A hybrid gaussian–wavelet multiscale algorithm for zero localization in oscillatory functions

We propose a scale-normalized Gaussian–wavelet algorithm for zero localization in oscillatory functions. Unlike pointwise sign-change or residual-based strategies, the proposed approach identifies candidate zeros as minimizers of an integral multiscale functional that couples local amplitude and derivative-sensitive variation. Under standard smoothness assumptions, we prove a local consistency result: near an isolated simple zero, minimizers converge to the true zero as the scale parameter tends to zero. Under an additional structural assumption, we obtain a conditional second-order localization estimate. Numerical experiments on representative oscillatory functions, including the Riemann -function and residual functions arising from Sturm–Liouville eigenvalue problems, indicate consistent localization behavior in the tested noisy sampling regimes. These results support the proposed method as a practical preprocessing stage for basin identification and initialization of classical root-finding methods when reliable initialization is not available.