  We aim to optimize the functional form of the compactly supported smooth
(CSS) regulator within the functional renormalization group (RG), in the
framework of bosonized two-dimensional Quantum Electrodynamics (QED_2) and of
the three-dimensional O(N=1) scalar field theory in the local potential
approximation (LPA). The principle of minimal sensitivity (PMS) is used for the
optimization of the CSS regulator, recovering all the major types of regulators
in appropriate limits. Within the investigated class of functional forms, a
thorough investigation of the CSS regulator, optimized with two different
normalizations within the PMS method, confirms that the functional form of a
regulator first proposed by Litim is optimal within the LPA. However, Litim's
exact form leads to a kink in the regulator function. A form of the CSS
regulator, numerically close to Litim's limit while maintaining infinite
differentiability, remains compatible with the gradient expansion to all
orders. A smooth analytic behaviour of the regulator is ensured by a small, but
finite value of the exponential fall-off parameter in the CSS regulator.
Consequently, a compactly supported regulator, in a parameter regime close to
Litim's optimized form, but regularized with an exponential factor, appears to
have favorable properties and could be used to address the scheme dependence of
the functional renormalization group, at least within the the approximations
employed in the studies reported here.
