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Accurate and efficient reconstruction of discontinuous functions from truncated series expansions

Author: Knut S. Eckhoff
Journal: Math. Comp. 61 (1993), 745-763
MSC: Primary 65T20; Secondary 65D10
MathSciNet review: 1195430
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Abstract: Knowledge of a truncated Fourier series expansion for a discontinuous $ 2\pi $-periodic function, or a truncated Chebyshev series expansion for a discontinuous nonperiodic function defined on the interval $ [-1, 1]$, is used in this paper to accurately and efficiently reconstruct the corresponding discontinuous function. First an algebraic equation of degree M for the M locations of discontinuities in each period for a periodic function, or in the interval $ (-1, 1)$ for a nonperiodic function, is constructed. The M coefficients in that algebraic equation of degree M are obtained by solving a linear algebraic system of equations determined by the coefficients in the known truncated expansion. By solving an additional linear algebraic system for the M jumps of the function at the calculated discontinuity locations, we are able to reconstruct the discontinuous function as a linear combination of step functions and a continuous function.

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Article copyright: © Copyright 1993 American Mathematical Society

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