Detection of Discontinuity Points in one Variable Functions using Spaces of Trigonometric Functions

Abstract

Given a set of Lagrange type data in two dimensions and assuming that the points of the data set are associated with the graph of an explicit function with discontinuities, one wants to determine the points at which the function exhibits discontinuities. This is a concrete problem that appears in the area of approximation of curves with discontinuities and is present in different scientific areas. To do this, it is necessary to recognize discontinuity placement at the function. This need to characterize the placement of discontinuity points is fundamental for the development of mathematical models that take into account the discontinuities of functions. In this work, a new methodology is proposed to determine the points where the discontinuities of a function occur using an approximation space constructed from continuous trigonometric functions. The approach used to locate the discontinuity points of the function is based on the Gibbs phenomenon which is related to the oscillations found at the points of discontinuity when the discontinuous function is represented by a continuous function. Results will be presented and show the numerical process to approximate the placements of discontinuity points is successful.

Keywords

• approximation
• detection of discontinuities
• numerical methods
• gibbs phenomenon
• applied mathematics