# TMA4170 Fourieranalyse våren 2016

Kursbeskrivelse finnes i studiehåndboka.

## Beskjeder

** OFFICE HOURS MONDAY 6 JUNE 15-16, TUESDAY 7 JUNE 14-16 IN 1152 SB II**

***Extra exercises during the lecture Monday 25.IV.2016***

* THE FIRST LECTURE WILL TAKE PLACE ON MONDAY THE 11th of January 2016

* THE FIRST EXERCISE SESSION ON TUESDAY the 19th of January 2016

## Kursinformasjon

#### LECTURES

- Monday 10.15-12, Auditorium KJL24
- Friday 14.15-16, Auditorium KJL24

#### EXERCISES

- Tuesday 17.15-18. Auditorium KJE24

#### EXAM.

#### SYLLABUS

From the book by Boggess& Narcowich

- Chapter 1

- Chapter 2

- Section 3.1

- Chapter 4

- Chapter 5
- Chapter 6
- Section 7.4

Moreover, the syllabus contains:

- the Fourier Transform of Distributions
- the Poisson Summation Formula
- the Fourier Transform in several variables (basic knowledge). Radial functions.
- all the EXERCISES!

___

#### TEXT BOOK

Most of the material will be found in

**A. Boggess & F. Narcowich: A first course in Wavelets with Fourier Analysis.**

The Fourier Series and Transforms have to be complemented with other sources. The Fourier transforms of distributions have to be added.

### Teacher

- Peter Lindqvist: Office 1152 in SB II, lqvist [at] math [dot] ntnu [dot] no

Please write TMA4170 in the subject line on all e-mail related to the course.

- Office hours: ???day 1?:15-1?:00

#### EXERCISES

Week 3 | Exercises | 2016 Example 5 postponed |
---|---|---|

Week 4 | Exercises | 2016 |

week 5 | Exercises | 2016 |

week 6 | Exercises | 2016. Do example 5 for many functions |

week 7 | Exercises | 2016 |

week 8 | Exercises | 2016 MOVED TO G038 23.II.2016 |

week 9 | Exercises | 2016. Some helpdistribution.pdf. .Sign(x) |

week 10 | Exercises | 2016. Not related to the examples:Laplace Eqn |

week 11 | Exercises | 2016. Some solutions Appendix: some formulas in several variables |

week 14 | Chapter V: 4,5,7,8,18 | 2016 See Shannon's wavelet below! |

week 15 | Exercises | 2016. Obvious missprint in the range of integration of the Four.coeff.ex.4 |

week 16 | Exercises | 2016 |

week 17 | Extra exercises | Monday 25.IV.2016, 10-12 o"clock. |

week 17 | Exercises | Tuesday 26th of April |

### Notes

More "advanced" theory:

### Additional Literature

- E. Stein, R. Shakarchi: "Fourier Analysis", Princeton.
- "A Guide to Distribution Theory and Fourier Transforms" by R. Strichartz, World Scientific.–Easy to read. Useful information.
- "Wavelets -A Primer" by Ch. Blatter

## Tentative lecture plan

usually according to Boggess & Narcowich

Week 3 | Ch. 0 | The Hilbert Space. Orthonormal bases. Fourier Coefficients. The Trigonometric System.Bessel's ineq. | |

Week 4 | Ch 1 | Convergence. Dirichlet's Kernel, Fejer's Kernel | Se notes on conv. below! |

week 5 | Ch 2 | (Parseval's formula Ch 1), Gibb's phenomenon. Fourier transform | |

week 6 | Ch 2 | Fourier transform in L2, Plancherel's formula, convolutions. Sampling Theorem. | |

week 7 | Ch 2 | Oversampling. Heisenberg's Principle. Poisson's summation formula. | Exercise 2.14 Oversampling |

——–7 | Ch 3 | Discrete Fourier Transform. | |

week 8 | Ch 3 | Fast Fourier Transform. Theory of Distributions | Not in the text book. |

week 9 | Notes | Fourier Transform of Distributions. The Schwartz class. Haar basis | not in the text book. |

week 10 | Ch 4 | Haar Basis. Wavelets. | |

week 11 | Ch 5 | Wavelets. | |

week 13 | Ch 6 | Daubechies' Wavelets. Shannon's Wavelet | Calculation of 2nd moment Last 5 lines irrelevant. One 1/2 missing inside Q(w) |

week 14 | Ch 6 | Daubechies's wavelet. Continuous Wavelet Transform. | |

week 15 | … | Random walks, various things, H. Weyl's theorem (from the book by Stein-Shakarchi) | |

week 16 | … | Repetition | |

week 17 | … | Extra exercises on Monday the 25.IV.2016 |

## Referansegruppe

- Erik Jørgensen ….. erijorg@stud.ntnu.no
- Are Austad ….. aresa@stud.ntnu.no