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The FourierSeries Method For Inverting Transforms Of Probability Distributions
, 1991
"... This paper reviews the Fourierseries 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 Fourierseries method are remar ..."
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Cited by 192 (52 self)
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This paper reviews the Fourierseries 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 Fourierseries method are remarkably easy to use, requiring programs of less than fifty lines. The Fourierseries 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 Fourierseries 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...
Lower Bounds for Norms of Inverses of Interpolation Matrices for Radial Basis Functions
, 1994
"... : Interpolation of scattered data at distinct points x 1 ; . . . ; x n 2 IR d by linear combinations of translates \Phi(kx \Gamma x j k 2 ) of a radial basis function \Phi : IR 0 ! IR requires the solution of a linear system with the n by n distance matrix A := (\Phi(kx i \Gamma x j k 2 ). Recent ..."
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Cited by 11 (5 self)
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: Interpolation of scattered data at distinct points x 1 ; . . . ; x n 2 IR d by linear combinations of translates \Phi(kx \Gamma x j k 2 ) of a radial basis function \Phi : IR 0 ! IR requires the solution of a linear system with the n by n distance matrix A := (\Phi(kx i \Gamma x j k 2 ). Recent results of Ball, Narcowich and Ward, using Laplace transform methods, provide upper bounds for kA \Gamma1 k 2 , while Ball, Sivakumar, and Ward constructed examples with regularly spaced points to get special lower bounds. This paper proves general lower bounds by application of results of classical approximation theory. The bounds increase with the smoothness of \Phi. In most cases, they leave no more than a factor of n \Gamma2 to be gained by optimization of data placement, starting from regularly distributed data. This follows from comparison with results of Ball, Baxter, Sivakumar, and Ward for points on scaled integer lattices and supports the hypothesis that regularly spaced data a...
Ultraspherical GaussKronrod Quadrature is not possible for lambda > 3
"... . With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter has only few real zeros for > 3 and suciently large n . Since the nodes of the GaussKronrod ..."
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Cited by 11 (1 self)
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. With the help of a new representation of the Stieltjes polynomial it is shown by using Bessel functions that the Stieltjes polynomial with respect to the ultraspherical weight function with parameter has only few real zeros for > 3 and suciently large n . Since the nodes of the GaussKronrod quadrature formulae subdivide into the zeros of the Stieltjes polynomial and the Gaussian nodes, it follows immediately that GaussKronrod quadrature is not possible for > 3 . On the other hand, for = 3 and suciently large n , even partially positive GaussKronrod quadrature is possible. Key Words. GaussKronrod quadrature, Stieltjes polynomials, orthogonal polynomials, Bessel functions AMS subject classication. 33C10, 33C45, 42C05, 65D32 1. Introduction and Main Results Let be a nonnegative nontrivial measure and let p n (x; d) := p n (x) = x n + : : : , n 2 N , be the monic polynomials orthogonal with respect to , i.e., (1:1) Z R x j p n (x) d(x) = 0 for j = 0; : : : ; ...
The Spectral Structure of TES Processes
 Stochastic Models
, 1994
"... TES (TransformExpandSample) is a versatile class of stochastic sequences consisting of marginally uniform autoregressive schemes with modulo1 reduction, followed by various transformations. TES modeling aims to fit a TES model to empirical records by simultaneously capturing both firstorder and ..."
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Cited by 6 (6 self)
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TES (TransformExpandSample) is a versatile class of stochastic sequences consisting of marginally uniform autoregressive schemes with modulo1 reduction, followed by various transformations. TES modeling aims to fit a TES model to empirical records by simultaneously capturing both firstorder and secondorder properties of the empirical data. In this paper we study the spectral properties of general TES processes and their component innovation sequences, thus generalizing the results reported in Jagerman and Melamed [9]. We derive formulas for the power spectral density function and the spectral distribution function which are suitable for efficient numerical computation, and exemplify them for TES processes with uniform and exponential marginals. The results contribute to the understanding of TES sequences as models of autocorrelated sequences, particularly in a Monte Carlo simulation context. Keywords and Phrases: TES Processes, Stochastic Processes, Spectrum, Spectral Density, Spe...
On the "Favard theorem" and its extensions
, 2000
"... In this paper we present a survey on the \Favard theorem" and its extensions. Key words: Favard Theorem, recurrence relations 1 Introduction. Given a sequence fP n g 1 n=0 of monic polynomials satisfying a certain recurrence relation, we are interested in nding a general inner product, if one ..."
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Cited by 5 (0 self)
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In this paper we present a survey on the \Favard theorem" and its extensions. Key words: Favard Theorem, recurrence relations 1 Introduction. Given a sequence fP n g 1 n=0 of monic polynomials satisfying a certain recurrence relation, we are interested in nding a general inner product, if one exists, such that the sequence fP n g 1 n=0 is orthogonal with respect to it. The original \classical" result in this direction is due to J. Favard [10] even though his result seems to be known by dierent mathematicians. The rst who obtained a similar result was Stieltjes in 1894 [23]. In fact, from the point of view of J continued fractions obtained from the contraction of an S continued fraction with positive coecients, Stieltjes proved the existence of a positive linear functional such that the denominators of the approximants are orthogonal with respect to it [23, x11]. Later on, Stone gave another approach using the spectral resolution of a selfadjoint operator associated to a Jacobi...
Approximation of Univariate SetValued Functions  an Overview
"... The paper is an updated survey of our work on the approximation of univariate setvalued functions by samplesbased linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for realvalued functions to setvalued functions, by replacing ..."
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Cited by 3 (1 self)
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The paper is an updated survey of our work on the approximation of univariate setvalued functions by samplesbased linear approximation operators, beyond the results reported in our previous overview. Our approach is to adapt operators for realvalued functions to setvalued functions, by replacing operations between numbers by operations between sets. For setvalued functions with compact convex images we use Minkowski convex combinations of sets, while for those with general compact images metric averages and metric linear combinations of sets are used. We obtain general approximation results and apply them to Bernstein polynomial operators, Schoenberg spline operators and polynomial interpolation operators.
Two results on Polynomial Interpolation . . .
, 1991
"... We present two results that quantify the poor behavior of polynomial interpolation in n equally spaced points. First, in bandlimited interpolation of complex exponential functions e‘li (c ( E Iw), the error decreases to 0 as n + a, if and only if d ( is small enough to provide at least six points p ..."
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Cited by 1 (1 self)
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We present two results that quantify the poor behavior of polynomial interpolation in n equally spaced points. First, in bandlimited interpolation of complex exponential functions e‘li (c ( E Iw), the error decreases to 0 as n + a, if and only if d ( is small enough to provide at least six points per wavelength. Second, the Lebesgue constant,4. (supremum norm of the nth interpolation operator) satisfies lim, j cc A,!,“‘ = 2. Both of these results are more than 50 years old, but they are generally unknown to approximation theorists.
COMPLEXIFICATIONS OF POLYNOMIALS AND MULTILINEAR MAPS ON REAL BANACH SPACES.
"... Abstract. This paper is intended to be a survey on complexifications of real Banach spaces. Using a general complexification procedure, we present a number of known results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. In some ca ..."
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Abstract. This paper is intended to be a survey on complexifications of real Banach spaces. Using a general complexification procedure, we present a number of known results for comparison of the norms of real polynomials and multilinear mappings with the norms of their complex extensions. In some cases we provide improvements of previously obtained inequalities. INTRODUCTION AND NOTATION The richness of complex analysis is an obvious motivation for the development of a formal theory of complexifications of real Banach spaces. The idea of complexifying a real Banach space has been considered in the past by many authors trying to solve specific problems in a real setting by using complex techniques. The investigation
Numerical differentiation Legendre polynomials
"... ing f 0ð1Þ from given data of the form ðtj; f ðtjÞÞMj1 where the points tj are in the interval on of derivatives posing a method els based on preiable funct ftjgj1 1;1. Given information of the form fðtj; f ðtjÞÞgj1, find approximately the value of f ðtÞ. In practical proble data is often noisy, o ..."
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ing f 0ð1Þ from given data of the form ðtj; f ðtjÞÞMj1 where the points tj are in the interval on of derivatives posing a method els based on preiable funct ftjgj1 1;1. Given information of the form fðtj; f ðtjÞÞgj1, find approximately the value of f ðtÞ. In practical proble data is often noisy, or at least given up to a fixed accuracy d. This situation can be described by the socalled determ noise model. In this model, the noise intensity level is measured by a small positive number d, and the available infor has the form fðtj; fdðtjÞÞgMj1, where fd is a continuous function on 1;1 such that
Filtered Legendre Expansion Method for Numerical Differentiation at the Boundary Point with Application to Blood Glucose Predictions
"... Abstract Let f: [−1,1] → R be continuously differentiable. We consider the question of approximating f ′ (1) from given data of the form (tj, f(tj)) M j=1 where the points tj are in the interval [−1,1]. It is well known that the question is ill–posed, and there is very little literature on the subj ..."
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Abstract Let f: [−1,1] → R be continuously differentiable. We consider the question of approximating f ′ (1) from given data of the form (tj, f(tj)) M j=1 where the points tj are in the interval [−1,1]. It is well known that the question is ill–posed, and there is very little literature on the subject known to us. We consider a summability operator using Legendre expansions, together with high order quadrature formulas based on the points tj’s to achieve the approximation. We also estimate the effect of noise on our approximation. The error estimates, both with or without noise, improve upon those in the existing literature, and appear to be unimprovable. The results are applied to the problem of short term prediction of blood glucose concentration, yielding better results than other comparable methods.