Accurate calculations of the thermodynamic properties of a system near its critical point are now possible using finite-lattice approximations to renormalization groups. In order to investigate some features of such approximations, we introduce a linear-renormalization-group transformation appropriate to a system of continuous spins like the Gaussian model of Berlin and Kac. We solve the renormalization-group equations for the Gaussian model exactly, and then study in detail a finite-lattice approximation to the renormalization group for the three-dimensional case. The renormalization-group transformation contains a parameter
b. The exact transformation has fixed points only for $b={2}^{\ensuremath{-}\frac{5}{2}}$. The approximate transformation has fixed points for a range of values of
b around ${2}^{\ensuremath{-}\frac{5}{2}}$. Eigenvalues of the transformation, which determine critical exponents, depend on the value of
b used in the calculation. We study the problem of identifying the correct value of
b.
We present a new analysis on Monte Carlo Renormalization Group (MCRG) results obtained earlier by means of the Delft Ising System Processor (DISP). The MCRG data involve a total of 57 coupling constants, 36 even and 21 odd. Simulations were carried out for simple cubic lattices with 643, 323 and 163 spins. The RG transformation is assumed to be analytic. A number of relations exist between correlation functions at different renormalization levels. Some of these involve the derivatives of the stability matrix. These correlation functions enable an analysis of the so-called regular part of the RG transformation. If the Hamiltonian of the original lattice only contains nearest-neighbour couplings then the regular contributions to the specific heat and the magnetic susceptibility can be easily determined. These contributions must depend only weakly on the initial lattice size, at least if the RG transformation is analytic. We investigated whether this is indeed true when the majority-rule is applied. New simulations involving higher-order correlations will enable us to study the analytic contributions in more detail.