This paper introduces a family of semi-norms defined as $ \|u\|_{m,s} = (f_{\mathbb{R}^{n}}|\tau|^{2s}|\mathcal{D}^{n}u(\tau)|^{2}d\tau)^{1/2} $, and shows that minimizing these semi-norms under interpolation conditions leads to functions with simple forms. These functions preserve polynomials of degree $ \leq m-1 $, commute with similarities, translations, and rotations, and converge in Sobolev spaces $ H^{m+s}(\Omega) $. Examples include "thin plate" functions, "multi-conic" functions, pseudo-cubic splines, and polynomial splines. The paper also discusses the use of convolution expressions for splines, involving kernels like $ \mu \star |t|^{2m+2s-n} $ or $ \mu \star |t|^{2m+2s-n} \log |t| $, plus polynomials. The paper presents an alternative approach to constructing multidimensional splines, inspired by the physical interpretation of cubic splines as equilibrium positions of beams. It shows that minimizing a functional similar to the bending energy of a thin plate leads to "thin plate" functions in $ \mathbb{R}^2 $, which are also called "surface splines." The paper notes that these functionals are invariant under translations and rotations, and scale by a power of $ |\lambda| $ under similarity transformations. This implies that the corresponding interpolation methods commute with similarities. The paper also explores the use of a weighting function $ w(\tau) = |\tau|^{\theta} $ to create other interpolation methods. It shows that minimizing $ \int_{\mathbb{R}^{n}}|\tau|^{\theta}|\mathcal{D}^{\mathbb{R}^{n}}v(\tau)|^{2}d\tau $ is possible under certain conditions on $ \theta $, and that the resulting functionals are invariant under translations and rotations. The paper introduces Sobolev-type spaces such as $ \tilde{H}^{s} $, $ D^{-m\tilde{H}^{s}} $, and $ H_{loc}^{m+s} $, and discusses their relevance to the problem.This paper introduces a family of semi-norms defined as $ \|u\|_{m,s} = (f_{\mathbb{R}^{n}}|\tau|^{2s}|\mathcal{D}^{n}u(\tau)|^{2}d\tau)^{1/2} $, and shows that minimizing these semi-norms under interpolation conditions leads to functions with simple forms. These functions preserve polynomials of degree $ \leq m-1 $, commute with similarities, translations, and rotations, and converge in Sobolev spaces $ H^{m+s}(\Omega) $. Examples include "thin plate" functions, "multi-conic" functions, pseudo-cubic splines, and polynomial splines. The paper also discusses the use of convolution expressions for splines, involving kernels like $ \mu \star |t|^{2m+2s-n} $ or $ \mu \star |t|^{2m+2s-n} \log |t| $, plus polynomials. The paper presents an alternative approach to constructing multidimensional splines, inspired by the physical interpretation of cubic splines as equilibrium positions of beams. It shows that minimizing a functional similar to the bending energy of a thin plate leads to "thin plate" functions in $ \mathbb{R}^2 $, which are also called "surface splines." The paper notes that these functionals are invariant under translations and rotations, and scale by a power of $ |\lambda| $ under similarity transformations. This implies that the corresponding interpolation methods commute with similarities. The paper also explores the use of a weighting function $ w(\tau) = |\tau|^{\theta} $ to create other interpolation methods. It shows that minimizing $ \int_{\mathbb{R}^{n}}|\tau|^{\theta}|\mathcal{D}^{\mathbb{R}^{n}}v(\tau)|^{2}d\tau $ is possible under certain conditions on $ \theta $, and that the resulting functionals are invariant under translations and rotations. The paper introduces Sobolev-type spaces such as $ \tilde{H}^{s} $, $ D^{-m\tilde{H}^{s}} $, and $ H_{loc}^{m+s} $, and discusses their relevance to the problem.