# Mat4, Spring 2011

## Integration and Fourier Theory

The course is given by Horia Cornean, Institut for Matematiske Fag.
*ECTS*: 3 ECTS-point.

*Prerequisites*: one should have previously followed Analysis 1 and 2 on MAT1, and
Differentiable curves and manifolds on MAT3.

*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

### The structure

Each lecture is organized as follows:

Time |
wednesday |

08:15--09:00 |
Repetition |

09:00-10:30 |
Lecture |

10:30--12:00 |
Exercise session |

One should read the indicated material in advance, and try to solve the given exercises beforehand.

### Book

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

Written by
Horia Cornean.