Sumários
27th Class - Extra class
11 junho 2010, 13:00 • Jorge Filipe Drumond Pinto da Silva
Sketch of the proof of Riesz's theorem for the conjugation problem in L p ( T), (1<p< ∞). Distribution function, weak L w p spaces. The Chebyshev inequality: L p ⊂ L w p . The (diagonal) Marcinkiewicz interpolation theorem. Maximal operators and pointwise convergence approximations of the identity: the Hardy-Littlewood maximal function and Lebesgue's differentiation theorem.
26th Class
9 junho 2010, 13:30 • Jorge Filipe Drumond Pinto da Silva
Hardy spaces h p ( D) and H p ( D). Boundary values of Hardy space functions. Representation theorem for h p ( D) functions as convolutions of Borel regular measures (p=1), or L p ( T) functions (p>1), with the Poisson kernel. The conjugation problem in terms of Hardy spaces.
25th Class
7 junho 2010, 13:00 • Jorge Filipe Drumond Pinto da Silva
The conjugation and the projection problems in L p ( T). Equivalence between these two problems, and equivalence with norm convergence of Fourier series in L p ( T). Example in L 1 ( T) of a function that does not admit conjugation. The conjugation problem and the Hilbert transform.
24th Class
2 junho 2010, 13:30 • Jorge Filipe Drumond Pinto da Silva
Analitic and harmonic functions on the complex unit disc. The Cauchy and Poisson kernels. The Poisson integral formula for the solution of the Dirchlet problem for Laplace's equation, on the unit disc. The Poisson conjugate kernel and conjugation in L p ( T).
23rd Class
31 maio 2010, 13:00 • Jorge Filipe Drumond Pinto da Silva
Baire's theorem, the uniform boundedness theorem and their consequences for the proof of existence of dense sets of continuous functions with divergent Fourier series at dense, non-countable, sets of points in T .