Applications of a General Comparison Theory for Convolution Integrals

摘要

Some Tauberian theorems with applications to approximation theory are extended. Let X be a Banach space and B a Banach algebra of tempered distributions on R sup d. For F belongs to B put F sub (lambda) (x) = (lambda sup d) F (lambda X). The space of all test functions such that (double modulus F sub (lambda) * test function; Omega) sub X = the convolution integral of (lambda to the minus s power) as lambda tends to infinity is characterized when Omega is all of R sup d or a bounded subset of R sup d. The proofs are based on the Shapiro comparison theorem.