  Representing a signal as a linear combination of a set of basis functions is
central in a wide range of applications, such as approximation, de-noising,
compression, shape correspondence and comparison. In this context, our paper
addresses the main aspects of signal approximation, such as the definition,
computation, and comparison of basis functions on arbitrary 3D shapes. Focusing
on the class of basis functions induced by the Laplace-Beltrami operator and
its spectrum, we introduce the diffusion and Laplacian spectral basis
functions, which are then compared with the harmonic and Laplacian
eigenfunctions. As main properties of these basis functions, which are commonly
used for numerical geometry processing and shape analysis, we discuss the
partition of the unity and non-negativity; the intrinsic definition and
invariance with respect to shape transformations (e.g., translation, rotation,
uniform scaling); the locality, smoothness, and orthogonality; the numerical
stability with respect to the domain discretisation; the computational cost and
storage overhead. Finally, we consider geometric metrics, such as the area,
conformal, and kernel-based norms, for the comparison and characterisation of
the main properties of the Laplacian basis functions.
