  This paper develops a general asymptotic theory of series estimators for
spatial data collected at irregularly spaced locations within a sampling region
$R_n \subset \mathbb{R}^d$. We employ a stochastic sampling design that can
flexibly generate irregularly spaced sampling sites, encompassing both pure
increasing and mixed increasing domain frameworks. Specifically, we focus on a
spatial trend regression model and a nonparametric regression model with
spatially dependent covariates. For these models, we investigate
$L^2$-penalized series estimation of the trend and regression functions. We
establish uniform and $L^2$ convergence rates and multivariate central limit
theorems for general series estimators as main results. Additionally, we show
that spline and wavelet series estimators achieve optimal uniform and $L^2$
convergence rates and propose methods for constructing confidence intervals for
these estimators. Finally, we demonstrate that our dependence structure
conditions on the underlying spatial processes include a broad class of random
fields, including L\'evy-driven continuous autoregressive and moving average
random fields.
