Zur Konvergenzbeschleunigung von Fourier‐Reihen

Tasche, Manfred

doi:10.1002/mana.19790900110

Cited by 14 publications

(1 citation statement)

References 8 publications

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“…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.

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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.

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Section: Introductionmentioning

confidence: 99%

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

Poghosyan

Barkhudaryan

2021

Armen. J. Math.

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Fourier Series and Right Invertible Operators

Delvos¹

1985

Math. Nachr.

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In several papers [S, 9, 101 M. TASCHE has applied the algebraic theory of right invertible operators [ . ; I to problems of interpolat.ion and approximation. In a recent paper [2] we used the method of parametric extension to construct right invertible operators in spaces of bivariate continuous functions. right invertible operator in the linear space C(J2) of bivariate continuous functions on the square JZ (J=[O, %I) which is related to the expansion of bivariate functions into double FOURIER series. In particuIar, we will derive a decomposition formula for smooth functions in C'J(J2) which yields as D special application the absolute convergence of the double FOURIER series of (t periodic function in C'.'(J2).It is the objective of this paper to construct Constriiction of right invertible operatorsIn this section we will describe a method for constructing right invertible operators in a real or complex linear space X (see also [2], section 3). Assume that P nncl R are linear operators on X such that (1.1) ker (R) = (0) , (1.2) p2 = P , (1.3) P R = O . The subset X, defined by is a linear subspace of X. For any f EX, condition (1.1) guarantees the existence of a unique Bf in Y such that (1.4) The linearity of P, R, XI and (1.1) imply that B is a linear operator in X with X1={fcX : f-Pfcim (R)} f = P ( f ) +R(Bf) * dom @)=XI. 7.

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Error Estimates for Sine Series Expansions

Baszenski

Franz-Jürgen

1988

Mathematische Nachrichten

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Iii troduc tioiiThe expansion of smooth functions into trigonometric sine series is descrjbed in a detailed way in JONES/HARDY [GI, SHAW/JOHNSON/RIESS [lo], and TASCHE [ll].If a function x ( t ) is defined on [0, 11 and is sufficiently smooth, then the asymptotic magnitude of its FOCRIER coefficients x : in the expansion(1)(2) and the order of convergence of its associated series expansion (1) depend only on the boundary ralues of x(t,, more explicitly on the largest integer p with 1 6 (3) x(?k)(O)=x(2'~)(1)=0 ( k = O , . . . , p -1) .If x ( t ) is smooth but (3) fails to be satisfied, then a polynomial y ( t ) can be determined which corrects the boundary values of x(t). This polynomial P-1(4 solves the Lidstone interpolation problem, where the Lidstone polynomials Ak(t) in (4) are recursively defined by ( 5 ) Ao(t) =t , A;+l(t) =a&) , Ak+l(O) =&+I( 1) = 0 .TAWHE gives an &-error estimate for the partial sums of the corrected function z ( t ) -y ( t ) . JONES/HARDY state an order of convergence in the TCHEBYCHEV norm.In the sequel we mill prove sharper e-estimates than JONES/HARDY'S (which are

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Zur Konvergenzbeschleunigung von Fourier‐Reihen

Cited by 14 publications

References 8 publications

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

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

Fourier Series and Right Invertible Operators

Error Estimates for Sine Series Expansions

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