Lacunary boehmians

Abstract

A series of the form , where is a sequence of positive integers for which inf n≥1 (λ n+1/λ n )>1 and λ−n =−λ n for all n ∈ ℕ, is called a lacunary series. We investigate a space of generalized functions βΛ such that each member of βΛ has a lacunary Fourier series representation. βΛ contains all lacunary functions and lacunary Schwartz distributions, as well as all lacunary hyperfunctions. We show that the space βΛ with Δ-convergence is a Fréchet space. We also show that if F ∈ βΛ is well behaved locally, then F is well behaved globally.