On the restriction to unitarity for rational approximations to the exponential function

TL;DR

This paper compares Chebyshev and unitary best rational approximations to the exponential function, showing that restricting to unitarity doubles the approximation error but remains practical, with both methods sharing similar asymptotic errors near zero.

Contribution

It demonstrates that the unitarity constraint does not significantly impair approximation quality and clarifies the relationship between Chebyshev and unitary best approximations for the exponential.

Findings

Chebyshev approximants are not unitary.

Unitary best approximation has at most twice the error of Chebyshev approximation.

Both approximations share the same asymptotic error near the origin.

Abstract

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.