Accelerating convergence of eigenfunction expansions

Shaw, J. K.; Johnson, L. Clark; Riess, R. D.

doi:10.1090/s0025-5718-1976-0418404-4

Cited by 10 publications

(2 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recovery of spectral accuracy in the presence of shocks and other singularities has become a "big business". A partial list of other efforts to accelerate Fourier series include [31,9,32,30,29,28,27,24,25,23,22,21,20,17,36,19,40,39,41,42,43,44,51,45,2,3,47,48,61,54,52,70,71,57,67,68,11,10] It therefore seems worthwhile to give a simple proof that this "spectral recovery" is founded on stone instead of sand.…”

Section: Definition 2 (Filter Order)mentioning

confidence: 99%

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

View full text Add to dashboard Cite

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)

show abstract

Section: Definition 2 (Filter Order)mentioning

confidence: 99%

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

View full text Add to dashboard Cite

show abstract

“…A similar idea was used in the Richardson extrapolation process [77]. Investigations and generalizations of the Krylov-Lanczos and the Eckhoff methods see also in [78][79][80][81][82][83][84][85][86][87][88][89]. Polynomial corrections applied to trigonometric expansions for the approximation/interpolation of the piecewise-smooth functions completely eliminate the Gibbs phenomenon and accelerate convergence both away from the singularities and on the entire interval.…”

Section: Introductionmentioning

confidence: 99%

On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

Poghosyan

Barkhudaryan

2021

Armen. J. Math.

View full text Add to dashboard Cite

We investigate the convergence of the quasi-periodic approximations in different frameworks and reveal exact asymptotic estimates of the corresponding errors. The estimates facilitate a fair comparison of the quasi-periodic approximations to other classical well-known approaches. We consider a special realization of the approximations by the inverse of the Vandermonde matrix, which makes it possible to prove the existence of the corresponding implementations, derive explicit formulas and explore convergence properties. We also show the application of polynomial corrections for the convergence acceleration of the quasi-periodic approximations. Numerical experiments reveal the auto-correction phenomenon related to the polynomial corrections so that utilization of approximate derivatives surprisingly results in better convergence compared to the expansions with the exact ones.

show abstract

Zur Konvergenzbeschleunigung von Fourier‐Reihen

Tasche

1979

Mathematische Nachrichten

View full text Add to dashboard Cite

EinfiihrungBekanntlich beeinflussen die Glattheit und das Randverhalten einer auf [0, 11 definierten reellwertigen Funktion z(t) die Konvergenz der zugehorigen FOURIER-Reihe 1 bk = 2 J z(s) sin 2 k~s ds, (k = I, 2, . . .) 0 wesentlich. z ( p ) ( t ) sei beschdnkt und integrierbar in [0, 11. Gilt aul3erdemEs sei x ( t ) beispielsweise p-ma1 differenzierbar ( p = 1, 2, . . .) und die p-te Ableitung (2) z(k)(O) = x(k)(l), (k = 0, 1, ..., p -l ) , so existiert nach einem Ergebnis von S. N. BERNSTEIN eine nichtnegative Konstante a~ derart, da13 fur alle n = 1, 2, ... und jedes t E [0, 11 die Ungleichung (3) J z ( t ) -.sn(t)l 5 un-p In n erfiillt ist. Dabei bedeutet ~,,(t) die n-te Teilsumme von (1) (siehe 2;. B. [3], S. 494ff.).Genugt die Funktion x(t) den Randbedingungen (2) nicht, so gilt die AbschLtzung (3) i. a. nicht und die Konvergenz von (1) erfolgt langsamer. Urn auch in diesem Pall zu einer schnell konvergierenden Fomrm-Reihe zu gelangen, bildet man die Differenz s p ( t ) = z(t) -hP(t) mit einem geeigneten Polynoin hp(t), so dalj gp(t) den Randbedingungen (2) genugt. Somit lafit sich gp(t) in eine schnell konvergierende FomER-Reihe entwickeln und x(t) ist als Summe eines Polynoms und einer schnell konvergierenden Fomm-Reihe darstellbar.

show abstract

Accelerating convergence of eigenfunction expansions

Cited by 10 publications

References 5 publications

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

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

On the Convergence of the Quasi-Periodic Approximations on a Finite Interval

Zur Konvergenzbeschleunigung von Fourier‐Reihen

Contact Info

Product

Resources

About