  It is known that adaptive Fourier decomposition (AFD) offers efficient
rational approxima- tions to functions in the classical Hardy H2 spaces with
significant applications. This study aims at rational approximation in Bergman,
and more widely, in weighted Bergman spaces, the functions of which have more
singularity than those in the Hardy spaces. Due to lack of an effective inner
function theory, direct adaptation of the Hardy-space AFD is not performable.
We, however, show that a pre-orthogonal method, being equivalent to AFD in the
classical cases, is available for all weighted Bergman spaces. The theory in
the Bergman spaces has equal force as AFD in the Hardy spaces. The methodology
of approximation is via constructing the rational orthogonal systems of the
Bergman type spaces, called Bergman space rational orthog- onal (BRO) system,
that have the same role as the Takennaka-Malmquist (TM) system in the Hardy
spaces. Subsequently, we prove a certain type direct sum decomposition of the
Bergman spaces that reveals the orthogonal complement relation between the span
of the BRO system and the zero-based invariant spaces. We provide a sequence of
examples with different and ex- plicit singularities at the boundary along with
a study on the inclusion relations of the weighted Bergman spaces. We finally
present illustrative examples for effectiveness of the approximation.
