Direct and inverse computation of Jacobi matrices of infinite iterated function systems
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Numerische Mathematik
Direct and inverse computation of Jacobi matrices of infinite iterated function systems Giorgio Mantica
Received: 24 June 2011 / Revised: 21 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract We introduce a new set of algorithms to compute the Jacobi matrices associated with invariant measures of infinite iterated function systems, composed of one– dimensional, homogeneous affine maps. We demonstrate their utility in the study of theoretical problems, like the conjectured almost periodicity of such Jacobi matrices, the singularity of the measures, and the logarithmic capacity of their support. Since our technique is based on a reversible transformation between pairs of Jacobi matrices, it can also be applied to solve an inverse/approximation problem. The proposed algorithms are tested in significant, highly sensitive cases: they perform in a stable fashion, and can reliably compute Jacobi matrices of large order. Mathematics Subject Classification (2000) 81Q10
42C05 · 47B36 · 65D32 · 65J22 ·
1 Introduction Every non-negative probability measure μ, supported on a compact set in R, is uniquely associated with a Jacobi matrix Jμ : Supported by MIUR-PRIN “Nonlinearity and disorder in classical and quantum transport processes”. G. Mantica (B) Center for Complex and Nonlinear Systems, Dipartimento di Scienze ed Alta Tecnologia, Università dell’Insubria, Via Valleggio 11, 22100 Como, Italy e-mail: [email protected] G. Mantica INFN sezione di Milano, Milan, Italy G. Mantica CNISM unità di Como, Como, Italy
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G. Mantica
⎛
⎞ a0 b1 ⎜ b1 a1 b2 ⎟ ⎜ ⎟ Jμ := ⎜ b2 a2 b3 ⎟, ⎝ ⎠ .. .. .. . . .
(1)
where an and bn > 0 are real numbers. This symmetric tridiagonal matrix encodes the three-term recurrence relation of the orthonormal polynomials { pn (μ; s)}n∈N of μ: spn (μ; s) = bn+1 pn+1 (μ; s) + an pn (μ; s) + bn pn−1 (μ; s),
(2)
initialized by b0 p−1 (μ; s) = 0 and p0 (μ; s) = 1. As it is well known, the above follows from the defining fact that the integral pn (μ; s) pm (μ; s)dμ(s) is equal to one when n = m and is null in all other cases. The Jacobi matrix, as well as the sequence of orthogonal polynomials, is infinite whenever the cardinality of the support of μ is such. According to Gautschi [29,30,32], the computation of the Jacobi matrix associated with a given measure is a fundamental problem of numerical analysis. In fact, when the moment problem is determined [2] (this is always the case under the compactness hypothesis above), the Jacobi matrix Jμ uniquely identifies the measure μ and enables the computation of its orthogonal polynomials, via Eq. (2), and the evaluation of orthogonal sums, via Clenshaw’s algorithm [13]. Moreover, Jμ is a cornerstone of differentiation and integration using orthogonal polynomials [18] and of Gaussian quadratures [16,30,31,44], constructed via the linear algebra technique of Golub and Welsch [34,35]. Jacobi matrices play a major role also in mathematical physics, since they can be seen as discrete Schrödinger operators acting in
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