Summary
The problem of generating the recurrence coefficients of orthogonal polynomials from the moments or from modified moments of the weight function is well understood for positive weight distributions. Here we extend this theory and the basic algorithms to the case of an indefinite weight function. While the generic indefinite case is formally not much different from the positive definite case, there exist nongeneric degenerate situations, and these require a different more complicated treatment. The understanding of these degenerate situations makes it possible to construct a stable approximate solution of an ill-conditioned problem.
The application to adaptive iterative methods for linear systems of equations is anticipated.
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Dedicated to R.S. Varga on the occasion of his sixtieth birthday
G.H. Golub (USA, Canada), M.H. Gutknecht (other countries)
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Golub, G.H., Gutknecht, M.H. Modified moments for indefinite weight functions. Numer. Math. 57, 607–624 (1990). https://doi.org/10.1007/BF01386431
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DOI: https://doi.org/10.1007/BF01386431