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I.The Basic Information of the Course

Course Number:202032023008

The EnglishName of theCourse：Foundation of Modern Mathematics

The ChineseName of theCourse：近世数学基础

In-classHours andAllocation(total class hours:48, classroom teaching:48classhours.)

Credit(s):3

Semester:1

AppliedDiscipline(ProfessionalDegreeCategory)：science and engineering majors

CourseObjectOriented：Masteranddoctoral student

EvaluationMode:Closed-bookExamination

TeachingMethod:Seminar-styleTeaching

CourseOpening Department:Collegeof Mathematical Sciences

II.Prerequisite Course

Mathematical analysis, Real variable functions

III. The Objectives and Requirements of the Course

Through the study of this course, so that the students comprehensive grasp of the basic theory and method of modern mathematics, try to mathematical rigor and establishing a balance between the practical application, pay attention to use examples to explain all kinds of abstract concepts and theorem, does not emphasize mathematical theory system of rigorous and complete, so that readers can more easily to learn the basic knowledge of modern mathematics, improve the modern mathematics accomplishment. Cultivate students' theoretical thinking ability, expand students' research ideas, improve students' practical ability, and lay a solid theoretical foundation for further study of each branch of mathematics, professional research and application.

IV. The Content of the Course

1. Basic spatial structure: Including range space, Banach space, Hilbert space, Fourier analysis on Hilert space, variational principle and orthogonal decomposition theorem.

2. Basic theory of linear operators: including basic concepts and properties of linear operators, some important fundamental theorems, spectrum theory of linear operators, generalized functions and Sobolev space, framework and representation of signals.

3. Foundation of nonlinear functional analysis: Gateaux differential, Frrchet differential, Taylor's formula, implicit function theorem and inverse function theorem.

4. Basis of the variational method: including the basic lemma, the variational problem with fixed boundaries, the variational problem with functional functions containing multiple functions, the variational problem with higher derivatives of unknown functions, and the functional extremum of functions of multiple variables.

5.Time-frequency analysis and Fractional Fourier Transform: Including Fourier series, Fourier Transform, Gabor transform, continuous wavelet transform, fractional Fourier transform.

6.Basis of wavelet analysis: including Haar wavelet analysis, multi-resolution analysis, wavelet structure, lifting wavelet, wavelet packet, multi-wavelet, binary wavelet analysis.

In terms of curriculum ideology and politics, by introducing Chinese mathematicians, the difficult course of overcoming academic cutting-edge problems, and the rich research achievements made by Chinese mathematicians in recent years in linear system theory, the patriotic spirit is promoted, the original intention of scientific research is never forgotten, and the mission of rejuvenating the country through scientific innovation is always kept in mind.

V. Reference Books, Reference Literatures, and Reference Materials

1. Xu Tianzhou, Li Bingzhao. Foundation of modern mathematics. Beijing Institute of Technology Press, 2009.

2. Zhang Gongqing, Lin Yuanqu. Handout on Functional Analysis (I). Peking University Press, 1987.

3. Sun Jiong, WangWanyi, HeJianwen. Functional analysis. Higher Education Press, 1998.

Outline Writer (Signature):冯国峰

Leader in charge of teaching at the College (Signature)：

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