  These are the lecture notes based on [dD23] for the (upcoming) lecture
"T-systems with a special emphasis on sparse moment problems and sparse
Positivstellens\"atze" in the summer semester 2024 at the University of
Konstanz. The main purpose of this lecture is to prove the sparse Positiv- and
Nichtnegativstellens\"atze 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great care finding the correct historical references where the results appeared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.
