Interpolation by \(D^m\)-splines and bases in Sobolev spaces. (English. Russian original) Zbl 0934.41008
Sb. Math. 189, No. 11, 1657-1684 (1998); translation from Mat. Sb. 189, No. 11, 75-102 (1998).
Let \(\Omega\subset \mathbb{R}^n\) be a Lipschitz domain, and \(\Delta\subset\overline\Omega\) is an irregular grid. A \(D^m\)-spline \(s= s(f,\Delta, m,\Omega)\) is a solution to the variational problem
\[
\min\Biggl\{\int_\Omega \sum_{|\alpha|= m} {m!\over \alpha!} (D^\alpha g)^2 dx;\;g\in ((\overline\Omega)\cap W^m_2(\Omega)),\;f(x)= g(x),\;x\in\Delta\Biggr\},
\]
where \(f:\Omega\to \mathbb{R}\) is an interpolated function and \(m> n/2\).
Under some assumptions on \(f\) and \(\Delta\) the author gives sharp in order estimates of \[ \|D^\ell(f- s(f))\|_{L_q(\Omega)} \] in terms of moduli of smoothness in \(L_p(\Omega)\) of the corresponding derivatives of \(f\).
Under some assumptions on \(f\) and \(\Delta\) the author gives sharp in order estimates of \[ \|D^\ell(f- s(f))\|_{L_q(\Omega)} \] in terms of moduli of smoothness in \(L_p(\Omega)\) of the corresponding derivatives of \(f\).
Reviewer: Y.A.Brudnyi (Haifa)
MSC:
41A15 | Spline approximation |
41A28 | Simultaneous approximation |
46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
41A63 | Multidimensional problems |