Advanced Real Analysis-II
This course will enable the students to -
- Understand the advanced concept of real analysis and give sufficient knowledge of the subject which can be used by students for further applications in their respective domains of interest.
- An introduction to derivative and integrability of absolutely continuous functions, general measure and integration, Fourier series of functions of class L, distribution theory.
Course Outcomes (COs):
|
Course |
Learning outcomes (at course level) |
Learning and teaching strategies |
Assessment Strategies |
|
|---|---|---|---|---|
|
Course Code |
Course Title |
|||
|
MAT 424C |
Advanced Real Analysis-II (Theory)
|
The students will be able to –
CO182: Analyze properties derivative and integrability of absolutely continuous functions. CO183: Describe properties of measure and integration. CO184: Understand the concept of Fourier series of functions of class L and prove related theorem. CO185: Discuss operation , properties and convergence of distribution’s CO186: Apply differentiation on distribution and define direct product of distributions. CO187: Analyse the concept of Fourier series of functions of class L and related theorem.
|
Approach in teaching: Interactive Lectures, Discussion, Power Point Presentations, Informative videos Learning activities for the students: Self learning assignments, Effective questions, presentations, Field trips |
Quiz, Poster Presentations, Power Point Presentations, Individual and group projects, Open Book Test, Semester End Examination
|
Derivative: Banach-Zarecki theorem, Derivative and integrability of absolutely continuous functions, Lebesgue point of a function, Determining a function by its derivative.
General Measure and Integration: Additive set functions, Measure and signed measure, Limit theorems, Jordan and Hahn decomposition theorems, Complete measures, Integrals of non- negative functions, Integrable functions, Absolute continuous and singular measures, Radon-Nikodym theorem, Radon- Nikodyn derivative in a measure space.
Fourier series: Fourier series of functions of class L, Fejer-Lebesgue theorem, Integration of Fourier series, Cantor- Lebesgue theorem on trigonometric series, Riemann’s theorem on trigonometric series, Uniqueness of trigonometric series.
Distribution Theory: Test functions, Compact support functions, Distributions, Operation on distributions, Local properties of distributions, Convergence of distributions.
Differentiation of distributions and some examples, Derivative of locally integrable functions, Distribution of compact support, Direct product of distributions and its properties, Convolution and properties of convolutions.
- A.M. Bruckner, J.B. Bruckner and B.S. Thomson, Real Analysis, Prentice-Hall, N.Y.2001.
- H.S. Gakill and P.P. Narayanswami, Elements of Real Analysis, Prentice-Hall India, 1997.
- W.P. Parzynski and P.W. Zipse, Introduction to Mathematical Analysis, Mc Graw-Hill Company, 1982.
- C. Goffman, Real Functions, Rinehart Company, N.Y. 1953.
- J.F. Randolph, Basic Real and Abstract Analysis, Academic Press, N.Y. 2014.
- Robert G. Bartle and Donald R. Sherbert, Introduction to Real Analysis, John Wiley & Sons Canada, 2011.
- A.J. Kosmala, Introductory Mathematical Analysis, WCB Company, 1995.
