on simple oversampled a d conversion in shift invariant spaces it has been found that the quantization error e for a conventional oversampled analog to digital a d conversion behaves like spl par e spl par sup 2 o spl tau sup 2 with respect to the sampling rate spl tau recently conventional a d conversion has been extended to a d conversion based on shift invariant spaces as consequences of such extension it offers rich choices to build a nonideal a d conversion system of high accuracy and low computational complexity as well as reduces the noise sensitivity and computational complexity in digital to analog d a conversion therefore it is necessary to establish the estimate of quantization error for the extended a d conversion based on shift invariant spaces in this paper we introduce a constructive method to establish an estimate of the quantization error as e sup 2 o spl tau sup 2 for oversampled a d conversion in shift invariant spaces meanwhile we demonstrate that the bit rate required to encode the converted digital signal in such a d conversion scheme only increases as the logarithm of the sampling ratio therefore the quantization error is an exponentially decaying function of the bit rate in order to establish such an estimate we need the nonuniform sampling theorem for shift invariant spaces which as the necessary preparation is studied prior to introducing the constructive method

on the approximation power of convolution based least squares versus interpolation there are many signal processing tasks for which convolution based continuous signal representations such as splines and wavelets provide an interesting and practical alternative to the more traditional sine based methods the coefficients of the corresponding signal approximations are typically obtained by direct sampling interpolation or quasi interpolation or by using least squares techniques that apply a prefilter prior to sampling we compare the performance of these approaches and provide quantitative error estimates that can be used for the appropriate selection of the sampling step h specifically we review several results in approximation theory with a special emphasis on the strang fix 1971 conditions which relate the general o h sup l behavior of the error to the ability of the representation to reproduce polynomials of degree n l 1 we use this theory to derive pointwise error estimates for the various algorithms and to obtain the asymptotic limit of the l sub 2 error as h tends to zero we also propose a new improved l sub 2 error bound for the least squares case in the process we provide all the relevant bound constants for polynomial splines some of our results suggest the existence of an intermediate range of sampling steps where the least squares method is roughly equivalent to an interpolator with twice the order we present experimental examples that illustrate the theory and confirm the adequacy of our various bound and limit determinations

splines are universal solutions of linear inverse problems with generalized tv regularization splines come in a variety of flavors that can be characterized in terms of some differential operator l the simplest piecewise constant model corresponds to the derivative operator likewise on

cardinal exponential splines part i theory and filtering algorithms causal exponentials play a fundamental role in classical system theory starting from those elementary building blocks we propose a complete and self contained signal processing formulation of exponential splines defined on a uniform grid we specify the corresponding b spline basis functions and investigate their reproduction properties green function and exponential polynomials we also characterize their stability riesz bounds we show that the exponential b spline framework allows an exact implementation of continuous time signal processing operators including convolution differential operators and modulation by simple processing in the discrete b spline domain we derive efficient filtering algorithms for multiresolution signal extrapolation and approximation extending earlier results for polynomial splines finally we present a new asymptotic error formula that predicts the magnitude and the nth order decay of the l sub 2 approximation error as a function of the knot spacing t

multiplicity of positive radial solutions for an elliptic inclusion system on an annulus in this paper we present the sufficient conditions for existence and multiplicity of positive radial solutions for elliptic inclusion systems our results are obtained by utilizing the generalization of leggett and williams s fixed point theorem established in this paper for the norm type cone expansion and compression of multivalued operators

on strict system equivalence for multidimensional systems strict system equivalence describes the connection between least order realizations of a transfer function in terms of a general linear constant system rosenbrock h h 1970 state space and multivariable theory london nelson the paper generalizes this and related results to the case of multidimensional systems i e systems of linear partial difference or differential equations with constant coefficients

cardinal exponential splines part ii think analog act digital by interpreting the green function reproduction property of exponential splines in signal processing terms we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts the link is that the latter are the b spline coefficients of the former which happen to be exponential splines motivated by this observation we introduce an extended family of cardinal splines the generalized e splines to generalize the concept for all convolution operators with rational transfer functions we construct the corresponding compactly supported b spline basis functions which are characterized by their poles and zeros thereby establishing an interesting connection with analog filter design techniques we investigate the properties of these new b splines and present the corresponding signal processing calculus which allows us to perform continuous time operations such as convolution differential operators and modulation by simple application of the discrete version of these operators in the b spline domain in particular we show how the formalism can be used to obtain exact discrete implementations of analog filters finally we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real world analog to digital a to d and d to a conversion systems

continuous domain solutions of linear inverse problems with tikhonov versus generalized tv regularization we consider one dimensional 1 d linear inverse problems that are formulated in the continuous domain the object of recovery is a function that is assumed to minimize a convex objective functional the solutions are constrained by imposing a continuous domain regularization we derive the parametric form of the solution representer theorems for tikhonov quadratic and generalized total variation gtv regularizations we show that in both cases the solutions are splines that are intimately related to the regularization operator in the tikhonov case the solution is smooth and constrained to live in a fixed subspace that depends on the measurement operator by contrast the gtv regularization results in a sparse solution composed of only a few dictionary elements that are upper bounded by the number of measurements and independent of the measurement operator our findings for the gtv regularization resonates with the minimization of the ell 1 norm which is its discrete counterpart and also produces sparse solutions finally we find the experimental solutions for some measurement models in one dimension we discuss the special case when the gtv regularization results in multiple solutions and devise an algorithm to find an extreme point of the solution set which is guaranteed to be sparse

lp approximation by certain projection operators abstract we estimate the lp r d approximation rate 1 p provided dilates of an orthogonal projection operator of l2 r d onto a space generated by shifts of a function which has a polynomial decay rate at infinity and has stable shifts to this end we employ a quasi interpolation scheme and invoke wiener s lemma in particular this paper provides further substantiation to one of the fundamental properties of refinable functions the connection between smoothness and approximation order

on the connections between two classical notions of multidimensional system equivalence in this paper we study the connections between two classical notions of equivalence used in the literature of linear multidimensional nd systems on one hand we have the notion of zero coprime

b spline signal processing ii efficiency design and applications for pt i see ibid vol 41 no 2 p 821 33 1993 a class of recursive filtering algorithms for the efficient implementation of b spline interpolation and approximation techniques is described in terms of simplicity of realization and reduction of computational complexity these algorithms compare favorably with conventional matrix approaches a filtering interpretation low pass filter followed by an exact polynomial spline interpolator of smoothing spline and least squares approximation methods is proposed these techniques are applied to the design of digital filters for cubic spline signal processing an efficient implementation of a smoothing spline edge detector is proposed it is also shown how to construct a cubic spline image pyramid that minimizes the loss of information in passage from one resolution level to the next in terms of common measures of fidelity this data structure appears to be superior to the gaussian laplacian pyramid

fixed point theorem of leggett williams type and its application we establish a norm type cone expansion and compression fixed point theorem for completely continuous operators our theorem is then applied to prove the existence of positive solution of second order three point boundary value problem

leggett williams norm type fixed point theorems for multivalued mappings this paper presents several norm type versions of the leggett williams fixed point theorem for multivalued maps with nonempty convex compact values

positive periodic solutions of second order nonlinear differential systems with two parameters by employing the deimling fixed point index theory we consider a class of second order nonlinear differential systems with two parameters l m r 2 0 0 we show that there exist three nonempty subsets of r 2 0 0 c d 1 and d 2 such that r 2 0 0 c d 1 d 2 and the system has at least two positive periodic solutions for l m d 1 one positive periodic solution for l m c and no positive periodic solutions for l m d 2 meanwhile we find two straight lines l 1 and l 2 such that c lies between l 1 and l 2

fast b spline transforms for continuous image representation and interpolation efficient algorithms for the continuous representation of a discrete signal in terms of b splines direct b spline transform and for interpolative signal reconstruction indirect b spline transform with an expansion factor m are described expressions for the z transforms of the sampled b spline functions are determined and a convolution property of these kernels is established it is shown that both the direct and indirect spline transforms involve linear operators that are space invariant and are implemented efficiently by linear filtering fast computational algorithms based on the recursive implementations of these filters are proposed a b spline interpolator can also be characterized in terms of its transfer function and its global impulse response cardinal spline of order n the case of the cubic spline is treated in greater detail the present approach is compared with previous methods that are reexamined from a critical point of view it is concluded that b spline interpolation correctly applied does not result in a loss of image resolution and that this type of interpolation can be performed in a very efficient manner

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

continuous domain signal reconstruction using l_ p norm regularization we focus on the generalized interpolation problem there one reconstructs continuous domain signals that honor discrete data constraints this problem is infinite dimensional and ill posed we make it well posed by imposing that the solution balances data fidelity and some l p norm regularization more specifically we consider p geq 1 and the multi order derivative regularization operator mathrm l mathrm d n 0 we reformulate the regularized problem exactly as a finite dimensional one by restricting the search space to a suitable space of polynomial splines with knots on a uniform grid our splines are represented in a b spline basis which results in a well conditioned discretization for a sufficiently fine grid our search space contains functions that are arbitrarily close to the solution of the underlying problem where our constraint that the solution must live in a spline space would have been lifted this remarkable property is due to the approximation power of splines we use the alternating direction method of multipliers along with a multiresolution strategy to compute our solution we present numerical results for spatial and fourier interpolation through our experiments we investigate features induced by the l p norm regularization namely sparsity regularity and oscillatory behavior

modified b splines for the sampling of bandlimited functions it is shown that there is an optimal finite linear combination of b splines denominated modified b splines such that a pertinent low frequency condition called m flatness is satisfied a profound relationship of the modified b splines with the beta distribution implies an asymptotic sampling theorem with exact reconstruction requiring only small oversampling

a transformation for 2 d linear systems and a generalization of a theorem of rosenbrock the transformation of zero coprime system equivalence z c s e with its various characterizations is shown to have at least one important role for two dimensional linear systems theory this paper shows that it is z c s e which forms the basis of the generalization of rosenbrock s characterization of all least order polynomial realizations of a transfer function matrix for the case of 2 d systems the definition of what consistutes the least order is shown to be crucial

existence of three periodic positive solutions for a class of integral equations with parameters abstract in this paper we establish the existence of three periodic positive solutions for a class of abstract integral equations by leggett williams fixed point theorem using the existence results for abstract integral equations the population models are also considered

interpolation revisited medical images application based on the theory of approximation this paper presents a unified analysis of interpolation and resampling techniques an important issue is the choice of adequate basis functions the authors show that contrary to the common belief those that perform best are not interpolating by opposition to traditional interpolation the authors call their use generalized interpolation they involve a prefiltering step when correctly applied the authors explain why the approximation order inherent in any basis function is important to limit interpolation artifacts the decomposition theorem states that any basis function endowed with approximation order ran be expressed as the convolution of a b spline of the same order with another function that has none this motivates the use of splines and spline based functions as a tunable way to keep artifacts in check without any significant cost penalty the authors discuss implementation and performance issues and they provide experimental evidence to support their claims

wavelet theory demystified we revisit wavelet theory starting from the representation of a scaling function as the convolution of a b spline the regular part of it and a distribution the irregular or residual part this formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory including some new extensions for fractional orders n a self contained accessible fashion in particular we prove that the b spline component is entirely responsible for five key wavelet properties order of approximation reproduction of polynomials vanishing moments multiscale differentiation property and smoothness regularity of the basis functions we also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions this allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal additional results include a new peeling theory of smoothness leading to the extended notion of wavelet differentiability in the l sub p sense and a sharper theorem stating that smoothness implies order