymmetric length and maximum regularity the better overall performances compared to the classical qmf subband filters are explained the asymmetry of the filter lengths results in a better compaction of the energy especially in the highpass subbands moreover the quantization error is reduced due to the short lowpass synthesis filter the undesirable ringing effect is considerably reduced due to the good step response of the synthesis lowpass filter the proposed design takes into account the statistics of natural images and the effect of quantization errors in the reconstructed images which explains the better coding performance

the circle method and bounds for l functions iii t aspect subconvexity for gl 3 l functions letbe a hecke maass cusp form for sl 3 z in this paper we will prove the following subconvex bound

the circle method and bounds for l functions i let f be a hecke maass or holomorphic primitive cusp form of arbitrary level and nebentypus and let chi be a primitive character of conductor m for the twisted l function l s f otimes chi we establish the hybrid subconvex bound begin aligned l left frac 1 2 it f otimes chi right ll m 3 t frac 1 2 frac 1 18 varepsilon end aligned for t in mathbb r the implied constant depends only on the form f and varepsilon

discrete biorthogonal wavelet transforms as block circulant matrices abstract a complete characterization of banded block circulant matrices that have banded inverse is derived by factorizations similar to those used for orthogonal matrices of this kind the nonorthogonal part of these matrices is shown to be related to block companion type matrices that are nilpotent it also follows from this characterization that the width of the band of the inverse is bounded in terms of the number and size of the blocks generating the block circulant matrix matrices of this type appear in the description of the action of perfect reconstruction filter banks as well as in applications of discrete biorthogonal higher multiplicity wavelet transforms

an additive problem in the fourier coefficients of cusp forms we establish an estimate on sums of shifted products of fourier coefficients coming from holomorphic or maass cusp forms of arbitrary level and nebentypus these sums are analogous to the binary additive divisor sum which has been studied extensively as an application we derive extending work of duke friedlander and iwaniec a subconvex estimate on the critical line for l functions associated to character twists of these cusp forms

the genlot generalized linear phase lapped orthogonal transform the general factorization of a linear phase paraunitary filter bank lppufb is revisited from this new perspective a class of lapped orthogonal transforms with extended overlap generalized linear phase lapped orthogonal transforms genlots is developed as a subclass of the general class of lppufb in this formulation the discrete cosine transform dct is the order 1 genlot the lapped orthogonal transform is the order 2 genlot and so on for any filter length that is an integer multiple of the block size the genlots are based on the dct and have fast implementation algorithms the implementation of genlots is explained including the method to process finite length signals the degrees of freedom in the design of genlots are described and design examples are presented along with image compression tests

lagrange multiplier approaches to the design of two channel perfect reconstruction linear phase fir filter banks the authors present two approaches to the design of two channel perfect reconstruction linear phase finite impulse response fir filter banks both approaches analyze and design the impulse responses of the analysis filter bank directly the synthesis filter bank is then obtained by simply changing the signs of odd order coefficients in the analysis filter bank the approach deals with unequal length filter banks by designing the lower length filters first one can take advantage of the fact that the number of variables for designing the higher length filters is more than the number of perfect reconstruction constraint equations the second approach generalizes the first and covers the design for all parts of linear phase perfect reconstruction constraint equations

polyphase networks block digital filtering lptv systems and alias free qmf banks a unified approach based on pseudocirculants the relationship between block digital filtering and quadrature mirror filter qmf banks is explored necessary and sufficient conditions for alias cancellation in qmf banks are expressed in terms of an associated matrix derived from the polyphase components of the analysis and synthesis filters these conditions called the pseudocirculant conditions make it possible to unite qmf banks with the framework of block digital filtering directly absence of amplitude distortion in an alias free qmf bank translates into the losslessness property of the pseudocirculant matrix involved

theory and design of m channel maximally decimated quadrature mirror filters with arbitrary m having the perfect reconstruction property based on the concept of losslessness in digital filter structures this paper derives a general class of maximally decimated m channel quadrature mirror filter banks that lead to perfect reconstruction the perfect reconstruction property guarantees that the reconstructed signal hat x n is a delayed version of the input signal x n i e hat x n x n n 0 it is shown that such a property can be satisfied if the alias component matrix ac matrix for short is unitary on the unit circle of the z plane the number of channels m is arbitrary and when m is two the results reduce to certain recently reported 2 channel perfect reconstruction qmf structures a procedure based on recently reported fir cascaded lattice structures is presented for optimal design of such fir m channel filter banks design examples are included

the circle method and bounds for l functions ii subconvexity for twists of gl 3 l functions let pi be a rm sl 3 bbb z hecke maass cusp form let chi chi 1 chi 2 be a dirichlet character with chi i primitive modulo m i suppose m 1 m 2 are primes such that sqrt m 2 m 4 delta m 1 m 2m 3 delta where m m 1m 2 and 0 delta 1 28 in this paper we will prove the following subconvex bound l left 1 over 2 pi otimes chi right ll pi varepsilon m 3 over 4 delta varepsilon

multiresolution standard form of a matrix we present a new method to compress matrices based on interpolatory multiresolution the matrices so compressed allow all kinds of algebraic operations including matrix matrix multiplication we illustrate through examples the performance of the compressed matrices we call the form devised here standard in contrast to the nonstandard form introduced by harten and yad shalom in previous papers because they are the interpolatory multiresolution equivalent to the standard and nonstandard wavelet forms respectively

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

lattice structures for optimal design and robust implementation of two channel perfect reconstruction qmf banks a lattice structure and an algorithm are presented for the design of two channel qmf quadrature mirror filter banks satisfying a sufficient condition for perfect reconstruction the structure inherently has the perfect reconstruction property while the algorithm ensures a good stopband attenuation for each of the analysis filters implementations of such lattice structures are robust in the sense that the perfect reconstruction property is preserved in spite of coefficient quantization the lattice structure has the hierarchical property that a higher order perfect reconstruction qmf bank can be obtained from a lower order perfect reconstruction qmf bank simply by adding more lattice sections several numerical examples are provided in the form of design tables

an adaptive wavelet vaguelette algorithm for the solution of pdes the paper first describes a fast algorithm for the discrete orthonormal wavelet transform and its inverse without using the scaling function this approach permits to compute the decomposition of a function into a lacunary wavelet basis i e a basis constituted of a subset of all basis functions up to a certain scale without modification the construction is then extended to operator adapted biorthogonal wavelets this is relevant for the solution of certain nonlinear evolutionary pdes where a priori information about the significant coefficients is available we pursue the approach described in j frohlich and k schneider europ j mech b fluids13 439 1994 which is based on the explicit computation of the scalewise contributions of the approximated function to the values at points of hierarchical grids here we present an improved construction employing the cardinal function of the multiresolution the new method is applied to the helmholtz equation and illustrated by comparative numerical results it is then extended for the solution of a nonlinear parabolic pde with semi implicit discretization in time and self adaptive wavelet discretization in space results with full adaptivity of the spatial wavelet discretization are presented for a one dimensional flame front as well as for a two dimensional problem

the design of uniformly and nonuniformly spaced pseudoquadrature mirror filters in the past quadrature mirror filters qmf s have been used to derive both uniformly and nonuniformly spaced filterbanks a pair of qmf s divides a signal into two equal bands which can be decimated at 2 1 and subsequently combined to reconstruct the original signal in order to derive filterbanks with more than two bands qmf s are combined in a binary tree structure pseudoquadrature mirror filters are similar to qmf s but can be designed to split a signal directly into any number of equally spaced bands thus generalizing the qmf concept in this paper the theory of pseudoquadrature mirror filters is reviewed these filters retain the desirable property that the channel signals from a uniformly spaced bank of m filters can be decimated by m 1 then interpolated and reassembled to reproduce the original signal an extension is made to the theory to allow a set of nonuniformly spaced filters to be derived from a uniformly spaced set and still retain all the desirable characteristics another extension to the theory is the derivation of a family of different sized filterbanks all derived from the same original prototype potential applications for the new filterbanks include improvements in subband coding of speech and music and analog scrambling of speech

m ary wavelet transform and formulation for perfect reconstruction in m band filter bank the binary wavelet transform is generalized and extended to the m ary biorthonormal case the computational equivalence between the discrete wavelet analysis and the m band multirate signal filtering is indicated the equivalence allows the perfect reconstruction requirement in a filter bank to be investigated from the vector space decomposition reconstruction in wavelet analysis from the construction of the biorthonormal wavelet bases the necessary and sufficient condition for the filters in a perfect reconstruction filter bank is formulated under this formulation an additional optimization procedure is then used to model the frequency domain requirement in filter bank design

hybrid bounds for twisted l functions the aim of this paper is to derive bounds on the critical line s 1 for l functions attached to twists f of a primitive cusp form f of level n and a primitive character modulo q that break convexity simultaneously in the s and q aspects if f has trivial nebentypus it is shown that

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

quadrature mirror filter design for an arbitrary number of equal bandwidth channels the two channel quadrature mirror filter structure of croisier and esteban may be extended to an arbitrary number of equal bandwidth channels given certain restrictions on the bandpass filters the most serious restriction is that the stopband attenuation of eacli band pass filter must be high for all frequencies outside twice the nominal 3 db bandwidth of the filter this restriction is not really a limiting factor for speech subband waveform coding since high adjacent channel attenuation is a necessity for the confinement of quantization noise a key feature of our filter structure is that the number of multiplies adds and stored coefficients required for implementation is significantly less than those needed for the conventional qmf structure given the same number of channels fortran code for a 16 channel filter structure is listed as an example of efficient implementation

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points keywords rankin selberg l function subconvexity heegner points reference tan article 2004 002doi 10 4007 annals 2004 160 185 record created on 2008 11 14 modified on 2017 05 12

multiresolution algorithms for the numerical solution of hyperbolic conservation laws given any scheme in conservation form and an appropriate uniform grid for the numerical solution of the initial value problem for one dimensional hyperbolic conservation laws we describe a multiresolution algorithm that approximates this numerical solution to a prescribed tolerance in an efficient manner to do so we consider the grid averages of the numerical solution for a hierarchy of nested diadic grids in which the given grid is the finest and introduce an equivalent multiresolution representation the multiresolution representation of the numerical solution consists of its grid averages for the coarsest grid and the set of errors in predicting the grid averages of each level of resolution in this hierarchy from those of the next coarser one once the numerical solution is resolved to our satisfaction in a certain locality of some grid then the prediction errors there are small for this particular grid and all finer ones this enables us to compress data by setting to zero small components of the representation which fall below a prescribed tolerance therefore instead of computing the time evolution of the numerical solution on the given grid we compute the time evolution of its compressed multiresolution representation algorithmically this amounts to computing the numerical fluxes of the given scheme at the points of the given grid by a hierarchical algorithm which starts with the computation of these numerical fluxes at the points of the coarsest grid and then proceeds through diadic refinements to the given grid at each step of refinement we add the values of the numerical flux at the center of the coarser cells the information in the multiresolution representation of the numerical solution is used to determine whether the solution is locally well resolved when this is the case we replace the costly exact value of the numerical flux with an accurate enough approximate value which is obtained by an inexpensive interpolation from the coarser grid the computational efficiency of this multiresolution algorithm is proportional to the rate of data compression for a prescribed level of tolerance that can be achieved for the numerical solution of the given scheme