Set Of Orthonormal Polynomials Research Articles

POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR FRACTAL PROCESSES

Context. We consider a Kolmogorov-Wiener filter for fractal random processes, which, for example, may take place in moderninformation-telecommunication systems and in control of complex technological processes. The weight function of the consideredfilter may be applied to data forecast in the corresponding systems.Objective. As is known, in the continuous case the Kolmogorov-Wiener filter weight function obeys the Fredholm integralequation of the first kind. The aim of the work is to obtain the Kolmogorov-Wiener filter weight function as an approximate solutionof the corresponding integral equation.Method. The truncated orthogonal polynomial expansion method for approximate solution of the Fredholm integral equation ofthe first kind is used. A set of orthonormal polynomials is used.Results. We obtained approximate results for the Kolmogorov-Wiener weight function for fractal processes with a power-lawstructure function. The weight function is found as an approximate solution of the Fredholm integral equation of the first kind thekernel of which is the correlation function of the corresponding fractal random process. Analytical results for the one-, two-, three-,four- and five-polynomial approximations are obtained. A numerical comparison of the left-hand and right-hand sides of the integralequation for the obtained weight functions is given for different values of the parameters. The corresponding numerical investigationis made up to the 18-polynomial approximation on the basis of the Wolfram Mathematica package. The applicability of the obtainedsolutions is discussed.Conclusions. The Kolmogorov-Wiener weight function for fractal processes is obtained approximately in the form of a truncatedorthogonal polynomial series. The validity of the obtained weight functions is discussed. The obtained results may be applied to thedata forecast in a wide variety of different systems where fractal random processes take place.

Orthogonal polynomials generated by random vectors

ABSTRACTEvery random q-vector with finite moments generates a set of orthonormal polynomials. These are generated from the basis functions xn = xn11…xnqq using Gram–Schmidt orthogonalization. One can cycle through these basis functions using any number of ways. Here, we give results using minimum cycling. The polynomials look simpler when centered about the mean of X, and still simpler form when X is symmetric about zero. This leads to an extension of the multivariate Hermite polynomial for a general random vector symmetric about zero. As an example, the results are applied to the multivariate normal distribution.

Multidimensional generalization of the Korous inequality

ABSTRACTThis paper deals with some qualitative properties of orthogonal polynomials in several variables. The boundedness and relations between two sets of orthonormal polynomials associated with an arbitrary weight function and its extension are investigated. It presents an analogy to Korous' result for general orthogonal polynomials in one variable.

Robust Design Optimization of Gas Turbine Compression Systems

Gas turbine compression systems are required to perform adequately over a range of operating conditions. Complexity has encouraged the conventional design process for compressors to focus initially on one operating point, usually the most commonor arduous, to draw up an outline design. Generally, only as this initial design is refined is its offdesign performance assessed in detail. Not only does this necessarily introduce a potentially costly and timeconsuming extra loop in the design process, but it also may result in a design whose offdesign behavior is suboptimal. Aversion of nonintrusive polynomial chaos was previously developed in which a set of orthonormal polynomials was generated to facilitate a rapid analysis of robustness in the presence of generic uncertainties with good accuracy. In this paper, this analysis method is incorporated in real time into the design process for the compression system of a three-shaft gas turbine aeroengine. This approach to robust optimization is shown to lead to designs that exhibit consistently improved system performance with reduced sensitivity to offdesign operation.

Orthonormal polynomials in wavefront analysis: error analysis

Zernike circle polynomials are in widespread use for wavefront analysis because of their orthogonality over a circular pupil and their representation of balanced classical aberrations. However, they are not appropriate for noncircular pupils, such as annular, hexagonal, elliptical, rectangular, and square pupils, due to their lack of orthogonality over such pupils. We emphasize the use of orthonormal polynomials for such pupils, but we show how to obtain the Zernike coefficients correctly. We illustrate that the wavefront fitting with a set of orthonormal polynomials is identical to the fitting with a corresponding set of Zernike polynomials. This is a consequence of the fact that each orthonormal polynomial is a linear combination of the Zernike polynomials. However, since the Zernike polynomials do not represent balanced aberrations for a noncircular pupil, the Zernike coefficients lack the physical significance that the orthonormal coefficients provide. We also analyze the error that arises if Zernike polynomials are used for noncircular pupils by treating them as circular pupils and illustrate it with numerical examples.

Image reconstruction from limited range projections using orthogonal moments

A set of orthonormal polynomials is proposed for image reconstruction from projection data. The relationship between the projection moments and image moments is discussed in detail, and some interesting properties are demonstrated. Simulation results are provided to validate the method and to compare its performance with previous works.

Vibration analysis of twisted plates using first order shear deformation theory

Based on general shell theory and the first order shear deformation theory, an accurate relationship between strains and displacements of a twisted plate is derived by the Green strain tensor. An equation of equilibrium for free vibration is given by the principle of virtual work and the governing equation is solved by using the Rayleigh–Ritz method with sets of orthonormal polynomials in which only the first polynomials are defined according to the geometric boundary conditions of a plate and the others are generated by the Gram–Schmidt process. The numerical verification is carried out by comparing with previous results of cantilever plates. Vibration characteristics of cantilever twisted plates such as frequency parameters and corresponding mode shapes are obtained by the present numerical method, and the effects of the twist angle, the aspect ratio and the thickness ratio on them are studied.

Uniform Asymptotic Formula for Orthogonal Polynomials with Exponential Weight

Let {pn(x)}_{n\ge 0}$ be the set of orthonormal polynomials with respect to the exponential weight w(x)=e-v(x) , where v(x)=x2m + ... is a monic polynomial of degree 2m with $m \ge 2$ and is even. An asymptotic approximation is obtained for pn(x), as $n \to \infty$, which holds uniformly for $0 \le x \le O(n^{1/2m})$. As a corollary, a three-term asymptotic expansion is also derived for the zeros of these polynomials.

Multiple Mathematical Corrections Before Quantitative Analysis of Biological Infrared Spectra

Abstract Infrared reflectance spectroscopy is often used to predict the composition of various food products. However the precision of the analysis could be improved. For this purpose, several treatments can be used; (i) baseline correction. Baseline deformation is one of the major problem that is associated with infrared spectra; (ii) spectra can be expanded in terms of a set of orthonormal polynomials derived from the Legendre polynomials, and the leading terms of the expansion, which contains most of the baseline variation, can be removed; (iii) the classification of samples prior to analysis is one way to obtain more homogenuous families. Hence, with the aim of improving the precision of the values obtained during quantitative determination of constituants from biological samples, we have sucessively applied different mathematical treatments on their infrared spectra: baseline correction, Legendre polynomials decomposition and classification procedure. The application of these mathematical treatments ...

A unified approach to the helioseismic forward and inverse problems of differential rotation

view Abstract Citations (153) References (38) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS A Unified Approach to the Helioseismic Forward and Inverse Problems of Differential Rotation Ritzwoller, Michael H. ; Lavely, Eugene M. Abstract A general, degenerate perturbation theoretic treatment of the helioseismic forward and inverse problem for solar differential rotation is presented. For the forward problem, differential rotation is represented as the axisymmetric component of a general toroidal flow field using velocity spherical harmonics. This approach allows each degree of differential rotation to be estimated independently from all other degrees. In the inverse problem, the splitting caused by differential rotation is expressed as an expansion in a set of orthonormal polynomials that are intimately related to the solution of the forward problem. The combined use of vector spherical harmonics as basis functions for differential ratio and the Clebsch-Gordon coefficients to represent splitting provides a unified approach to the forward and inverse problems of differential rotation which greatly simplify inversion. Publication: The Astrophysical Journal Pub Date: March 1991 DOI: 10.1086/169785 Bibcode: 1991ApJ...369..557R Keywords: Clebsch-Gordan Coefficients; Helioseismology; Solar Oscillations; Solar Rotation; Angular Momentum; Perturbation Theory; Spherical Harmonics; Stellar Models; Solar Physics; SUN: OSCILLATIONS; SUN: ROTATION full text sources ADS |

On Nevai's characterization of measures with almost everywhere positive derivative

It is known that if dμ is a finite positive Borel measure on the unit circle ∂Δ := {z ϵ C:¦z¦ = 1} with μ′ > 0 a.e. in [0, 2π], then lim n→∞ ∫ 2π 0 |ϑ n(z)| 2 | n+l(z)| 2 −1 dθ=0, z=e iθ (∗) uniformly in l ⩾ 0, where { ϑ n ( z)} n = 0 ∞ is the set of orthonormal polynomials with respect to dμ. Recently, Nevai proved that if (∗) holds uniformly in l ⩾ 0, then μ′ > 0 a.e. in [0, 2π]. In this note we establish another characterization of such measures in terms of Turán-type inequalities and we utilize it to give an alternative proof of Nevai's result.

Discrete ordinate method of solution of a Fokker—Planck equation with a bistable potential

The solution of a Fokker—Planck (FP) equation describing a bifurcation system is obtained with a recently developed discrete ordinate method. Eigenvalues and corresponding eigenfunctions are calculated, and employed in describing the time evolution of the system. The results with the discrete ordinate method are compared with the results obtained with a set of orthonormal polynomials, constructed with the equilibrium distribution as a weight function.

Neutron transport half-range orthogonal polynomial sets and related quadrature sums and properties

The half-range orthogonality property for neutron transport eigenfunctions leads to the definition of two weight functions. Each of these yields a set of orthonormal polynomials and related nodes and quadrature coefficients. The polynomials of different set, same order, are here shown to be linked by singular integral transformations. As a consequence, they obey almost identical recursion formulae. The transformations also generate cross-set relations involving nodes and quadrature coefficients of different set and equal order. Calculation procedures are described and examples of numerical tables for nodes and quadrature coefficients presented. Appendices deal with the reduction to the fullrange case and the limit behaviour of nonasymptotic quadrature sums.

Transmission Line Models for Use with Circuit/System Analysis Programs

Techniques for modeling multiconductor transmission lines for use with the SCEPTRE computer program are presented. The transmission line models developed can be modified for compatibility with other circuit/ system transient analysis programs and are amenable to modification to include nuclear weapon effects. The general modeling approach has been to develop computationally efficient and accurate terminal models which can be arbitrarily loaded at the source and load ends and which can be used in conjunction with nonlinear electronic circuit models using either simplified or discrete modeling techniques. The concept of the method is to derive a set of transfer functions in the Laplace domain relating forward and backward traveling waves on the line to voltages and currents at the source and load ends of the line, approximate the transfer functions with a set of orthonormal polynomials, and represent the resulting rational polynomials in the time domain with state variable differential equations. For the multiconductor case, the orthogonal characteristics of wave propagation are used to decouple the modes of propagation except at the source and load boundary condition circuits.

The Boundedness of Certain Sets of Orthonormal Polynomials in One, Two, and Three Variables

The Boundedness of Certain Sets of Orthonormal Polynomials in One, Two, and Three Variables

The boundedness of certain sets of orthonormal polynomials in one, two, and three variables

The boundedness of certain sets of orthonormal polynomials in one, two, and three variables