# TMA4170 Fourieranalyse våren 2013

Kursbeskrivelse finnes i studiehåndboka.

## Beskjeder

- EXAM 2013 + Solutions below*
- There is a link to a short note on Shannon's Wavelet.
- THE MATERIAL ON WAVELETS CAN BE EXTRACTED FROM A. Boggess & F. J. Narcowich: "A First Course in WAVELETS with Fourier Analysis",Prentice Hall. It is electronically available at the University Library. Also found on "google". Chapters 4, 5, and 6 are relevant.
- A link with the proof of the completeness of the trigonometric system has been added.
- THE FIRST LECTURE IS TUESDAY 15 January 2013, aud 732 SBII
- PERMITTED AIDS DURING THE EXAM.: One A4-sized sheet of paper stamped by the Department of Mathematical Sciences. (You may in advance write what pleases you on the sheet.) Calculator HP30S or Citizen SR-270X.

- 4. januar 2013: Første øving blir maandag 21. januar.

## Kursinformasjon

### LECTURES

- Tuesday 14.15-16, Auditorium 732 SB II
- Thursday 12.15-14, Auditorium S23

### Teacher

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

### EXERCISES

- Monday, 10.15–11.00 in aud. Kjl22

Week | Problems | Comments |
---|---|---|

4 | Exercise 1 | The exercises in Chapter 1 are a good repetition. Not included now. -Minor misprint in ex. 2 |

5 | Exercise 2 | |

6 | Exercise 3 | |

7 | No exercises on Monday 11.II.2013 | |

8 | Exercise 4 | |

9 | Exercise 5 | Black-Scholes, Poisson Summation Ex. |

10 | Exercise 6 | It was Exercise 8a (misprint)! |

11 | Exercise 7 | |

12 | Exercise 8 | |

15 | Exercise 9 | |

16 | Exercise 10 | |

17 | Exercise 11 | This is the last one. Second moment |

#### All the exercises are included in the syllabus (pensum)!

### Textbook

- "Fourier Analysis" by E. Stein & R. Shakarchi, Princeton University Press.

### Literature

- "A Guide to Distribution Theory and Fourier Transforms" by R. Strichartz, World Scientific.–Easy to read. Useful information.
- "A First Course on Wavelets with Fourier Analysis" by A. Boggeness, F. Narcowich, Prentice Hall.–An accessible introduction to wavelets (the Fourier Analysis is not well presented).
- "Wavelets -A Primer" by Ch. Blatter
- "Fourier Analysis and Applications" by G. Gasquet & P. Witomski, Springer.
- "Fourier Series and Orthogonal Functions" by H. Davis, Dover. Clear examples and good proofs. (No wavelets.)

### Pensum

**ALL THE EXERCISES !!!**

**FOURIER SERIES**

*Definitions

*the Riemann-Lebesgue Lemma

*Dirichlet's Kernel, Partial Sums

*Fejer Kernel, Cesaro Means

*Pointwise Convergence. Functions of Bounded Variation.

*Riemann's localization Principle

*Gibbs' Phenomenon

*Weierstrass approximation theorem

**HILBERT SPACES**

*The best L² approximation*

*Convergence in L²*

*Bessel's Inequality*

*Parseval's Formula*

*Riesz-Ficher's Theorem about completeness*

**THE FOURIER TRANSFORM**

*The L¹ theory*

*The Inverse Transform*

*The L^2 theory, Plancherel, Heisenberg*

*The DISCRETE FOURIER TRANSFORM*

*The FAST FOURIER TRANSFORM*

**SAMPLING**

*Shannon's Formula*

*Poisson's Summation Formula*

**DISTRIBUTIONS**

*The classes S, D*

*Distributions in S', D'*

*Principal values, Dirac*s delta*

*The Fourier Transform as a Distribution*

**VARIAE**

*Radial functions, Radon transforms*

*Hausdorff-Young's Inequality (interpolation). Optional!*

*Weyl's Equidistribution Theorem*

*Kirchhoff's formula for the Wave Equation*

**WAVELETS**

*Haar basis*

*Multiresolution Analysis*

*Daubechies's wavelets*

*Shannon's wavelet*

### Examination

The date of the exam is 21 May 2013 med tillatte hjelpemidler C. Aids: One A4-sized sheet of paper stamped by the Department of Mathematical Sciences. HP30S or Citizen SR-270X

EXAM 2012 Text Solutions to exam 2012 EXAM 2013 Text Solutions to exam. 2013 Cont. ex. 2013 Solutions cont. ex. 8 Aug. 2013

## Forelopig Forelesningsplan

Week | Chapter | Topic |
---|---|---|

3 | 1, 2 | Introduction. Basic Properties of Fourier Series. Hilbert Space. |

4 | 2, 3 | Hilbert Space. Fejer's theorem. Completeness of trig. system |

5 | 3, 4 | Pointwise Convergence. Gibbs" Phenomenon |

6 | 4 | Weyl's equidistribution theorem. Riemann's integral. Weierstrass' approx. thm. |

7 | 5 | The Fourier Transform. The Schwartz class. Heisenberg's Principle |

8 | 6 | The L2-theory. Sampling (Shannon's Theorem) |

9 | 6 | Several variables. Radial Functions. Radon transform |

10 | 7 | Discrete FT, Fast FT. Distributions |

11 | Distributions and their Fourier Transforms | |

12 | 6 | Kirchhoff's Formula, Wave Equation, Haar Basis, Wavelets |

14 | Multiresolution analysis | |

15 | Daubechie's wavelets, Shannon's wavelet | |

16 | Moments. Continuous wavelet transform. | |

17 | REPETITION |

**Remark** The Wavelets can be found in A. Boggess & F. J. Narcowich: "A first course in Wavelets with Fourier Analysis", Chapters 4, 5, and 6. Electronically available from UBIT

### NOTES, LINKS

*Pointwise Convergence. The Dirichlet and Fejer Kernels.

*Completeness of the Trigonometric System

*Lebesgue's Integral. A synopsis

## Referansegruppe

- N. N.
- N. N.