Error-Correcting Sparse Interpolation in the Chebyshev Basis
Andrew Arnold, Erich Kaltofen · 2015
We present an error-correcting interpolation algorithm for a univariate black-box polynomial that has a sparse representation using Chebyshev polynomials as a term basis. Our algorithm assumes that an upper bound on the number of erroneous evaluations is given as input, and is a generalization of the algorithm by Lakshman and Saunder [SIAM J. Comput., vol. 24 (1995)] for interpolating sparse Chebyshev polynomials and the techniques in error-correcting sparse interpolation in the usual basis of consecutive powers of the variable due to Comer, Kaltofen, and Pernet [Proc. ISSAC 2012 and 2014]. We prove the correctness of our list-decoder-based algorithm with a Descartes-rule-of-signs-like property for sparse polynomials in Chebyshev basis. We also give a new algorithm that reduces the sparse interpolation in Chebyshev basis to that in power basis, thus making the many techniques for the sparse interpolation in power basis, for instance, supersparse (lacunary) interpolation over large finite fields, available to interpolation in Chebyshev basis. Furthermore, we can customize the randomized early termination algorithms from Kaltofen and Lee [J. Symb. Comput., vol. 36 (2003)] to our new approach.