It is devoted mainly to the problems of convergence and summability of spectral decompositions on the fundamental functions of self-adjoint second-order elliptic operators .. For arbitrary self-adjoint non-negative extensions of such operators and their average Riess, the conditions of uniform convergence are established, final in each of the classes of functions of Sobolev - Liouville, Nikolsky, Besov and Sigmund-Gelder .. Develops a new method that does not use the traditional Carlemann technique and the apparatus of tauber theorems to evaluate the residual member of the spectral function. The exact conditions of both uniform equivalency and equivalency of almost everywhere two arbitrary spectral decompositions. are established. For researchers, graduate students, senior students - mathematicians and physicists using the spectral theory of differential operators .