The paper introduces a family of semi-norms $\|u\|_{m,s}$ defined on Sobolev spaces, which are used to minimize rotation-invariant functionals. These semi-norms are given by $(\int_{\mathbb{R}^n} |t|^{2s} |\mathcal{G}'|^{m} u(\tau)^2 d\tau)^{1/2}$. Minimizing these norms subject to certain interpolating conditions yields functions that preserve polynomials of degree $\leq m-1$, commute with translations, rotations, and similarities, and converge in Sobolev spaces $H^{m,s}(\mathbb{R})$. Examples of such splines include "thin plate" functions, "multi-conic" functions, and pseudo-cubic splines. The paper also discusses the construction of multi-dimensional splines from one-dimensional ones and the use of Fourier transforms to introduce weighting functions, leading to other interpolation methods. The author emphasizes the invariance properties of the functionals under translations and rotations, which simplify the computation and ensure that the interpolation methods commute with these transformations.The paper introduces a family of semi-norms $\|u\|_{m,s}$ defined on Sobolev spaces, which are used to minimize rotation-invariant functionals. These semi-norms are given by $(\int_{\mathbb{R}^n} |t|^{2s} |\mathcal{G}'|^{m} u(\tau)^2 d\tau)^{1/2}$. Minimizing these norms subject to certain interpolating conditions yields functions that preserve polynomials of degree $\leq m-1$, commute with translations, rotations, and similarities, and converge in Sobolev spaces $H^{m,s}(\mathbb{R})$. Examples of such splines include "thin plate" functions, "multi-conic" functions, and pseudo-cubic splines. The paper also discusses the construction of multi-dimensional splines from one-dimensional ones and the use of Fourier transforms to introduce weighting functions, leading to other interpolation methods. The author emphasizes the invariance properties of the functionals under translations and rotations, which simplify the computation and ensure that the interpolation methods commute with these transformations.