  We establish inverse and direct theorems on best approximations in
quasi-normed Abelian groups through bilateral Bernstein-Jackson inequalities
with exact constants. Using integral representations for quasi-norms of
functions $f$ in Lebesgue's spaces by decreasing rearrangements $f^*$ with the
help of approximation $E$-functionals, error estimates are found. Examples of
numerical calculations and spectral approximations of self-adjoint operators
obtained by obtained estimates are given.
