In this thesis, we will look at the Hardy spaces H ^P and the disc algebra A. We will review some results concerning the relations between functions in these spaces and their Taylor coefficients. These include Paley's theorem, which asserts that the lacunary Taylor coefficients of an H ¹ function are square summable, and Rudin's theorem, which asserts that subsequences of the absolute Taylor coefficients of a function in A may have divergent sums. Then we will give two proofs of the converse of Paley's theorem for the disc algebra, namely, given a square summable sequence, there always exists a function in A with the given sequence as its lacunary Taylor coefficients. Regarding general Taylor coefficients, we also strengthened Rudin's theorem, following Anderson's constructive method....[ Read more ]

In this thesis, we will look at the Hardy spaces H ^P and the disc algebra A. We will review some results concerning the relations between functions in these spaces and their Taylor coefficients. These include Paley's theorem, which asserts that the lacunary Taylor coefficients of an H ¹ function are square summable, and Rudin's theorem, which asserts that subsequences of the absolute Taylor coefficients of a function in A may have divergent sums. Then we will give two proofs of the converse of Paley's theorem for the disc algebra, namely, given a square summable sequence, there always exists a function in A with the given sequence as its lacunary Taylor coefficients. Regarding general Taylor coefficients, we also strengthened Rudin's theorem, following Anderson's constructive method.