On Spheroidal Wave Functions of Order Zero

Bouwkamp, C.J.

doi:10.1002/sapm194726179

Cited by 85 publications

(42 citation statements)

References 23 publications

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“…We also notice that for all c > 0, So far, intensive work has been devoted to developing efficient methods for computing the PSWFs and their eigenvalues (see, e.g., [4,29,8,13,16]). A package of Matlab programs for manipulating PSWFs is also available (cf.…”

Section: Lemma 22 For Anymentioning

confidence: 99%

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Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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Abstract. In this paper, the approximation properties of the prolate spheroidal wave functions of order zero (PSWFs) are studied, and a set of optimal error estimates are derived for the PSWF approximation of non-periodic functions in Sobolev spaces. These results serve as an indispensable tool for the analysis of PSWF spectral methods. A PSWF spectral-Galerkin method is proposed and analyzed for elliptic-type equations. Illustrative numerical results consistent with the theoretical analysis are also presented.

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Section: Lemma 22 For Anymentioning

confidence: 99%

“…Boyd [8]). Following Bouwkamp [4] (also see, e.g., [33]), we expand ψ c n (x) in terms of normalized Legendre polynomials:…”

Section: Lemma 22 For Anymentioning

confidence: 99%

Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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“…There are two standard procedures for finding such solutions. The first was developed independently by Bouwkamp [14] and Blanch [15], and makes use of a fundamental equivalence between three-term recurrences and continued fractions [16]. This provides a method for determining the eigenvalues numerically to high precision, although it relies on the availability of a sufficiently accurate starting estimate for the eigenvalue.…”

Section: Numerical Computation 31 the Spheroidal Eigenvalues L mentioning

confidence: 99%

Theory and computation of spheroidal wavefunctions

Falloon

Abbott

Wang

2003

J. Phys. A: Math. Gen.

123

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“…In X, the equations for the polynomial expansion coefficients can be solved analytically [2,3,12,13] and in T 2 r they have been solved numerically in [11]. In both cases, the residual of the Sturm-Liouville equation will decrease exponentially fast as the number of terms in the polynomial truncation is increased [11], and thus the truncated polynomial approximation can be made very accurate.…”

Section: Computing Generalized Eigenfunctions Of Sturm-liouville Equamentioning

confidence: 99%

“…For details about computing polynomial approximations to PSWFs, see [2,3,12]. In this work, all one-dimensional results are computed using the Maple(TM) 1 symbolic manipulation software and taking M = 30 + 3N.…”

Section: Computing Generalized Eigenfunctions Of Sturm-liouville Equamentioning

confidence: 99%

Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

Taylor

2006

J Eng Math

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The spatial resolution of eigenfunctions of Sturm-Liouville equations in one-dimension is frequently measured by examining the minimum distance between their roots. For example, it is well known that the roots of polynomials on finite domains cluster like O(1/N 2 ) near the boundaries. This technique works well in one dimension, and in higher dimensions that are tensor products of one-dimensional eigenfunctions. However, for non-tensor-product eigenfunctions, finding good interpolation points is much more complicated than finding the roots of eigenfunctions. In fact, in some cases, even quasi-optimal interpolation points are unknown. In this work an alternative measure, , is proposed for estimating the characteristic length scale of eigenfunctions of Sturm-Liouville equations that does not rely on knowledge of the roots. It is first shown that is a reasonable measure for evaluating the eigenfunctions since in one dimension it recovers known results. Then results are presented in higher dimensions. It is shown that for tensor products of one-dimensional eigenfunctions in the square the results reduce trivially to the one-dimensional result. For the non-tensor product Proriol polynomials, there are quasi-optimal interpolation points (Fekete points). Comparing the minimum distance between Fekete points to shows that is a reasonably good measure of the characteristic length scale in two dimensions as well. The measure is finally applied to the non-tensor product generalized eigenfunctions in the triangle proposed by Taylor MA, Wingate BA [(2006) J Engng Math, accepted] where optimal interpolation points are unknown. While some of the eigenfunctions have larger characteristic length scales than the Proriol polynomials, others show little improvement.

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On Spheroidal Wave Functions of Order Zero

Cited by 85 publications

References 23 publications

Analysis of spectral approximations using prolate spheroidal wave functions

Analysis of spectral approximations using prolate spheroidal wave functions

Theory and computation of spheroidal wavefunctions

Measuring characteristic length scales of eigenfunctions of Sturm–Liouville equations in one and two dimensions

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