1. Full name:     Lai Tien Minh                                         2. Sex: Male

3. Date of birth: 25 – 06 – 1984                                      4. Place of birth: Bac Giang

5. Admission decision number: Decision 4982/QĐ-ĐHKHTN, dated on 27 November 2013 by Rector of VNU University of Science.

6. Changes in academic process: Decision 1033/QĐ-ĐHKHTN, dated on 25 April 2017  and Decision 736/QĐ-ĐHKHTN, dated 20 March 2018 by Rector of VNU University of Science on extending the study

7. Thesis title: Convolutions associated with the offset linear canonical transform and the canonical Hartley – type transform

8. Major: Applied Mathematics                                       9. Code: 9460112.01

10. Supervisors: Assoc.Prof.Dr. Nguyen Minh Tuan

    Assoc.Prof.Dr. Nguyen Huu Dien

11. Summary of new findings of the thesis

In this thesis, some basic properties of the integral operator OLCT and CHTT have been provided.  Moreover, some properties and applications in signal processing of convolutions associated with OLCT and CHTT will be also investigated.

+ Proving some basic properties of the integral operator OLCT and CHTT. This will include its mapping properties, as well as corresponding inversion formulas, Plancherel’s extensions and Parseval identities.  Moreover, the relationship between the Hermite functions and the OLCT has been also analyzed.

+ Proposing some convolutions associated with the offset linear canonical transform and the canonical Hartley – type transform. It includes convolutions associated with OLCT with Hermite - type weights, Chirp - type weights, Gauss - type weights and convolution for Hartley – type transform.

+ Proving some basic properties of these convolutions such as product theorems, commutative, combined, distribution properties and Young’s convolution inequalities. Beside this, the solvability of classes of convolution integral equations will be also provided.

+ We also obtained some Heisenberg - type uncertainty principle for OLCT and CHTT which, in particular cases, turn out to be Heisenberg uncertainty principle for the linear canonical transform, fractional Hartley transform and Hartley transform.

+ We introduce some applications in signal processing. Proposing method to derive  Shannon sampling theorem by using new convolutions. The way to design multiplicative filter in the time domain such as multiplicative filters, Gaussian filter and dual filter. Some numerical experiments have been presented to illustrate the theoretical results.    

12. Practical applicability, if any: The methods can be used to solve real-life problems which arise in many scientific and engineering areas such as signal processing, mechanics, quantum mechanics…

13. Further research directions, if any:

+ Constructing the general convolutions associated with the offset linear canonical transform and the canonical Hartley – type transform. Therefore, broaden their applications in designing filters.

+ Extending the applications of convolutions in other fields such as: image processing and Neural network.

+ Constructing uniform sampling theorem and Shannon sampling theorem in finite intervals, as well.

14. Thesis-related publications:

[1] L. P. Castro, L. T. Minh, N. M. Tuan, (2018), ``New convolutions for quadratic-phase Fourier integral operators and their applications", Mediterranean Journal of Mathematics, 15(1) (SCIE, IF1.0, Q2, H-index 18).

[2] L. P. Castro, L. T. Minh, N. M. Tuan, (2018), ``Convolutions and applications for the Offset linear canonical transform via Hermite weight", American Institute of Physics, AIP Proceedings, 2046 (020014) (SCOPUS).

[3] T. Q. Ha, L. T. Minh, N. M. Tuan, (2019), ``Convolution for the offset linear canonical transform with Gaussian weight and its application'', VNU Journal of Science: Mathematics - Physics, 35 (1).