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 Rn 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.


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

Written by Horia Cornean.