Clustered-index asymptotics for the KEG-D method

Establish the boundary-value and \(L_2\)-error asymptotics of the KEG-D method for clustered configurations in which the limiting ratios \(n_s/N\) have multiplicity, under the regularity condition \(f\in C^{q+\alpha-1}([-1,1];\mathbb R^2)\) with \(f^{(q+\alpha-1)}\) absolutely continuous and with \(p\) and \(r\) sufficiently smooth to support the required higher-order eigenfunction asymptotic expansions.

Background

The main asymptotic results assume that the limiting ratios $c_s=n_s/N$ are pairwise distinct within each parity class. The remark considers the excluded case in which several ratios converge to the same limit, with maximal multiplicity $\alpha$. In analogy with the trigonometric case, the authors expect the boundary-value asymptotics and the associated $L_2$-error asymptotics to remain valid after increasing the regularity assumptions and incorporating the additional reduced boundary values $g_{q+1}^{\pm},\ldots,g_{q+\alpha-1}^{\pm}$ into the interpolation system. This extension is explicitly left unproved.