Våren 2002
Foreleser:
Ola Bratteli
Kontor: B715
Telefon: 22 85 58 95
E-post bratteli@math.uio.no
Hjemmeside: http://www.math.uio.no/~bratteli/
Tid og sted:
Tirsdager 1015 – 12.00 og Torsdager 12.15 – 14.00 i B 63
Hovedboken vil være:
[BJ] Wavelets through a Looking Glass: The World of the Spectrum
by Ola Bratteli and Palle E. T. Jorgensen
Birkhauser, March 2002.
Denne boken er ennå ikke trykt, så i første omgang vil det bli brukt forelesningsnotater
og tex versjonen. Boka kan bestilles allerede nå, se:
http://www.math.uio.no/~bratteli/wavebook/info.html
Den første delen av kurset vil bli tatt fra:
[DW], A course in Real Analysis,
by John N. McDonald and Neil A.Weiss
Academic Press 1999
Spesielt vil de delene av seksjonene 11.1 – 11.3 og 11.5 som ikke er dekket i Ma 134
bli forelest. Deretter vil seksjonenee 1.1, 1.2 og kapittel 2 og 3 I hovedreferensen bli
gjennomgått.
29/1 [BJ], Chap. 1, Ouverture
31/1 [BJ], Chap. 1 unto 1.21
5/2 og 7/2 [BJ], rest of introduction to Chap. 1, Exercise 1-6
12/2 [DW], Section 11.1 and 11.2 through Dirichlets teorem, Teorem 11.3
14/2 [BJ], Exercise 1-10
19/2 [BJ], Exercise 1-11
21/2 [DW], Rest of Section 11,2, Sections 11.3 and 11.5
26/2 [DW], Exercises 11.32 (Poisson summation formula) and 11.58 (Shannon sampling theorem).
28/2 Section 2.2 i [BJ] until Lemma 2.2.3, with details about tight frames.
5/3 og 7/3 Rest of Section 2.2 in [BJ] until (,2.2.67)
6/3 Extra session on Exercise 2-18 in [BJ]
12/3 Final part of Section 2.2, including Exercises 2-4 and 2-19
14/3 [BJ] Section 2.1 through (2.1.3);
Section 3 in the paper [O.Bratteli and P.E.T.Jorgensen, Wavelet filters and infinite-dimensional unitary groups, in Proceedings of the International Conference on Wavelet Analysis and Applications, Donggao Deng, Daren Huang, Rong-Qing Jia, Wei Lin, Jianzhong Wang (eds.), International Press, Boston (2001), pp. 35—64] on decomposition of polynomial loops into linear factors;
[BJ] Continuing Section 2.1: Proposition 2.1.1 and its proof through (2.1.10)
19/3 [BJ] Continuing Section 2.1 through (2.1.22)
Section 3.2 from the beginning until (3.2.20)
4/4 [BJ] Remainder of Section 2.1 from (2.1.23) to the end.
9/4 [BJ] Exercise 2-21 (Finally 2-22), Cursory treatment of Sections 2.3 and 2.4
11/4 [BJ] Section 3.4, (3.4.1) – (3.4.43)
18/4 Cursory about the strong ergodic theorem and the connection to Tulcea-Marinescus Theorem (from Dunford-Schwartz, Linear Operators 1, VII.7.1-4 and
VIII.5.1-3, and M. Frank Norman, Markov Processes and Learning Models, Academic 1972, Sections 3.1 – 3.2). Section 4.3, (4.3.1) – (4.3.25).
23/4 [BJ] Exercises 2-3 – 2-6 (Exercise 2-4 (a)-(c) were done 12/3)
25/4 [BJ] Section 2.5 (Note that it is a mistake in Theorem 2.5.8 in the March 9 version of the book. This has been corrected in the published version of May 22).
Pensum i kurset.
Det som har blitt forelest I følge framdriftsoversikten ovenfor. (forelesningsnotater er tilgjengelige). Som alternativt pensum godkjennes:
1. Fourieranalyse: Gerald B. Folland ”Real Analysis”, Wiley 1984
Kap. 8.1, 8.2, 8.3, 8.4
+ (6.19) Minkowski´s ulikhet for integraler
+ (6.27) Riesz-Thorin interpolation thm (u/bevis)
2. Wavelet-teori: P.Wojtaszczyk ”A mathematical introduction to wavelets”,
Cambridge 1997
Chap. 1, 2 and 3, and Section 4.3
Eksamen.
Det blir muntlig eksamen i kurset tirsdag 4 juni 2002 kl. 13.00.