a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

the pseudospectral method for solving differential eigenvalue problems abstract a simple pseudospectral method is presented for the numerical solution of linear differential eigenvalue problems the method does not produce the spurious eigenvalues which generally occur when such problems are solved by the spectral tau method the pseudospectral method is presented using two model problems and the presentation contains a useful algorithm for the computation of the spectral differentiation matrices at general collocation points numerical results are offered for a set of benchmark problems including the orr sommerfeld equation for stability of plane poiseuille flow between parallel plates the results indicate that the new pseudospectral method is comparable in accuracy to the tau method

blur identification by the method of generalized cross validation the point spread function psf of a blurred image is often unknown a priori the blur must first be identified from the degraded image data before restoring the image generalized cross validation gcv is introduced to address the blur identification problem the gcv criterion identifies model parameters for the blur the image and the regularization parameter providing all the information necessary to restore the image experiments are presented which show that gvc is capable of yielding good identification results a comparison of the gcv criterion with maximum likelihood ml estimation shows the gcv often outperforms ml in identifying the blur and image model parameters

regularized iterative image restoration with ringing reduction linear space invariant image restoration algorithms often introduce ringing effects near sharp intensity transitions it is shown that these artifacts are attributable to the regularization of the ill posed image restoration problem two possible methods to reduce the ringing effects in restored images are proposed the first method incorporates deterministic a priori knowledge about the original image into the restoration algorithm the second method locally regulates the severity of the noise magnification and the ringing phenomenon depending on the edge information in the image a regularized iterative image restoration algorithm is proposed in which both ringing reduction methods are included by making use of the theory of the projections onto convex sets and the concept of norms in a weighted hilbert space both the numerical performance and the visual evaluation of the results are improved by the use of ringing reduction

new pocs algorithms for regularization of inverse problems abstract ill posed problems described by first kind fredholm equations appear in many interesting practical cases in engineering or mathematical physics such as deconvolution and require regularization techniques to get adequate solutions this paper presents under a projection operators onto convex sets pocs framework to introduce the needed regularization operations a series of non adaptive and adaptive pocs regularization algorithms which offer as main advantages besides including previously proposed methods a lower computational load and the possibility of including any kind of constraints linear and nonlinear a series of simulation examples serve to appreciate the high performance of the proposed techniques in a typical problem recovering the discontinuites of a sequence showing sharp edges

on certain zeta functions attached to two hilbert modular forms ii the case of automorphic forms on a quaternion algebra as to the meaning of the symbols the reader is referred to the introduction of part i we recall here only that f is a totally real algebraic number field of degree n and w denotes the fourier coefficients of an elliptic modular form 52 z e wq a e2q iaz in part i we investigated d assuming that ex a n q s is an l function of a cm field with an algebraic valued hecke character in the present part ii we treat the case where x a can be obtained as eigenvalues of hecke operators t a on a quaternion algebra b over f thus we begin our study by recalling the theory of automorphic forms on the product o r of r copies of the upper half plane with respect to congruence subgroups of b x here r is the number of archimedean primes of f unramified in b then we introduce in section 2 the notion of arithmeticity or q rationality of such automorphic forms and show that the space of automorphic forms can be spanned by the arithmetic ones our main theorems will be stated in section 3 and proved in section 6 sections 4 and 5 are preliminaries to the proof the central result specialized to the case r 1 asserts that if g is a q rational eigenform such that gj t a x a g for all integral ideals a of f then d s0 for certain integers so are algebraic numbers times 7tdg g where d is an integer determined by the weight of g and is the petersson inner product

on spherical convergence convexity and block iterative projection algorithms in hilbert space abstract we call a sequence x n in hilbert space spherical if there exists u such that lim x n u exists and is finite if moreover u is a weak accumulation point of the sequence we call the sequence spherically convergent we demonstrate that for large classes of nonexpansive possibly nonstationary discrete time processes in hilbert space the iterates are spherically convergent basic identities and orthogonality relations pertinent to this type of convergence are exhibited sufficient conditions for weak and spherical convergence in terms of the fullness of the set of fixed points of the process are established and compared spherical convergence of the general block iterative projection scheme in hilbert space is established

on mod p properties of siegel modular forms we prove two results on mod p properties of siegel modular forms first we use theta series in order to construct of a siegel modular form of weight p 1 which is congruent to 1 mod p second we define a theta operator vartheta on q expansions and show that the algebra of siegel modular forms mod p is stable under vartheta by exploiting the relation between vartheta and generalized rankin cohen brackets

spatially adaptive wavelet based multiscale image restoration in this paper we present a new spatially adaptive approach to the restoration of noisy blurred images which is particularly effective at producing sharp deconvolution while suppressing the noise in the flat regions of an image this is accomplished through a multiscale kalman smoothing filter applied to a prefiltered observed image in the discrete separable 2 d wavelet domain the prefiltering step involves constrained least squares filtering based on optimal choices for the regularization parameter this leads to a reduction in the support of the required state vectors of the multiscale restoration filter in the wavelet domain and improvement in the computational efficiency of the multiscale filter the proposed method has the benefit that the majority of the regularization or noise suppression of the restoration is accomplished by the efficient multiscale filtering of wavelet detail coefficients ordered on quadtrees not only does this lead to potential parallel implementation schemes but it permits adaptivity to the local edge information in the image in particular this method changes filter parameters depending on scale local signal to noise ratio snr and orientation because the wavelet detail coefficients are a manifestation of the multiscale edge information in an image this algorithm may be viewed as an edge adaptive multiscale restoration approach

arithmetic properties of fourier jacobi expansions of automorphic forms in several variables in this paper we establish some basic arithmetic results concerning automorphic forms in several variables our results are in part analogues of the classical facts for elliptic modular forms and results of shimura on hilbert and siegel modular forms 9 and 10 in essence this involves relating rationality properties of fourier jacobi coefficients to rationality properties in algebraic geometry our automorphic forms are c valued holomorphic functions on certain cartesian powers hnd of siegel upper half spaces hn we consider appropriate arithmetic groups r acting on hnd such that r h d is noncompact in particular we reduce our study to that of congruence subgroups r of reductive linear algebraic groups g such that the derived group g is almost simple over q further let k be a maximal compact subgroup of g r the connected component of the identity in the real

a modified tau spectral method that eliminated spurious eigenvalues abstract a modified tau spectral method is presented which eliminates the spurious eigenvalues produced by the usual tau method the modified tau method essentially involves an appropriate factorization of the differential operators in the eigenvalue problem it is developed for eigenvalue problems posed as single differential equations or as systems of such equations several eigenvalue problems are solved using both the usual and modified chebyshev tau methods including the orr sommerfeld stability equation for plane poiseuille flow the convergence of the modified tau method is shown to be at least as rapid as that of the usual tau method the use of the tau coefficients in indicating convergence is also discussed

resolution enhancement of tomographic images using the row action projection method the row action projection rap method is used to increase the spatial resolution of images reconstructed via the filtered back projection fbp algorithm an implementation of rap is introduced which is computationally efficient and facilitates local adaptation of the projection operators the local mean value as well as minimum and maximum bounds are used as constraints the method is proposed to provide zoom in capability which yields a high resolution estimate of a specified region of the image computer simulations demonstrate the new method to be very effective in recovering high order spectral components of designated regions of the reconstructed image

the feasible solution in signal restoration the feasible solution to the signal restoration problem is defined as the one which satisfies all constraints which can be imposed on the true solution a very important set of constraints can be obtained by examining the statistics of the noise these and other constraints can be described as closed convex sets thus projection onto closed convex sets is the numerical method used to obtain a feasible solution examples of this method demonstrate its usefulness in one and two dimensional signal restoration the limitations of the method are discussed

image restoration by convex projections using adaptive constraints and the l sub 1 norm a new projection method based on the row action projection rap algorithm for image restoration is presented the new implementation is computationally attractive and facilitates local adaption of projection operators the local mean and the least l sub 1 norm set of solutions are used as constraints computer simulations illustrate the new methods to be very competitive in restoring missing spectral components of a degraded image

the adelic zeta function associated with the space of binary cubic forms with coefficients in a function field in this paper we study the adelic zeta function associated with the prehomogeneous vector space of binary cubic forms defined over a function field we establish its rationality find its poles and residues and a simple functional equation that this zeta function satisfies introduction the subject of integral equivalence of binary cubic forms although not as popular or well developed as the theory of integral equivalence of binary quadratic forms has received a fair amount of attention the first paper on this topic 9 written by eisenstein appeared in 1844 later on several other first rate mathematicians tried their hand at the subject among others the list includes arndt cayley hermite and davenport the classical method of attacking this problem is to draw upon a one to one discriminant preserving relation between binary cubic and binary quadratic forms which assigns to a cubic form its hessian and then use the reduction theory for binary quadratic forms to obtain results on class numbers of binary cubic forms the best of the classical results are due to davenport 5 who showed that e h ir n 2 36x o x15 16 nax e h ir n v2 12x o x 5116 nax where h ir n is the class number of irreducible binary cubic forms of discriminant n the credit for a new surge of interest in the study of integral equivalence of binary cubic forms must indisputably go to shintani for it was his 1972 paper 25 that put this study on modern footing in that paper shintani introduced a certain zeta function which as it turns out decomposes into a linear combination of products of dirichlet series whose coefficients are weighted class numbers of binary cubic forms and certain mellin transforms by studying analytic properties of the zeta function shintani was able to improve the error term in davenport s result to o x576 it is of interest to note that the zeta function and the dirichlet series it received by the editors september 3 1985 1980 mathematics subject classification 1985 revision primary 12a70 secondary iocio 12a90 1987 american mathematical society 0002 9947 87 1 00 25 per page

prototype image constraints for set theoretic image restoration constraints based on prototype images are developed and used in set theoretic image restoration a prototype can be obtained as a result of applying a predetermined operator to the observed image in this case the operator and the bound which limits the variation of the restored image from the prototype are the two defining quantities of a prototype constraint general guidelines for rigorously estimating the defining bound of a prototype constraint under certain simplifying conditions are discussed the authors provide two examples of prototype constraints where the prototypes are obtained by the wiener filtering operator and a local averaging operator the projection onto convex sets algorithm using the prototype constraints is applied to both monochrome and color images degraded by out of focus blur at different noise levels the results show significant improvement over the wiener restoration in reducing the restoration artifacts

set theoretic signal restoration using an error in variables criterion the restoration of a signal degraded by a stochastic impulse response is formulated as a problem with uncertainties in both the measurements and the impulse response the method of total least squares and variants thereof are effective techniques for solving this class of problems however unlike set theoretic estimation schemes these methods do not allow the incorporation of other a priori information in the estimate in this correspondence two new sets motivated by total least squares are introduced for set theoretic estimation the convexity of these sets is established and the projection operators onto these sets are given through simulations the advantages of the new technique over conventional and older set theoretic schemes for restoration are demonstrated

the origin and nature of spurious eigenvalues in the spectral tau method the chebyshev tau spectral method for approximating eigenvalues of boundary value problems may produce spurious eigenvalues with large positive real parts even when all true eigenvalues of the problem are known to have negative real parts we explain the origin and nature of the spurious eigenvalues in an example problem the explanation will demonstrate that the large positive eigenvalues are an approximation of infinite eigenvalues in a nearby generalized eigenvalue problem

correction to poles and residues of standard l functions attached to siegel modular forms correction to my paper on the poles of standard l functions attached to siegel modular forms

estimation of the parameters of a multipath channel using set theoretic deconvolution a method for the estimation of the parameters of multipath communication channels is developed the method makes use of matched filtering set theoretic deconvolution and autoregressive modeling pulse compression waveforms are used as channel probing signals the results of a feasibility study in which the proposed method is applied to specular multipath channels to estimate the multipath parameters at a number of different signal to noise ratios snr are presented

on the arithmetic of siegel hilbert cuspforms petersson inner products and fourier coefficients the latter is a sort of l indistinguishability result concerning the presumably transcendental petersson norms squared of hecke eigenfunctions the general assertion is essential in discussion of special values of l functions obtained as inner products incidental to the proof of this fact we obtain a very short proof of the important known result that the space of holomorphic cuspforms of such weight and with respect to a principal congruence subyroup is spanned by those with rational fourier coefficients for weights as above rather than starting from the theory of canonical models we begin with consideration of the arithmetic of the fourier coefficients of siegel s eisenstein series this is a relatively elementary issue amenable to direct calculation although ironically the best reference currently available seems to be h3 wherein canonical models results are invoked by contrast previous proofs of results concerning rationality properties of fourier coefficients have relied essentially upon the theory of canonical models the paradigms are the two papers sh2 and sh3 of shimura which depend upon the sequence of his papers culminating in shl which developed the necessary theory of canonical models

theta dichotomy for unitary groups some recent work of gross and prasad 14 suggests that the root numbers attached to certain symplectic representations of the weil deligne group of a local field f control certain branching rules for representations of orthogonal groups over f on a global level this local phenomenon should have implications for the structure of the value at the center of symmetry for certain l functions of arithmetic interest 12 this conjectural picture is based to some extent on the now classic work of tunnell 47 and waldspurger 49 as well as on the case of the triple product l function 13 34 17 in all of these examples the local root number detects the existence of a certain type of invariant linear functional it is possible to set up analogous conjectures for the isometry groups of hermitian or quaternion hermitian forms it turns out that root numbers also play a role in the local and global theta correspondence roughly speaking certain local root numbers should control the occurrence of representations in the local theta correspondence between groups of the same size e g for dual pairs of the form sp n o 2n 1 and u n u n as will be seen the theta correspondence for such pairs is connected with a certain induced representation in s at the point s 0 on the unitary axis by contrast the correspondence for the pairs sp n o 2n and sp n o 2n 2 discussed by prasad 35 is connected to the behavior of a similar induced representation at the points 12 in which case no epsilon factor arises in this paper we consider the local theta correspondence for unitary groups in the non archimedean case let f be a non archimedean local field of characteristic not equal to 2 and let e be a quadratic extension of f for further notation see the notation section at the end of this introduction let v and w be e vector spaces of dimensions m and n equipped respectively with a hermitian form v v e and a skew hermitian form w w e then the isometry groups g v and g w of the spaces v and w form a dual reductive pair in the symplectic group sp w where w v e w viewed as an f vector space of dimension 2mn and equipped with the symplectic form

zeta functions and eisenstein series on classical groups we construct an euler product from the hecke eigenvalues of an automorphic form on a classical group and prove its analytic continuation to the whole complex plane when the group is a unitary group over a cm field and the eigenform is holomorphic we also prove analytic continuation of an eisenstein series on another unitary group containing the group just mentioned defined with such an eigenform as an application of our methods we prove an explicit class number formula for a totally definite hermitian form over a cm field

the selberg trace formula for sl 3 ℤ sl 3 ℝ so 3 ℝ in this paper we compute the trace formula for sl 3 z in detail and refine it to a greater extent than has previously been done we show that massive cancellation occurs in the parabolic terms leading to a far simpler formula than had been thought possible

an adaptive regularized method for deconvolution of signals with edges by convex projections a new adaptive deconvolution method based on the projection operators onto convex sets pocs is presented a minimum norm least squares mnls is obtained for signals with edges by means of an estimation detection protection scheme the regularized differentiation technique is necessary for a reasonable detection of the signal edges the improvement introduced with this method is illustrated through a simulation example finally a discussion of the wide series of possibilities open along these lines closes this article

derivative of the standard p adic l function associated with a siegel form this work has greatly bene ted from an excellent long stay at columbia university founded by a fwo travel grant v4 260 14n while the author was a fwo phd fellow at ku leuven

zeta functions and eisenstein series on metaplectic groups the main objectives of this paper are 1 to construct an euler product from a hecke eigenform of half integral weight 2 to prove its meromorphic continuation to the whole plane 3 to examine its relationship with the fourier coefficients of the form and 4 to apply these results to the investigations of eisenstein series of half integral weight the automorphic forms are holomorphic ones considered on the metaplectic cover m of g where g n sp n f with a totally real algebraic number field f the first three are generalizations of the corresponding results in the elliptic and hilbert modular cases obtained in 2 and 6 at the same time all four closely parallel similar results in the case of integral weight in 9 and s10 to describe our results let us take for simplicity f to be q in which case the weight k is half an odd positive integer we consider a holomorphic form f of weight k with respect to a principal congruence subgroup of sp n z of level n which is an eigenform in the sense that f it m 2 m f with a complex number 2 m where n is a multiple of 4 and t m is a hecke operator defined for each positive integer m in a natural way as our first main result we shall prove theorem 4 4 that

capacitance matrix technique for avoiding spurious eigenmodes in the solution of hydrodynamic stability problems by chebyshev collocation method we present a simple technique for avoiding physically spurious eigenmodes that often occur in the solution of hydrodynamic stability problems by the chebyshev collocation method the method is demonstrated on the solution of the orr sommerfeld equation for plane poiseuille flow following the standard approach the original fourth order differential equation is factorised into two second order equations using a vorticity type auxiliary variable with unknown boundary values which are then eliminated by a capacitance matrix approach however the elimination is constrained by the conservation of the structure of matrix eigenvalue problem it can be done in two basically different ways a straightforward application of the method results in a couple of physically spurious eigenvalues which are either huge or close to zero depending on the way the vorticity boundary conditions are eliminated the zero eigenvalues can be shifted to any prescribed value and thus removed by a slight modification of the second approach

elimination of spurious eigenvalues in the chebyshev tau spectral method spectral methods have been used to great advantage in hydrodynamic stability calculations the concepts are described in orszag s seminal application of the chebyshev tau method to the orr sommerfeld equation for plane poiseuille flow in 1971 orszag discusses both the chebyshev galerkin and the chebyshev tau methods but presents results for the tau method which is easier to implement than the galerkin method the tau method has the disadvantage that two unstable eigenvalues are produced that are artifacts of the discretization an extremely simple modification to the chebyshev tau method is presented which eliminates the spurious eigenvalues first a simplified model of the orr sommerfeld equation discussed by gottlieb and orszag was studied then the chebyshev tau method is considered which has two spurious eigenvalues and then a modification which eliminates them is described finally results for the orr sommerfeld equation are considered where the modified tau method also eliminates the spurious eigenvalues the simplicity of the modification makes it a convenient alternative to other approaches to the problem

on the bloch kato conjecture for elliptic modular forms of square free level let kappa ge 6 be an even integer m an odd square free integer and f in s 2 kappa 2 gamma 0 m a newform we prove that under some reasonable assumptions that half of the lambda part of the bloch kato conjecture for the near central critical value l kappa f is true we do this by bounding the ell valuation of the order of the appropriate bloch kato selmer group below by the ell valuation of algebraic part of l kappa f we prove this by constructing a congruence between the saito kurokawa lift of f and a cuspidal siegel modular form

on certain zeta functions attached to two hilbert modular forms i the case of hecke characters to make our exposition smooth let us first introduce some notational conventions for an algebraic number field k of finite degree we denote by jk the set of all embeddings of k into c and by ik the free z module generated by the elements of jk we then put rik ik0 r and cik ik0 c if p eapau e ik with a e jk and po e z we put xp fl xa pfor 0 x e k this is meaningful for p e cik if xa are all real and positive throughout the paper we denote by d the complex upper half plane and by f a totally real algebraic number field of degree n now the zeta function to be studied in this paper when suitably specialized has the form

confluent hypergeometric functions on tube domains our problem specialized to the case of the siegel upper half space h of degree m concerns the fourier expansion of a series s z fl det z a det a a l here z is a variable on h l is a lattice in the space v of all real symmetric matrices of size m and fl ec it can be shown that this is convergent if re f l m and also that if re m 2 and r e f l m 2 it has a fourier expansion of the form

incorporation of a priori moment information into signal recovery and synthesis problems abstract the problem addressed is that of synthesizing a function subject to a priori moment constraints each moment constraint requires that the function to be synthesized lies in a closed convex set for r moment constraints the function must lie at the intersection of r convex sets the method of projections onto convex sets pocs is used to synthesize a function consistent with the moment constraints the algorithm has application in medical imaging especially in computer aided tomography

adaptive image restoration with artifact suppression using the theory of convex projections the authors develop an iterative adaptive space variant restoration incorporating both regularization and artifact suppression constraints the algorithm is formulated using the method of projections onto convex sets pocs they introduce a closed convex regularization constraint set called the partial wiener solution set projection onto this set forces the solution to be equal to the wiener solution over a predetermined set of frequencies that lie outside the neighborhoods of the zeros of the degradation transfer function in the neighborhoods of the zeros the wiener solution contains significant errors that contribute to artifacts hence in these neighborhoods the wiener solution is discarded and the missing frequency components are determined so that the entire solution is consistent with the artifact suppression constraints and other a priori information the proposed artifact suppression constraints bound the norm of the variation of the image from its local mean using regionally adaptive bounds the high quality of the resulting restorations is noteworthy

removal of spurious modes encountered in solving stability problems by spectral methods a technique based on the galerkin approximation is developed to remove spurious roots arising when chebyshev spectral methods are used to solve eigenvalue problems in hydrodynamic stability the derivation of galerkin chebyshev approximations is explained and numerical results for the orr sommerfeld equations of plane poiseuille flow and a blasius profile are presented in tables and compared with those obtained by the method of zebib 1984 it is pointed out that the present method does not increase the size of the algebraic system to be solved

concealment of damaged block transform coded images using projections onto convex sets an algorithm for lost signal restoration in block based still image and video sequence coding is presented problems arising from imperfect transmission of block coded images result in lost blocks the resulting image is flawed by the absence of square pixel regions that are notably perceived by human vision even in real time video sequences error concealment is aimed at masking the effect of missing blocks by use of temporal or spatial interpolation to create a subjectively acceptable approximation to the true error free image this paper presents a spatial interpolation algorithm that addresses concealment of lost image blocks using only intra frame information it attempts to utilize spatially correlated edge information from a large local neighborhood of surrounding pixels to restore missing blocks the algorithm is a gerchberg 1974 type spatial domain spectral domain constraint satisfying iterative process and may be viewed as an alternating projections onto convex sets method

set theoretic estimation by random search an adapted random search algorithm is shown to be a feasible method for the synthesis of set theoretic estimates it circumvents the theoretical and computational shortcomings of existing deterministic methods and does not place any geometrical restrictions on the sets the proposed method can handle arbitrarily complex property sets in applications for which the number of unknown parameters is typically low e g parametric multidimensional spectral estimation system identification blur identification and filter design