On approximation by translates of globally supported functions
The author is motivated by the paper of \textit{R.-Q. Jia} and \textit{Ch. A. Micchelli} [Curves and surfaces, Pap. Int.Conf., Chamonix-Mont-Blanc/Fr. 1990, 209-246 (1991; Zbl 0777.41013)], who considered the density of multiresolutions generated by a function \(\varphi\) that satisfies a decay condition weaker than \[ \bigl | \widehat \varphi (x) \bigr | = O \bigl( \| x \|^{- (x + h + \lambda)} \bigr) \tag{1} \] as \(x \to \infty\), where \(\| x \|\) denotes the uniform norm of \(x = (x_ 1, \dots, x_ n)\in\mathbb{R}^ n\) given by \(\max 1 \leq j \leq n | x_ j |\). In fact, Jia and Micchelli introduced the subspace \(L_ p\) of \(L_ p = L_ p (\mathbb{R}^ n)\) for \(1 \leq p \leq \infty\) for a function defined on \(\mathbb{R}^ n\) by \[ \varphi^ 0 : = \sum_{j \in\mathbb{Z}^ n} - \bigl | (\cdot - j) \bigr | \quad \text{and} \quad | \varphi |_ p : = \| \varphi \|_ p \bigl( (0,1]^ n \bigr). \tag{2} \] Then \(L_ p\) is defined to be the linear space of all functions for which \(| \varphi |_ p < \infty\). Equipped with the norm \(| \cdot |_ p L_ p\) becomes a Banach space and proved the theorem: If \(\varphi \subset L_ p (0 \leq p < \infty)\) and \(\sum_{j \in\mathbb{Z}^ n} \varphi (\cdot - j) = 1\); then \(\varphi\) provides \(L_ p\) approximation of order at least \(o(1)\) for the functions in \(L_ p\). Observing that if \(\varphi\) satisfies only the weaker decay condition \(\varphi \in L_ p\), then the approximation order guaranteed by the property \(\sum_{j \in \mathbb{Z}^ n} \varphi (\cdot - j) = 1\) is only \(o(1)\) and not \(\varphi (h)\) as one may expect, the author is led to investigate the case when the decay condition (1) is weakened to be \[ \bigl( 1 + \| \cdot \| \bigr)^{k - 1} \varphi\in L_ p,\;1 \leq p \leq \infty. \tag{3} \] The author points out that this work extends the work of \textit{Jia} and \textit{Lei} [Approximation by multiinteger translates of a function having global support, to appear in J. Approximation Theory), and the result concerning density of multiresolutions in the paper of Jia and Micchelli.
scientific article; zbMATH DE number 597641