Distributionally regulated functions

Jasson Vindas¹ ; Ricardo Estrada¹

¹Department of Mathematics Louisiana State University Baton Rouge, LA 70803-4918, U.S.A.

Studia Mathematica, Tome 181 (2007) no. 3, pp. 211-236

Résumé

We study the class of distributions in one variable that have distributional lateral limits at every point, but which have no Dirac delta functions or derivatives at any point, the “distributionally regulated functions." We also consider the related class where Dirac delta functions are allowed. We prove several results on the boundary behavior of functions of two variables $F(x,y),$ $x\in\mathbb{R},$ $y>0,$ with $F(x,0^{+}) =f(x) $ distributionally, both near points where the distributional point value exists and points where the lateral distributional limits exist. We give very general formulas for the jumps, in terms of $F$ and related functions. We prove that the set of singular points of a distributionally regulated function is always countable at the most. We also characterize the Fourier transforms of tempered distributionally regulated functions in two ways.

Détail

DOI : 10.4064/sm181-3-2

Keywords: study class distributions variable have distributional lateral limits every point which have dirac delta functions derivatives point distributionally regulated functions consider related class where dirac delta functions allowed prove several results boundary behavior functions variables mathbb distributionally near points where distributional point value exists points where lateral distributional limits exist general formulas jumps terms related functions prove set singular points distributionally regulated function always countable characterize fourier transforms tempered distributionally regulated functions ways