T系统导论——重点关注稀疏矩问题、稀疏正项定理（Positivstellensätze）和稀疏非负定理（Nichtnegativstellensätze）
An Introduction to T-Systems -- with a special Emphasis on Sparse Moment Problems, Sparse Positivstellensätze, and Sparse Nichtnegativstellensätze
AI总结：
本讲义为康斯坦茨大学2024年夏季学期课程配套资料，引入T系统框架证明Karlin的稀疏正项与非负定理并应用于一元多项式场景，同时明确其因一维限制无法推广的局限性，为研究者提供参考。
摘要：
本讲义基于文献[dD23]编写，对应康斯坦茨大学2024年夏季学期即将开设的课程“T系统：重点关注稀疏矩问题与稀疏正项定理”。课程的核心目标是证明Samuel Karlin于1963年提出的稀疏正项定理与稀疏非负定理，并将其应用于代数场景。具体而言，给定有限个单项式（例如$1, x^2, x^3, x^6, x^7, x^9$），需要刻画这些单项式的所有线性组合中，在区间$[a,b]$或$[0,\infty)$上严格正或非负的函数形式，即描述并写出所有满足在$[a,b]$上$f(x)>0$、或在$[0,\infty)$上$f(x)\geq 0$的函数$f(x) = a_0 + a_1 x^2 + a_2 x^3 + a_3 x^6 + a_4 x^7 + a_5 x^9$。为解决这一问题，本文引入了可回答该问题的理论框架：T系统。我们通过研究T系统推导出Karlin的正项定理与非负定理，同时也不回避T系统方法的局限性。其核心局限来自Curtis–Mairhuber–Sieklucki定理，该定理本质上表明所有T系统都是一维的，因此这些结果仅能应用于一元多项式场景。这一点也可视为一种启示甚至警示：该方法已被研究并证实存在适用边界，了解这些结果与局限性可帮助学生和研究者避免重走已被证明走不通的老路。我们尽力考证了结果首次出现的正确历史文献，但也清楚地意识到，和前人一样，我们并非总能做到完全准确。
英文摘要：
These are the lecture notes based on [dD23] for the (upcoming) lecture "T-systems with a special emphasis on sparse moment problems and sparse Positivstellensätze" in the summer semester 2024 at the University of Konstanz. The main purpose of this lecture is to prove the sparse Positiv- and Nichtnegativstellensätze of Samuel Karlin (1963) and to apply them to the algebraic setting. That means given finitely many monomials, e.g. $1, x^2, x^3, x^6, x^7, x^9,$ how do all linear combinations of these look like which are strictly positive or non-negative on some interval $[a,b]$ or $[0,\infty)$, e.g. describe and even write down all $f(x) = a_0 + a_1 x^2 + a_2 x^3 + a_3 x^6 + a_4 x^7 + a_5 x^9$ with $f(x)>0$ or $f(x)\geq 0$ on $[a,b]$ or $[0,\infty)$, respectively. To do this we introduce the theoretical framework in which this question can be answered: T-systems. We study these T-systems to arrive at Karlin's Positiv- and Nichtnegativstellensatz but we also do not hide the limitations of the T-systems approach. The main limitation is the Curtis$-$Mairhuber$-$Sieklucki Theorem which essentially states that every T-system is only one-dimensional and hence we can only apply these results to the univariate polynomial case. This can also be understood as a lesson or even a warning that this approach has been investigated and found to fail, i.e., learning about these results and limitations shall save students and researchers from following old footpaths which lead to a dead end. We took great care finding the correct historical references where the results appeared first but are perfectly aware that like people before we not always succeed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
