Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions (Q879130)

Let \(S_r'\) denote the space of tempered distributions of order \(r\) and \(\widetilde S_r\) denote those elements \(f\) of \(S_r\) which satisfy the inequality \(\{f^{(q)}(x)\}\leq C_m(1+ x^2)^{-m/2}\), \(x\in\mathbb{R}\), \(0\leq q\leq r\). Let \(\{V_j\}_{j\in\mathbb{Z}}\) denote a multiresolution analysis of \(L^2\) (MRA of \(L^2\)) and \(\phi\) its scaling function. The main result of this article is Theorem~3 which reads as follows: Let there be given an \(r\)-regular MRA of \(L^2\) such that \(\phi\in\widetilde S_r\). Let \(E_j(x, y)\), \(j\in\mathbb{Z}\), denote the function \(E_j(x, y)= 2^j E(2^j x, 2^j y)\), where \(E(x, y)= \sum_{k\in\mathbb{Z}} \phi(x- k)\overline\phi(y- k)\), \(x,y\in\mathbb{R}\). If \(\sigma\in S_{r+1}\), then the sequence \(\langle E_j(x, y),\sigma(x)\rangle\) converges to \(\sigma(y)\) in \(S_r\) as \(j\to \infty\). The consequences of Theorem 3 are: a result on the multiresolution expansion of \(f\in S'\) in \(S_r'\), some approximation results of order \(k\) in \(S_r'\), and the quasi-asymptotic behaviour of an \(f\in S'\) through its projections \(f_j\).