Simultaneous approximation from PSI spaces
For a principal shift invariant space \(S (\varphi)\), generated by \(\varphi \in W_ 2^ m (\mathbb{R}^ s)\), a necessary and sufficient condition is found for determining whether \(S (\varphi)\) provides simultaneous approximation of order \((m,k)\). For a certain type of generating function \(\varphi\) it is proved that this necessary and sufficient condition is equivalent to the existence of a constant \(c\) such that \(| \widehat \varphi (\cdot + 2 \pi \beta) | \leq c | \cdot |^ k | \widehat \varphi |\) for all \(\beta \in \mathbb{Z}^ s \backslash \{0\}\) on a neighborhood of the origin.
scientific article; zbMATH DE number 756612