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Favard interpolation from subsets of a rectangular lattice. (English) Zbl 1242.41005

For \(\Omega\subset \mathbb R^k\) let \(\mathbb Z_\Omega=\Omega\cap \mathbb Z^k.\) For \(z\in \mathbb Z^k\) and \(\alpha\in\mathbb Z^k_+\) denote by \(\lozenge^\alpha_z\) the tensor product difference acting componentwise on \(k\) variable functions. For a positive diagonal matrix \(M\) put \(\lozenge^\alpha_{M,z}: f\mapsto \) diag\((M)^{-\alpha}\lozenge^\alpha_z\left(f(M\cdot)\right).\)
The author studies the following Favard type interpolation problem: Suppose that \(n,k\) are positive integers, \(\Omega \) is a connected union of cells in \(\mathbb R^k\) (i.e., sets of the form \([z,z+\mathbf{1}],\, z\in \mathbb Z^k,\, \mathbf{1}=(1,\dots,1)\)), and \(M\) a positive diagonal matrix. Find an operator \(F_{\Omega,M}\) mapping functions \(f\) defined on \(M\mathbb Z_\Omega\) to \(F_{\Omega,M}f\) possessing all derivatives of total order \(n\) on \(M\Omega\) so that \(F_{\Omega,M}f =f\) on \(M\mathbb Z_\Omega\) and \(\|D^\alpha F_{\Omega,M}f\|_{L_\infty(M\Omega)}\leq C\max\{\|\lozenge^\alpha_{M,z}f\| : \alpha\in\mathbb Z^k_+,\, |\alpha|=n,\, [z,z+\alpha]\subset \Omega\},\) where \(C\) is a constant independent of \(f.\) In some instances, depending on the geometry of the set \(M\Omega\), one constructs a Favard interpolant operator \( F_{\Omega,M},\) while in other instances the interpolant could not exist. Some examples are considered.

MSC:

41A05 Interpolation in approximation theory
41A25 Rate of convergence, degree of approximation
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65D05 Numerical interpolation
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References:

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