# Mat6, Spring 2013

## Integration and Fourier Theory

The course is shared by Horia Cornean and Morten Grud Rasmussen.

*ECTS*: 5 ECTS-point.

*Prerequisites*: one should have previously followed Analysis
1 and 2 on MAT3 and MAT4, and Geometry on MAT5.

*Content*: Integration of positive measurable functions; the
monotone and dominated convergence theorems. The Lebesgue measure on
R^{n} and its invariance. Product-measure, Tonelli and Fubini
theorems. Hölder and Minkowski inequalities; Lebesgue-spaces and
their completeness. Approximation by smooth functions. Convolution
and approximation of identity. Fourier series, Riesz-Fischer theorem;
orthogonal expansions in Hilbert spaces. The Fourier transform and
the inversion theorem; Plancherel's identity.

*Exam*: oral examination, with intern censorship.

*Grade*: Pass/fail

One should read the indicated material in advance, and try to
solve the given exercises beforehand. The book is wonderful, but
rather concise and difficult to read.

### Literature

Walter Rudin: *Real and Complex
Analysis*. McGraw-Hill Education, Third edition. 1987

H.D. Cornean: Noter om bl.a.
Banach, Brouwer, Schauder, Kakutani fikspunktsætninger, Hairy
Ball, Jordans kurvesætning, Nash ligevægtspunkt for
endelige spil. (link)

H.D. Cornean: Notes for the course *Operatorer i Hilbertrum*
(link)

Written by Horia
Cornean.