robust prediction of band limited signals from past samples corresp for a complex valued deterministic signal of finite energy band limited to the normalized frequency band w leq pi explicit coefficients a kn are found such that for any t satisfying 0 left f t sum 2n k 1 a kn f t kt right leq e f cdot beta n where e f is the signal energy and beta doteq 0 6863 thus the estimate of f t in terms of 2n past samples taken at a rate equal to or in excess of twice the nyquist rate converges uniformly at a geometric rate to f t on infty infty the suboptimal coefficients a kn have the desirable property of being pure numbers independent of both the particular band limited signal and of the selected sampling rate 1 t it is also shown that these same coefficients can be used to estimate the value of x t of a wide sense stationary random process in terms of past samples

typical cluster size for two dimensional percolation processes the typical cluster size for two dimensional percolation models is discussed it is shown that forw 0 x z 2 0 x lim n 1 n logp p w 0 n 1 p p c asp p c provided thate p w 0 2 e p w 0 p p c asp p c furthermore we introduce a new quantityf s p which may be thought of as the singular part of the free energy and show thatf s p p p c 2v provided that the correlation length p p c v asp p c

robust identification from band limited data consider the problem of identifying a scalar bounded input bounded output stable transfer function from pointwise measurements at frequencies within a bandwidth we propose an algorithm which consists of building a sequence of maps from data to models converging uniformly to the transfer function on the bandwidth when the number of measurements goes to infinity the noise level to zero and asymptotically meeting some gauge constraint outside error bounds are derived and the procedure is illustrated by numerical experiments

tree graph inequalities and critical behavior in percolation models various inequalities are derived and used for the study of the critical behavior in independent percolation models in particular we consider the critical exponent associated with the expected cluster sizex and the structure of then site connection probabilities n x1 xn it is shown that quite generally 1 the upper critical dimension above which attains the bethe lattice value 1 is characterized both in terms of the geometry of incipient clusters and a diagramatic convergence condition for homogeneousd dimensional lattices with x y o x y d 2 atp p c our criterion shows that 1 if 6 d 3 the connectivity functions n are generally bounded by tree diagrams which involve the two point function we conjecture that above the critical dimension the asymptotic behavior of n in the critical regime is actually given by such tree diagrams modified by a nonsingular vertex factor other results deal with the exponential decay of the cluster size distribution and the function 2 x y

hyperscaling inequalities for the contact process and oriented percolation the contact process and oriented percolation are expected to exhibit the same critical behavior in any dimension above their upper critical dimension dc they exhibit the same critical behavior as the branching process assuming existence of the critical exponents we prove a pair of hyperscaling inequalities which together with the results of refs 16 and 18 implies dc 4

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

u invariant sampling extrapolation and causal interpolation from generalized samples causal processing of a signal s samples is crucial in on line applications such as audio rate conversion compression tracking and more this paper addresses the problems of predicting future samples and causally interpolating deterministic signals we treat a rich variety of sampling mechanisms encountered in practice namely in which each sampling function is obtained by applying a unitary operator on its predecessor examples include pointwise sampling at the output of an antialiasing filter and magnetic resonance imaging mri which correspond respectively to the translation and modulation operators from an abstract hilbert space viewpoint such sequences of functions were studied extensively in the context of stationary random processes we thus utilize powerful tools from this discipline although our problems are deterministic by nature in particular we provide necessary and sufficient conditions on the sampling mechanism such that perfect prediction is possible for cases where perfect prediction is impossible we derive the predictor minimizing the prediction error we also derive a causal interpolation method that best approximates the commonly used noncausal solution finally we study when causal processing of the samples of a signal can be performed in a stable manner

the predictability of continuous time bandlimited processes in communications and signal processing we can find examples of applications that could benefit from the prediction of a bandlimited random process we consider a continuous time linear predictor applied to a bandlimited process we show that if the past values of the process are known over an interval of arbitrary positive length the mean squared prediction error may be made arbitrarily small regardless of how far in the future we wish to make the prediction we also show that this is no longer true when a certain energy constraint is applied to the predictor furthermore we discuss what this means for the case in which the prediction is based on past values that are corrupted by estimation errors

some general results concerning the critical exponents of percolation processes in this paper we will give some results concerning the critical exponents of percolation processes which are valid for any model these results show that in several respects the behavior which occurs for percolation on the binary tree provides bounds on one side for what happens in general these results and their proofs are closely related to their analogues for the ising model

correlation length and its critical exponent for percolation processes some critical exponent inequalities are given involving the correlation length of site percolation processes on d in particular it is shown thatv 2 d which implies that the critical exponentv cannot take its mean field value for the three dimensional percolation processes

on the prediction of a class of wide sense stationary random processes we prove that under suitable conditions a multi band wide sense stationary stochastic process can be linearly predicted at time with arbitrarily small error using past samples taken at uniform rate this result generalizes previous similar results for band limited signals moreover we prove that the prediction problem from uniform past samples is equivalent to a disjoint translates condition on the spectrum together with the divergence of a logarithmic integral we also show that for the band limited case under similar conditions non uniform samples can be taken

linear prediction from samples of a function and its derivatives some formulas for the prediction of the values of a band limited function based on its samples from the past are generalized by including past samples of its first derivative the new sums developed by an approach based on newton series make it possible to double the distance between the sample points the resulting formulas are shown to apply to the prediction problem for a large class of entire functions of exponential type in addition a related prediction formula which uses past samples of successively higher derivatives is shown to behave similarly to the taylor series approximation again for a class of functions that includes the band limited functions

mean field critical behavior for the contact process the contact process is a model of spread of an infectious disease combining with the result of ref 1 we prove that the critical exponents take on the mean field values for sufficiently high dimensional nearest neighbor models and for sufficiently spread out models with d 4 c as c and c 1 as c where and are the spread probability and the susceptibility of the infection respectively and c is the critical infection rate our results imply that the upper critical dimension for the contact process is at most 4

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

computationally efficient linear prediction from past samples of a band limited signal and its derivative formulas for linear prediction of a band limited signal are developed where the signal may be either deterministic or wide sense stationary the prediction formulas are based on finite differences modified by two or more parameters and finite differences allow the formulas to be easily adapted to changes in order of the prediction it is shown that a formula to predict the next signal value from a set of past equally spaced values is a formula that can be extended to provide prediction at points even beyond that in addition the formula is extended to a difference scheme involving an arbitrary number of parameters as well as to a formula that includes samples of the derivative of the signal this approach differs from that of solving the normal or yule walker equations but it has the advantage that the suboptimal prediction coefficients are independent of the particular signal spectrum or autocorrelation function

geometric critical exponent inequalities for general random cluster models a set of new critical exponent inequalities d 1 1 2 dv 1 1 andd 1 is proved for a general class of random cluster models which includes independent or dependent percolations lattice animals with any interactions and various stochastic cluster growth models the inequalities imply that the critical phenomena in the models are inevitably not mean field like in the dimensions one two and three

difference methods for the predictions of band limited signals various formulas have been given that allow the prediction of a band limited signal f t from its past samples at f t kt k 1 2 cdots in this paper we show that these samples are enoug

extrapolating a band limited function from its samples taken in a finite interval a band limited signal of finite energy can be reconstructed from its samples taken at the nyquist rate moreover the reconstruction is stable a feature crucial for implementation a small error in the sample values generates only a correspondingly small error in the resulting signal the nyquist sample values are mutually independent so that knowledge of them in a given interval t leq t leq 0 provides hardly any information about the behavior of the signal outside the interval however when the samples are taken at a greater rate a case referred to as over sampling they are interrelated and this redundancy can be exploited in various ways to improve the behavior of the reconstruction procedure a natural question is whether it can also be used to form accurate estimates of the signal outside the interval of observation this problem is relevant as well to prediction theory with oversampling when t infty so that the samples are known on the entire half line t they determine the signal everywhere although the reconstruction is now no longer stable here we examine the case of finite t of course a finite amount of data can yield only limited accuracy we prove that the samples can be used to form an approximation to the signal outside the sampling interval with an error which asymptotically as t rightarrow infty decreases exponentially in t over a range which grows linearly with t however as in the limiting case these approximations are not useful in practice since they require the sample values to be known exactly in the presence of measurement error the nature of the results changes good approximations are now available for only a bounded distance outside the interval of observation regardless of its length and their accuracy and range of validity can be increased only by improving the precision of sample reading since physical measurements are never perfect it is this conclusion which counts for applications the same results hold for the extrapolation of bounded band limited signals

b spline signal processing i theory the use of continuous b spline representations for signal processing applications such as interpolation differentiation filtering noise reduction and data compressions is considered the b spline coefficients are obtained through a linear transformation which unlike other commonly used transforms is space invariant and can be implemented efficiently by linear filtering the same property also applies for the indirect b spline transform as well as for the evaluation of approximating representations using smoothing or least squares splines the filters associated with these operations are fully characterized by explicitly evaluating their transfer functions for b splines of any order applications to differentiation filtering smoothing and least squares approximation are examined the extension of such operators for higher dimensional signals such as digital images is considered

minimax approximation of representation coefficients from generalized samples many sources of information are of analog or continuous time nature however digital signal processing applications rely on discrete data we consider the problem of approximating l2 inner products i e representation coefficients of a continuous time signal from its generalized samples adopting a robust approach we process these generalized samples in a minimax optimal sense specifically we minimize the worst approximation error of the desired representation coefficients by proper processing of the given sample sequence we then extend our results to criteria which incorporate smoothness constraints on the unknown function finally we compare our methods with the piecewise constant approximation technique commonly used for this problem and discuss the possible improvements by the suggested schemes

hyperscaling inequalities for percolation a set of critical exponent inequalities for independent percolation which saturate under the hyperscaling hypothesis is proved one of the consequences of the inequalities is the lower bounddc 6 for the upper critical dimension the proof is based on a rigorous version of the finite size scaling argument which extends easily to other systems such as ising ferromagnets

a note on difference methods for the prediction of band limited signals from past samples recent results are extended on difference methods for the prediction of bandlimited signals obtained by d h mugler and w splettstosser 1986 1987 and by d h mugler 1990 are extended to generalized bandlimited signals defined by m zakai 1965 and a j lee 1975 1976 this makes the difference method applicable to a wider class of signals

on an elliptic operator with discontinuous nonlinearity abstract a family of elliptic operators with discontinuous nonlinearity affected by the presence of a positive parameter is considered a critical value c of and the zero set ec of the corresponding critical solution u c play a special role in the treatment of these operators sharp estimates for c as a functional of boundary functions are given and the structure of ec is investigated an asymptotic formula for the solution u and its free boundary is presented as

analyticity properties and power law estimates of functions in percolation theory we consider percolation on the sites of a graphg e g a regulard dimensional lattice all sites ofg are occupied vacant with probabilityp respectively q 1 p independently of each other w denotes the cluster of occupied sites containing a fixed site which will usually be taken to be the origin andw the cardinality ofw the percolation probability is the probability that w i e p pp w some critical values ofp ph andpt are defined respectively as the smallest value ofp for which p 0 and for which the expectation of w is infinite formally ph inf p p 0 andpt inf p ep w we show for fairly general graphsgthat ifp pt thenpp w n decreases exponentially inn for the special casesg g0 the simple quadratic lattice andg1 the graph which corresponds to bond percolation on 2 we obtain upper and lower bounds for p of the formc p ph and bounds forep w of the formc p ph we also investigate smoothness properties of p ep number of clusters per site ep w 1 w 1 this function was introduced by sykes and essam who assumed that has exactly one singularity namely atp ph for the graphsg0 andg1 i e site or bond percolation on 2 we show that p is analytic atp ph and has two continuous derivatives atp ph the emphasis is on rigorous proofs

truncation error bounds for the cardinal sampling expansion of band limited signals bounds are derived for the truncation error of the cardinal sampling expansion for a large class of band limited deterministic and random signals these bounds extend and improve upon the bounds available in the literature

prediction by samples from the past with error estimates covering discontinuous signals there are several reasons why the classical sampling theorem is rather impractical for real life signal processing first the sinc kernel is not very suitable for fast and efficient computation it decays much too slowly second in practice only a finite number n of sampled values are available so that the representation of a signal f by the finite sum would entail a truncation error which decreases rather slowly for n due to the first drawback third band limitation is a definite restriction due to the nonconformity of band and time limited signals further the samples needed extend from the entire past to the full future relative to some time t t0 this paper presents an approach to overcome these difficulties the sinc function is replaced by certain simple linear combinations of shifted b splines only a finite number of samples from the past need be available this deterministic approach can be used to process arbitrary not necessarily bandlimited nor differentiable signals and even not necessarily continuous signals best possible error estimates in terms of an lp average modulus of smoothness are presented several typical examples exhibiting the various problems involved are worked out in detail

on causal estimation from bandlimited stationary sequences this paper considers the problem of estimating a stationary sequence y from the observation of a stationary correlated sequence x by means of a causal linear filter thereby it is assumed that the spectral density x of x vanishes on a subset of the unit circle of positive lebesgue measure such that the classical derivation of the estimation filter based on the spectral factorization of x can not be applied the paper derives the transfer function of such an estimation filter discusses its stability behavior and applies the result to the causal reconstruction of deterministic signals from its samples

a general sampling theory for nonideal acquisition devices the authors first describe the general class of approximation spaces generated by translation of a function spl psi x and provide a full characterization of their basis functions they then present a general sampling theorem for computing the approximation of signals in these subspaces based on a simple consistency principle the theory puts no restrictions on the system input which can be an arbitrary finite energy signal bandlimitedness is not required in contrast to previous approaches this formulation allows for an independent specification of the sampling analysis and approximation synthesis spaces in particular when both spaces are identical the theorem provides a simple procedure for obtaining the least squares approximation of a signal they discuss the properties of this new sampling procedure and present some examples of applications involving bandlimited and polynomial spline signal representations they also define a spectral coherence function that measures the similarity between the sampling and approximation spaces and derive a relative performance bound for the comparison with the least squares solution

on the prediction of band limited signals from past samples abstract a new method is used to construct coefficients a kn such that any band limited signal function with highest frequency w can be reconstituted from its samples from the past via t lim n k 1 n a kn t kt w although the existence of such predictor coefficients can be proved for each positive t t 0 built1 2 by j l brown jr 1 using the mathematical tool of power series expansions in the complex domain t can be extended to at least t 1 2 that is the sampling rate has to exceed three halves the nyquist rate again the predictor coefficients a kn do not depend on the structure of or on the bandwidth parameter w

more inequalities for critical exponents a variety of rigorous inequalities for critical exponents is proved most notable is the low temperature josephson inequalitydv 2 2 others are 1 1 v 1 1 1 d 1 1 for d dv 3 for d 4 and 2m 2m 2 form 2 the hypotheses vary all inequalities are true for the spin 1 2 ising model with nearest neighbor ferromagnetic pair interactions