asymptotic behavior of solutions for the damped wave equation with slowly decaying data abstract we consider the cauchy problem for the damped wave equation u t t u u t u 1 u t x r r n and the heat equation t 1 t x r r n if the data is small and slowly decays likely c 1 1 x k n 0 k 1 then the critical exponent is c k 1 2 k n for the semilinear heat equation in this paper it is shown that in the supercritical case there exists a unique time global solution to the cauchy problem for the semilinear heat equation in any dimensional space r n whose asymptotic profile is given by 0 t x r n e x y 2 4 t 4 t n 2 c 1 1 y 2 k n 2 d y provided that the data 0 satisfies lim x x k n 0 x c 1 0 even in the semilinear damped wave equation in the supercritical case a time global solution u with the data u u t 0 x u 0 u 1 x is shown in low dimensional spaces r n n 1 2 3 to have the same asymptotic profile 0 t x provided that lim x x k n u 0 u 1 x c 1 0 those proofs are given by elementary estimates on the explicit formulas of solutions

on the critical nonlinear damped wave equation with large initial data abstract we study the nonlinear damped wave equation 0 1 u t t u t u u u x r n t 0 u 0 x u 0 x u t 0 x u 1 x x r n in the critical case of 2 n our aim is to prove the large time asymptotic formulas for solutions of the cauchy problem 0 1 without any restriction on the size of the initial data and on the spatial dimension

existence and non existence of global solutions of the cauchy problem for higher order semilinear pseudo hyperbolic equations abstract we consider the cauchy problem for a higher order pseudo hyperbolic equation using the l p l q type estimation for the corresponding linear problem the existence and non existence criteria of global solutions are found the existence and uniqueness of smooth global solutions are also investigated we also establish the behavior of solutions and their derivatives as t

efficient reconstruction of band limited sequences from nonuniformly decimated versions by use of polyphase filter banks an efficient polyphase structure for the reconstruction of a band limited sequence from a nonuniformly decimated version is developed theoretically the reconstruction involves the implementation of a bank of multilevel filters and it is shown that how all these reconstruction filters can be obtained at the cost of one mth band low pass filter and a constant matrix multiplier the resulting structure is therefore more general than previous schemes in addition the method offers a direct means of controlling the overall reconstruction distortion t z by appropriate design of a low pass prototype filter p z extension of these results to multiband band limited signals and to the case of nonconsecutive nonuniform subsampling are also summarized along with generalizations to the multidimensional case design examples are included to demonstrate the theory and the complexity of the new method is seen to be much lower than earlier ones

critical exponents for semilinear dissipative wave equations in rn abstract we shall present new critical exponents 1 2m n with m 1 2 to the cauchy problem utt u ut u p 1u with the initial data u 0 u 1 h 1 r n l m r n l 2 r n l m r n that is the small data global existence property can be derived to the cauchy problem above with power 1 2m n

nonuniform sampling of periodic bandlimited signals digital processing techniques are based on representing a continuous time signal by a discrete set of samples this paper treats the problem of reconstructing a periodic bandlimited signal from a finite number of its nonuniform samples in practical applications only a finite number of values are given extending the samples periodically across the boundaries and assuming that the underlying continuous time signal is bandlimited provides a simple way to deal with reconstruction from finitely many samples two algorithms for reconstructing a periodic bandlimited signal from an even and an odd number of nonuniform samples are developed in the first the reconstruction functions constitute a basis while in the second they form a frame so that there are more samples than needed for perfect reconstruction the advantages and disadvantages of each method are analyzed specifically it is shown that the first algorithm provides consistent reconstruction of the signal while the second is shown to be more stable in noisy environments next we use the theory of finite dimensional frames to characterize the stability of our algorithms we then consider two special distributions of sampling points uniform and recurrent nonuniform and show that for these cases the reconstruction formulas as well as the stability analysis are simplified significantly the advantage of our methods over conventional approaches is demonstrated by numerical experiments

efficient recovery of sub nyquist sampled sparse multi band signals using reconfigurable multi channel analysis and modulated synthesis filter banks sub nyquist cyclic nonuniform sampling cnus of a sparse multi band signal generates a nonuniformly sampled signal assuming that the corresponding uniformly sampled signal satisfies the nyquist sampling criterion the sequence obtained via cnus can be passed through a reconstructor to recover the missing uniform grid samples at present these reconstructors have very high design and implementation complexity that offsets the gains obtained due to sub nyquist sampling in this paper we propose a scheme that reduces the design and implementation complexity of the reconstructor in contrast to the existing reconstructors which use only a multi channel synthesis filter bank fb the proposed reconstructor utilizes both analysis and synthesis fbs which makes it feasitble to achieve an order of magnitude reduction of the complexity the analysis filters are implemented using polyphase networks whose branches are allpass filters with distinct fractional delays and phase shifts in order to reduce both the design and the implementation complexity of the synthesis fb the synthesis filters are implemented using a cosine modulated fb in addition to the reduced complexity of the reconstructor the proposed multi channel recovery scheme also supports online reconfigurability which is required in flexible multi mode systems where the user subband locations vary with time

exact interpolation and iterative subdivision schemes we examine the circumstances under which a discrete time signal can be exactly interpolated given only every mth sample after pointing out the connection between designing an m fold interpolator and the construction of an m channel perfect reconstruction filter bank we derive necessary and sufficient conditions on the signal under which exact interpolation is possible bandlimited signals are one obvious example but numerous others exist we examine these and show how the interpolators may be constructed a main application is to iterative interpolation schemes used for the efficient generation of smooth curves we show that conventional bandlimited interpolators are not suitable in this context a better criterion is to use interpolators that are exact for polynomial functions we demonstrate that these interpolators converge when iterated and show how these may be designed for any polynomial degree n and any interpolation factor m this makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium

classical sampling theorems in the context of multirate and polyphase digital filter bank structures the recovery of a signal from so called generalized samples is a problem of designing appropriate linear filters called reconstruction or synthesis filters this relationship is reviewed and explored novel theorems for the subsampling of sequences are derived by direct use of the digital filter bank framework these results are related to the theory of perfect reconstruction in maximally decimated digital filter bank systems one of the theorems pertains to the subsampling of a sequence and its first few differences and its subsequent stable reconstruction at finite cost with no error the reconstruction filters turn out to be multiplierless and of the fir finite impulse response type these ideas are extended to the case of two dimensional signals by use of a kronecker formalism the subsampling of bandlimited sequences is also considered a sequence x n with a fourier transform vanishes for mod omega mod or l pi m where l and m are integers with l

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

filter design for mimo sampling and reconstruction we address the problem of finite impulse response fir filter design for uniform multiple input multiple output mimo sampling this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases the input signals are modeled as either continuous time or discrete time multiband input signals with different band structures we present conditions on the channel and the sampling rate that allow perfect inversion of the channel additionally we provide a stronger set of conditions under which the reconstruction filters can be chosen to have frequency responses that are continuous we also provide conditions for the existence of fir perfect reconstruction filters and when such do not exist we address the optimal approximation of the ideal filters using fir filters and a minmax l sub 2 end to end distortion criterion the design problem is then reduced to a standard semi infinite linear program an example design of fir reconstruction filters is given

critical exponent for the cauchy problem to the weakly coupled damped wave system abstract in this paper we consider a system of weakly coupled semilinear damped wave equations we determine the critical exponent for any space dimensions our proof of the global existence of solutions for supercritical nonlinearities is based on a weighted energy method whose multiplier is appropriately modified in the case where one of the exponent of the nonlinear term is less than the so called fujita s critical exponent we also give estimates of the lifespan of solutions from above for subcritical nonlinearities

sampling theorems for uniform and periodic nonuniform mimo sampling of multiband signals we examine a multiple input multiple output mimo sampling scheme for a linear time invariant continuous time mimo channel the input signals are modeled as multiband signals with different spectral supports and the channel outputs are sampled on either uniform or periodic nonuniform sampling sets with possibly different but commensurate intervals on the different outputs this scheme encompasses papoulis generalized sampling and several nonuniform sampling schemes as special cases we derive necessary and sufficient conditions on the channel and the sampling rate that allow stable perfect reconstruction of the inputs or equivalently perfect inversion of the channel from an implementation viewpoint we note that it is desirable that the reconstruction filters have continuous frequency responses we derive necessary and sufficient conditions that guarantee this continuity property the frequency responses of the reconstruction filters are specified as solutions to a system of linear equations finally we demonstrate that perfect reconstruction may be possible even when the channel outputs are sampled at an average rate that does not allow the reconstruction of any output from its samples alone in certain instances this average rate can achieve the recently presented fundamental bounds on mimo sampling density

fast reconstruction methods for bandlimited functions from periodic nonuniform sampling a well known generalization of shannon s sampling theorem states that a bandlimited function can be reconstructed from its periodic nonuniformly spaced samples if the effective sampling rate is at least the nyquist rate analogous to shannon s sampling theorem this generalization requires that an infinite number of samples be available which however is never the case in practice most existing reconstruction methods for periodic nonuniform sampling yield very low order often not even first order accuracy when only a finite number of samples is given in this paper we propose a fast numerically robust root exponential accurate reconstruction method the efficiency and accuracy of the algorithm is obtained by fully exploiting the sampling structure and utilizing localized fourier analysis we discuss applications in analog to digital conversion where nonuniform periodic sampling arises in various situations finally we demonstrate the performance of our algorithm by numerical examples

damped wave equation in the subcritical case abstract we study large time asymptotics of small solutions to the cauchy problem for the one dimensional nonlinear damped wave equation 1 v tt v t v xx v 1 0 x r t 0 v 0 x v 0 x v t 0 x v 1 x in the sub critical case 2 3 2 we assume that the initial data v 0 1 x 1 v 1 l l 1 a a 0 1 where l 1 a l 1 l 1 a a l 1 x 1 x 2 also we suppose that the mean value of initial data r v 0 x v 1 x dx 0 then there exists a positive value such that the cauchy problem 1 has a unique global solution v t x c 0 l l 1 a satisfying the following time decay estimate v t l c t 1 for large t 0 here 2 3 2

global asymptotics of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u 1 u 0 t x r r n the behavior of u as t is expected to be same as that for the corresponding heat equation t 1 0 t x r r n in the subcritical case 1 c n 1 2 n there exists a similarity solution w b t x with the form t 1 1 f x t depending on b lim x x 2 1 f x 0 our first aim is to show the decay rates u t l 2 u t l 1 u t l 2 o t 1 1 n 4 t 1 1 n 2 1 t 1 1 1 2 n 4 provided that the initial data without initial data size restriction spatially decays with reasonable polynomial order the decay rates are sharp in the sense that they are same as those of the similarity solution the second aim is to show that the gauss kernel is the asymptotic profile in the supercritical case which has been shown in case of one dimensional space by hayashi kaikina and naumkin n hayashi e i kaikina p i naumkin asymptotics for nonlinear damped wave equations with large initial data preprint 2004 we show this assertion in two and three dimensional space to prove our results both the weighted l 2 energy method and the explicit formula of solutions will be employed the weight is an improved one originally developed in y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

existence non existence and asymptotic behavior of global solutions to the cauchy problem for systems of semilinear hyperbolic equations with damping terms abstract we consider the cauchy problem for systems of semilinear hyperbolic equations using the l p l q type estimation for the corresponding linear parts the existence and uniqueness of weak global solutions are investigated we also established the behavior of solutions and their derivatives as t using the method of test functions developed in the works mitidieri and pokhozhaev 2001 11 veron and pohozaev 2001 12 and caristi 2000 23 we obtain the analogue of the fujita hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping it follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense

higher order sampling and recovering of lowpass signals a higher order sampling function applied to lowpass signals is described to recover the original signal from its periodic nonuniform samples a formulation of the sampling function is introduced based on the occurrence of local gaps in the sampled signal spectrum as a result the interpolant is greatly simplified

cyclic lti systems in digital signal processing cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer l in such cases the frequency domain is defined as a uniform discrete grid as in l point dft this offers more freedom in theoretical as well as design aspects while circular convolution has been the centerpiece of many algorithms in signal processing for decades such freedom especially from the viewpoint of linear system theory has not been studied in the past in this paper we introduce the fundamentals of cyclic multirate systems and filter banks presenting several important differences between the cyclic and noncyclic cases cyclic systems with allpass and paraunitary properties are studied the paraunitary interpolation problem is introduced and it is shown that the interpolation does not always succeed state space descriptions of cyclic lti systems are introduced and the notions of reachability and observability of state equations are revisited it is shown that unlike in traditional linear systems these two notions are not related to the system minimality in a simple way throughout the paper a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

mmse design of interpolation and downsampling fir filters in the context of periodic nonuniform sampling the generalized sampling theorem states that any analog signal whose spectrum is limited to 1 t can be exactly recovered from n sequences of samples taken at a rate 2 nt and all having a different sampling phase when n 2 the exact interpolation formula can be derived quite easily the ideal interpolation filters have infinite impulse responses iirs this paper addresses first the question of recovering from the two initial sequences any other sequence taken at the same rate 1 t and with a different sampling phase the design problem is dealt with for finite length filters and the criterion is the minimization of the mean squared interpolation error next the problem of computing from the two initial sequences a third one at a lower rate is addressed fir decimation filters are also designed for an mmse criterion these problems are illustrated for two typical covariance functions

minimum mean squared error reconstruction for generalized undersampling of cyclostationary processes the generalized sampling problem is considered using a filter bank model with a stochastic process framework a signal reconstruction solution minimizing the time averaged mean squared error is found using a filter bank optimization technique for cyclostationary signals the use of a stochastic model approach to find a minimized error solution rather than a perfect reconstruction solution allows consideration of sampling problems that deterministic approaches cannot solve for instance a particular sampling density is not required allowing reconstruction optimization in cases of undersampling an extension is also presented for optimal reconstruction in the presence of additive noise and interference the result is a generalized sampling model which can be used for an extensive variety of scenarios combining aspects of undersampling multiple input multiple output mimo sampling and periodic nonuniform sampling

a boundary integral approach in the geoelectrical cavity prospecting abstract the fredholm integral equation of the second kind with respect to electrical potential is derived associated with the cavity concealed within the earth in this approach the positioning constant to be attached to the governing equation for the fundamental solution is of great significance and we formulate the positioning constant at an unsmooth point on the boundary interface through the spherical geometry it is shown that the numerical solutions are in sufficient agreement with the experimental data

asymptotic behaviors of solutions for time dependent damped wave equations abstract in this paper we consider the cauchy problem for the wave equation with time dependent damping b t u t and absorbed semilinear term u 1 u here b t b 0 1 t with 1 1 and b 0 0 using the weighted energy method we obtain the l 1 and l 2 decay rates of the solution which coincide to those for self similar solutions to the corresponding parabolic equation when 1 f n 1 2 n

perfect reconstruction formulas and bounds on aliasing error in sub nyquist nonuniform sampling of multiband signals we examine the problem of periodic nonuniform sampling of a multiband signal and its reconstruction from the samples this sampling scheme which has been studied previously has an interesting optimality property that uniform sampling lacks one can sample and reconstruct the class spl bscr spl fscr of multiband signals with spectral support spl fscr at rates arbitrarily close to the landau 1969 minimum rate equal to the lebesgue measure of spl fscr even when spl fscr does not tile r under translation using the conditions for exact reconstruction we derive an explicit reconstruction formula we compute bounds on the peak value and the energy of the aliasing error in the event that the input signal is band limited to the span of spl fscr the smallest interval containing spl fscr which is a bigger class than the valid signals spl bscr spl fscr band limited to spl fscr we also examine the performance of the reconstruction system when the input contains additive sample noise

some general lagrange interpolations over simplex finite elements with reference to derivative singularities abstract for simplex finite elements the native lagrange family with arbitrarily placed nodes is presented in hierarchy ranking expressions it includes the well known complete lagrange family as well as the mid edge lagrange family to be proposed in this paper this new family enables us to utilize harmonious combinations of interpolations of different orders in finite element analysis as an application of developed simplex interpolations to fracture mechanics where some derivative singularities are needed we then describe the semi radial singularity mapping with examinations of peculiar trial function spaces

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

optimal sub nyquist nonuniform sampling and reconstruction for multiband signals we study the problem of optimal sub nyquist sampling for perfect reconstruction of multiband signals the signals are assumed to have a known spectral support spl fscr that does not tile under translation such signals admit perfect reconstruction from periodic nonuniform sampling at rates approaching landau s 1967 lower bound equal to the measure of spl fscr for signals with sparse spl fscr this rate can be much smaller than the nyquist rate unfortunately the reduced sampling rates afforded by this scheme can be accompanied by increased error sensitivity in a previous study we derived bounds on the error due to mismodeling and sample additive noise adopting these bounds as performance measures we consider the problems of optimizing the reconstruction sections of the system choosing the optimal base sampling rate and designing the nonuniform sampling pattern we find that optimizing these parameters can improve system performance significantly furthermore uniform sampling is optimal for signals with spl fscr that tiles under translation for signals with nontiling spl fscr which are not amenable to efficient uniform sampling the results reveal increased error sensitivities with sub nyquist sampling however these can be controlled by optimal design demonstrating the potential for practical multifold reductions in sampling rate

reconstruction of nonuniformly sampled bandlimited signals by means of digital fractional delay filters this paper considers the problem of reconstructing a class of nonuniformly sampled bandlimited signals of which a special case occurs in e g time interleaved analog to digital converter adc systems due to time skew errors to this end we propose a synthesis system composed or digital fractional delay filters the overall system i e nonuniform sampling and the proposed synthesis system can be viewed as a generalization of time interleaved adc systems to which the former reduces as a special case compared with existing reconstruction techniques our method has major advantages from an implementation point of view to be precise 1 we can perform the reconstruction as well as desired in a certain sense by properly designing the digital fractional delay filters and 2 if properly implemented the fractional delay filters need not be redesigned in case the time skews are changed the price to pay for these attractive features is that we need to use a slight oversampling it should be stressed however that the oversampling factor is less than two as compared with the nyquist rate the paper includes error and quantization noise analysis the former is useful in the analysis of the quantization noise and when designing practical fractional delay filters approximating the ideal filters

singularity and positioning constants in poisson s equation with a point source of unit intensity abstract the singularity constants and positioning constants are described in the potential field governed by poisson s equation with a point source of unit intensity it is then demonstrated that the singularity constants play an important role in the integral equation formulations utilizing rather complicated fundamental solutions

new quadrature mirror filter structures this paper introduces new quadrature mirror filter qmf structures for the frequency domain analysis and synthesis of digital signals the conventional qmf technique is first extended to cover complex quadrature mirror filters cqmf in which a digital signal is split into n adjacent complex subbands where the real and imaginary parts are subsampled by 1 2n with respect to the original signal the computational complexity of qmf banks is then analyzed and a new scheme which reduces the computational complexity by about a factor of two over conventional qmf implementations is proposed finally the filter design tradeoffs are discussed and the microprogramed implementation of qmf banks is evaluated

decay property of solutions for damped wave equations with space time dependent damping term abstract we consider the cauchy problem for the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 using the weighted energy method we can obtain the l 2 decay rate of the solution which is almost optimal in the case c n 1 2 n combining this decay rate with the result that we got in the paper j lin k nishihara j zhai l 2 estimates of solutions for damped wave equations with space time dependent damping term j differential equations 248 2010 403 422 we believe that c n is a critical exponent note that when 0 c n coincides to the fujita exponent f n 1 2 n the new points include the estimate in the supercritical exponent and for not necessarily compactly supported data

damped wave equation with a critical nonlinearity in higher space dimensions abstract we study the cauchy problem for nonlinear damped wave equations with a critical defocusing power nonlinearity u 2 n u where n denotes the space dimension for n 1 2 3 global in time existence of small solutions was shown in 4 in this paper we generalize the results to any spatial dimension via the method of decomposition of the equation into the high and low frequency components under the assumption that the initial data are small and decay rapidly at infinity furthermore we present a sharp time decay estimate of solutions with a logarithmic correction

damped wave equation with a critical nonlinearity we study large time asymptotics of small solutions to the cauchy problem for nonlinear damped wave equations with a critical nonlinearity 2 t u t u u u 1 2 n 0 x r n t 0 u 0 x eu 0 x t u 0 x eu 1 x x r n where e 0 and space dimensions n 1 2 3 assume that the initial data uo h 0 n h 0 u 1 e h 1 0 n h 1 where n 2 weighted sobolev spaces are h l m e l2 m l x l 2 1 x 2 also we suppose that 2 n 0 u 0 x dx 0 where then we prove that there exists a positive e 0 such that the cauchy problem above has a unique global solution u c 0 oo h 0 satisfying the time decay property u t e g t x e t lp ce 1 2 n g 1 n 2 t n 2 1 1 p for all t 0 1 p where e 0 e 0

existence and nonexistence of time global solutions to damped wave equation on half line on the half line r 0 infinity the initial boundary value problems with null dirichlet boundary for both the semilinear heat equation and damped wave equation are considered the critical exponent p c n k of semilinear term for the existence and nonexistence about the semilinear heat equation on the halved space d n d k r k rn k is given by p c n k 1 2 n k j appl math phys 39 1988 135 149 arch rational mech anal 109 1990 63 71 since the damped wave equation is expected to be close to the heat equation j differential equations 191 2003 445 469 math z 244 2003 631 649 the critical exponent for the semilinear damped wave equation is expected to be same as that of the semilinear heat equation however there is no blow up result on the halved space for the damped wave equation in this paper the exponent pc l 1 2 is shown to be critical for the existence and nonexistence of time global solution to both the semilinear heat equation and damped wave equation on the half line r together with the derivation of the blow up time for the proof the explicit formulas of solutions are used in a similar fashion to those in li and zhou discrete continuous dynamic systems 1 1995 503 520 c 2005 elsevier ltd all rights reserved

small data global existence for the semilinear wave equation with space time dependent damping abstract in this paper we consider the critical exponent problem for the semilinear wave equation with space time dependent damping when the damping is effective it is expected that the critical exponent agrees with that of only the space dependent coefficient case we shall prove that there exists a unique global solution for small data if the power of nonlinearity is larger than the expected exponent moreover we do not assume that the data are compactly supported however it is still open whether there exists a blow up solution if the power of nonlinearity is smaller than the expected exponent

minimum rate sampling and reconstruction of signals with arbitrary frequency support we examine the question of reconstruction of signals from periodic nonuniform samples this involves discarding samples from a uniformly sampled signal in some periodic fashion we give a characterization of the signals that can be reconstructed at exactly the minimum rate once a nonuniform sampling pattern has been fixed we give an implicit characterization of the reconstruction system and a design method by which the ideal reconstruction filters may be approximated we demonstrate that for certain spectral supports the minimum rate can be approached or achieved using reconstruction schemes of much lower complexity than those arrived at by using spectral slicing as in earlier work previous work on multiband signals have typically been those for which restrictive assumptions on the sizes and positions of the bands have been made or where the minimum rate was approached asymptotically we show that the class of multiband signals which can be reconstructed exactly is shown to be far larger than previously considered when approaching the minimum rate this freedom allows us in certain cases to have a far less complex reconstruction system

lp lq estimates of solutions to the damped wave equation in 3 dimensional space and their application it has been asserted that the damped wave equation has the diffusive structure as t in this paper we consider the cauchy problem in 3 dimensional space for the linear damped wave equation and the corresponding parabolic equation and obtain the lp lq estimates of the difference of each solution which represent the assertion precisely explicit formulas of the solutions are analyzed for the proof the second aim is to apply the lp lq estimates to the semilinear damped wave equation with power nonlinearity if the power is larger than the fujita exponent then the time global existence of small weak solution is proved and its optimal decay order is obtained

quadrature mirror filter design in the time domain a new technique for designing quadrature mirror filters is described the formulation carried out in the time domain is shown to result in an optimization problem requiring minimization of a quartic multinomial an iterative solution is suggested which involves computation of the eigenvector of a matrix with a dimensionality equal to one half the number of filter taps our experiments show that convergence to the optimum tap weights is stable and the accuracy of the final solution is limited only by the accuracy of the eigenvalue eigenvector routine as in an earlier technique the user can specify the stop band frequency the relative weights of the passband ripple energy and the stopband residual energy and of course the number of filter taps

subband coder design incorporating recursive quadrature filters and optimum adpcm coders this paper presents the results of an analytical and experimental study of the use of iir quadrature mirror filters and forward adapting adpcm with optimum quantizers for octave band subband coders for speech it is shown that the underlying analysis synthesis systems can be designed such that there is no interband aliasing distortion in the coded speech and such that the analysis synthesis transfer function has no frequency distortion or no phase distortion but not both the experimental study showed that subband coders based on a mixture of infinite impulse response iir and finite duration impulse response fir filters resulted in higher quality coding systems using fewer multiplies than for systems based on fir filters alone it was also found that a signal to noise ratio snr gain of about 4 db could be obtained using optimum quantizers for forward adapting adpcm coders over linear quantizers in backward adapting apcm codes but no perceptual quality gain was observed

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

decay properties of solutions to the cauchy problem for the damped wave equation with absorption abstract we consider the cauchy problem for the damped wave equation with absorption u t t u u t u p 1 u 0 t x r r n the behavior of u as t is expected to be the same as that for the corresponding heat equation t p 1 0 t x r r n which has the similarity solution w a t x with the form t 1 p 1 f x t depending on a lim x x 2 p 1 f x 0 provided that p is less than the fujita exponent p c n 1 2 n in this paper as a first step if 1 p p c n and the data u 0 u 1 x decays exponentially as x without smallness condition the solution is shown to decay with rates as t u t l 2 u t l p 1 u t l 2 o t 1 p 1 n 4 t 1 p 1 n 2 p 1 t 1 p 1 1 2 n 4 those of which seem to be reasonable because the similarity solution w a t x has the same decay rates as for the proof the weighted l 2 energy method will be employed with suitable weight similar to that in todorova and yordanov y todorova b yordanov critical exponent for a nonlinear wave equation with damping j differential equations 174 2001 464 489

l2 estimates of solutions for damped wave equations with space time dependent damping term abstract in this paper we consider the damped wave equation with space time dependent potential b t x and absorbing semilinear term u 1 u here b t x b 0 1 x 2 2 1 t with b 0 0 0 and 0 1 based on the local existence theorem we obtain the global existence and the l 2 decay rate of the solution by using the weighted energy method the decay rate coincides with the result of nishihara k nishihara decay properties for the damped wave equation with space dependent potential and absorbed semilinear term preprint in the case of 0 and coincides with the result of nishihara and zhai k nishihara j zhai asymptotic behaviors of time dependent damped wave equations preprint in the case of 0