comparison between dpcm and hadamard transform coding in the composite coding of the ntsc color tv signal since differential pulse code modulation dpcm and orthogonal transform coding otc are the most fundamental methods of high efficiency coding bit reduction method it is important to clarify the basic coding characteristics of these methods and the difference between them in order to utilize the bit reduction method effectively this paper theoretically as well as experimentally compares the coding efficiency of a dpcm having a two dimensional predictor with that of a two dimensional hadamard transform coding method htc in the intrafield coding of the ntsc composite signal the comparison evidenced that the distinctive difference in coding characteristics between dpcm and htc depends greatly on the power level of carrier chrominance signals that is it is confirmed theoretically and experimentally that the coding efficiency of the htc is far lower than that of the dpcm in the case of a signal having a high power level carrier chrominance signal such as a color bar signal

on the analytical summation of fourier series and its relation to the asymptotic behaviour of fourier transforms forms of the poisson summation formula psf appropriate for the summation of semi infinite and infinite fourier series are derived application of these results to the acceleration of convergence of various types of series with monotonically decreasing coefficient functions yields transformed series with terms that decay either exponentially or with the inverse first or second power of the index variable these two very different convergence properties are explained in terms of the asymptotic properties of the relevant fourier transforms which are in turn related to the power series expansions of the summand functions in the original fourier series the result is that the poisson summation formula works best for fourier cosine series in which the summand functions are expansible in even powers and for fourier sine series in which the summand functions have power series with odd powers here application of the psf produces series of terms that decay exponentially with increasing argument x in contrast application of the semi infinite version of the psf to fourier cosine series of terms with odd power expansions or to fourier sine series of terms with even power expansions yields transformed series involving functions of the form which decay approximately as if the summand function in the fourier series has a power series with both even and odd powers the transformed series involves sine and cosine integral functions which decay approximately as fourier series of these last three types in general require additional acceleration for example by application of the kummer transformation

on the gibbs phenomenon i recovering exponential accuracy from the fourier partial sum of a nonperiodic analytic function it is well known that the fourier series of an analytic and periodic function truncated after 2n 1 terms converges exponentially with n even in the maximum norm it is also known that if the function is not periodic the rate of convergence deteriorates in particular there is no convergence in the maximum norm although the function is still analytic this is known as the gibbs phenomenon in this paper we show that the first 2n 1 fourier coefficients contain enough information about the function so that an exponentially convergent approximation in the maximum norm can be constructed the proof is a constructive one and makes use of the gegenbauer polynomials c n x it consists of two steps in the first step we show that the first m coefficients of the gegenbauer expansion based on c n x for 0 n m of any l2 function can be obtained within exponential accuracy provided that both and m are proportional to but smaller than n in the second step we construct the gegenbauer expansion based on c n 0 n m from the coefficients found in the first step we show that this series converges exponentially with n provided that the original function is analytic though nonperiodic thus we prove that the gibbs phenomenon can be completely overcome

on a high order numerical method for functions with singularities by splitting a given singular function into a relatively smooth part and a specially structured singular part it is shown how the traditional fourier method can be modified to give numerical methods of high order for calculating derivatives and integrals singular functions with various types of singularities of importance in applications are considered relations between the discrete and the continuous fourier series for the singular functions are established of particular interest are piecewise smooth functions for which various important applications are indicated and for which numerous numerical results are presented

rate bounds on ssim index of quantized images in this paper we derive bounds on the structural similarity ssim index as a function of quantization rate for fixed rate uniform quantization of image discrete cosine transform dct coefficients under the high rate assumption the space domain ssim index is first expressed in terms of the dct coefficients of the space domain vectors the transform domain ssim index is then used to derive bounds on the average ssim index as a function of quantization rate for uniform gaussian and laplacian sources as an illustrative example uniform quantization of the dct coefficients of natural images is considered we show that the ssim index between the reference and quantized images fall within the bounds for a large set of natural images further we show using a simple example that the proposed bounds could be very useful for rate allocation problems in practical image and video coding applications

a lag averaged generalization of euler s method for accelerating series abstract euler s method is an efficient accelerator for an alternating series that is one whose terms a j are such that sign a j sign a j 1 however its effectiveness deteriorates rapidly as the period p of the oscillations in degree j increases to reduce this nonuniformity we introduce an algorithm which begins with the recursive computation of a triangular array of numbers the diagonal elements define a generalization of euler s method each horizontal row defines a delayed euler method which is even more effective the key element is that the array is generated by repeatedly averaging partial sums of the original series with a lag that is a distance in degree between the partial sums that are combined which optimally is half the period p of the oscillations in the series we illustrate the algorithm with the fourier series for a function with a shock wave jump discontinuity the terms a j 1 j sin jx vary from strictly alternating p 2 at x 0 to monotonic p as x the lag averaged euler scheme with and without delay is much more efficient than its classical counterpart especially near the discontinuities we present an analytical convergence theory for both the standard and generalized euler methods

a united approach to accelerating trigonometric expansions abstract the fourier series of a smooth function on a compact interval usually has slow convergence due to the fact that the periodic extension of the function has jumps at the interval endpoints for various symmetry conditions polynomial interpolation methods have been developed for performing a boundary correction the resulting variants of krylov approximants are a sum of a correction polynomial and a fourier sum of the corrected function 1 8 in this paper we review these methods and derive estimates in the maximum norm we further show that derivatives of the krylov approximants are again krylov approximants of derivatives of the considered function this enables us to give a unified treatment of the problem of simultaneous approximation

spurious waves in discrete computation of wave phenomena and flow problems in the present work we focus on spurious propagating disturbances q waves to establish the existence of q waves in computations we compare properties of different numerical methods drawn from finite difference finite volume and finite element methods existence and properties of q waves are demonstrated with propagation of wave packets following one dimensional 1d convection equation skewed wave propagation and by solution of linearized rotating shallow water wave equation lrswe specific numerical experiments are performed with parameters that convert a wave packet into a q wave we also show the case where q waves are created additionally to physical disturbances those propagate downstream formation of q waves are shown in the case of a discrete shielded vortex in the uniform flow and incompressible transitional flow past an aerofoil by solving the navier stokes equation in performing this exercise we establish critical wavenumber range beyond which q waves are created relevance of this information for dns and les is discussed we have further discussed the case of spurious caustics in discrete computing

distributions of the two dimensional dct coefficients for images for a two dimensional discrete cosine transform dct image coding system there have been different assumptions concerning the distributions of the transform coefficients this paper presents results of distribution tests that indicate that for many images the statistics of the coefficients are best approximated by a gaussian distribution for the dc coefficient and a laplacian distribution for the other coefficients furthermore from a simulation of the dct coding system it is shown that the assumption that the coefficients are laplacian yields a higher actual output signal to noise ratio and a much better agreement between theory and simulation than the gaussian assumption

an integral transform approach for a mixed boundary problem involving a flowing partially penetrating well with infinitesimal well skin 1 a flowing partially penetrating well with infinitesimal well skin is a mixed boundary because a cauchy condition is prescribed along the screen length and a neumann condition of no flux is stipulated over the remaining unscreened part an analytical approach based on the integral transform technique is developed to determine the laplace domain solution for such a mixed boundary problem in a confined aquifer of finite thickness first the mixed boundary is changed into a homogeneous neumann boundary by substituting the cauchy condition with a neumann condition in terms of well bore flux that varies along the screen length and is time dependent despite the well bore flux being unknown a priori the modified model containing this homogeneous neumann boundary can be solved with the laplace and the finite fourier cosine transforms to determine well bore flux screen length is discretized into a finite number of segments to which the cauchy condition is reinstated this reinstatement also restores the relation between the original model and the solutions obtained for a given time the numerical inversion of the laplace domain solution yields the drawdown distributions well bore flux and the well discharge this analytical approach provides an alternative for dealing with the mixed boundary problems especially when aquifer thickness is assumed to be finite

the asymptotic chebyshev coefficients for functions with logarithmic endpoint singularities mappings and singular basis functions abstract when a function is singular at the ends of its expansion interval its chebyshev coefficients a n converge very poorly we analyze three numerical strategies for coping with such singularities of the form 1 x k log 1 x and in the process make some modest additions to the theory of chebyshev expansions the first two numerical methods are the convergence improving changes of coordinate x sin 2 y and x tanh ly 1 y 2 1 2 we derive the asymptotic chebyshev coefficients in the limit n for both mappings and for the original untransformed chebyshev series for the original function the asymptotic approximation for general k is augmented by the exact chebyshev coefficients for integer k numerical tests show that the sine mapping is excellent for k 1 increasing the rate of convergence to b n o 1 n 4 k 1 although the tanh transformation is guaranteed to be better for sufficiently large n we offer both theoretical and numerical evidence to explain why the sine mapping is usually better in practice sufficiently large n is usually huge instead of mapping one may use a third strategy supplementing the chebyshev polynomials with singular basis functions simple experiments show that this approach is also successful

efficient calculation of the free space periodic green s function electromagnetic scattering from periodic structures can be formulated in terms of an integral equation that has as its kernel a periodic green s function the periodic green s function can be derived as a response to an array of line point sources spatial domain or as a response to series of current sheets spectral domain these responses are a fourier transform pair and are slowly convergent summations the convergence problems in each domain arise from unavoidable singularities in the reciprocal domain a method is discussed to overcome the slow convergence by using the poisson summation formula and summing in a combination of spectral and spatial domains a parameter study is performed to determine an optimum way to weigh the combination of domains simple examples of scattering from a one dimensional array of strips and two dimensional array of plates are used to illustrate the concepts

on the gibbs phenomenon and its resolution the nonuniform convergence of the fourier series for discontinuous functions and in particular the oscillatory behavior of the finite sum was already analyzed by wilbraham in 1848 this was later named the gibbs phenomenon this article is a review of the gibbs phenomenon from a different perspective the gibbs phenomenon as we view it deals with the issue of recovering point values of a function from its expansion coefficients alternatively it can be viewed as the possibility of the recovery of local information from global information the main theme here is not the structure of the gibbs oscillations but the understanding and resolution of the phenomenon in a general setting the purpose of this article is to review the gibbs phenomenon and to show that the knowledge of the expansion coefficients is sufficient for obtaining the point values of a piecewise smooth function with the same order of accuracy as in the smooth case this is done by using the finite expansion series to construct a different rapidly convergent approximation

design of linear equalizers optimized for the structural similarity index we propose an algorithm for designing linear equalizers that maximize the structural similarity ssim index between the reference and restored signals the ssim index has enjoyed considerable application in the evaluation of image processing algorithms algorithms however have not been designed yet to explicitly optimize for this measure the design of such an algorithm is nontrivial due to the nonconvex nature of the distortion measure in this paper we reformulate the nonconvex problem as a quasi convex optimization problem which admits a tractable solution we compute the optimal solution in near closed form with complexity of the resulting algorithm comparable to complexity of the linear minimum mean squared error mmse solution independent of the number of filter taps to demonstrate the usefulness of the proposed algorithm it is applied to restore images that have been blurred and corrupted with additive white gaussian noise as a special case we consider blur free image denoising in each case its performance is compared to a locally adaptive linear mse optimal filter we show that the images denoised and restored using the ssim optimal filter have higher ssim index and superior perceptual quality than those restored using the mse optimal adaptive linear filter through these results we demonstrate that a designing image processing algorithms and in particular denoising and restoration type algorithms can yield significant gains over existing in particular linear mmse based algorithms by optimizing them for perceptual distortion measures and b these gains may be obtained without significant increase in the computational complexity of the algorithm

invariance theorems concerning reflection at numerical boundaries abstract downwind computational boundaries in the numerical approximation of hyperbolic equations are in general not transparent and they create spurious reflection a useful measure of this is given by the energy or usual sum of squares of the reflected solution in response to an arbitrary solution which originates from within the computing domain we prove in that respect a somewhat unexpected property namely that for those full discretizations which are obtained by applying to a space discretization of the equations an energy conservative discrete time marching method the energy reflected at the boundary is independent of the value of t and is strictly equal to the energy reflected in the semidiscrete case this is verified in numerical experiments optimal boundary equations may be defined in the semidiscrete case of those which maximize the rate of convergence to zero of the reflected energy when h 0 a corollary of the preceding result is that those boundary equations remain optimal in the same sense when used in an energy conservative full discretization moreover this convergence result continues to hold when a nonconservative but stable i e dissipative time discretization is used this is verified numerically with the first order adams predictor corrector method the results of this paper are derived with the mathematics of the simple three point central difference discretization of spatial derivatives obvious generalizations to other cases are mentioned at the end of the paper

on discontinuous solutions of hyperbolic equations abstract a modified fourier method which is applicable to numerical simulations of discontinuous solutions of hyperbolic equations is presented truncated fourier series representations are de truncated at every timestep by utilising step functions in the reconstruction of the discontinuous solutions the de truncated expansions are used to eliminate the numerical dispersion and diffusion which normally ruin the discontinuous solutions in a few timesteps when traditional fourier methods are applied

absorbing boundary conditions for free surface waves in this paper the use of absorbing boundary conditions is investigated for the numerical simulation of gravity waves on an incompressible inviscid fluid in three dimensions a review of existing methods is given for linear and nonlinear waves after which first and second order partial differential equations are introduced as absorbing boundary conditions for the linearized model well posedness is investigated and it is shown that the reflection properties of the second order equation are superior to those of the first order equation

acceleration of algebraically converging fourier series when the coefficients have series in powers of 1 n if the coefficients in a fourier cosine series f x f email protected n 0 a ncos nx decrease as a small negative power of n then one may need millions of terms to sum the series to high accuracy we show that if the a n are known analytically and have a power series in 1 n then it is straightforward to approximate f x as a series of what we shall the lanczos krylov lk functions we describe the similar methodology for sine series general fourier series are merely the sum of a cosine series with a sine series and thus are implicitly handled too for cosine coefficients that involve only even powers of n and sine coefficients that are functions of odd powers of n the lk functions may be expressed in terms of bernoulli polynomials the lk functions for cosine coefficients involving odd powers of n and for sine coefficients in even powers of n are not known explicitly these are also known as clausen functions we provide rapidly convergent series to compute these clausen functions to high accuracy our method includes the endpoint subtraction ideas of lanczos and krylov but is more general the sum n 1 1 n 1 1 email protected cos nx where l 0 is a constant arises in phase transitions in absorbed monolayers on metal surfaces it is easily summed by our method which correctly incorporates the logarithmic singularities at x email protected

analysis and application of fourier gegenbauer method to stiff differential equations the fourier gegenbauer fg method introduced by gottlieb shu solomonoff and vandeven icase report 92 4 hampton va 1992 is aimed at removing the gibbs phenomenon that is recovering the point values of a nonperiodic function from its fourier coefficients in this paper we discuss some numerical aspects of the fg method related to its em pseudospectral implementation in particular we analyze the behavior of the gegenbauer series with a moderate several hundred number of terms suitable for computations we also demonstrate the ability of the fg method to get a spectrally accurate approximation on small subintervals for rapidly oscillating functions or functions having steep profiles bearing on the previous analysis we suggest a high order spectral fourier method for the solution of nonperiodic differential equations it includes a polynomial subtraction technique to accelerate the convergence of the fourier series and the fg algorithm to evaluate derivatives on the boundaries of nonperiodic functions the present hybrid fourier gegenbauer hfg method possesses better resolution properties than the original fg method the precision of this method is demonstrated by solving stiff elliptic problems with steep solutions

accurate reconstructions of functions of finite regularity from truncated fourier series expansions kowledge of a truncated fourier series expansion for a 2 periodic function of finite regularity which is assumed to be piecewise smooth in each period is used to accurately reconstruct the corresponding function an algebraic equation of degree m is constructed for the m singularity locations in each period for the function in question the m coefficients in this algebraic equation are obtained by solving an algebraic system of m equations determined by the coefficients in the known truncated expansion if discontinuities in the derivatives of the function are considered in addition to discontinuities in the function itself that algebraic system will be nonlinear with respect to the m unknown coefficients the degree of the algebraic system will depend on the desired order of accuracy for the reconstruction i e a higher degree will normally lead to a more accurate determination of the singularity locations by solving an additional linear algebraic system for the jumps of the function and its derivatives up to the arbitrarily specified order at the calculated singularity locations we are able to reconstruct the 2 periodic function of finite regularity as the sum of a piecewise polynomial function and a function which is continuously differentiab1e up to the specified order

a visually weighted quantization scheme for image bandwidth compression at low data rates source encoding of images for bandwidth compression has become attractive in recent years because of decreasing hardware costs by combining the source encoding approach with transform coding techniques it is possible to obtain good image quality at low data rates the general aspects of such a system are presented the design of the quantizer for transform coefficients which is the major source of error associated with the compression process is considered using a visual fidelity criterion and subject to the constraint that the entropy of the quantizer be a prespecified quantity a visually weighted suboptimal quantization scheme is developed to take into account the relative importance of different transform coefficients to the human visual system

the calculation of trigonometric fourier coefficients a method for the numerical evaluation of trigonometric fourier coefficients is described this method is suitable for sufficiently smooth functions and is essentially a modification of the standard method based on trapezoidal rule sums using optionally the fast fourier transform

numerical comparisons of nonlinear convergence accelerators as part of a continuing program of numerical tests of convergence accelerators we have compared the iterated aitken s a2 method wynn s algorithm brezinski s 0 algorithm and levin s u transform on a broad range of test problems linearly convergence alternating monotone and irregular sign series logarithmically convergent series power method and bernoulli method sequences alternating and monotone asymptotic series and some perturba tion series arising in applications in each category either the algorithm or the u transform gives the best results of the four methods tested in some cases differences among methods are slight and in others they are quite striking 1 introduction we reported in 19 on a numerical test program that compared eleven different scalar convergence accelerators on a broad range of infinite series problems in an attempt to find an across the board accelerator that would handle both linear and logarithmic convergence and both alternating and monotone series or sequences the best candidates to emerge from that study were levin s u transform 14 and brezinski s 0 algorithm 6 we also provided some theoretical support for that conclusion in terms of the wide range of series summed exactly by these methods and by proving that 02 and the levin transforms all accelerate linear convergence the other levin transforms t and v produced essentially equivalent numerical results to u except that t fails on logarithmic convergence we have continued the test program started with 19 to include monotone linearly convergent series series with irregular sign patterns specifically fourier series and sequences whose terms do not depend explicitly on the previous term and or the index e g bernoulli s method since convergence accelerators are often known to transform divergent series to convergent ones we have also included in our test program divergent power series asymptotic series and a few perturbation series the methods considered in this paper are the u and 0 algorithms noted above and

high order numerical sommerfeld boundary conditions theory and experiments abstract we develop high order non reflecting boundary equations for a semidiscrete approximation of the simple hyperbolic advection equation u 1 cu x 0 these boundary equation are based on a discrete interpretation of sommerfeld s radiation condition for a second order wave equation which is associated with the semi discrete equation the performance of these schemes is expressed by an exact measure of the energy reflected at the boundary for low order cases the discrete sommerfeld boundary equations are identical with the standard finite difference equations but for high orders of approximation starting with 4 points the discrete sommerfeld schemes differ from standard finite differences it is shown and verified experimentally that the discrete sommerfeld schemes are optimal in the sense that they produce the least amount of reflected energy moreover it is known theoretically and we verify experimentally that the reflected energy remains invariat when the semi discrete equation are time discretized with the trapezoidal crank nicolson method the corresponding fully discrete boundary equations are thus also optimal in the sense that they minimize the reflected energy

absorbing boundary conditions for the discretization schemes of the one dimensional wave equation when computing a partial differential equation it is often necessary to introduce artificial boundaries here we explain a systematic method to obtain boundary conditions for the wave equation in one dimension fitting to the discretization scheme and stable moreover we give error estimates on the reflected part

detecting the singularities of a function of vp class by its integrated fourier series in the present paper we pursue the general idea suggested in our previous work namely we utilize the truncated fourier series as a tool for the approximation of the points of discontinuities and the magnitudes of jumps of a 2 periodic bounded function earlier we used the derivative of the partial sums while in this work we use integrals first we obtain new identities which determine the jumps of a 2 periodic function of vp 1 p 2 class with a finite number of discontinuities by means of the tails of its integrated fourier series next based on these identities we establish asymptotic expansions for the approximations of the location of the discontinuity and the magnitude of the jump of a 2 periodic piecewise smooth function with one singularity by an appropriate linear combination obtained via integrals of different order we significantly improve the accuracy of the initial approximations then we apply richardson s extrapolation method to enhance the approximation results for a function with multiple discontinuities we use simple formulae which eliminate all discontinuities of the function but one then we treat the function as if it had one singularity finally we give the description of a programmable algorithm for the approximation of the discontinuities investigate the stability of the method study its complexity and present some numerical results

the influence of the mass matrix on the dispersive nature of the semi discrete second order wave equation abstract the application of discrete solution methods to the second order wave equation can yield a dispersive representation of the non dispersive wave propagation problem resulting in a phase speed that depends not only upon the wavelength of the signal being propagated but also upon the direction of propagation in this work the dependence of the dispersive errors on the wave propagation direction mesh aspect ratio and wave number is investigated with the goal of understanding and hopefully reducing the phase and group errors associated with the two dimensional bilinear finite element an analysis of the dispersive effects associated with the consistent row sum lumped and higher order mass matrices has led to a reduced coupling penta diagonal mass matrix that yields improved phase and group errors with respect to wavelength and propagation direction the influence of row sum lumping the finite element mass matrix is demonstrated to always introduce lagging phase and group error while a linear combination of the lumped and consistent mass matrices i e a higher order mass matrix is shown to improve the dispersion characteristics of both the reduced and full integration element using a 2 5 phase error metric the higher order mass matrix can reduce the mesh resolution requirements relative to the lumped mass matrix by nearly a factor of two in one dimension a factor of 4 in two dimensions similarly the reduced coupling mass matrix can reduce the grid requirements by a factor of approximately four relative to the lumped mass matrix a factor of 16 in two dimensions although the reduced coupling mass matrix requires additional storage relative to the lumped mass the factor of 16 reduction in grid requirements promises tremendous overall savings for computing a fixed wavelength spectrum or alternatively for resolving shorter wavelengths with a fixed grid resolution

spectrally accurate solution of nonperiodic differential equations by the fourier gegenbauer method it is well known that the fourier partial sum of an analytic nonperiodic function supported on a finite interval converges slowly inside the interval and exhibits o 1 spurious oscillations near the boundaries the gibbs phenomenon an effective algorithm which allows one to completely overcome the gibbs phenomenon was developed in j comput appl math 43 1992 pp 81 92 the basic concept of this approach consists of the re expansion of the fourier partial sums into the rapidly convergent gegenbauer series in this paper we extend the fourier gegenbauer f g method of j comput appl math 43 1992 pp 81 92 to the evaluation of the spatial derivatives of a piecewise analytic function also we apply this method to the solution of nonperiodic boundary value problems although the derivatives of a discontinuous function are not in l2 the exponential convergence of the truncated gegenbauer series can be proved and the rate of convergence can be estimated the solution of differential equations is accomplished in two steps first a particular solution with arbitrary boundary conditions is constructed using the f g method this particular solution is then corrected to satisfy the prescribed boundary conditions of the problem by adding a proper linear combination of homogeneous solutions for boundary layer problems the intermediate particular solution has steep profiles near the boundaries these steep regions introduce a large error into the final solution which presumably has a smooth profile on the whole interval a method which compensates for this loss of accuracy by using the appropriately constructed homogeneous solutions is proposed

computation of spherical harmonic expansion coefficients via fft s abstract a method for numerically expanding an arbitrary function on the sphere in a series of spherical harmonics which makes use of the speed of a fast fourier transform is described discussions of the operation count storage requirements accuracy and an algebraic and a numerical example are included a comparison with straightforward integration is made throughout also a new method for evaluating the spherical harmonics is discussed

local spline approximation of discontinuous functions and location of discontinuities given low order fourier coefficient information in recent years several workers have published methods for accurately approximating a function with discontinuities given fourier coefficient information our own contribution concentrates on robustness and use of low order coefficients in this paper we present a version of our method which is local in that it approximates the orthogonal projection of the function onto a space of splines which in a period interval is supported on a proper subinterval the approximation is only slightly influenced by behavior of the function outside the subinterval in particular discontinuity locations need not be known outside the subinterval we establish modest bounds for the approximation operator which proves robustness in the presence of noise one advantage of this local method is that compared with the earlier nonlocal version it can be less expensive the cost is related to the dimension of the space of splines and this dimension can be small if the subinterval is small another advantage is that this version can be used to locate discontinuities we choose the support of the spline space so that only one discontinuity in a period interval can lie in the support we then shift the support systematically and monitor quantities related to the approximation as functions of the shift the spline space and monitored quantities are chosen so that easily identified values indicate the location of the discontinuity

a robust method for accurately representing nonperiodic functions given fourier coefficient information when a function is smooth but not smoothly periodic with a particular period and nonetheless is represented by partial sums of a fourier series calculated using that period the well known gibbs phenomenon defeats uniform convergence of the sums and convergence is slow in recent years several workers have developed methods for recovering accurate and fast converging representations for functions in this situation these efforts have not concentrated on bounds for the operators corresponding to the methods and thus have not explicitly proven robustness in the presence of noise in this paper we present a method for which explicit bounds are established for the operator the method is in effect least squares fitting of the given fourier coefficients by the coefficients of polynomial splines with appropriate discontinuities we obtain bounds by exact calculations of projections in spline spaces using a computer algebra system we give examples of the method and two other published methods working with noisy data

convergence acceleration of fourier series by analytical and numerical application of poisson s formula a form of the poisson summation formula appropriate for the transformation of infinite fourier series is derived and shown to be applicable to slowly converging series arising in the solution of laplace s equation in a rectangular electrochemical cell the result is a rapidly convergent fourier series which is particularly useful in giving the potential close to the electrode surface results obtained from the analytical expression are found to be in close agreement with those resulting from numerical evaluation of the fourier integrals using simpson s rule and the euler transformation and direct summation by using goertzel s 1958 algorithm

generalization of the inverse polynomial reconstruction method in the resolution of the gibbs phenomenon the finite fourier representation of a function f x exhibits oscillations where the function or its derivatives are nonsmooth this is known as the gibbs phenomenon a robust and accurate reconstruction method that resolves the gibbs oscillations was proposed in a previous paper j comput appl math 161 2003 41 based on the inversion of the transformation matrix which represents the projection of a set of basis functions onto the fourier space if the function is a polynomial this inverse polynomial reconstruction method iprm is exact in this paper we develop the iprm by requiring that the proper error be orthogonal to the fourier or polynomial space the iprm is generalized to any set of basis functions the primitive basis polynomials nonclassical orthogonal polynomials and the gegenbauer polynomials are used to illustrate the wide validity of the iprm it is shown that the iprm yields a unique reconstruction irrespective of the basis set for any analytic function and yields spectral convergence the ill posedness of the transformation matrix due to the exponential growth of the condition number of the matrix is also discussed

one and two dimensional lattice sums for the three dimensional helmholtz equation the accurate and efficient computation of lattice sums for the three dimensional helmholtz equation is considered for the cases where the underlying lattice is one or two dimensional we demonstrate using careful numerical computations that the reduction method in which the sums for a two dimensional lattice are expressed as a sum of one dimensional lattice sums leads to an order of magnitude improvement in performance over the well known ewald method in the process we clarify and improve on a number of results originally formulated by twersky in the 1970s

adaptive image coding with perceptual distortion control this paper presents a discrete cosine transform dct based locally adaptive perceptual image coder which discriminates between image components based on their perceptual relevance for achieving increased performance in terms of quality and bit rate the new coder uses a locally adaptive perceptual quantization scheme based on a tractable perceptual distortion metric our strategy is to exploit human visual masking properties by deriving visual masking thresholds in a locally adaptive fashion the derived masking thresholds are used in controlling the quantization stage by adapting the quantizer reconstruction levels in order to meet the desired target perceptual distortion the proposed coding scheme is flexible in that it can be easily extended to work with any subband based decomposition in addition to block based transform methods compared to existing perceptual coding methods the proposed perceptual coding method exhibits superior performance in terms of bit rate and distortion control coding results are presented to illustrate the performance of the presented coding scheme

optimal filter and mollifier for piecewise smooth spectral data we discuss the reconstruction of piecewise smooth data from its pseudo spectral information spectral projections enjoy superior resolution provided the function is globally smooth while the presence of jump discontinuities is responsible for spurious o 1 gibbs oscillations in the neighborhood of edges and an overall deterioration of the convergence rate to the unacceptable first order classical filters and mollifiers are constructed to have compact support in the fourier frequency and physical time spaces respectively and are dilated by the projection order or the width of the smooth region to maintain this compact support in the appropriate region here we construct a noncompactly supported filter and mollifier with optimal joint time frequency localization for a given number of vanishing moments resulting in a new fundamental dilation relationship that adaptively links the time and frequency domains not giving preference to either space allows for a more balanced error decomposition which when minimized yields an optimal filter and mollifier that retain the robustness of classical filters yet obtain true exponential accuracy