Spectral approximations of strongly degenerate elliptic differential operators - Dmytryshyn, M.I.; Lopushansky, O.V. (Carpathian Mathematical Publications, 2019)
We establish analytical estimates of spectral approximations errors for strongly degenerate elliptic differential operators in the Lebesgue space L_q(Ω) on a bounded domain Ω. Elliptic operators have coefficients with strong degeneration near boundary. Their spectrum consists of isolated eigenvalues of finite multiplicity and the linear span of the associated eigenvectors is dense in L_q(Ω). The received results are based on an appropriate generalization of Bernstein-Jackson inequalities with explicitly calculated constants for quasi-normalized Besov-type approximation spaces which are associated with the given elliptic operator. The approximation spaces are determined by the functional E(t,u), which characterizes the shortest distance from an arbitrary function uL_q(Ω) to the closed linear span of spectral subspaces of the given operator, corresponding to the eigenvalues such that not larger than fixed t>0. Such linear span of spectral subspaces coincides with the subspace of entire analytic functions of exponential type not larger than t>0. The approximation functional E(t,u) in our cases plays a similar role as the modulus of smoothness in the functions theory.