aspg-34-4475-1791482498 20261008180138 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Prospects for Applied Mathematics and Data Analysis PAMDA 2836-4449 2025 5 2 Agnes Osagie Cape Peninsula University of Technology, Faculty of Applied Science, South Africa Mohammad Abobala Department of Mathematics, Faculty of Science, Tishreen University, Syria Contiguous observation gaps are difficult to reconstruct when a count process is simultaneously periodic, weathersensitive, and subject to abrupt operating-state changes. A single isotropic penalty treats these components as if they had the same spectral roughness; weak regularization then leaves high-frequency coefficients unstable, whereas strong regularization suppresses legitimate diurnal peaks. This paper formulates hourly reconstruction as an anisotropic periodic inverse problem. The response is represented on a square-root scale by exogenous covariates, nested daily, weekly, and annual Fourier systems, and modulated daily harmonics. A block-diagonal Tikhonov operator assigns separate regularization levels to these mechanisms and weights the kth periodic mode by k4, the Fourier form of a squared-curvature penalty. The estimator is available in closed form. Conditions for uniqueness, a perturbation bound for reconstructed gaps, a generalized spectral-filter representation, and monotonicity of the effective dimension are derived. Numerical reconstruction on one year of hourly bicycle demand was evaluated under independent missingness, eight-hour blocks, and complete-day outages using 24 repeated masks per regime. The proposed estimator reduced RMSE relative to a periodic-profile reconstruction by 29.75%, 26.85%, and 29.24%, respectively. Against an isotropic ridge model using the same 139-dimensional basis, the anisotropic model was slightly better for random and full-day gaps, statistically indistinguishable for eight-hour blocks, and attained an effective dimension of 113.57, illustrating that the principal gain is controlled multiscale regularization rather than an increase in model size. 2025 09 14 4475 https://creativecommons.org/licenses/by/4.0/ 10.54216/PAMDA.050202 https://www.americaspg.com/journal/34/article/4475 https://www.americaspg.com/storage/articles/manuscripts/01M2RNBQCRA30JBWEV7QK106T6.pdf