MATS4300 Analysis and X-ray tomography

Course material

The course is based on lecture notes. Exercises are included in the notes; there are no separate exercise sheets.

Completing the course

The course is completed by solving a sufficient amount of exercises and returning them at the end of the course. Each exercise is graded (0–2 points) and the course grade is calculated from the total points. The exercises must be given to the instructor (by email, in person, or in the mail slot) by Thursday, October 26, 2017. Later dates are possible, but this will delay grading significantly, and you should contact the instructor for details. Electronic submissions are preferred.

In addition to the normal points from exercises, one can earn bonus points:

There are a total of 125 exercises in the first version of the lecture notes, so 250 points are available. The limits for grades are as follows:

The limits can be decreased if adjustment is needed. They will not be increased. If more exercises are published, there is more to choose from but the limits will remain.

The weekly exercise sessions are a forum to discuss the problems and verify the answers. The solutions are not graded in or before the exercise sessions.

Each section ends in a question asking about confusing points, feedback, or any related questions. Mentioning one smaller thing gives one point, mentioning two or more things or something more substantial gives two points. You can also use questions you asked during the lecture. The answers to these special exercise problems must be given to the instructor at or before the relevant exercise session in writing. This ensures that any confusions can be cleared quickly. You can also include them in the final returned exercise set; it will help keep everything one place but will not affect grading.

Attendance has no effect on grading. The weekly compulsory exercises (see above) can be returned by email or to the mail slot. The only exception concerns the bonus points talk, which has to be given in person.


There will be lectures and exercises as indicated in Korppi. However, there are slight alterations to the roles of the events. Each lecture (45 + 45 minutes) will cover a single section of the lecture notes, and each excercise session will cover two sections. No separate exercise sheets will be given; the problems are contained in the notes.

Preliminary plan:

The talk schedule is as follows. Each talk is about 20 minutes long. The times are approximate, but talks will be interrupted forcibly after 25 minutes.

Tuesday, October 3
14:15–14:35 Covi Riesz potentials
14:35–14:55 Hörmann Radon's inversion formula
14:55–15:15 Railo The attenuated X-ray transform
Tuesday, October 10
14:15–14:35 Zhu Fourier transform
14:35–16:00 Lecture 11
Thursday, October 12
10:15–10:55 Rasimus & Uusluoto Distributions
11:15–11:35 Mönkkönen Fourier series

Questions and answers

Here are your questions and my answers to them. Some of the comments were typos, and they will be corrected in future versions of the notes. They are not included here but are much appreciated. $\newcommand{\R}{\mathbb R}\newcommand{\C}{\mathbb C}\newcommand{\Z}{\mathbb Z}\newcommand{\T}{\mathbb T}\newcommand{\N}{\mathbb N}\newcommand{\xrt}{\mathcal I}\newcommand{\ft}{\mathcal F}\newcommand{\abs}[1]{\lvert #1 \rvert}\newcommand{\A}{\mathcal A}\newcommand{\der}{\mathrm{d}}\newcommand{\dd}{\,\der}$

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Section 3

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Section 6

Section 7

Section 8

Section 9

Section 10

Section 11

Section 12

Section 13