Tests for completeness of root subspaces of a differentiation operator with abstract boundary conditions

Hereditary completeness for systems of exponentials and reproducing kernels

“…It was shown by Young [24] that, in contrast to the general situation, for any exact system of exponentials its biorthogonal system is complete. Another approach to this problem was suggested in [12], where it is shown that any exact system of exponentials is the system of eigenfunctions of the differentiation operator i d dx in L 2 (−a, a) with a certain generalized boundary condition. Applying the Fourier transform F one reduces the problem for exponential systems in L 2 (−π, π) to the same problem for systems of reproducing kernels in the Paley-Wiener space PW π = F L 2 (−π, π).…”

The Young type theorem in weighted Fock spaces

“…Young showed that every complete and minimal system of exponentials in (or, equivalently, any complete and minimal system of reproducing kernels in the Paley–Wiener space ) has a complete biorthogonal system. This result was also obtained independently and by different methods by Gubreev and Kovalenko . In the first two authors studied completeness of biorthogonal systems in the context of de Branges spaces (or model subspaces of the Hardy space) and showed that in these spaces there exist complete and minimal systems of reproducing kernels whose biorthogonal system is incomplete with any given (even infinite) defect.…”

On nonlocal problems for ordinary differential equations and on a nonlocal parabolic transmission problem

“…Completeness of the system of root functions for differential operators with functional boundary conditions is analyzed in [4,6,11,14]. The coefficient of the high order derivative is assumed to be constant in [6,11], whereas it is assumed to be continuous in [14]. The …”