Prof. Diego Ribeiro Moreira

Argomenti del programma: Module 1 The topics of Module 1, taught by Diego Ribeiro Moreira, include: Orthogonality and Fourier series; Fourier sine and cosine series, even and odd functions, and half-range expansions; Basic convergence and approximation properties of Fourier series; Fourier integral, Fourier sine and cosine transforms, and Fourier transform on the real line; Basic properties of the Fourier transform; Discrete Fourier…

Argomenti del programma: Module 1 The topics of Module 1, taught by Diego Ribeiro Moreira, include: Orthogonality and Fourier series; Fourier sine and cosine series, even and odd functions, and half-range expansions; Basic convergence and approximation properties of Fourier series; Fourier integral, Fourier sine and cosine transforms, and Fourier transform on the real line; Basic properties of the Fourier transform; Discrete Fourier…

Argomenti del programma: Course contents A. Laplace Equation and Harmonic Functions Divergence theorem and Green’s identities; mollification tools. Subharmonic functions: mean-value inequality and monotonicity of averages. Maximum principles: weak and strong versions. Comparison principle; uniqueness for the Dirichlet problem; counterexamples in unbounded domains. Harnack inequality and the Harnack principle for monotone limits.

Argomenti del programma: 1. The measurement problem: Cells and boxes in Rn, volume, coverings, and the concept of outer measure. 2. Outer measure m*: Definition, monotonicity, countable subadditivity, and translation invariance. 3. Caratheodory measurability: Measurable sets, Lebesgue measure, and measure extension. 4. Basic properties of Lebesgue measure: Continuity from below and above; nested sequences of measurable sets. 5.