
On the infinitedimensional QR algorithm
Spectral computations of infinitedimensional operators are notoriously ...
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Computing spectral measures of selfadjoint operators
Using the resolvent operator, we develop an algorithm for computing smoo...
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Multidomain spectral approach for the Hilbert transform on the real line
A multidomain spectral method is presented to compute the Hilbert trans...
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Analysis of the spectral symbol associated to discretization schemes of linear selfadjoint differential operators
Given a linear selfadjoint differential operator L along with a discret...
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On the computation of recurrence coefficients for univariate orthogonal polynomials
Associated to a finite measure on the real line with finite moments are ...
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Lagrange approximation of transfer operators associated with holomorphic data
We show that spectral data of transfer operators given by holomorphic da...
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Convergence and perturbation theory for an infinitedimensional MetropolisHastings algorithm with selfdecomposable priors
We study a MetropolisHastings algorithm for target measures that are ab...
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Computing Spectral Measures and Spectral Types: New Algorithms and Classifications
Despite new results on computing the spectrum, there has been no general method able to compute spectral measures (as given by the classical spectral theorem) of infinitedimensional normal operators. Given a matrix representation, we show that if each matrix column decays at infinity at a known asymptotic rate, then it is possible to compute spectral measures of selfadjoint and unitary linear operators on separable Hilbert spaces. The central ingredient of the new algorithm is the computation of the resolvent operator with error control. Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of a problem through the number of limits needed to numerically compute the solution. We classify the computation of measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus and RadonNikodym derivatives in the SCI hierarchy for such operators. The new algorithms are demonstrated to be efficient and practical on examples taken from orthogonal polynomials on the real line and the unit circle, and are also applied to evolution equations on a twodimensional quasicrystal model.
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