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Topics in Analysis 1 -- Real functions, Spring 2008

homework

first homework assignment. Due: March 11, 2008 .

second homework assignment. Due: April 8, 2008

third homework assignment. Due: May 27, 2008

It may help to consult the lecture notes!

You can hand your answers in during the lectures, bring them to my office, put them in my mailbox, or send them by mail or email. Your answers will be corrected and graded. It is not obligatory to make the assignments, but good homework will be a substantial part of the final grade and will make the oral exam lighter. Succes!

program

The first part of the course is based on a few pages of the book of Van Rooij and Schikhof and the book of Royden, see the first and second handout. The entire contents of the course will become available as printed material, either as handouts or as lecture notes. The material will grow gradually when the course proceeds.

Feb 5.
1. Monotone functions. 1.1 Continuity. ---Th. 1.2, Cor. 1.3, Th. 7.5, Cor 7.6 of [Rooij] and Th. 15, Cor. 16 of [Royden].

Feb 12.
1. Monotone functions. 1.2 Differentiability. ---Section 5.1 of [Royden].

Feb 19.
1. Monotone functions. 1.3 Bounded variation. ---Section 5.2 of [Royden].
2. Fundamental theorem of calculus. 2.2 The Cantor function. ---15.9 of [Rooij].

Feb 26.
2. Fundamental theorem of calculus. 2.1 Indefinite integrals. ---Section 5.3 of [Royden].
2.3 Absolute continuity. ---Section 5.4 of [Royden].

Mar 4.
2. Fundamental theorem of calculus. 2.3 Absolute continuity (continued). ---Section 5.4 of [Royden].
3. Sobolev spaces. 3.1 Sobolev spaces on intervals. ---lecture notes

Mar 11.
3. Sobolev spaces. 3.2 Application to differential equations. ---lecture notes

Mar 18.
3. Sobolev spaces. 3.2 Application to differential equations. ---lecture notes

Apr 1.
3. Sobolev spaces. 3.3 Distributions. ---lecture notes

Apr 8.
3. Sobolev spaces. 3.4 Fourier transform and fractional derivative. ---lecture notes

Apr 15.
4. Stieltjes integral. ---lecture notes

Apr 22.
4. Stieltjes integral (continued). ---lecture notes
4. Stieltjes integral. 4.2 Langevin equation. ---lecture notes
4. Stieltjes integral. 4.3 Wiener integral (Optional!). ---lecture notes

May 6.
5. Convex functions. --- Section 1.2 of [Rooij].

May 13.
Discussion of homework assignment 1 and 2.

Here are the lecture notes (Version of May 21. The optional section 4.3 is now included and several layout errors have been corrected.)

topic

We intend to study a part of analysis often refered to as "theory of real functions". The main themes are continuity, derivatives, and integrals of real functions depending on one real variable and their relations. In Analysis 1 these topics have been considered under sufficient regularity conditions to allow for a smooth theory. If one tries to minimize such conditions, the theory becomes more difficult and even more intriguing.

For instance, the following issues will be addressed.

If a function is differentiable and its derivative is integrable, is the function a primitive of its derivative? Or, more generally, if a function is differentiable at almost every point and the almost everywhere defined derivative is integrable, is the function a primitive of that derivative? In contrast to the continuously differentiable case, these questions are surprisingly deep.

There are continuous functions that are nowhere differentiable. At `how many' points can monotone continuous functions be non-differentiable?

The difference quotients of a differentiable function converge pointwise to the derivative. If the function has additional properties, do these quotients converge uniformly, or in some suitable norm?

From a theoretical point of view, the theory of real functions examines the heart of the central operations in analysis (limits, derivatives, integrals). From a more applied point of view, the theory provides essential insights and tools for applied analysis and stochastics. For instance, the use of Sobolev spaces in the analysis of partial differential equations, the use of convex functions in optimization, and the study of path properties of continuous time stochastic processes.

general

Teacher: Onno van Gaans.

Books etc: The course will be based on parts of several books. The reading material will be a combination of handouts and downloadable course notes. Participants need not purchase a book for the course.
Here are some of the books:

[Rooij] A.C.M. van Rooij and W.H. Schikhof, A second course on real functions, Cambridge University Press, Cambridge, 1982.
[Royden] H.L. Royden, Real Analysis, Macmillan, New York, 1963.

Examination: Oral exam and assignments. The assignments will be announced during the lecures as well as on this webpage.

Credit points: 6 ECTS

Lectures: on Tuesday 9:00-10:45 a.m. in Room 401 of the Snellius building (i.e., the building of the Mathematical Institute).

For any more information, please feel free to contact me (Onno van Gaans), Phone: 071 5277122, Email: my email address