TMA4170 Fourieranalyse våren 2013
Kursbeskrivelse finnes i studiehåndboka.
- 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.
- Tuesday 14.15-16, Auditorium 732 SB II
- Thursday 12.15-14, Auditorium S23
- Peter Lindqvist. –Office 1152 in SB II. –email@example.com
- Monday, 10.15–11.00 in aud. Kjl22
|4||Exercise 1||The exercises in Chapter 1 are a good repetition. Not included now. -Minor misprint in ex. 2|
|7||No exercises on Monday 11.II.2013|
|9||Exercise 5||Black-Scholes, Poisson Summation Ex.|
|10||Exercise 6||It was Exercise 8a (misprint)!|
|17||Exercise 11||This is the last one. Second moment|
All the exercises are included in the syllabus (pensum)!
- "Fourier Analysis" by E. Stein & R. Shakarchi, Princeton University Press.
- "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.)
ALL THE EXERCISES !!!
*the Riemann-Lebesgue Lemma
*Dirichlet's Kernel, Partial Sums
*Fejer Kernel, Cesaro Means
*Pointwise Convergence. Functions of Bounded Variation.
*Riemann's localization Principle
*Weierstrass approximation theorem
*The best L² approximation*
*Convergence in L²*
*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*
*Poisson's Summation Formula*
*The classes S, D*
*Distributions in S', D'*
*Principal values, Dirac*s delta*
*The Fourier Transform as a Distribution*
*Radial functions, Radon transforms*
*Hausdorff-Young's Inequality (interpolation). Optional!*
*Weyl's Equidistribution Theorem*
*Kirchhoff's formula for the Wave Equation*
|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|
|15||Daubechie's wavelets, Shannon's wavelet|
|16||Moments. Continuous wavelet transform.|
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
- N. N.
- N. N.