Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.

Positivity of polynomials, as a key notion in real algebra, is one of the oldest topics. In a given context, some polynomials can be represented in a form that reveals their positivity immediately, like sums of squares. A large body of literature deals with the question which positive polynomials can be represented in such a way.<p>The milestone in this development was Schm"udgen's solution of the moment problem for compact semi-algebraic sets. In 1991, Schm"udgen proved that if the associated basic closed semi-algebraic set $K_{S}$ is compact, then any polynomial which is strictly positive on $K_{S}$ is contained in the preordering $T_{S}$.<p>Putinar considered a further question: when are `linear representations' possible? He provided the first step in answering this question himself in 1993. Putinar proved if the quadratic module $M_{S}$ is archimedean, any polynomial which is strictly positive on $K_{S}$ is contained in $M_{S}$, i.e., has a linear representation.<p>In the present thesis, we concentrate on the linear representations in the one variable polynomial ring. We first investigate the relationship of the two conditions in Schm"udgen's Theorem and Putinar's Criterion: $K_{S}$ compact and $M_{S}$ archimedean. They are actually equivalent. We find another proof for this result and hereby we can improve Schm"udgen's Theorem in the one variable case.<p>Secondly, we investigate the relationship of $M_{S}$ and $T_{S}$. We use elementary arguments to prove in the one variable case when $K_{S}$ is compact, they are equal.<p>Thirdly, we present Scheiderer's Main Theorem with a detailed proof. Scheiderer established a local-global principle for the polynomials non-negative on $K_{S}$ to be contained in $M_{S}$ in 2003. This principle which we call Scheiderer's Main Theorem here extends Putinar's Criterion.<p>Finally, we consider Scheiderer's Main Theorem in the one variable case, and give a simplified version of this theorem. We also apply this Simple Version of the Main Theorem to give some elementary proofs for existing results.