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On the infinite-dimensional QR algorithm
Spectral computations of infinite-dimensional operators are notoriously ...
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Computing spectral measures of self-adjoint operators
Using the resolvent operator, we develop an algorithm for computing smoo...
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Multi-domain spectral approach for the Hilbert transform on the real line
A multi-domain spectral method is presented to compute the Hilbert trans...
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Analysis of the spectral symbol associated to discretization schemes of linear self-adjoint differential operators
Given a linear self-adjoint 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 infinite-dimensional Metropolis-Hastings algorithm with self-decomposable priors
We study a Metropolis-Hastings 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 infinite-dimensional 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 self-adjoint 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 Radon--Nikodym 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 two-dimensional quasicrystal model.
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