Expansion of analytic functions of an operator in series of Faber polynomials

Hasson, Maurice

doi:10.1017/s0004972700031051

Cited by 4 publications

(3 citation statements)

References 10 publications

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“…see [28]. Here • E denotes the supremum norm on E. Estimates for f − n k=0 a k F k E for fixed n are given, e.g., in [14,32], [51, p. 142], and [54,Ch.…”

Section: For the Intervalmentioning

confidence: 99%

Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Ramos¹,

Sète²,

Nabben³

2021

etna

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We introduce a new polynomial preconditioner for solving the discretized Helmholtz equation preconditioned with the complex shifted Laplace (CSL) operator. We exploit the localization of the spectrum of the CSL-preconditioned system to approximately enclose the eigenvalues by a non-convex 'bratwurst' set. On this set, we expand the function 1/z into a Faber series. Truncating the series gives a polynomial, which we apply to the Helmholtz matrix preconditioned by the shifted Laplacian to obtain a new preconditioner, the Faber preconditioner. We prove that the Faber preconditioner is nonsingular for degrees one and two of the truncated series. Our numerical experiments (for problems with constant and varying wavenumber) show that the Faber preconditioner reduces the number of GMRES iterations.

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“…see [28]. Here • E denotes the supremum norm on E. Estimates for f − n k=0 a k F k E for fixed n are given, e.g., in [14,32], [51, p. 142], and [54,Ch.…”

Section: For the Intervalmentioning

confidence: 99%

Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Ramos¹,

Sète²,

Nabben³

2021

etna

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“…[8], p. 42), F p (z) = 2 cos[p · arccos z] (see e.g. [9], p. 307). In this case, by the formulas cos(t) = e it +e −it 2 , cot(t) = cos t sin t , we get …”

Section: Applicationsmentioning

confidence: 99%

Approximation in compact sets by q-Stancu–Faber polynomials, q>1

Gal

2011

Computers & Mathematics with Applications

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“…We follow essentially three sources: [16,18,20]. In [14,10] complex approximation is used in different contexts.…”

Section: The Fundamental Approximation Theorem Of Bernsteinmentioning

confidence: 99%

The degree of approximation by polynomials on some disjoint intervals in the complex plane

Hasson

2007

Journal of Approximation Theory

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Let f (z) be a continuous function defined on the compact set K ⊂ C and let E n (f ) = E n (f, K) be the degree of approximation to f, for the supremum norm on K, by polynomials of degree (at most) n. ThusHere P n denotes the space of polynomials of degree at most n and . is the supremum norm on K.For a positive integer s and for 0 < a < b, letWe show that for a large class of piecewise analytic functions f defined on K lim sup n→∞ (E n (f, K)) 1 n = s b s 2 −a s 2 b s 2 +a s 2 , thus recovering several classical results.The proof of this error estimate is then translated into an algorithm that finds the polynomial of near best approximation.

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Expansion of analytic functions of an operator in series of Faber polynomials

Cited by 4 publications

References 10 publications

Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Preconditioning the Helmholtz equation with the shifted Laplacian and Faber polynomials

Approximation in compact sets by q-Stancu–Faber polynomials, q>1

The degree of approximation by polynomials on some disjoint intervals in the complex plane

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