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 \(f\).