TMA4170 Fourieranalyse våren 2013

Kursbeskrivelse finnes i studiehåndboka.

Beskjeder

====*Office Hours before the exam.: Friday 10.V. from 13 to 14, Monday 13.V from 14 to 15 and Wed. 15.V. from 15 to 16 ====
  • 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
5Exercise 2
6Exercise 3
7 No exercises on Monday 11.II.2013
8Exercise 4
9Exercise 5Black-Scholes, Poisson Summation Ex.
10Exercise 6 It was Exercise 8a (misprint)!
11Exercise 7
12Exercise 8
15Exercise 9
16Exercise 10
17Exercise 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

*Pointwise Convergence. The Dirichlet and Fejer Kernels.

*Completeness of the Trigonometric System

*Gibbs's Phenomenon

*Lebesgue's Integral. A synopsis

*Preliminary calculation for FFT

*The Earth's Temperature

*The Shannon Wavelet

Referansegruppe

  • N. N.
  • N. N.
2013-08-27, Lars Peter Lindqvist