Area rearrangement operator in discrete and continuous calculus

Abstract

By interpreting sums as area we construct the area rearrangement operator which looks like Φ = −x d/dx in the continuous case and φ = −x∆ in the discrete case. We explore the properties of these operators, among them how they create a sequence of linearly independent functions all of which integrate/sum to the same value. Using the discrete operator, we discover a family of functions that satisfies those two properties, as well as one regarding their “finite diagonals”. These three properties becomes the criteria for the main problem we will explore in this paper, where we search for a way to find other families of functions that satisfies this. This leads us to the “main solution”, which itself can be seen as an operator, which exists both in discrete and continuous calculus, with its own interesting properties.