  In the context of functional data analysis, probability density functions as
non-negative functions are characterized by specific properties of scale
invariance and relative scale which enable to represent them with the unit
integral constraint without loss of information. On the other hand, all these
properties are a challenge when the densities need to be approximated with
spline functions, including construction of the respective spline basis. The
Bayes space methodology of density functions enables to express them as real
functions in the standard $L^2$ space using the centered log-ratio
transformation. The resulting functions satisfy the zero integral constraint.
This is a key to propose a new spline basis, holding the same property, and
consequently to build a new class of spline functions, called compositional
splines, which can approximate probability density functions in a consistent
way. The paper provides also construction of smoothing compositional splines
and possible orthonormalization of the spline basis which might be useful in
some applications. Finally, statistical processing of densities using the new
approximation tool is demonstrated in case of simplicial functional principal
component analysis with anthropometric data.
