  The analysis of branching problems for restriction of representations brings
the concept of symmetry breaking transform and holographic transform. Symmetry
breaking operators decrease the number of variables in geometric models,
whereas holographic operators increase it. Various expansions in classical
analysis can be interpreted as particular occurrences of these transforms. From
this perspective, we investigate two remarkable families of differential
operators: the Rankin-Cohen operators and the holomorphic Juhl conformally
covariant operators. Then we establish for the corresponding symmetry breaking
transforms the Parseval-Plancherel type theorems and find explicit inversion
formul{\ae} with integral expression of holographic operators.
  The proof uses the F-method which provides a duality between symmetry
breaking operators in the holomorphic model and holographic operators in the
$L^2$-model, leading us to deep links between special orthogonal polynomials
and branching laws for infinite-dimensional representations of real reductive
Lie groups.
