Multiresolution expansion and approximation order of generalized tempered distributions (Q1950006)

Summary: Let \(\mathcal K^{r\prime}_M(\mathbb R)\) be the generalized tempered distributions of \(e^{M(x)}\)-growth with restricted order \(r \in \mathbb N_0\), where the function \(M(x)\) grows faster than any linear functions as \(|x| \to \infty\). We show the convergence of multiresolution expansions of \(\mathcal K^{r\prime}_M(\mathbb R)\) in the test function space \(\mathcal K^r_M(\mathbb R)\) of \(\mathcal K^{r\prime}_M(\mathbb R)\). In addition, we show that the kernel of an integral operator \(K : \mathcal K^{r\prime}_M(\mathbb R) \to \mathcal K^{r\prime}_M(\mathbb R)\) provides approximation order in \(\mathcal K^{r\prime}_M(\mathbb R)\) in the context of shift-invariant spaces.