Classification of bounded Baire class $\xi $ functions

Kechris and Louveau showed that each real-valued bounded Baire class 1 function defined on a compact metric space can be written as an alternating sum of a decreasing countable transfinite sequence of upper semi-continuous functions. Moreover, the length of the shortest such sequence is essentially the same as the value of certain natural ranks they defined on the Baire class 1 functions. They also introduced the notion of pseudouniform convergence to generate some classes of bounded Baire class 1 functions from others. The main aim of this paper is to generalize their results to Baire class $\xi$ functions. For our proofs to go through, it was essential to first obtain similar results for Baire class 1 functions defined on not necessary compact Polish spaces. Using these new classifications of bounded Baire class $\xi$ functions, one can define natural ranks on these classes. We show that these ranks essentially coincide with those defined by Elekes et. al.