Transfer Operators in the Context of Orthogonal Polynomials
Sina Straub · mediaTUM – the media and publications repository of the Technical University Munich (Technical University Munich) · 2012
We use transfer operators, a standard tool in dynamical systems theory, together with the theory of orthogonal polynomials, polynomial hypergroups and harmonic analysis to define a new transfer operator. We define this transfer operator via the preimages of certain orthogonal polynomials, namely the Chebyshev polynomials of the first kind. This transfer operator acts on various function spaces which are determined by orthogonal polynomials. We find that the defined transfer operator is bounded on the function spaces that we study and that the transformation operator coincides with the adjoint operator and the right inverse, respectively. Using a quadratic transformation which is given by the second order Chebyshev polynomial of the first kind, we construct an orthogonal polynomial sequence which generates a polynomial hypergroup the transfer operator acts on. Similarly, we construct an orthogonal polynomial sequence using a cubic transformation. However, in the cubic case, the orthogonal polynomial sequence does not generate a hypergroup. Furthermore, a brief investigation of the inverse branches of the Chebyshev polynomials shows that these are infinite. The concepts we established can serve as a starting point for an orthogonal polynomial point of view in transfer operator theory and can be transferred to the classical transfer operator theory in dynamical systems and wavelet theory.