<i>Spheroidal Wave Functions</i>

Stratton, J. A.; Morse, Philip Μ.; Chu, Lan Jen; Little, John D. C.; Corbató, F. J.; Abramowitz, Milton

doi:10.1063/1.3060000

Cited by 155 publications

(32 citation statements)

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“…Now, we consider a sequence of approximations by truncating (25). In the case of genic selection with no recurrent mutation, the eigenfunctions B n (x) of the backward generator L are also known as the oblate spheroidal functions in mathematical physics, and they have received considerable amounts of attention previously (e.g., see Stratton et al 1941). Note that the algorithm presented in this section provides an efficient way to evaluate these functions, a problem that remained difficult in the past.…”

Section: Algorithm 1 (Genic Selection)mentioning

confidence: 99%

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

2012

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The transition density function of the Wright-Fisher diffusion describes the evolution of population-wide allele frequencies over time. This function has important practical applications in population genetics, but finding an explicit formula under a general diploid selection model has remained a difficult open problem. In this article, we develop a new computational method to tackle this classic problem. Specifically, our method explicitly finds the eigenvalues and eigenfunctions of the diffusion generator associated with the Wright-Fisher diffusion with recurrent mutation and arbitrary diploid selection, thus allowing one to obtain an accurate spectral representation of the transition density function. Simplicity is one of the appealing features of our approach. Although our derivation involves somewhat advanced mathematical concepts, the resulting algorithm is quite simple and efficient, only involving standard linear algebra. Furthermore, unlike previous approaches based on perturbation, which is applicable only when the population-scaled selection coefficient is small, our method is nonperturbative and is valid for a broad range of parameter values. As a by-product of our work, we obtain the rate of convergence to the stationary distribution under mutation-selection balance. DIFFUSION processes, which are continuous-time Markov processes with almost surely continuous sample paths, have been successfully applied in various population genetic analyses in the past. Examples include finding the stationary distribution of allele frequencies and approximating fixation times and probabilities (see Karlin and Taylor 1981;Ewens 2004;Durrett 2008 for other applications of diffusion processes). This success is largely due to the fact that the diffusion approximation captures the key features of an evolutionary model while ignoring unimportant details, thereby arriving at a simpler process that facilitates computation. However, when a reasonably complex model of evolution is considered, one is faced with unwieldy equations even under the diffusion approximation. In particular, for Wright-Fisher diffusions with general diploid selection, finding an explicit analytic transition density function, which characterizes the evolution of population-wide allele frequencies over time, has remained a challenging open problem. The diffusion theory allows one to write down a partial differential equation (PDE) satisfied by the transition density, but solving the PDE analytically has proved to be difficult.The transition density has several practical applications, including the following: Recently, there has been growing interest in analyzing samples taken from the same or related populations at different time points. For example, such data arise from experimental evolution of model organisms in the laboratory (e.g., bacteria, see Lenski 2011), from viral/ phage populations (Shankarappa et al. 1999;Wichman et al. 1999), or from ancient DNA (Hummel et al. 2005); see also Bollback et al. (2008) and references therein. In part...

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Section: Algorithm 1 (Genic Selection)mentioning

confidence: 99%

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

2012

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“…The paper considers integral mode numbers m and n, and real spheroidal parameter c. There is a large literature: [1] is the original paper; [2][3][4] are standard works; [5] provides readily-accessible tables; [6] has extensive references; and [7] provides accurate values for comparison. However, published results are usually limited to small values of m and n (typically less than 10).…”

Section: Introductionmentioning

confidence: 99%

Calculation of spheroidal wave functions

Kirby¹

2006

Computer Physics Communications

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“…The classical theory of PSWFs is based primarily on their connection with Legendre polynomials (see [2,18,19]): the coefficients of the Legendre series for a PSWF are the coordinates of an eigenvector of a certain tridiagonal matrix; the latter becomes diagonally dominant when the order of the function is large compared to the band-limit. Historically known as the Bouwkamp algorithm and formulated in terms of three-term recursions, this apparatus leads to an effective numerical scheme for the evaluation of the PSWFs, and yields a number of analytical properties of PSWFs.…”

Section: Introductionmentioning

confidence: 99%

High-Frequency Asymptotic Expansions for Certain Prolate Spheroidal Wave Functions

Xiao

Rokhlin

2003

Journal of Fourier Analysis and Applications

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Prolate Spheroidal Wave Functions (PSWFs) are a well-studied subject with applications in signal processing, wave propagation, antenna theory, etc. Originally introduced in the context of separation of variables for certain partial differential equations, PSWFs became an important tool for the analysis of band-limited functions after the famous series of articles by Slepian et al. The popularity of PSWFs seems likely to increase in the near future, as band-limited functions become a numerical (as well as an analytical) tool.The classical theory of PSWFs is based primarily on their connection with Legendre polynomials: the coefficients of the Legendre series for a PSWF are the coordinates of an eigenvector of a certain tridiagonal matrix, and the latter becomes diagonally dominant when the order of the function is large compared to the band-limit. This apparatus (historically formulated in terms of three-term recursions, and referred to as the Bouwkamp algorithm) leads to an effective numerical scheme for the evaluation of the PSWFs, and yields a number of analytical properties of PSWFs. When the order of the PSWF is not large compared to the band-limit, the scheme still can be used as a numerical tool (though it becomes less efficient), but does not supply very much analytical information.In this article, we observe that the coefficients of the Hermite expansion of a PSWF also satisfy a three-term recursion; the latter becomes diagonally dominant when the band-limit is large compared to the order of the function (i.e., in the regime where the classical recursion loses its simplicity), and leads to asymptotic (for large band-limits) expressions for the PSWFs, their corresponding eigenvalues, and a number of related quantities.We present several such asymptotic expressions, and illustrate their behavior with numerical examples.Math Subject Classifications. 33E10, 35C20, 42C10, 35S30.

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Spheroidal Wave Functions

Cited by 155 publications

References 0 publications

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

A Simple Method for Finding Explicit Analytic Transition Densities of Diffusion Processes with General Diploid Selection

Calculation of spheroidal wave functions

High-Frequency Asymptotic Expansions for Certain Prolate Spheroidal Wave Functions

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