Large mode number eigenvalues of the prolate spheroidal differential equation

Boyd, John P.

doi:10.1016/s0096-3003(03)00280-7

Cited by 12 publications

(8 citation statements)

References 7 publications

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“…Although there is an increasing number of papers devoted to analytic properties and numerical evaluation of the PSWFs (see, e.g., [33,5,6,21,22,13,34] and the references therein), the approximability of the PSWFs is studied in a limited number of papers. Some error analysis for approximation of bandlimited functions by PSWFs was carried out in the note [24] (also see Xiao's thesis [32]).…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Analysis of spectral approximations using prolate spheroidal wave functions

Wang

2009

Math. Comp.

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“…Boyd showed in that

\sqrt{χ_{n}} \sim \frac{π}{2} (n + 1 / 2) \frac{1}{{\bar{E} (1, m)}} .

Recall ,

\sqrt{χ_{n}} = \frac{c}{\sqrt{m}}

. It follows that

c / \sqrt{m}

is indeed equal to

(n + 1 / 2) (π / 2) / \bar{E} (1; m)

and therefore the prolate asymptotics can be written

\begin{matrix} true \\ left ψ_{n} (cos (t); c) \sim prefixsin {(t)}^{- 1 / 2} \sqrt{scriptE (prefixcos (t); m)} {(1 - m prefixcos {(t)}^{2})}^{- 1 / 4} J_{0} \\ left ((), \frac{π}{2}, (n + 1 / 2), \frac{1}{\bar{E} (1, m)}, E, (x; m)) . \end{matrix}

The mapped Legendre polynomials are asymptotically

P_{n} (prefixcos (t)) \sim sin {(τ^{E})}^{- 1 / 2} \sqrt{τ^{E}} J_{0} ([n + 1 / 2] ([], \frac{π}{2}, \frac{1}{\bar{E} (1; m)}, E, (x; m))) .

…”

Section: Asymptotic Approximation Of Prolate Spheroidal Functions By mentioning

confidence: 98%

Quasi‐Uniform Spectral Schemes (QUSS), Part I: Constructing Generalized Ellipses for Graphical Grid Generation

Boyd

2015

Stud Appl Math

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Chebyshev and Legendre polynomial spectral methods are bedeviled by highly nonuniform grids. The separation between nearest neighbors of an N -point grid at the center of the interval is π/2 larger than the spacing of a uniform grid with the same number of points. Quasi-Uniform Spectral Schemes (QUSS) redistribute grid points and choose basis functions in order to recover this factor of π/2 as nearly as possible while retaining a high density of points near the endpoints to avoid the horrors of the Gibbs or Runge Phenomenon. Here, we introduce a systematic approach, dubbed "mapped cosine bases," that embraces the widely used Kosloff/Tal-Ezer functions as a special case. The mapped cosine approach uses grid points x j that are the images of a uniform grid under the coordinate mapping x = τ inv (t). Here, we show how to generalize the well-known graphical construction of the Chebyshev grid using a circle to QUSS mappings using a generalized ellipse. This provides a way to visualize the maps and grids and the subtle differences between different mappings of the mapped cosine family. We illustrate and compare the Kosloff/Tal-Ezer map with two new maps that use elliptic integrals and Jacobian theta functions, respectively. We show that the elliptic integral grid is an asymptotic approximation to the usual grid for prolate spheroidal functions. This suggests the conjecture that one can obtain the benefits of a prolate basis without the complications of prolate functions by using mapped polynomials instead.

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“…where x is in the reference interval Λ:=(−1,1), and c=πWT is also called the bandwidth parameter. Using the fundamental identity: 6) one verifies readily that…”

Section: )mentioning

confidence: 99%

“…• The first is to pursue the highest DOP over the 2N-dimensional space V c 2N−1 (cf. [6,7]). More precisely, we fix x 0 = −1,x N = 1, and search for quadrature nodes {x j } N−1 j=1 and weights…”

Section: Prolate Points and Prolate Quadraturementioning

confidence: 99%