  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.
