sum rules and constraints on passive systems a passive system is one that cannot produce energy a property that naturally poses constraints on the system a system in convolution form is fully described by its transfer function and the class of herglotz functions holomorphic functions mapping the open upper half plane to the closed upper half plane is closely related to the transfer functions of passive systems following a well known representation theorem herglotz functions can be represented by means of positive measures on the real line this fact is exploited in this paper in order to rigorously prove a set of integral identities for herglotz functions that relate weighted integrals of the function to its asymptotic expansions at the origin and infinity the integral identities are the core of a general approach introduced here to derive sum rules and physical limitations on various passive physical systems although similar approaches have previously been applied to a wide range of specific applications this paper is the first to deliver a general procedure together with the necessary proofs this procedure is described thoroughly and exemplified with examples from electromagnetic theory

optimal domain splitting for interpolation by chebyshev polynomials polynomial interpolants defined using chebyshev extreme points as nodes converge uniformly at a geometric rate when sampling a function that is analytic on an interval however the convergence rate can be arbitrarily close to unity if the function has a singularity close to the interval when extended to the complex plane in such cases splitting the interval and doing piecewise interpolation may be more efficient in the total number of nodes than the global interpolant because the convergence rate is determined by bernstein ellipses obtained through a joukowski conformal map relative efficiency of splitting at any point in the interval can be calculated and then optimized over the interval the optimal splitting may be applied recursively the chebfun software project uses a simple rule of thumb without prior singularity information to create a binary search that can be shown to do an excellent job of finding the optimal splitting in most cases however the process can use a large number of intermedi

on nonsmooth solutions of linear hyperbolic systems initial value problems for linear hyperbolic systems with smooth 2 periodic coefficients are solved numerically by a modified fourier galerkin method when the initial values are nonsmooth the described approach is seen to give substantially improved accuracy compared to more traditional methods the discontinuities are accurately resolved already on coarse grids and the fine structure of structured solutions is resolved on relatively coarse grids as well the accuracy is seen to be of high order and even for very long term integrations the global error can be kept very small if the grid is sufficiently refined

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

approximating the jump discontinuities of a function by its fourier jacobi coefficients in the present paper we generalize eckhoff s method i e the method for approximating the locations of discontinuities and the associated jumps of a piecewise smooth function by means of its fourier chebyshev coefficients a new method enables us to approximate the locations of discontinuities and the associated jumps of a discontinuous function which belongs to a restricted class of the piecewise smooth functions by means of its fourier jacobi coefficients for arbitrary indices approximations to the locations of discontinuities and the associated jumps are found as solutions of algebraic equations it is shown as well that the locations of discontinuities and the associated jumps are recovered exactly for piecewise constant functions with a finite number of discontinuities in addition we study the accuracy of the approximations and present some numerical examples

bounds on herglotz functions and fundamental limits of broadband passive quasistatic cloaking using a sum rule we derive new bounds on herglotz functions that generalize those given in bernland et al j phys a math theor 44 14 145205 2011 and gustafsson and sjoberg new j phys 12 4 043046 2010 these bounds apply to a wide class of linear passive systems such as electromagnetic passive materials among these bounds we describe the optimal ones and also discuss their meaning in various physical situations like in the case of a transparency window where we exhibit sharp bounds then we apply these bounds in the context of broadband passive cloaking in the quasistatic regime to refute the following challenging question is it possible to construct a passive cloaking device that cloaks an object over a whole frequency band our rigorous approach although limited to quasistatics gives quantitative limitations on the cloaking effect over a finite frequency range by providing inequalities on the polarizability tensor associated with the cloaking device we emphasize that our resul

detection of edges in spectral data ii nonlinear enhancement we discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities where f x f x f x neq 0 our approach is based on two main aspects localization using appropriate concentration kernels and separation of scales by nonlinear enhancement to detect such edges one employs concentration kernels k epsilon cdot depending on the small scale epsilon it is shown that odd kernels properly scaled and admissible in the sense of having small w 1 infty moments of order cal o epsilon satisfy k epsilon f x f x cal o epsilon thus recovering both the location and amplitudes of all edges as an example we consider general concentration kernels of the form k sigma n t sum sigma k n sin kt to detect edges from the first 1 epsilon n spectral modes of piecewise smooth f s here we improve in generality and simplicity over our previous study in a gelb and e tadmor appl comput harmon anal 7 1999 pp 101 135 both periodic and nonperiodic spectral projections are considered we identify in particular a new family of exponential factors sigma exp cdot with superior localization properties the other aspect of our edge detection involves a nonlinear enhancement procedure which is based on separation of scales between the edges where k epsilon f x sim f x neq 0 and the smooth regions where k epsilon f cal o epsilon sim 0 numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors

variational principles for the static electric and magnetic polarizabilities of anisotropic media with perfect electric conductor inclusions we present four variational principles for the electric and magnetic polarizabilities for a structure consisting of anisotropic media with perfect electric conductor pec inclusions from these principles we derive monotonicity results and upper and lower bounds on the electric and magnetic polarizabilities when computing the polarizabilities numerically the bounds can be used as error bounds the variational principles demonstrate important differences between electrostatics and magnetostatics when pec bodies are present

exact computation of amplification for a class of nonlinear systems arising from cellular signaling pathways a commonly employed measure of the signal amplification properties of an input output system is its induced l 2 norm sometimes also known as h gain in general however it is extremely difficult to compute the numerical value for this norm or even to check that it is finite unless the system being studied is linear this paper describes a class of systems for which it is possible to reduce this computation to that of finding the norm of an associated linear system in contrast to linearization approaches a precise value not an estimate is obtained for the full nonlinear model the class of systems that we study arose from the modeling of certain biological intracellular signaling cascades but the results should be of wider applicability

convergence acceleration of modified fourier series in one or more dimensions modified fourier series have recently been introduced as an adjustment of classical fourier series for the approximation of nonperiodic functions defined on d variate cubes such approximations offer a number of advantages including uniform convergence however like fourier series the rate of convergence is typically slow in this paper we extend eckhoff s method to the convergence acceleration of multivariate modified fourier series by suitable augmentation of the approximation basis we demonstrate how to increase the convergence rate to an arbitrary algebraic order moreover we illustrate how numerical stability of the method can be improved by utilising appropriate auxiliary functions in the univariate setting it is known that eckhoff s method exhibits an auto correction phenomenon we extend this result to the multivariate case finally we demonstrate how a significant reduction in the number of approximation coefficients can be achieved by using a hyperbolic cross index set

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

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

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

a stability barrier for reconstructions from fourier samples we prove that any stable method for resolving the gibbs phenomenon that is recovering high order accuracy from the first m fourier coefficients of an analytic and nonperiodic function can converge at best root exponentially fast in m any method with faster convergence must also be unstable and in particular exponential convergence implies exponential ill conditioning this result is analogous to a recent theorem of platte trefethen and kuijlaars concerning recovery from pointwise function values on an equispaced m grid the main step in our proof is an estimate for the maximal behavior of a polynomial of degree n with bounded m term fourier series this estimate is related to a conjecture of hrycak and grochenig in the final part of the paper we discuss the implications of our main theorem to polynomial based interpolation and least squares approaches for overcoming the gibbs phenomenon

uniform approximation of piecewise r smooth and globally continuous functions we study the uniform chebyshev approximation of continuous and piecewise r smooth r ge2 functions f 0 t to mathbb r with a finite number of singular points the approximation algorithms use only n function values at adaptively or nonadaptively chosen points we construct a nonadaptive algorithm mathcal a r n rm non that for the functions with at most one singular point enjoys the best possible convergence rate n r this is in sharp contrast to results concerning discontinuous functions for r ge3 this optimal rate of convergence holds only in the asymptotic sense i e it occurs only for sufficiently large n that depends on f in a way that is practically impossible to verify however it is enough to modify mathcal a r n rm non by using r 1 lfloor r 1 2 rfloor extra 1pt function evaluations to obtain an adaptive algorithm mathcal a r n rm ada with error satisfying f mathcal a r n rm ada f c le c rt r f r l infty n r for all n ge n 0 and n 0 independent of f this result cannot be achieved for functions with more than just one singular point however the convergence rate n r can be recovered asymptotically by a nonadaptive algorithm overline mathcal a r n rm non that is a slightly modified mathcal a r n rm non specifically limsup n to infty f overline mathcal a r n rm non f c cdot n r le c rt r f r l infty for all r smooth functions f with finitely many singular points

reduction of the gibbs phenomenon for smooth functions with jumps by the e algorithm recently brezinski has proposed to use wynn s e algorithm in order to reduce the gibbs phenomenon for partial fourier sums of smooth functions with jumps by displaying very convincing numerical experiments in the present paper we derive analytic estimates for the error corresponding to a particular class of hypergeometric functions and obtain the rate of column convergence for such functions possibly perturbed by another sufficiently differentiable function we also analyze the connection to pade fourier and pade chebyshev approximants including those recently studied by kaber and maday

some studies on nonpolynomial interpolation and error analysis abstract in this paper we propose nonpolynomial and hermite nonpolynomial interpolation with multiple parameters and present method to determine optimal value of parameters which generate minimum error in approximation the generalized error analysis results for nonpolynomial and hermite nonpolynomial interpolations are derived we establish theoretical relationship among nonpolynomial polynomial interpolation and the fourier series and propose solution to runge s phenomenon through nonpolynomial interpolation the hermite nonpolynomial cubic spline nonpolynomial cubic spline interpolation methods and their error analysis are presented numerical simulations are carried out for the analysis of error in cubic spline interpolations proposed method is applied to the analysis of various time series to show comparison in errors between polynomial and nonpolynomial spline interpolations and to empirical mode decomposition emd to illustrate practical usefulness of the results

time domain approach to the forward scattering sum rule the forward scattering sum rule relates the extinction cross section integrated over all wavelengths with the polarizability dyadics it is useful for deriving bounds on the interaction between scatterers and electromagnetic fields antenna bandwidth and directivity and energy transmission through sub wavelength apertures the sum rule is valid for linearly polarized plane waves impinging on linear passive and time translational invariant scattering objects in free space here a time domain approach is used to clarify the derivation and the used assumptions the time domain forward scattered field defines an impulse response energy conservation shows that this impulse response is the kernel of a passive convolution operator which implies that the fourier transform of the impulse response is a herglotz function the forward scattering sum rule is finally constructed from integral identities for herglotz functions

approximation of the discontinuities of a function by its classical orthogonal polynomial fourier coefficients in the present paper we generalize the method suggested in an earlier paper by the author and overcome its main deficiency first we modify the well known prony method which subsequently will be utilized for recovering exactly the locations of jump discontinuities and the associated jumps of a piecewise constant function by means of its fourier coefficients with respect to any system of the classical orthogonal polynomials next we will show that the method is applicable to a wider class of functions namely to the class of piecewise smooth functions for functions which piecewise belong to c 2 1 1 the locations of discontinuities are approximated to within o 1 n by means of their fourier jacobi coefficients unlike the previous one the generalized method is robust since its success is independent of whether or not a location of the discontinuity coincides with a root of a classical orthogonal polynomial in addition the error estimate is uniform for any c d c 1 1 to the end we discuss the accuracy stability and complexity of the method and present numerical examples

illustrations of new physical bounds on linearly polarized antennas a recent approach to physical bounds on antennas of arbitrary shape is numerically illustrated in particular physical bounds for antennas circumscribed by the rectangular parallelepiped finite cylinders and planar rectangles are presented the bounds are verified against numerical results for various small antennas with excellent agreement

on the non uniqueness of the electromagnetic instantaneous response the instantaneous response models the rapid part of the interaction between electromagnetic fields and materials from a physical point of view the instantaneous response is non observable and from a mathematical point of view it constitutes the principal part of the maxwell equations in this paper it is shown that the instantaneous response is not uniquely determined from a given set of values of the constitutive map the given values only give an upper bound on the size of the instantaneous response a numerical example illustrates the non uniqueness of the instantaneous response