This book explores Tchebycheff systems (T-systems) and their profound influence on real algebraic geometry and its dual counterpart, the moment problem. Non-negative polynomials play a fundamental role in real algebraic geometry, with important applications in optimization, mathematical programming, and algorithm design. Although the moment problem and positivity theory have been studied extensively, a key part of the literature has remained largely incomplete, namely, Samuel Karlin's Positivstellensätze and Nichtnegativstellensätze. The current book closes this missing gap by  providing a systematic and self-contained treatment of Karlin's Positivstellensätze and Nichtnegativstellensätze as well as demonstrating the significance of these results for both real algebraic geometry and moment theory. The book focuses on three main topics:

• T-systems;
• their impact on real algebraic geometry, particularly in the study of non-negative polynomials; and
• their applications to moment problems.

The heart of the book is the comprehensive study of T-systems and the main aim is to give a complete proof of Karlin's theorem, which describes all positive and non-negative functions in these T-systems.

This book serves both as a research reference and as a textbook for advanced courses and seminars on T-systems, moment theory, real algebraic geometry, and related areas of mathematics.

Philipp J. di Dio is a research fellow and serves as substitute professor at the University of Konstanz. He studied Chemistry (B.Sc.) and Mathematics (Dipl.-Math.) at Leipzig University. He later completed his doctoral studies at Leipzig University and the Max Planck Institute for Mathematics in the Sciences where he specialized in the mathematical theory of moments and its functional-analytic foundations. His research lies at the intersection of real algebraic geometry, functional analysis, optimization, probability, and partial differential equations.