Journal Article

The Transform: A High-Resolution Integral Transform and Its Key Properties

Eduardo Trutié-Carrero, Damian Valdés-Santiago, Laura Carballo-Sigler, Diego Seuret-Jiménez, José M. Nieto-Jalil, Jorge Cantó

2023

Overview

In this paper, we present six new contributions: two novel definitions and four groundbreaking theorems related to the theoretical foundations of the integral T e transform, with a specific focus on analyzing functions with integrable modulus. The definitions referred to the T e window and the T e transform in two parameters, respectively. The theorems provide the main theoretical basis for the T e transform: the existence of the T e transform in two parameters, the T e transform ∈ L 1( R ) , the existence of the inverse T e transform, and uniqueness of the T e transform. These results reveal the importance of the fact that the T e transform only depends on two parameters (translation and dyadic frequency), obtaining its inverse transformation more directly; hence, breaking through a new approach in function analysis by representing a function in the scale-frequency plane. The theoretical results presented in this paper are supported by the previous works of the authors.

• Publisher: MDPI AG
• Subject: T e transform / T e window / existence and uniqueness of T e transform / inverse T e transform / multi-sensitive spectrum