Asymptotic Estimates of Fourier Coefficients

Elliott, David; Tuan, P. D.

doi:10.1137/0505001

Cited by 16 publications

(14 citation statements)

References 3 publications

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“…As we shall see, these results are often overestimated. For special cases of the Gegenbauer spectral expansion such as Chebyshev and Legendre expansions, the estimates of their expansion coefficients have been studied extensively in the past few decades (see [3,5,8,10,31,32,33] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

Wang¹

2016

SIAM J. Numer. Anal.

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In this paper, we study optimal estimates and comparison of the coefficients in the Gegenbauer series expansion. We propose an alternative derivation of the contour integral representation of the Gegenbauer expansion coefficients which was recently derived by Cantero and Iserles [SIAM J. Numer. Anal., 50 (2012), pp. . With this representation, we show that optimal estimates for the Gegenbauer expansion coefficients can be derived, which in particular includes Legendre coefficients as a special case. Further, we apply these estimates to establish some rigorous and computable bounds for the truncated Gegenbauer series. In addition, we compare the decay rates of the Chebyshev and Legendre coefficients. For functions whose singularity is outside or at the endpoints of the expansion interval, asymptotic behaviour of the ratio of the nth Legendre coefficient to the nth Chebyshev coefficient is given, which provides us an illuminating insight for the comparison of the spectral methods based on Legendre and Chebyshev expansions.

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Section: Introductionmentioning

confidence: 99%

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

Wang¹

2016

SIAM J. Numer. Anal.

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“…The method of steepest descent is ancient; its application to Fourier and Chebyshev series coefficient integrals began with the pioneering work of D. Elliott and his students: Elliott [24,25], Elliott and Szekeres [26], Elliott and Tuan [27], Nemeth [37] and Miller [36] in the mid-60's. Boyd has used the method to optimize a variety of spectral algorithms [7-10, 12, 13].…”

Section: The Methods Of Steepest Descentsmentioning

confidence: 99%

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

Boyd¹

2005

J Sci Comput

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In constructing local Fourier bases and in solving differential equations with nonperiodic solutions through Fourier spectral algorithms, it is necessary to solve the Fourier Extension Problem. This is the task of extending a nonperiodic function, defined on an interval x ∈ [−χ, χ], to a functionf which is periodic on the larger interval x ∈ [−Θ, Θ]. We derive the asymptotic Fourier coefficients for an infinitely differentiable function which is one on an interval x ∈ [−χ, χ], identically zero for |x| > Θ, and varies smoothly in between. Such smoothed "top-hat" functions are "bells" in wavelet theory. Our bell is (for x ≥ 0) T (x; L, χ, Θ) = (1 + erf(z))/2 where z = Lξ/ 1 − ξ 2 where ξ ≡ −1 + 2(Θ − x)/(Θ − χ). By applying steepest descents to approximate the coefficient integrals in the limit of large degree j , we show that when the width L is fixed, the Fourier cosine coefficients a j of T on x ∈ [−Θ, Θ] are proportional to a j ∼ (1/j) exp(−Lπ 1/2 2 −1/2 (1 − χ/Θ) 1/2 j 1/2)Λ(j) where Λ(j) is an oscillatory factor of degree given in the text. We also show that to minimize error in a Fourier series truncated after the N th term, the width should be chosen to increase with N as L = 0.91 1 − χ/ΘN 1/2. We derive similar asymptotics for the function f (x) = x as extended by a more sophisticated scheme with overlapping bells; this gives an even faster rate of Fourier convergence.

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“…The Fourier-ClootWeideman basis, orthogonal rational functions, Laguerre functions and Hermite expansions for f (x) that decay faster than exp(−Ax 2 ) ("super-Gaussians") are similarly analyzed in [62,77,78] and [31], respectively. To illustrate the ideas, however, we shall concentrate on a simple example: Hermite expansions of functions that decay for large x in proportional to exp(−A|x| k ) for some k < 2 ("sub-Gaussian functions") [31].…”

Section: Steepest Descent For Hermite Coefficients Of Sub-gaussian Fumentioning

confidence: 99%

“…Furthermore, asymptotic methods like steepest descent have historically been used to bypass the use of Darboux's Principle and the Method of Model Functions to compute the consequences of a pole or branch point directly, especially in the work of David Elliott and his collaborators [58][59][60][61][62][63], Jet Wimp [64,65], and Géza Németh [56].…”

Section: Asymptotics-by-examplementioning

confidence: 99%

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

Boyd

2008

J Eng Math

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When a function is expanded asfor some set of basis functions φ j (x), its spectral coefficients a n generally have an asymptotic approximation, as n → ∞, in the form of an inverse power series plus terms that decrease exponentially with n. If f (x) is analytic on the expansion interval, then all the coefficients of the inverse power series are zero and the problem becomes one of "beyond-all-orders" or "exponential" asymptotics. The method of steepest descent for integrals and other complexpath integration techniques can successfully connect the exponentially small behavior of the spectral coefficients to the singularities of f (x) off the expansion interval. Many examples are given in both one and two dimensions.

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Asymptotic Estimates of Fourier Coefficients

Cited by 16 publications

References 3 publications

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

On the Optimal Estimates and Comparison of Gegenbauer Expansion Coefficients

Asymptotic Fourier Coefficients for a C∞ Bell (Smoothed-“Top-Hat”) & the Fourier Extension Problem

Large-degree asymptotics and exponential asymptotics for Fourier, Chebyshev and Hermite coefficients and Fourier transforms

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