TMA4170 Fourieranalyse våren 2012

Kursbeskrivelse finnes i studiehåndboka.

Beskjeder

  • THE LECTURES ARE IN ENGLISH
  • LAST EXERCISE 24th of APRIL. LAST LECTURE 26th of APRIL!!
  • 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 2012: Første forelesning blir mandag 9. januar kl. 10.
  • The exam. 2012 with solutions below.
  • 26'th of January: The Fourier transform in L¹
  • 4. januar 2012: Første øving blir tirsdag 17. januar kl. 16.
  • §§ 1-4 are assumed as known (or to be read soon).

Kursinformasjon

LECTURES

  • Monday 10.15-12, Auditorium F6
  • Thursday 12.15-14, Auditorium F4

Teacher

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

EXERCISES

  • Tuesday, 16.15–17.00 in aud. F6.
Week Problems Comments
3 Exercise 1In prob. 4 the last norm should be abs. value. Prob. 2 is geom. ser.
4 Exercise 2 In prob. 1 the exponent should be 2inx 3.1415…/a
5 Exercise 3
6 Exercise 4
7Exercise 5
8Exercise 6Misprint in 2, -aaf = (f missing).sign(x)
9Exercise 7
10Exercise 8 Ex.4, multiresolution has to be assumed
11Exercise 9The number 3/2 is wrong
12Exercise10Ex. 4, the sincfunction should be squared
13Exercise11
16Exercise12
17Exercise13The last one!

Textbook

  • "Fourier Analysis and Applications" by C. Gasquet & P. Witomski, Springer.

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).
  • "Fourier Analysis, An Introduction" by E. Stein, R. Shakarchi, Princeton. –Avoids Lebesgue's Integral but is advanced. Many interesting topics.
  • "Wavelets -A Primer" by Ch. Blatter

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, and E*

*Distributions in S', D', and E'*

*Principal values, Dirac*s delta*

*The Fourier Transform as a Distribution*

VARIAE

*Radial functions, Radon transforms*

*Hausdorff-Young's Inequality (interpolation)*

*Weyl's Equidistribution Theorem*

WAVELETS

*Haar basis*

*Multiresolution Analysis*

*Daubechies's wavelets*

*Shannon's wavelet*

Examination

The date of the exam is 8 June 2012 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

Forelesningsplan

Week Chapter Topic
2 1-4, 16 Introduction. Trigonometric polynomials. Hilbert's Space
3 6, 5 Also the Fejer Kernel and …
4 Fejers theorem. Gibbs phenomenon. Riemann's localization principle
" 17 The Fourier Transform of Integrable Functions
5 18, 19 The Inverse Fourier Transform. The Space S.
6 22, 26, .. Distributions. The L2 theory.
7 28, 29, 30, Distributions
8 8.1, 8.2, 9.1, 10.1 Discrete FT, Fast Fourier Transform
9 38 Shannon's Formula. Sampling. Poisson's summation formula
10 Haar wavelet. Multiresolution Analysis
11 Daubechies Wavelet, Shannon's Wavelet
12 Helson Hausdorff-Young inequality. Radial functions. X-ray tr.
13 Stein-Shakarchi Radon transform. Weyl's Equidistribution Theorem. Heat Eqn.
16 Weierstrass Approximation Thm. Comments on Lebesgue's Integral. REPETITION
17 REPETITION

*Pointwise Convergence. The Dirichlet and Fejer Kernels.

*Lebesgue's Integral. A synopsis

*Preliminary calculation for FFT

*The Earth's Temperature

Referansegruppe

  • N. N.
  • N. N.
2012-06-12, Lars Peter Lindqvist