In this work, we deal with two approximation problems in a finite-dimensional generalized Wendland space of compactly supported radial basis functions. Namely, we present an interpolation method and a smoothing variational method in this space. Next, the theory of the presented method is justified by proving the corresponding convergence result. Likewise, to illustrate this method, some graphical and numerical examples are presented in ${\mathbb{R}}^2$, and a comparison with another work is analyzed.

We introduce a new methodological approach for the approximation of generalized offset surfaces using smoothing radial basis functions (RBFs). Existing offset surface generation methods often exhibit limitations such as self-intersections and singularities, which affect their robustness and accuracy. To address these issues, we employ generalized Wendland radial basis functions, which are compactly supported and provide enhanced stability. We establish the existence and uniqueness of the solution of the proposed method, analyze its computational aspects, and prove convergence results. Finally, numerical experiments are presented to demonstrate its effectiveness as well as to compare it with an existing method from the literature.