A convergence acceleration method of Fourier series

Oleksy, Czesław

doi:10.1016/0010-4655(96)00044-6

Cited by 16 publications

(10 citation statements)

References 9 publications

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“…, 0.8 probably the best approximant of its sum has respectively 15.5, 13.9, 14.3, 12.9, 11.3, 9.4, 7.5 exact digits. For series (11) (which very slowly converges at x = 1) accuracy is a little greater. Also for these x methods from Section 3 are more efficient.…”

Section: Theoremmentioning

confidence: 96%

“…After k iterations of such procedure we obtain a linear combination of k + 1 series. Oleksy [11] suggests using a standard convergence acceleration method (e.g. the ε-algorithm or Levin's t-transform) to each of these series and not to the original series.…”

Section: Example 4 the Last Relation Implies Thatmentioning

confidence: 99%

“…Two series (15) for q = 1 represent the indirect interaction energy of a periodic configuration of adsorbed atoms (for details, see [11]). …”

Section: Test Seriesmentioning

confidence: 99%

“…The algorithm was verified, among others, for series (10)(11)(12). In three different cases, namely for x 0.1, x 0.2 and x −0.2, the sum of each series after its transform was computed with at least 17 accurate digits.…”

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confidence: 99%

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Convergence acceleration of orthogonal series

Paszkowski

2007

Numer Algor

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The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin's and Weniger's methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler-Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.

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Section: Theoremmentioning

confidence: 96%

Section: Example 4 the Last Relation Implies Thatmentioning

confidence: 99%

“…Two series (15) for q = 1 represent the indirect interaction energy of a periodic configuration of adsorbed atoms (for details, see [11]). …”

Section: Test Seriesmentioning

confidence: 99%

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confidence: 99%

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Convergence acceleration of orthogonal series

Paszkowski

2007

Numer Algor

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“…which arises in the phase transitions of absorbed submonolayers on metal surfaces [40], and is also the real part of a special case of the ''Lerch zeta-function" in mathematics. If the series is truncated after the N-th term, the error falls only as fast as 1=N over the entire interval except in zones of width Oð1=NÞ near x ¼ AEp where the error is always Oð1Þ, the Gibbs Phenomenon.…”

Section: The Basic Ideamentioning

confidence: 99%

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

Boyd

2009

Journal of Computational Physics

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A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon

Boyd

2010

Lecture Notes in Computational Science and Engineering

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When a function f (x) is singular at a point x s on the real axis, its Fourier series, when truncated at the N-th term, gives a pointwise error of only O(1/N) over the entire real axis. Such singularities spontaneously arise as "fronts" in meteorology and oceanography and "shocks" in other branches of fluid mechanics. It has been previously shown that it is possible to recover an exponential rate of convegence at all points away from the singularity in the sense that |f (x)

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A convergence acceleration method of Fourier series

Cited by 16 publications

References 9 publications

Convergence acceleration of orthogonal series

Convergence acceleration of orthogonal series

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

A Proof, Based on the Euler Sum Acceleration, of the Recovery of an Exponential (Geometric) Rate of Convergence for the Fourier Series of a Function with Gibbs Phenomenon

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