Print Email Facebook Twitter Boundary values of analytic functions on the disc Title Boundary values of analytic functions on the disc Author dos Santos Pinto Leite, Henrique (TU Delft Electrical Engineering, Mathematics and Computer Science; TU Delft Delft Institute of Applied Mathematics) Contributor Frey, Dorothee (mentor) Amenta, Alex (graduation committee) van Elderen, Emiel (graduation committee) Ruszel, Wioletta (graduation committee) Degree granting institution Delft University of Technology Programme Applied Mathematics Date 2018-07-02 Abstract In this bachelor's thesis we will solve the Dirichlet problem with an Lp(T) boundary function. First, we will focus on the holomorphic version of the Dirichlet problem and introduce Hardy space theory, from which will follow a sufficient condition on the Fourier coefficients of the boundary function. Then we will prove the Marcinkiewicz interpolation theorem. After that we introduce the conjugate function "tilde f", which equals the Hilbert transform of f, and use functional analysis to prove an important duality argument of the Hilbert transform. Finally, we will give several different proofs for the boundedness of the map f ↦ tilde f using the Marcinkiewicz interpolation theorem and the duality argument: the last proof will be done rigorously from scratch, i.e. without relying on (unproved) arguments from other literature. Subject Dirichlet problemHardy spaceConjugate functionfourier analysis To reference this document use: http://resolver.tudelft.nl/uuid:7adf9c85-e3c4-49be-beae-ae89d8bad516 Part of collection Student theses Document type bachelor thesis Rights © 2018 Henrique dos Santos Pinto Leite Files PDF BEP_Riki.pdf 654.46 KB Close viewer /islandora/object/uuid:7adf9c85-e3c4-49be-beae-ae89d8bad516/datastream/OBJ/view