Generalized Pompeiu equation in distributions

Abstract

We consider a generalized Pompeiu equation in the space of Schwartz distributions and as an application we find the locally integrable solutions of the equation.

Sine functional equation in several variables

“…As in [5], [6], for any z ∈ R n there exist x, y ∈ R n such that x − y = z and x, x + y ∈ E, which implies f (z) = 0 for all z ∈ R n . Thus in view of (5) g is a smooth function since we exclude the trivial case f = 0.…”

On the Initial Values of Solutions of a General Functional Equation

“…, m, of the equation (1.1) are smooth functions in which a usual convolutional technique was used. Convolving in (1.2) some regularizing functions in both variables x and y as in [4,5,7], which is a different approach as in [1,2], the equation (1.2) and its generalization in [2] are converted to the equations of the form (1.1).…”