Mathematics (Oct 2023)
The <inline-formula><math display="inline"><semantics><mrow><msub><mrow><mi mathvariant="bold-script">T</mi></mrow><mrow><mi mathvariant="bold">e</mi></mrow></msub></mrow></semantics></math></inline-formula> 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 Te transform, with a specific focus on analyzing functions with integrable modulus. The definitions referred to the Te window and the Te transform in two parameters, respectively. The theorems provide the main theoretical basis for the Te transform: the existence of the Te transform in two parameters, the Te transform ∈L1(R), the existence of the inverse Te transform, and uniqueness of the Te transform. These results reveal the importance of the fact that the Te 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.
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