2024/10/09 by Tobias Jawecki, Jawecki, Tobias
Mathematics · #33B10 #41A20 #41A50 #Approximation Theory and Sequence Spaces #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2410.06903
openalex publication_date 2024/10/09 · openalex created_date 2024/10/12 · openalex updated_date 2026/07/28
In the present work we consider rational best approximations to the exponential function that minimize a uniform error on a subset of the imaginary axis. Namely, Chebyshev approximation and unitary best approximation where the latter is subject to further restriction to unitarity, i.e., requiring that the imaginary axis is mapped to the unit circle. We show that Chebyshev approximants are not unitary, and consequently, distinct to unitary best approximants. However, unitary best approximation attains at most twice the error of Chebyshev approximation, and thus, the restriction to unitarity is not a severe restriction in a practical setting. Moreover, Chebyshev approximation and unitary best approximation attain the same asymptotic error as the underlying domain of approximation shrinks to the origin.