Rigorous Stability Analysis and Convergence Proof for Variational and Weighted Least-Squares Kernel-Based Methods

The paper provides a rigorous proof of the stability estimates for variational least-squares kernel-based methods, filling a significant theoretical gap in previous work. It also establishes another stability inequality involving weighted-discrete norms and demonstrates that the exact quadrature weights are not necessary for the weighted least-squares kernel-based collocation method to converge.