This paper reviews the Fourier-series method for calculating cumulative distribution functions (cdf's) and probability mass functions (pmf's) by numerically inverting characteristic functions, Laplace transforms and generating functions. Some variants of the Fourier-series method are remarkably easy to use, requiring programs of less than fifty lines. The Fourier-series method can be interpreted as numerically integrating a standard inversion integral by means of the trapezoidal rule. The same formula is obtained by using the Fourier series of an associated periodic function constructed by aliasing; this explains the name of the method. This Fourier analysis applies to the inversion problem because the Fourier coefficients are just values of the transform. The mathematical centerpiece of the Fourier-series method is the Poisson summation formula, which identifies the discretization error associated with the trapezoidal rule and thus helps bound it. The greatest difficulty is approximately calculating the infinite series obtained from the inversion integral. Within this framework, lattice cdf's can be calculated from generating functions by finite sums without truncation. For other cdf's, an appropriate truncation of the infinite series can be determined from the transform based on estimates or bounds. For Laplace transforms, the numerical integration can be made to produce a nearly alternating series, so that the convergence can be accelerated by techniques such as Euler summation. Alternatively, the cdf can be perturbed slightly by convolution smoothing or windowing to produce a truncation error bound independent of the original cdf. Although error bounds can be determined, an effective approach is to use two different methods without elaborate error analysis. For this...

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Abstract. We discuss a general framework for recovering edges in piecewise smooth functions with finitely many jump discontinuities, where [f](x):=f(x+) − f(x−) ̸ = 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, Kɛ(·), depending on the small scale ɛ. Itis shown thatodd kernels, properly scaled, and admissible (in the sense of having small W −1,∞-moments of order O(ɛ)) satisfy Kɛ ∗ f(x) =[f](x) +O(ɛ), thus recovering both the location and amplitudes of all edges. As an example we consider general concentration kernels of the form Kσ ∑ N (t) = σ(k/N) sin kt to detect edges from the first 1/ɛ = 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, σexp (·), 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 Kɛ ∗ f(x) ∼ [f](x) ̸ = 0, and the smooth regions where Kɛ ∗ f = O(ɛ) ∼ 0. Numerical examples demonstrate that by coupling concentration kernels with nonlinear enhancement one arrives at effective edge detectors.

In this paper we construct, analyze and implement a new procedure for the spectral approximations of nonlinear conservation laws. It is well known that using spectral methods for nonlinear conservation laws will result in the formation of the Gibbs phenomenon once spontaneous shock discontinuities appear in the solution. These spurious oscillations will in turn lead to loss of resolution and render the standard spectral approximations unstable. The Spectral Viscosity (SV-) method [24] was developed to stabilize the spectral method by adding a spectrally small amount of high-frequencies diffusion carried out in the dual space. The resulting SV-approximation is stable without sacrificing spectral accuracy. The SV-method recovers a spectrally accurate approximation to the projectionof the entropy solution; the exact projection, however, is at best a first order approximation to the exact solution as a result of the formation of the shock discontinuities. The issue of spectral resolution is addressed by post-processing the SV-solution to remove the spurious oscillations at the discontinuities, as well as increase the first-order -- O(1/N) accuracy away from the shock discontinuities. Successful post-processing methods have been developed to eliminate the Gibbs phenomenon and recover spectral accuracy for the SV-approximation. However, such reconstruction methods require apriori knowledge of the locations of the shock discontinuities. Therefore, the detection of these discontinuities is essential to obtain an overall spectrally accurate solution. To this end, we employ the recently constructed enhanced edge detectors based on appropriate concentration factors [8], [9]. Once the edges of these discontinuities are identified, we can utilize a post-processing reconstruction met...

This paper addresses the recovery of piecewise smooth functions from their discrete data.

The problem of accurately reconstructing a piece-wise smooth, 2π-periodic function f and its first few derivatives, given only a truncated Fourier series representation of f, is studied and solved. The reconstruction process is divided into two steps. In the first step, the first 2N + 1 Fourier coefficients of f are used to approximate the locations and magnitudes of the discontinuities in f and its first M derivatives. This is accomplished by first finding initial estimates of these quantities based on certain properties of Gibbs phenomenon, and then refining these estimates by fitting the asymptotic form of the Fourier coefficients to the given coefficients using a least-squares approach. It is conjectured that the locations of the singularities are approximated to within O ( N −M−2) , and the associated jump of the k th derivative of f is approximated to within O ( N −M−1+k) , as N →∞,and the method is robust. These estimates are then used with a class of singular basis functions, which have certain “built-in ” singularities, to construct a new sequence of approximations to f. Each of these new approximations is the sum of a piecewise smooth function and a new Fourier series partial sum. When N is proportional to M, it is shown that these new approximations, and their derivatives, converge exponentially in the maximum norm to f, and its corresponding derivatives, except in the union of a finite number of small open intervals containing the points of singularity of f. The total measure of these intervals decreases exponentially to zero as M →∞.The technique is illustrated with

Abstract- A finite-by-infinite array of thin half-wave dipoles with H-plane scan is used to show the existence of a Gibbs’ phenomenon-type standing wave in scan impedance (normalized by the infinite array value) over the elements of the array. The period of this wave is.SA at broadside for X/2 array spacing and increases as the scan angle increases by a grating lobe-type expression. A simple empirical model based on Gibbs oscillations is fitted to the scan-impedance wave; the model predicts the 1/(1- sin&) period variation, and should be useful for systems trades and for preliminary design purposes.

. We present a formal framework based on metric temporal logic (MTL) for specifying and verifying real-time systems with a continuous environment. Metric temporal logic is extended to allow specification of properties about duration of system's states. We prove that every formula of the duration calculus (DC) can be translated into an equivalent MTL- R formula. Expressiveness of DC and MTL- R are compared. We formulate axioms for MTL- R and a sound rule for proving duration properties of timed transition systems. 1 Introduction In this paper, we extend metric temporal logic [16, 17, 3] (MTL for short) with a duration concept similar to the one in the duration calculus. We present a specification language MTL- R that extends MTL to allow the expression of properties about durations of system's states. Axioms for proving valid theorems about those properties are presented and proved sound. Furthermore, we extend the proof methodology presented in [12] by a proof rule for proving...

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The trigonometric network, introduced in this paper, is a multilayer feedforward neural network with sinusoidal activation functions. Unlike the N-dimensional Fourier series, the basis functions of the proposed trigonometric network have no strict harmonic relationship. An effective training algorithm for the network is developed. It is shown that the trigonometric network performs better than the sigmoidal neural network for some data sets. A pruning method based on the modified Gram-Schmidt orthogonalization procedure is presented to detect and prune useless hidden units. Other network architectures related to the trigonometric network, such as the sine network, are shown to be inferior to the network proposed in this paper. I. Introduction The trigonometric feedforward neural network proposed in this paper has both sine and cosine activations for each hidden layer net function. Since the derivative of the activations for the net function are never simultaneously zero, the hope is...