The T[sub.e] Transform: A High-Resolution Integral Transform and Its Key Properties

In this paper, we present six new contributions: two novel definitions and four groundbreaking theorems related to the theoretical foundations of the integral T[sub.e] transform, with a specific focus on analyzing functions with integrable modulus. The definitions referred to the T[sub.e] window and the T[sub.e] transform in two parameters, respectively. The theorems provide the main theoretical basis for the T[sub.e] transform: the existence of the T[sub.e] transform in two parameters, the T[sub.e] transform ∈L[sup.1] (R), the existence of the inverse T[sub.e] transform, and uniqueness of the T[sub.e] transform. These results reveal the importance of the fact that the T[sub.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.