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

Abstract

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