# TMA4170 Fourieranalyse våren 2016

Kursbeskrivelse finnes i studiehåndboka.

## Beskjeder

***NO EXERCISE SESSION TUESDAY 12.I.***

**No lectures on week 1**

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

* THE FIRST EXERCISE ON TUESDAY the 19th *

## Kursinformasjon

#### LECTURES

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

#### 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
- all the EXERCISES!

___

#### EXERCISES

- Tuesday 17.15-18, KJL24.

#### 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 book is too weak when it comes to some proofs).

### Teacher

- Peter Lindqvist: Office 1152 in SB II, lqvist@math.ntnu.no

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

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

#### EXERCISES

This is from 2015 and will gradually be updated.

Week 3 | Exercises | Text lost now |
---|---|---|

Week 4 | Exercises | |

week 5 | Exercises | |

week 6 | Exercises | |

week 7 | Exercises | Misprint: multiply dr by r in ex. 3 |

week 8 | Exercises | |

week 9 | Exercises | Some helpdistribution.pdf |

week 10 | Exercises | Laplace Eqn |

week 11 | Exercises | Ex. 2, misprint: square the norm. (use Thm for orthonorm. of translates) |

week 12 | Chapter V: 3,4,6,17,18 | |

week 13 | Exercises | Moved to April 10th. Obvious missprint in the range of integration of the Four.coeff. |

week 15 | See week 13 | |

week 16 | Exercises | |

week 17 | Extra Exercises | Compensates 27.III. Second part of 7 Solution?! In 9 misprint |

### 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
- "Fourier Series and Orthogonal Functions" by H. Davis, Dover. Clear examples and good proofs. (No wavelets.)

## Tentative lecture plan

===usually according to Boggess & Narcowich===

Week 2 | Ch. 0 | The Hilbert Space. Orthonormal bases. Fourier Coefficients. The Trigonometric System. | ||
---|---|---|---|---|

Week 3 | Ch 1 | Convergence. Dirichlet's Kernel, Fejer's Kernel. Gibb's Phenomenon | ||

week 4 | Ch 2 | (Parseval's formula Ch 1), The Fourier Transform | ||

week 5 | Ch 2 | Fourier transform in L2, Plancherel's formula, convolutions. Sampling Theorem. | ||

week 6 | Ch 2 | Oversampling. Heisenberg's Principle. Poisson's summation formula. | ||

——–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 | Continuous Wavelet Transform. Battle-Lemarie wavelet | ||

week 15 | … | Equidistributed numbers, various things, Repetition | ||

week 16 | … | Repetition | ||

week 17 | .. | ??? |

## Referansegruppe

- N. N.* * ???@stud.ntnu.no
- N. N. * * ???@stud.ntnu.no ?
- N. N.