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

The Early History of Moment Problems and Non-Negative Polynomials with  Gaps: Sparse Moment Problems, Sparse Positivstellens\"atze, and Sparse  Nichtnegativstellens\"atze from a T-System Point of View