Results 171 to 180 of about 44,437 (210)

Asymptotics of Orthonormal Polynomials

2001
In this chapter, we establish pointwise asymptotics of the orthonormal polynomials p n (W 2, x) for x in the interval of orthogonality, as well as asymptotics for the associated recurrence coefficients. We shall also reformulate some of the results of the previous chapters for this special case.
Eli Levin, Doron S. Lubinsky
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Orthonormal polynomials in segmented optical pupils

Optik, 2019
Abstract Segmented primary mirror is commonly adopted to construct the modern extremely large telescopes. For the sake of facilitating wavefront analysis for this complex aperture, we generalize the theory frame of obtaining orthonormal polynomials corresponding to the segmented pupil based on the non-recursive Gram-Schmidt process.
Yongfeng Zhang, Hao Xian
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Orthonormal polynomial bases in function spaces

Israel Journal of Mathematics, 1991
Let \(\Phi=(\Phi_ k)_{k=0}^ \infty\) be an orthonormal system of trigonometric polynomials on the unit circle \(T\) and let \(v_ n=\max_{k\leq n}\deg\Phi_ k\). The main problem consists in estimating the sequence \((v_ n)\) from above. It is shown by construction that there exists a system \(\Phi\) such that \(v_ n\leq{4\over 3}n\), and \(\Phi\) is a ...
Wojtaszczyk, P., Woźniakowski, K.
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Orthonormal polynomials for hexagonal pupils: addendum

Optics Letters, 2008
Remarks made in a recent paper [Opt. Lett.31, 24622006] regarding orthonormal polynomials for hexagon pupils are clarified.
Virendra N. Mahajan, Guang-ming Dai
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Risk Tails and General Orthonormal Polynomials

SSRN Electronic Journal, 2016
In order to characterize a statistical probability distribution p(x) of a variable x, the moments of the distribution are used; the first two of which are the mean and standard deviation. The z-score is often used to characterize data points of x (e.g. outliers with large z-scores).
Jan Dash, Harvey J. Stein, Mario Bondioli   +2 more
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An asymptotically orthonormal polynomial family

BIT, 1984
Es sei \(\Gamma\) eine Jordankurve in der z-Ebene, \(z=\Phi (w)\) sei die konforme Abbildung von \(\{\) \(w: | w| >1\}\) auf ext \(\Gamma\) mit \(\Phi (\infty)=\infty\). Bei Kenntnis von \(\Phi\) am Rande \(\partial {\mathbb{D}}\) führt der Verf. eine Folge von Polynomen \(p_ n\) vom genauen Grad n ein und studiert ihr asymptotisches Verhalten in ...
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Orthonormal Polynomials and Gram-Schmidt Orthonormalization

2013
In optical design, we trace rays from a point object through a system to determine the aberrations of the wavefront at its exit pupil. In optical testing, we determine the aberrations of a system or an element interferometrically. In both cases, we obtain aberration numbers at an array of points.
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Nonrecursive determination of orthonormal polynomials with matrix formulation

Optics Letters, 2006
A general theoretical approach has been developed for the determination of orthonormal polynomials over any integrable domain, such as a hexagon. This approach is better than the classical Gram-Schmidt orthogonalization process because it is nonrecursvie and can be performed rapidly with matrix transformations.
Guang-Ming, Dai, Virendra N, Mahajan
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n-Orthonormal Operator Polynomials

1988
If μ, is a positive Borel measure with infinite support on the unit circle T in the complex plane C, then the Gram-Schmidt process applied in L 2(μ) to the sequence of polynomials 1, z, z 2,..., yields an orthonormal sequence of polynomials in L 2(μ).
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Local Representation of the Geopotential by Weighted Orthonormal Polynomials

Journal of Guidance and Control, 1979
The finite-element model of the geopotential and its gradient as introduced by Junkins, is significantly improved by the use of orthonormal polynomials. The computational effort to obtain the modeling constants via the normal equations is greatly reduced, especially if the order of approximation is increased.
Engels, Remi C., Junkins, John L.
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