Splines minimizing rotation-invariant semi-norms in Sobolev spaces TeX Export

@incollection{citeulike:5773513, abstract = {We define a family of semi-norms ‖μ‖m,s=(∫ℝ n∣τ∣2s∣ℱ Dmu(τ)∣2 dτ)1/2 Minimizing such semi-norms, subject to some interpolating conditions, leads to functions of very simple forms, providing interpolation methods that: 1°) preserve polynomials of degree≤m−1; 2°) commute with similarities as well as translations and rotations of ℝn; and 3°) converge in Sobolev spaces Hm+s(Ω). Typical examples of such splines are: "thin plate" functions ( \$\$\mathop \Sigma \limits\_{a \in A} \lambda \_a |t - a|^2 Log|t - a| + \alpha .t + \beta\$\$ with Σ λa=0, Σ λa a=0), "multi-conic" functions (Σ λa|t−a|+C with Σ λa=0), pseudo-cubic splines (Σ λa|t−a|3+α.t+β with Σ λa=0, Σ λa a=0), as well as usual polynomial splines in one dimension. In general, data functionals are only supposed to be distributions with compact supports, belonging to H−m−s(ℝn); there may be infinitely many of them. Splines are then expressed as convolutions μ |t|2m+2s−n (or μ |t|2m+2s−n Log |t|) + polynomials.}, author = {Duchon, Jean}, citeulike-article-id = {5773513}, citeulike-linkout-0 = {http://dx.doi.org/10.1007/BFb0086566}, citeulike-linkout-1 = {http://www.springerlink.com/content/g27671q701166031}, doi = {10.1007/BFb0086566}, journal = {Constructive Theory of Functions of Several Variables}, keywords = {spline}, pages = {85--100}, posted-at = {2009-09-11 10:38:23}, priority = {2}, title = {Splines minimizing rotation-invariant semi-norms in Sobolev spaces}, url = {http://dx.doi.org/10.1007/BFb0086566}, year = {1977} }

TY - CHAP ID - citeulike:5773513 L3 - citeulike-article-id:5773513 N2 - We define a family of semi-norms ‖μ‖m,s=(∫ℝ n∣τ∣2s∣ℱ Dmu(τ)∣2 dτ)1/2 Minimizing such semi-norms, subject to some interpolating conditions, leads to functions of very simple forms, providing interpolation methods that: 1°) preserve polynomials of degree≤m−1; 2°) commute with similarities as well as translations and rotations of ℝn; and 3°) converge in Sobolev spaces Hm+s(Ω). Typical examples of such splines are: "thin plate" functions ( $$\mathop \Sigma \limits_{a \in A} \lambda _a |t - a|^2 Log|t - a| + \alpha .t + \beta$$ with Σ λa=0, Σ λa a=0), "multi-conic" functions (Σ λa|t−a|+C with Σ λa=0), pseudo-cubic splines (Σ λa|t−a|3+α.t+β with Σ λa=0, Σ λa a=0), as well as usual polynomial splines in one dimension. In general, data functionals are only supposed to be distributions with compact supports, belonging to H−m−s(ℝn); there may be infinitely many of them. Splines are then expressed as convolutions μ |t|2m+2s−n (or μ |t|2m+2s−n Log |t|) + polynomials. JF - Constructive Theory of Functions of Several Variables EP - 100 TI - Splines minimizing rotation-invariant semi-norms in Sobolev spaces SP - 85 KW - spline AU - Duchon, Jean PY - 1977/// UR - http://dx.doi.org/10.1007/BFb0086566 ER -