On the basis of concepts from the mathematical theory of approximation of functions, we propose a method of deriving microcanonical transition state theory rate coefficients, both as a function of the total energy and the total angular momentum, from thermal data, namely, the limiting high-pressure rate coefficients. The method does not require the knowledge of the frequencies and degeneracies of the transition state and is general in that it allows for non-Arrhenius forms of thermal data, but it only applies to reactions possessing an intrinsic energy barrier, It is shown that the derived microcanonical rate coefficient is almost identical to the computed Rice-Ramsperger-Kassel-Marcus (RRKM) microcanonical rate coefficient using explicit frequencies and degeneracies of the transition state, and furthermore, that the difference between the two is uniformly distributed over the entire range of total energy and the entire range of the total angular momentum. Comparison of the microcanonical coefficients from the proposed method with those from a standard nonvariational RRKM calculation is presented for the unimolecular decomposition of the ethyl radical and the unimolecular isomerization of methyl isocyanide. The agreement is shown to be excellent. A theoretical analysis of the fine structure of the microcanonical rate coefficient near the threshold of the reaction is enunicated and the difficulty of extending the method to obtain variational microcanonical rate coefficients is described. We also, briefly, speculate on the possible merits of certain theoretical methods of analysis for coping with the representation of thermal data, whose argument is the temperature which is of semiinfinite range.