Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of

Boyd, John P.

doi:10.1016/j.jcp.2008.10.039

Cited by 20 publications

(12 citation statements)

References 46 publications

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“…There are numerous other methods for resolving the Gibbs phenomenon, and we have not presented a complete list. The reader is referred to [17,34,49] and references therein for further information. We also mention that many methods designed for the related problem of recovering high accuracy from function values on equispaced grids (i.e.…”

Section: The Gibbs Phenomenon and Its Resolutionmentioning

confidence: 99%

A Stability Barrier for Reconstructions from Fourier Samples

Adcock¹,

Hansen²,

Shadrin³

2014

SIAM J. Numer. Anal.

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We prove that any stable method for resolving the Gibbs phenomenon -that is, recovering high-order accuracy from the first m Fourier coefficients of an analytic and nonperiodic function -can converge at best root-exponentially fast in m. Any method with faster convergence must also be unstable, and in particular, exponential convergence implies exponential ill-conditioning. This result is analogous to a recent theorem of Platte, Trefethen & Kuijlaars concerning recovery from pointwise function values on an equispaced m-grid. The main step in our proof is an estimate for the maximal behaviour of a polynomial of degree n with bounded m-term Fourier series. This bound is related to a conjecture of Hrycak & Gröchenig. In the second part of the paper we discuss the implications of our main theorem to polynomial-based interpolation and least-squares approaches for overcoming the Gibbs phenomenon. Finally, we propose the use of so-called Fourier extensions as an attractive alternative for this problem. We present numerical results demonstrating rapid convergence of the resulting approximation in a stable manner.

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Section: The Gibbs Phenomenon and Its Resolutionmentioning

confidence: 99%

A Stability Barrier for Reconstructions from Fourier Samples

Adcock¹,

Hansen²,

Shadrin³

2014

SIAM J. Numer. Anal.

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“…This formula is unsuitable for numerical computation when π < x ≤ 2π; instead we use the fact that Cl n (x) = (−1) n+1 Cl n (2π − x) when x is in this range. More on the computation of Clausen functions can be found in [10]. More on the computation of Clausen functions can be found in [10].…”

Section: Discussionmentioning

confidence: 99%

Lattice Sums for the Helmholtz Equation

Linton¹

2010

SIAM Rev.

133

138

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A survey of different representations for lattice sums for the Helmholtz equation is made. These sums arise naturally when dealing with wave scattering by periodic structures. One of the main objectives is to show how the various forms depend on the dimension d of the underlying space and the lattice dimension d Λ . Lattice sums are related to, and can be calculated from, the quasi-periodic Green's function and this object serves as the starting point of the analysis. LATTICE SUMS FOR THE HELMHOLTZ EQUATION 631to mathematically challenging problems. Attempts to evaluate the sum in (1.1) in closed form have so far been unsuccessful, though some remarkable identities have been derived in the process [23,113].The lattice sum in (1.1) arises from a sum of singular solutions of Laplace's equation. Here we will be concerned with related sums in which the underlying physics is governed by the Helmholtz equationwhich arises naturally in many contexts, particularly when investigating linear wave phenomena at a given frequency. For example, in acoustics u represents fluctuations in pressure, whereas in elasticity it would be some component of the displacement vector, and in electromagnetism a component of the electric or magnetic field. The quantity k (assumed real and positive) is the wavenumber, related to the frequency ω via some appropriate dispersion relation. In diffraction theory we are often faced with trying to solve (1.2) in a region exterior to some scatterer(s) (and maybe inside as well) subject to boundary conditions on the surface of the scatterer(s). The case where the scatterer is periodic represents an important class of such problems; examples include the study of diffraction gratings, scattering by periodic surfaces, and wave propagation through composite materials. Equation (1.2) is also equivalent to the time-independent Schrödinger equation in the presence of a constant potential, in which case k is related to the total energy of the particle under consideration, and as a result the Helmholtz equation finds application in the study of electron scattering in solids. A common simplification used is the so-called muffin-tin approximation (due to Slater [99]), in which the potential is supposed to be spherically symmetric within spheres surrounding each atom and constant in the region inbetween these spheres. This then leads naturally to the need to solve (1.2) on a periodic domain.The study of wave propagation in periodic structures has a long history and leads to a whole range of interesting mathematical problems. The classic text by Brillouin [11] arguably still serves as the best introduction to the subject, though many significant advances have been made since it was written. For example, the mathematical theory of diffraction gratings (based on functional analysis) is developed in books such as [120] and there is an ever-growing literature describing the rapidly developing field of photonic and phononic crystals; an excellent text is [39]. One of the consequences of periodicity is that a problem on...

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“…Note that, although we have used such functions as a theoretical tool, their practical application to the removal of the Gibbs phenomenon in certain Fourier series has been considered in [11].…”

Section: Interior Gibbs Phenomenonmentioning

confidence: 99%

“…It is a testament to the importance of the Gibbs phenomenon that the development of techniques for its amelioration, and indeed, complete removal, remains an active area of inquiry. The list of existing methods includes filtering [38], Gegenbauer reconstruction [20,21], techniques based on extrapolation [14][15][16], Padé methods [13] and Fourier extension/continuation methods [10,22], to name but a few (for a more comprehensive survey see [11,38] and references therein). All such methods rely on one common principle: the Gibbs phenomenon is so regular, and so well understood mathematically, that it is possible to devise techniques to circumvent it.…”

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confidence: 99%

Gibbs phenomenon and its removal for a class of orthogonal expansions

Adcock

2010

Bit Numer Math

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We detail the Gibbs phenomenon and its resolution for the family of orthogonal expansions consisting of eigenfunctions of univariate polyharmonic operators equipped with homogeneous Neumann boundary conditions. As we establish, this phenomenon closely resembles the classical Fourier Gibbs phenomenon at interior discontinuities. Conversely, a weak Gibbs phenomenon, possessing a number of important distinctions, occurs near the domain endpoints. Nonetheless, in both cases we are able to completely describe this phenomenon, including determining exact values for the size of the overshoot.Next, we demonstrate how the Gibbs phenomenon can be both mitigated and completely removed from such expansions using a number of different techniques. As a by-product, we introduce a generalisation of the classical Lidstone polynomials.

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Acceleration of algebraically-converging Fourier series when the coefficients have series in powers of

Cited by 20 publications

References 46 publications

A Stability Barrier for Reconstructions from Fourier Samples

A Stability Barrier for Reconstructions from Fourier Samples

Lattice Sums for the Helmholtz Equation

Gibbs phenomenon and its removal for a class of orthogonal expansions

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