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

Abstract

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.

The Spherical Harmonic Spectrum of a Function with Algebraic Singularities

“…An alternative and more pragmatic approach is simply to compare theoretical convergence results for specific basis sets, derived by comparison to the Fourier case [18; 17], contour integration [10; 16] or by applying asymptotic theory [22]. Expansions for common types of singularity, for a variety of basis sets, are summarised below [7]. However, before proceeding, we give the following definition:…”

“…Specific end point behaviour is unique to polynomial expansions: in the periodic case there is no such thing as an end-point: the integral is invariant to translations of the interval over which the transform is taken. Singularities of the form |x ± 1| p as x → ∓1 have associated (fully-normalised) Legendre coefficients that scale as n −(2p+3/2) [16] but Chebyshev coefficients that scale as n −(2p+1) [7]. Note also that the coefficients decay to zero twice as fast as a singularity of the same order at an interior point: the exponents being functions of 2p compared to p. In this sense, interior singularities are twice as severe as those at the end points.…”

“…The tail of the coefficient spectrum, showing how quickly the coefficients asymptotically decrease with increasing index, gives an important indication of how well any finitely truncated approximation is converged. Furthermore, this asymptotic scaling is often intimately linked with the location of singularities of the function: empirically determined spectra can therefore provide useful diagnostics about how and where the solution loses its differentiability, which may otherwise be difficult to obtain [7].…”

Preface to fourth Special Issue on Practical Asymptotics

“…In addition, there is a paper that falls within the discipline of solid mechanics, studying edge wrinkling of thin elastic plates [10]. Finally, the issue contains two papers of a theoretical nature, one considering uniformly valid polynomial solutions to boundary-value problems [11], and the other concerned with 'beyond-all-orders' approaches to studying the asymptotic behaviour of spectral coefficients [12].The analysis contained in this issue comprises a variety of perturbation methods, including repeated use of matched asymptotic expansions (with the usual suspects of inner and outer problems, corner regions, interior layers, etc. ), boundary-layer theory, quite sophisticated multi-scale approaches, WKB analysis, exponential asymptotics, and more.…”

“…In addition, there is a paper that falls within the discipline of solid mechanics, studying edge wrinkling of thin elastic plates [10]. Finally, the issue contains two papers of a theoretical nature, one considering uniformly valid polynomial solutions to boundary-value problems [11], and the other concerned with 'beyond-all-orders' approaches to studying the asymptotic behaviour of spectral coefficients [12].…”