Quadrature formulas associated with Rogers–Szegő polynomials

In this paper we study the computation of the moments associated to rational weight functions given as a power spectrum with known or unknown poles of any order in the interior of the unit disc. A recursive algebraic procedure is derived that computes the moments in a finite number of steps. We also study the associated interpolatory quadrature formulas with equidistant nodes on the unit circle. Explicit expressions are given for the positive quadrature weights in the case of a polynomial weight function. For rational weight functions with simple poles, mostly real or uniformly distributed on a circle in the open unit disc, we also obtain expressions for the quadrature weights and sufficient conditions that guarantee that they are positive. The Poisson kernel is a simple example of a rational weight function, and in a last section we derive an asymptotic expansion of the quadrature error.

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“…1. To compute m 0 take A(z) = h(z) and B(z) = z N in (9). For this selection observe that I = m 0 /(2π).…”

Section: Methodsmentioning

confidence: 99%

Properties of interpolatory quadrature with equidistant nodes on the unit circle

Bultheel

Santos-León

2017

Numer Algor

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“…However, the RMSE increases as the univariate quadrature points increase further from 6 to 8. This may be due to several possible reasons, such as the saturation of numerical approximation method, an inaccurate Gaussian being a closer approximation of the true pdf and numerical instability of the quadrature rule [28]. The relative computational time for N s being 2, 4, 6 and 8 are 1, 7.36, 25.71 and 55.53, respectively, which shows an exponentially increasing computational time as N s increases.…”

Section: A Problemmentioning

confidence: 99%

“…For the above monic and Rogers-Szegő polynomials, the univariate quadrature points and associated weights are derived in [28]…”

Section: B Integral Approximation Using Szegő Quadrature Rulementioning

confidence: 99%

Szegő Quadrature Kalman Filter for Oscillatory Systems

Singh

2020

IEEE Access

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Filtering problems with oscillatory system dynamics commonly appear in real-life. However, the existing Gaussian filters, like the unscented Kalman filter (UKF), cubature Kalman filter CKF, Gauss-Hermite filter (GHF) and cubature quadrature Kalman filter (CQKF), are accurate for the nonlinear systems with a particular order of polynomials only. This manuscript introduces a new Gaussian filter, which is accurate for oscillatory systems with 2π-period of oscillation. The proposed method is named as Szegő Quadrature Kalman Filter (SQKF). The SQKF transforms the intractable integrals that appear during the filtering over a unit circle. The transformed integral is approximated using the univariate Szegő quadrature rule. The univariate quadrature rule is extended in a multivariate domain using the product rule. Simulation results reveal an improved estimation accuracy for the SQKF in an oscillatory environment. The computational burden of the SQKF is similar to the GHF and higher than the UKF, CKF and CQKF. INDEX TERMS Nonlinear filtering, Gaussian filtering, Oscillatory system, Intractable integral, Szegő quadrature rule.

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“…Since then, these polynomials which bear the name of Szegő were extensively studied by many. We cite, for example, [5], [6], [7], [10], [19], [20], [22], [25] and [27] as some of the very recent contributions. The recent publications of the two excellent volumes [23] and [24] by Simon have given a boost to the interest in studying these polynomials.…”

Section: Introductionmentioning

confidence: 99%