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

Author

Listed:
  • Eduardo Trutié-Carrero

    (Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Ave. Universidad 1001, Chamilpa 62209, Mexico)

  • Diego Seuret-Jiménez

    (Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Ave. Universidad 1001, Chamilpa 62209, Mexico)

  • José M. Nieto-Jalil

    (Tecnologico de Monterrey, School of Engineering and Sciences, Reserva Territorial Atlixcáyotl, Puebla 72453, Mexico)

  • Jorge Cantó

    (Corrosión y Protección (CyP), Buffon 46, Mexico City 11590, Mexico)

  • Damian Valdés-Santiago

    (Facultad de Matemática y Computación, Universidad de La Habana, San Lázaro y L, Vedado, Plaza de la Revolución, La Habana 10400, Cuba)

  • Laura Carballo-Sigler

    (Centro de Investigación en Ingeniería y Ciencias Aplicadas, Universidad Autónoma del Estado de Morelos, Ave. Universidad 1001, Chamilpa 62209, Mexico)

Abstract

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.

Suggested Citation

  • Eduardo Trutié-Carrero & Diego Seuret-Jiménez & José M. Nieto-Jalil & Jorge Cantó & Damian Valdés-Santiago & Laura Carballo-Sigler, 2023. "The T e Transform: A High-Resolution Integral Transform and Its Key Properties," Mathematics, MDPI, vol. 11(21), pages 1-15, October.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:21:p:4495-:d:1271300
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