Numerical Methods Based on Sinc and Analytic Functions

Stenger, Frank

doi:10.1007/978-1-4612-2706-9

Cited by 594 publications

(651 citation statements)

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“…The first assertion is easy to check. Our proof of the second statement is similar to that in [22,Lemma 4.2.4]. We have to prove that…”

Section: Lemma 51 It Holds Thatsupporting

confidence: 51%

“…The Sinc-quadrature. Following [22,12,8], we construct a quadrature rule for the integral in (3.7) by using the Sinc approximation. Let…”

Section: Analyticity Of the Integrand Can Be Violated Only Ifmentioning

confidence: 99%

“…Convergence analysis. Adapting the ideas of [22,12,8], one can prove the following approximation results for functions from…”

Section: Analyticity Of the Integrand Can Be Violated Only Ifmentioning

confidence: 99%

“…The Dunford integral has previously been used in [23] (see also §4.6 of [22]) to achieve efficient inversions of differential operators, in particular, a method was derived and applied in [24] for solution of Burgers' equation. Although a different procedure was used in these references than those of the present paper, the above metioned results would be interesting for further comparisons of efficiency of the methods.…”

Section: Introductionmentioning

confidence: 99%

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Data-sparse approximation to the operator-valued functions of elliptic operator

Gavrilyuk¹,

Hackbusch

Khoromskij

2003

Math. Comp.

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Abstract. In previous papers the arithmetic of hierarchical matrices has been described, which allows us to compute the inverse, for instance, of finite element stiffness matrices discretising an elliptic operator L. The required computing time is up to logarithmic factors linear in the dimension of the matrix. In particular, this technique can be used for the computation of the discrete analogue of a resolvent (zI − L) −1 , z ∈ C.In the present paper, we consider various operator functions, the operator exponential e −tL , negative fractional powers L −α , the cosine operator function cos(t √ L)L −k and, finally, the solution operator of the Lyapunov equation. Using the Dunford-Cauchy representation, we get integrals which can be discretised by a quadrature formula which involves the resolvents (z k I − L) −1 mentioned above. We give error estimates which are partly exponentially, partly polynomially decreasing.

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“…The first assertion is easy to check. Our proof of the second statement is similar to that in [22,Lemma 4.2.4]. We have to prove that…”

Section: Lemma 51 It Holds Thatsupporting

confidence: 51%

“…The Sinc-quadrature. Following [22,12,8], we construct a quadrature rule for the integral in (3.7) by using the Sinc approximation. Let…”

Section: Analyticity Of the Integrand Can Be Violated Only Ifmentioning

confidence: 99%

“…Convergence analysis. Adapting the ideas of [22,12,8], one can prove the following approximation results for functions from…”

Section: Analyticity Of the Integrand Can Be Violated Only Ifmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Data-sparse approximation to the operator-valued functions of elliptic operator

Gavrilyuk¹,

Hackbusch

Khoromskij

2003

Math. Comp.

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“…If f decays rapidly enough at infinity, Shannon's sampling theorem asserts that C( f, h) = f for h sufficiently small if f is the restriction to IR of a function of exponential type (Paley-Wiener class, [13], p. 22 ff. ), while one has exponential convergence of C( f, h) when f is analytic in a horizontal strip about IR ( [13], p. 35 or [17], p. 136). These facts make C( f, h) unarguably the most important infinitely differentiable interpolant between equidistant points on the infinite line and on the circle, where it is the trigonometric interpolant [1].…”

Section: Introductionmentioning

confidence: 99%

A formula for the error of finite sinc-interpolation over a finite interval

Berrut

2007

Numer Algor

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Sinc collocation in quantum chemistry: Solving the planar coulomb Schr�dinger equation

Koures¹,

Harris²

1996

Int. J. Quantum Chem.

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Dirac bra-ket notation is introduced for the Whittaker cardinal (Sinc) functions and a previously unreported completeness relation for these quantities is presented and derived. With the use of this completeness relation it becomes simple to transform to a Sinc-basis the eigenvalue equations arising from a light-cone quantization of field theory or the similar equations occurring in nonrelativistic quantum mechanics. The simplicity and power of Sinc-function expansions is illustrated by computation of the eigenvalues and eigenfunctions of the position-space planar Coulomb equation, a problem for which convergence has not been achieved by a variety of other computational methods.

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Numerical Methods Based on Sinc and Analytic Functions

Cited by 594 publications

References 0 publications

Data-sparse approximation to the operator-valued functions of elliptic operator

Data-sparse approximation to the operator-valued functions of elliptic operator

A formula for the error of finite sinc-interpolation over a finite interval

Sinc collocation in quantum chemistry: Solving the planar coulomb Schr�dinger equation

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