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The product of a function and a Boehmian

Dennis Nemzer (1993)

Colloquium Mathematicae

Let A be the class of all real-analytic functions and β the class of all Boehmians. We show that there is no continuous operation on β which is ordinary multiplication when restricted to A.

The product of distributions on R m

Cheng Lin-Zhi, Brian Fisher (1992)

Commentationes Mathematicae Universitatis Carolinae

The fixed infinitely differentiable function ρ ( x ) is such that { n ρ ( n x ) } is a regular sequence converging to the Dirac delta function δ . The function δ 𝐧 ( 𝐱 ) , with 𝐱 = ( x 1 , ⋯ , x m ) is defined by δ 𝐧 ( 𝐱 ) = n 1 ρ ( n 1 x 1 ) ⋯ n m ρ ( n m x m ) . The product f ∘ g of two distributions f and g in 𝒟 m ' is the distribution h defined by error n 1 → ∞ ⋯ error n m → ∞ 〈 f 𝐧 g 𝐧 , φ 〉 = 〈 h , φ 〉 , provided this neutrix limit exists for all φ ( 𝐱 ) = φ 1 ( x 1 ) ⋯ φ m ( x m ) , where f 𝐧 = f * δ 𝐧 and g 𝐧 = g * δ 𝐧 .

The structure of quasiasymptotics of Schwartz distributions

Jasson Vindas (2010)

Banach Center Publications

In this article complete characterizations of the quasiasymptotic behavior of Schwartz distributions are presented by means of structural theorems. The cases at infinity and the origin are both analyzed. Special attention is paid to quasiasymptotics of degree -1. It is shown how the structural theorem can be used to study Cesàro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed. A condition for test functions in bigger spaces...

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