Asymptotics and bounds for the zeros of Laguerre polynomials: a survey
01 Jul 2002-Journal of Computational and Applied Mathematics (Elsevier Science Publishers B. V.)-Vol. 144, Iss: 1, pp 7-27
TL;DR: In this paper, two uniform asymptotic representations of the Bessel function Jα(x) and the Airy function Ai(x), respectively, are presented for the Laguerre polynomial Ln α(x).
About: This article is published in Journal of Computational and Applied Mathematics.The article was published on 2002-07-01 and is currently open access. It has received 43 citations till now. The article focuses on the topics: Laguerre polynomials & Hermite polynomials.
Citations
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TL;DR: In this paper, the interference pattern formed by the light beam and a reference field is analyzed to characterize the beam radial distribution, thus retrieving the entire information contained in the optical field.
Abstract: Light beams carrying orbital angular momentum are key resources in modern photonics. In many applications, the ability of measuring the complex spectrum of structured light beams in terms of these fundamental modes is crucial. Here we propose and experimentally validate a simple method that achieves this goal by digital analysis of the interference pattern formed by the light beam and a reference field. Our approach allows one to characterize the beam radial distribution also, hence retrieving the entire information contained in the optical field. Setup simplicity and reduced number of measurements could make this approach practical and convenient for the characterization of structured light fields.
82 citations
Cites background from "Asymptotics and bounds for the zero..."
...Here rmax is the maximum radius available on the sensor; rmin is the minimum radial distance where azimuthal oscillation associated with the OAM content of the LGp,m mode can be detected, before facing aliasing issues; r1 is a lower bound for the first root of the Laguerre polynomials contained in the expression of LG modes; similarly, rp is the upper bound for the p−th root, while r̃p, with rp < r̃p, delimits the oscillatory region of the Laguerre polynomials [53, 54]....
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TL;DR: In this paper, a unified fast time-stepping method for both fractional integral and derivative operators is proposed, where the fractional operator is decomposed into a local part with memory length and a history part, and the history part is approximated by a fast memory saving method.
Abstract: A unified fast time-stepping method for both fractional integral and derivative operators is proposed. The fractional operator is decomposed into a local part with memory length $$\varDelta T$$
and a history part, where the local part is approximated by the direct convolution method and the history part is approximated by a fast memory-saving method. The fast method has $$O(n_0+\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))$$
active memory and $$O(n_0n_T+ (n_T-n_0)\sum _{\ell }^L{q}_{\alpha }(N_{\ell }))$$
operations, where $$L=\log (n_T-n_0)$$
, $$n_0={\varDelta T}/\tau ,n_T=T/\tau $$
, $$\tau $$
is the stepsize, T is the final time, and $${q}_{\alpha }{(N_{\ell })}$$
is the number of quadrature points used in the truncated Laguerre–Gauss (LG) quadrature. The error bound of the present fast method is analyzed. It is shown that the error from the truncated LG quadrature is independent of the stepsize, and can be made arbitrarily small by choosing suitable parameters that are given explicitly. Numerical examples are presented to verify the effectiveness of the current fast method.
78 citations
20 Nov 2017
TL;DR: In this article, the interference pattern formed by the light beam and a reference field is used to characterize the beam radial distribution, thus retrieving the entire information contained in the optical field.
Abstract: Light beams carrying orbital angular momentum are key resources in modern photonics. In many applications, the ability to measure the complex spectrum of structured light beams in terms of these fundamental modes is crucial. Here we propose and experimentally validate a simple method that achieves this goal by digital analysis of the interference pattern formed by the light beam and a reference field. Our approach allows one to also characterize the beam radial distribution, hence retrieving the entire information contained in the optical field. Setup simplicity and reduced number of measurements could make this approach practical and convenient for the characterization of structured light fields.
54 citations
TL;DR: In this article, the spinless modular bootstrap for conformal field theories with current algebra U(1)c × U( 1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions, was studied.
Abstract: We carry out a numerical study of the spinless modular bootstrap for conformal field theories with current algebra U(1)c × U(1)c, or equivalently the linear programming bound for sphere packing in 2c dimensions. We give a more detailed picture of the behavior for finite c than was previously available, and we extrapolate as c → ∞. Our extrapolation indicates an exponential improvement for sphere packing density bounds in high dimen- sions. Furthermore, we study when these bounds can be tight. Besides the known cases c = 1/2, 4, and 12 and the conjectured case c = 1, our calculations numerically rule out sharp bounds for all other c < 90, by combining the modular bootstrap with linear programming bounds for spherical codes.
35 citations
TL;DR: The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial and achieves a complexity as low as O( √ n) operations.
Abstract: A fast and accurate algorithm for the computation of Gauss-Hermite and generalized Gauss-Hermite quadrature nodes and weights is presented. The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial. In the Gauss-Hermite case the initial guesses and evaluation scheme rely on explicit asymptotic formulas. For generalized Gauss-Hermite, the initial guesses are furnished by sampling a certain equilibrium measure and the associated polynomial evaluated via a Riemann-Hilbert reformulation. In both cases the n-point quadrature rule is computed in O(n) operations to an accuracy that is close to machine precision. For sufficiently largen, some of the quadrature weights have a value less than the smallest positive normalized floating-point number in double precision and we exploit this fact to achieve a complexity as low as O( √ n). a w(x)f(x)dx ≈ n X k=1 wkf(xk)
29 citations
Cites methods from "Asymptotics and bounds for the zero..."
...In order to compute Gatteschi’s initial guesses, the zeros of the Airy function are required....
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...Tricomi’s initial guesses for the nodes are accurate except for a handful near√ 2n+ 1, and for these nodes we use the asymptotic approximations derived by Gatteschi [9]: Lemma 3.2 (Gatteschi [9])....
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...In Figure 3.1 (left) we show the absolute error in Tricomi’s and Gatteschi’s initial guesses for n = 1,000....
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...Tricomi’s initial guesses for the nodes are accurate except for a handful near √ 2n+ 1, and for these nodes we use the asymptotic approximations derived by Gatteschi [9]: 4...
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...In practice, we use Tricomi’s initial guesses for k = 0, . . . , ⌊ρn⌋, where ρ = 0.4985, and Gatteschi’s otherwise....
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TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.
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TL;DR: The basic concepts of asymptotic expansions, Mellin transform techniques, and the distributional approach are explained.
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