This course, designed for third-year Mathematics degree students (semester 6), provides a rigorous introduction to three fundamental tools of modern analysis.
Chapter 1 develops the theory of Lp spaces (1 ≤ p ≤ ∞): Lebesgue integration review, definitions and norms, Hölder–Minkowski inequalities, completeness (Banach spaces), reflexivity and separability (for 1 < p < ∞), duality (L^{p'} as dual of L^p), and density of smooth functions.
Chapter 2 studies the Fourier transform: from L¹ functions (properties, inversion) to the Plancherel theorem and unitary operator on L², laying the foundation for harmonic analysis.
Chapter 3 covers the Laplace transform: definition and properties, computation of standard transforms, inversion techniques (Bromwich, residues), and applications to solving linear ODEs with constant coefficients.
The course emphasizes functional analysis foundations, key inequalities, transform techniques, and their interplay in solving differential equations and PDEs.
- Teacher: fares bensaid