  We deal with and investigate sparse univariate Positivstellens\"atze,
Nichtnegativstellens\"atze, and solutions to sparse moment problems. The paper
relies heavily on results on T-system by Karlin in 1963 and by Karlin and
Studden in 1966. We gain complete descriptions of all sparse strictly positive
and sparse non-negative algebraic polynomials on $[a,b]$ with $a\geq 0$ and
$[0,\infty)$. We extend, simplify, and solve the sparse Hausdorff and Stieltjes
moment problem with these results and the methods of adapted spaces and
T-systems.
