Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions

Multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces Sr(R), r∈N0. For a function h, which is smooth enough and of appropriate decay, it is shown that the derivatives of its projections to the corresponding spaces Vj, j∈Z, in a regular multiresolution analysis of L2(R), denoted by hj, multiplied by a polynomial weight converge in sup norm, i.e., hj→h in Sr(R) as j→∞. Analogous result for tempered distributions is obtained by duality arguments. The analysis of the approximation order of the projection operator within the framework of the theory of shift-invariant spaces gives a further refinement of the results. The order of approximation is measured with respect to the corresponding space of test functions. As an application, we give Abelian and Tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity.