A flexible method for estimating luminosity functions via Kernel Density Estimation -- III. Extending to Multiple Flux-Limited Samples

As the third paper in a series regarding the estimation of luminosity functions (LFs) via kernel density estimation (KDE), we present a further generalization of our framework by extending its applicability to multiple flux-limited samples. While our previous works addressed single flux-limited datasets, many practical applications involve surveys that cover disjoint fields of view with different flux limits. We introduce a piecewise estimation framework that partitions the luminosity-redshift plane into disjoint regions according to the staggered flux limits of the sub-samples. Within each region, we integrate data from all surveys capable of detecting sources into a combined sample and apply the transformation-reflection KDE method using the corresponding local flux threshold as the truncation boundary. This strategy allows for the full utilization of all available sources while maintaining rigorous statistical consistency. The robustness of this approach is validated through Monte Carlo simulations. Furthermore, application to SDSS DR7 and 2SLAQ quasar data shows overall agreement with parametric models, while a small residual discontinuity near a survey-transition boundary is discussed as a diagnostic of independent piecewise estimation and possible inter-survey systematics. The KDE calculations in each piecewise region are performed using our previously developed public Python package \texttt{kdeLF}.