Fourier analysis - extremal problems

In this work the first chapter, titled "Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem". First I veri_ed this counterexample for the three smallest cases (k = 0; 1; 2). In order to handle the calculation error, in the following three cases (k = 3; 4 and later even k = 5) I used more and more sophisticated processes and developed also an involved computer program to support the calculations. The second chapter is titled "Maximization problems for positive de_nite functions supported in a given subset of a locally compact group". Here I consider the extremal problem of maximizing a point value jf(z)j at a given point z in G by some positive definite and continuous function f on a locally compact group G, where z is in a given symmetric open set, and f vanishes outside this set, and is normalized by f(0) = 1.