Stokes’s Mathematical Work

2019 ◽  
pp. 129-140
Author(s):  
Richard B. Paris
Keyword(s):  
Fourier Series ◽  
Asymptotic Analysis ◽  
Asymptotic Behaviour ◽  
Uniform Convergence ◽  
Applied Mathematics ◽  
Mathematical Work ◽  
Generalized Hypergeometric Function ◽  
Moving Loads ◽  
Railway Bridges ◽  
Positive Argument

This chapter gives an account of the seven purely mathematical papers written by Stokes. The first is an account of his famous memoir on Fourier series in which he discussed modes of convergence and introduced the idea of uniform convergence. A second paper dealing with moving loads over railway bridges represents Stokes’s only foray into industrial applied mathematics. The remaining five papers are concerned with asymptotic analysis in which he considered an approximation for the zeros of an integral measuring the intensity of light in the neighbourhood of a caustic applied to the familiar rainbow. This eventually led him to resolving a paradox in asymptotics that is now known as the Stokes phenomenon. A final paper gives an estimate of the asymptotic behaviour of the generalized hypergeometric function for large positive argument.

Asymptotic analysis of the general stochastic epidemic with variable infectious periods

2001 ◽  
Vol 38 (01) ◽  
pp. 18-35 ◽  
Author(s):  
A. N. Startsev
Keyword(s):  
Asymptotic Analysis ◽  
Asymptotic Behaviour ◽  
Size Distribution ◽  
Random Variables ◽  
Independent Random Variables ◽  
Boundary Crossing ◽  
Total Size ◽  
Stochastic Epidemic ◽  
General Stochastic

A generalisation of the classical general stochastic epidemic within a closed, homogeneously mixing population is considered, in which the infectious periods of infectives follow i.i.d. random variables having an arbitrary but specified distribution. The asymptotic behaviour of the total size distribution for the epidemic as the initial numbers of susceptibles and infectives tend to infinity is investigated by generalising the construction of Sellke and reducing the problem to a boundary crossing problem for sums of independent random variables.

Eli’s Impact: A Case Study

2014 ◽  
Author(s):  
Charles Fefferman
Keyword(s):  
Fourier Series ◽  
Boltzmann Equation ◽  
Heat Kernel ◽  
Recent Work ◽  
Lie Group ◽  
Applied Mathematics ◽  
Compact Lie Group ◽  
Classical Study ◽  
The Right

This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. Before Eli, Littlewood–Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Stein, however, found the right viewpoint to develop Littlewood–Paley theory and went on to develop Littlewood–Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Afterward, the chapter discusses the remarkable recent work of Gressman and Strain on the Boltzmann equation, and explains in particular its connection to Stein's work.

scholarly journals Pointwise and Uniform Convergence of Fourier Extensions

2019 ◽  
Vol 52 (1) ◽  
pp. 139-175
Author(s):  
Marcus Webb ◽  
Vincent Coppé ◽  
Daan Huybrechs
Keyword(s):  
Fourier Series ◽  
Uniform Convergence ◽  
Pointwise Convergence ◽  
Gibbs Phenomenon ◽  
Lebesgue Function ◽  
Bernstein Type ◽  
Legendre Series ◽  
The Best Approximation ◽  
Open Questions ◽  
Algebraic Order

AbstractFourier series approximations of continuous but nonperiodic functions on an interval suffer the Gibbs phenomenon, which means there is a permanent oscillatory overshoot in the neighborhoods of the endpoints. Fourier extensions circumvent this issue by approximating the function using a Fourier series that is periodic on a larger interval. Previous results on the convergence of Fourier extensions have focused on the error in the $$L^2$$ L 2 norm, but in this paper we analyze pointwise and uniform convergence of Fourier extensions (formulated as the best approximation in the $$L^2$$ L 2 norm). We show that the pointwise convergence of Fourier extensions is more similar to Legendre series than classical Fourier series. In particular, unlike classical Fourier series, Fourier extensions yield pointwise convergence at the endpoints of the interval. Similar to Legendre series, pointwise convergence at the endpoints is slower by an algebraic order of a half compared to that in the interior. The proof is conducted by an analysis of the associated Lebesgue function, and Jackson- and Bernstein-type theorems for Fourier extensions. Numerical experiments are provided. We conclude the paper with open questions regarding the regularized and oversampled least squares interpolation versions of Fourier extensions.

Modifications of Fourier transforms and singular continuous measures

1980 ◽  
Vol 87 (3) ◽  
pp. 383-392
Author(s):  
Alan MacLean
Keyword(s):  
Fourier Transform ◽  
Fourier Series ◽  
Asymptotic Behaviour ◽  
Fourier Transforms ◽  
Sufficient Conditions ◽  
The Other ◽  
Absolutely Continuous ◽  
Necessary And Sufficient ◽  
Continuous Measures ◽  
Absolutely Continuous Measures

It has long been known, after Wiener (e.g. see (11), vol. 1, p. 108, (5), (8), §5·6)) that a measure μ whose Fourier transform vanishes at infinity is continuous, and generally, that μ is continuous if and only if is small ‘on the average’. Baker (1) has pursued this theme and obtained concise necessary and sufficient conditions for the continuity of μ, again expressed in terms of the rate of decrease of . On the other hand, for continuous μ, Rudin (9) points out the difficulty in obtaining criteria based solely on the asymptotic behaviour of by which one may determine whether μ has a singular component. The object of this paper is to show further that any such criteria must be complicated indeed. We shall show that the absolutely continuous measures on T = [0, 2π) whose Fourier transforms are the most well-behaved (namely, those of the form (1/2π)f(x)dx, where f has an absolutely convergent Fourier series) are such that one may modify their transforms on ‘large’ subsets of Z so that they become the transforms of singular continuous measures. Moreover, the singular continuous measures in question may be chosen so that their Fourier transforms do not vanish at infinity.

XLI.—On Sommerfeld's Form of Fourier's Repeated Integrals

1912 ◽  
Vol 31 ◽  
pp. 587-603
Author(s):  
W. H. Young
Keyword(s):  
Fourier Series ◽  
Applied Mathematics ◽  
Lebesgue Integral ◽  
General Hypothesis ◽  
Differential Coefficient ◽  
Right Hand ◽  
Fourier Sine ◽  
Repeated Integral ◽  
The Right

§ 1. In his treatise on Fourier Series and Integrals Carslaw quotes without proof Sommerfeld's theorem thatwhen the limit on the right-hand side exists. In applied mathematics, he remarks, it is this limit, rather than the corresponding Fourier repeated integral which occurs.In the present paper I propose to extend this result in various ways. After proving Sommerfeld's result on the general hypothesis, not considered by him, that the integral is a Lebesgue integral, I show that the limit in question is whenever the origin is a point at which f(u) is the differential coefficient of its integral, and I obtain the corresponding results for In all their generality these statements are only true when the interval (0, p) is a finite one. I then show how, under a variety of hypotheses with respect to the nature of f(x) at infinity, they can be extended so as to be still true when p = + ∞ . These hypotheses correspond precisely to those which have been proved f to be sufficient for the corresponding statements as to the Fourier sine and cosine repeated integrals in their usual forms.

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