Asymptotic Behavior of Generalized Functions

On Tauber's second Tauberian theorem

“…These relations are then particular cases of the so called quasiasymptotic behavior of distributions at finite points [10,27,43,46]. The distribution g in (5) must necessarily be homogeneous of degree α [26,27].…”

“…These notions have their obvious analogs at infinity [10,27,37,39,46]; indeed, one considers the behavior of f (λx) as λ → ∞ in the weak topology of a distribution space.…”

“…In such a case we write f (x) = U (x + i0). If f (x) = U (x + i0) and f (x 0 ) = γ, distributionally, then U (z) → γ as z → x 0 , in an angular (or nontangential) fashion [4,9,27,39]; however, the converse is not true in general [8]. In the case of boundary values of analytic functions, we have the following Tauberian theorem [41].…”

“…Moreover, the pointwise Fourier inversion formula holds in the (C, β) sense for any β ≥ β 1 + 1, that is, (27) f (x 0 ) = γ, distributionally, and 1 2π f (x), e ix 0 x = γ (C, β) .…”

“…Distributional Abelian and Tauberian theorems for other integral transforms are investigated in [22,25,26]. Other related results are found in [23,24,27]. Some Tauberian results for distributions have interesting consequences in the theory of Fourier series [11].…”