| Abstract: |
| We investigate the Gibbs overshoot for partial sums of symmetric Krawtchouk expansions of a centered sign function. By combining parity reduction, an exact coefficient identity on a neighboring lattice, and one-point WKB asymptotics, we obtain a continuum-limit description of the overshoot profile for degree cutoffs of the form $m\sim \delta N$, where $\delta$ is the truncation fraction. This leads to a one-parameter family of discrete Gibbs constants interpolating between the classical Fourier Gibbs constant as an upper bound and a recently numerically observed Krawtchouk Gibbs constant as a lower bound. We discuss how this framework may extend to other discrete orthogonal polynomial families. |
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