The Te 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 Te transform, with a specific focus on analyzing functions with integrable modulus. The definitions referred to the Te window and the Te tran...

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Bibliographic Details
Published inMathematics (Basel) Vol. 11; no. 21; p. 4495
Main Authors Trutié-Carrero, Eduardo, Seuret-Jiménez, Diego, Nieto-Jalil, José M., Cantó, Jorge, Valdés-Santiago, Damian, Carballo-Sigler, Laura
Format Journal Article
LanguageEnglish
Published Basel MDPI AG 01.11.2023
Subjects
Applied mathematics
Fourier transforms
Function analysis
Integral transforms
Mathematical functions
Parameters
System theory
Theorems
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Summary: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.
ISSN:2227-7390
2227-7390
DOI:10.3390/math11214495