sampling and reconstruction of signals in a reproducing kernel subspace of lp rd in this paper we consider sampling and reconstruction of signals in a reproducing kernel subspace of lp rd 1 p associated with an idempotent integral operator whose kernel has certain off diagonal decay and regularity the space of p integrable non uniform splines and the shift invariant spaces generated by finitely many localized functions are our model examples of such reproducing kernel subspaces of lp rd we show that a signal in such reproducing kernel subspaces can be reconstructed in a stable way from its samples taken on a relatively separated set with sufficiently small gap we also study the exponential convergence consistency and the asymptotic pointwise error estimate of the iterative approximation projection algorithm and the iterative frame algorithm for reconstructing a signal in those reproducing kernel spaces from its samples with sufficiently small gap

complexity of bezout s theorem v polynomial time we show that there are algorithms which find an approximate zero of a system of polynomial equations and which function in polynomial time on the average the number of arithmetic operations is cn 4s where n is the input size and c a universal constant

how many eigenvalues of a random matrix are real let a be an n x n matrix whose elements are independent random variables with standard normal distributions as n oo the expected number of real eigenvalues is asymptotic to v 7r we obtain a closed form expres sion for the expected number of real eigenvalues for finite n and a formula for the density of a real eigenvalue for finite n asymptotically a real normalized eigenvalue al ii of such a random matrix is uniformly distributed on the in terval 1 1 analogous but strikingly different results are presented for the real generalized eigenvalues we report on numerical experiments confirming these results and suggesting that the assumption of normality is not important for the asymptotic results department of mathematics massachusetts institute of technology cambridge massachusetts 02139 e mail address edelmanfmath mit edu arts and sciences kapi olani community college 4303 diamond head road honolulu hawaii 98616 e mail address kostlanfkahuna math hawaii edu tj watson rsearch center 32 2 ibm yorktown heights new york 10598 0218 e mail address shubqwatson ibm com this content downloaded from 157 55 39 29 on tue 12 apr 2016 08 54 01 utc all use subject to http about jstor org terms

average sampling and reconstruction of reproducing kernel signals in mixed lebesgue spaces abstract in this paper we mainly study the average sampling and reconstruction of signals in a reproducing kernel subspace of the mixed lebesgue space l p q r m n first the sampling stability for two kinds of average sampling functionals is considered then the corresponding iterative approximation projection algorithms with exponential convergence are established finally the asymptotic pointwise error estimates are given for reconstructing a signal from its average samples corrupted by additive random noise

nonuniform sampling and reconstruction in shift invariant spaces this article discusses modern techniques for nonuniform sampling and reconstruction of functions in shift invariant spaces it is a survey as well as a research paper and provides a unified framework for uniform and nonuniform sampling and reconstruction in shift invariant subspaces by bringing together wavelet theory frame theory reproducing kernel hilbert spaces approximation theory amalgam spaces and sampling inspired by applications taken from communication astronomy and medicine the following aspects will be emphasized a the sampling problem is well defined within the setting of shift invariant spaces b the general theory works in arbitrary dimension and for a broad class of generators c the reconstruction of a function from any sufficiently dense nonuniform sampling set is obtained by efficient iterative algorithms these algorithms converge geometrically and are robust in the presence of noise d to model the natural decay conditions of real signals and images the sampling theory is developed in weighted l p spaces

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire abstract the first order harmonic balance method via the first fourier coefficient is used to construct two approximate frequency amplitude relations for a conservative nonlinear oscillatory system in which the restoring force has an irrational form this system corresponds to the motion of a mass attached to a stretched wire two procedures are used to approximately solve the nonlinear differential equation in the first the differential equation is rewritten in a form that does not contain the square root expression while in the second the differential equation is solved directly the approximate frequency obtained using the second procedure is more accurate than the frequency obtained with the first due to the fact that in the second procedure application of the harmonic balance method produces an infinite set of harmonics while in the first procedure only two harmonics are produced both approximate frequencies are valid for the complete range of oscillation amplitudes and excellent agreement of the approximate frequencies with the exact one are demonstrated and discussed the discrepancy between the second approximate frequency and the exact one never exceeds 2 2

complexity of bezout s theorem iv probability of success extensions we estimate the probability that a given number of projective newton steps applied to a linear homotopy of a system of n homogeneous polynomial equations in n 1 complex variables of fixed degrees will find all the roots of the system we also extend the framework of our analysis to cover the classical implicit function theorem and revisit the condition number in this context further complexity theory is developed

nonuniform sampling in principal shift invariant subspaces of mixed lebesgue spaces lp q rd 1 abstract in this paper we study the nonuniform sampling and reconstruction problem in shift invariant subspaces of mixed lebesgue spaces we first show that shift invariant subspaces in mixed lebesgue spaces l p q r d 1 can be well defined then we propose that the sampling problem in shift invariant subspaces of mixed lebesgue spaces is well posed at last the nonuniform samples f x j y k k j j of a function f belonging to a shift invariant subspace of mixed lebesgue spaces are proposed and we give a fast reconstruction algorithm that allows exact reconstruction of f as long as the sampling set x x j y k k j j is sufficiently dense

on the worst case arithmetic complexity of approximating zeros of systems of polynomials let d 1 cdots d n be positive integers let mathcal p denote the set of systems of polynomials f mathbb c n to mathbb c n that have only finitely many zeros including those at infinity and that satisfy degree f i d i for all i let 0 varepsilon leqq r it is shown for fixed d 1 cdots d n that with respect to a certain model of computation the worst case computational complexity of obtaining varepsilon approximations to at least those zeros xi satisfying xi leqq r for arbitrary f in mathcal p is theta log log r varepsilon that is to say both upper and lower bounds are proved an algorithm for proving the upper bound is introduced the number of operations required by this algorithm is o left n mathcal d 4 log mathcal d log log r varepsilon n 2 mathcal d 4 left begin array 20 c 1 sigma d i n end array right 4 right quad text where mathcal d prod limits i 1

on a problem posed by steve smale the 17th of the problems proposed by steve smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial on the average in the size n of the input system a partial solution to this problem was given by carlos beltr an and luis miguel pardo who exhibited a randomized algorithm doing so in this paper we further extend this result in several directions firstly we exhibit a linear homotopy algorithm that eciently implements a nonconstructive idea of

smale s 17th problem average polynomial time to compute affine and projective solutions smale s 17th problem asks can a zero of n complex polynomial equations in n unknowns be found approximately on the average in polynomial time with a uniform algorithm we give a positive answer to this question namely we describe a uniform probabilistic algorithm that computes an approximate zero of systems of polynomial equations f cn cn performing a number of arithmetic operations which is polynomial in the size of the input on the average

on the determination of the limit cycles using the harmonic balance method several methods have been developed to search for limit cycles or bifurcation curves of nonlinear differential equations recently some non perturbative methods have been proposed however the majority of these methods are based on the so called harmonic balance method in this paper we will see how to use the harmonic balance method to obtain information of the number period and form of the limit cycles of certain nonlinear equations the method also gives a sequence of algebraic approximations of the limit cycles we will also compare our results with other non perturbative methods the results can be generalized to other nonlinear differential equations

error analysis of frame reconstruction from noisy samples this paper addresses the problem of reconstructing a continuous function defined on rd from a countable collection of samples corrupted by noise the additive noise is assumed to be i i d with mean zero and variance sigma2 we sample the continuous function on the uniform lattice 1 m zd and show for large enough m that the variance of the error between the frame reconstruction fepsiv m from noisy samples of f and the function f satisfy var fepsiv m x f x ap sigma2 md cx where cx is the best constant for every x isin rd we also prove a similar result in the case that our data are weighted average samples of corrupted by additive noise

rigid continuation paths i quasilinear average complexity for solving polynomial systems how many operations do we need on the average to compute an approximate root of a random gaussian polynomial system beyond smale s 17th problem that asked whether a polynomial bound is possible we prove a quasi optimal bound text input size 1 o 1 this improves upon the previously known text input size frac32 o 1 bound the new algorithm relies on numerical continuation along rigid continuation paths the central idea is to consider rigid motions of the equations rather than line segments in the linear space of all polynomial systems this leads to a better average condition number and allows for bigger steps we show that on the average we can compute one approximate root of a random gaussian polynomial system of n equations of degree at most d in n 1 homogeneous variables with o n 5 d 2 continuation steps this is a decisive improvement over previous bounds that prove no better than sqrt 2 min n d continuation steps on the average

on generalized newton algorithms quadratic convergence path following and error analysis abstract newton iteration is known under some precise conditions to convergence quadratically to zeros of non degenerate systems of polynomials this and other properties may be used to obtain theorems on the global complexity of solving systems of polynomial equations see shub and smale in 6 using a model of computability over the reals however it is not practical and not desirable to actually compute newton iteration exactly in this paper approximate newton iteration is investigated for several generalizations of the newton operator quadratic convergence theorems and a robustness theorem are extended to approximate newton iteration generalizing some of the results in 6 the results here can be used to prove complexity theorems on path following algorithms for solving systems of polynomial equations using a model of computation over the integers malajovich 3

nonideal sampling and interpolation from noisy observations in shift invariant spaces digital analysis and processing of signals inherently relies on the existence of methods for reconstructing a continuous time signal from a sequence of corrupted discrete time samples in this paper a general formulation of this problem is developed that treats the interpolation problem from ideal noisy samples and the deconvolution problem in which the signal is filtered prior to sampling in a unified way the signal reconstruction is performed in a shift invariant subspace spanned by the integer shifts of a generating function where the expansion coefficients are obtained by processing the noisy samples with a digital correction filter several alternative approaches to designing the correction filter are suggested which differ in their assumptions on the signal and noise the classical deconvolution solutions least squares tikhonov and wiener are adapted to our particular situation and new methods that are optimal in a minimax sense are also proposed the solutions often have a similar structure and can be computed simply and efficiently by digital filtering some concrete examples of reconstruction filters are presented as well as simple guidelines for selecting the free parameters e g regularization of the various algorithms

shannon sampling and function reconstruction from point values i first met rene at the well known 1956 meeting on topology in mexico city he then came to the university of chicago where i was starting my job as instructor for the fall of 1956 he suzanne clara and i became good friends and saw much of each other for many decades especially at ihes in paris thom s encouragement and support were important for me especially in my first years after my ph d i studied his work in cobordism singularities of maps and transversality gaining many insights i also enjoyed listening to his provocations for example his disparaging remarks on complex analysis 19th century mathematics and bourbaki there was also a stormy side in our relationship neither of us could hide the pain that our public conflicts over catastrophe theory caused rene thom was a great mathematician leaving his impact on a wide part of mathematics i will always treasure my memories of him steve smale

nonperturbative but approximate method for solving differential equations and finding limit cycles a nonperturbative method for solving differential equations and for finding limit cycles is proposed and is illustrated on the anharmonic oscillator and on the van der pol equation it is shown to give the amplitude period and equation of the limit cycle with a better accuracy than any perturbative results so far obtained

asymptotology by homotopy perturbation method in the paper an heuristical example is given to illustrate the basic idea of the homotopy perturbation method so that homotopy perturbation method has made all that is necessary simple and all that is complex unnecessary

fast computation of zeros of polynomial systems with bounded degree under finite precision a solution for smale s 17th problem for the case of systems with bounded degree was recently given this solution an algorithm computing approximate zeros of complex polynomial systems in average polynomial time assumed infinite precision in this paper we describe a finite precision version of this algorithm our main result shows that this version works within the same time bounds and requires a precision which on the average amounts to a polynomial amount of bits in the mantissa of the intervening floating point numbers

time sampling and reconstruction in weighted reproducing kernel subspaces abstract this paper mainly studies the time sampling and reconstruction of signals in weighted reproducing kernel subspaces of l p firstly a class of examples for weighted reproducing kernel subspaces are given then using samples produced by a crossing time encoding machine or an integrate and fire time encoding machine satisfying some density constraints iterative approximation algorithms are established for perfect recovery of signals in weighted reproducing kernel subspaces finally an integrate and fire sampling scheme is discussed and an approximate reconstruction procedure is proposed for signals in weighted reproducing kernel subspaces moreover we give detailed analysis for the approximate reconstruction method in the setting of shift invariant subspaces

a continuation method to solve polynomial systems and its complexity in a recent work shub found comput math 9 171 178 2009 shub obtained a new upper bound for the number of steps needed to continue a known zero 0 of a system f 0 to a zero t of an input system f t following the path of pairs f t t where f t t in 0 t is a polynomial system and f t t 0 he proved that if one can choose the step size in an optimal way then the number of steps is essentially bounded by the length of the path of f t t in the so called condition metric however the proof of that result in shub found comput math 9 171 178 2009 is not constructive we give an explicit description of an algorithm which attains that complexity bound including the choice of step size

reconstruction of band limited signals from local averages the sampling theorem says that every band limited signal is uniquely determined by its sampled values provided the sampling points satisfy certain conditions however sampled values obtained in practice may not be the exact values of a signal at sampling points but only averages of the signal near these points grochenig 1992 proved that band limited signals can be reconstructed exactly from local averages if the sampling density is large enough we study the reconstruction of band limited signals from local averages with symmetric averaging functions we study the aliasing error arising when a non band limited signal is reconstructed from local averages and give explicit error bounds since the classical point sampling can be viewed as a limiting case of average sampling we indeed give new aliasing error bounds for both regular and irregular sampling

postfiltering versus prefiltering for signal recovery from noisy samples we consider the extension of the whittaker shannon ws reconstruction formula to the case of signals sampled in the presence of noise and which are not necessarily band limited observing that in this situation the classical sampling expansion yields inconsistent reconstruction we introduce a class of signal recovery methods with a smooth correction of the interpolation series two alternative data smoothing methods are examined based either on a global postfiltering or a local data presmoothing we assess the accuracy of the methods by the global l sub 2 error both band limited and non band limited signals are considered a general class of correlated noise processes is taken into account the weak and strong rates of convergence of the algorithms are established and their relative efficiency is discussed the influence of noise memory and its moment structure on the accuracy is thoroughly examined

accurate approximate analytical solutions for nonlinear free vibration of systems with serial linear and nonlinear stiffness this paper deals with free vibration of a nonlinear system having combined linear and nonlinear springs in series the conservative oscillation system is formulated as a nonlinear ordinary differential equation having linear and nonlinear stiffness components the governing equation is linearized and associated with the harmonic balance method to establish new and accurate higher order analytical approximate solutions unlike the perturbation method which is restricted to nonlinear conservative systems with a small perturbed parameter and also unlike the classical harmonic balance method which results in a complicated set of algebraic equations the new approach yields simple approximate analytical expressions valid for small as well as large amplitudes of oscillation some examples are solved and compared with numerical integration solutions and published results new solutions to the nonlinear systems are also presented and discussed