Lambert-W Spectral Deflation with Tight-Frame Least Squares for Retarded Delay Differential Equations

Abstract Scalar retarded delay differential equations are infinite-dimensional even when their coefficients are constant, because the characteristic equation contains infinitely many roots. This paper develops a hybrid approximation in which the slowly decaying characteristic content is removed analytically before a localized numerical discretization is applied. For the characteristic roots are represented by Lambert-$W$ branches. Exact residues associated with selected simple roots are used to cancel those poles, leaving a residual DDE with a modified history but the same differential operator. Three analytical results support the construction. First, finite modal deflation is shown to cancel the selected Laplace poles exactly. Second, under an explicit contour-shift hypothesis, the residual decays at any exponential rate to the right of the first nondeflated spectral abscissa. Third, after the prescribed history is separated from the unknown, the least-squares operator satisfies a finite-horizon coercivity estimate in $H^1$, which yields quasi-optimality and an a posteriori residual bound. The residual is discretized in a redundant piecewise-linear B-spline framelet dictionary generated by the unitary extension principle, with the linear system solved by a rank-revealing least-squares factorization. A manufactured solution gives essentially second-order maximum and RMS convergence. For the weakly damped problem $u'=-u-2u(t-1)$, at the same effective dimension $257$, dominant-pair deflation reduces the maximum error on $[0,8]$ from $2.46\times10^{-3}$ to $7.19\times10^{-4}$ and the RMS error from $1.40\times10^{-3}$ to $1.45\times10^{-4}$. Deflating additional, more oscillatory pairs at the same resolution is less accurate, which motivates a resolution-aware mode-selection rule based on the computable residual. For a final horizon $T=50$, solving the numerical residual only to $T_w=8$ and then retaining the dominant Lambert pair gives a global maximum error $7.18\times10^{-4}$, while the omitted tail after $T_w$ is below $1.67\times10^{-6}$. The main benefit is therefore not matrix conditioning, which is unchanged for a fixed frame space, but accuracy per degree of freedom and a long-time representation whose numerical part can be confined to the transient window. MSC 2020: 34K06, 34K20, 65L03, 65L10, 42C40.