Interpolation of subspaces and applications to exponential bases

Precise conditions are given under which the real interpolation space $[Y_0,X_1]_{\theta,p}$ coincides with a closed subspace of $[X_0,X_1]_{\theta,p}$ when $Y_0$ is a closed subspace of codimension one. This result is applied to the study of nonharmonic Fourier series in the Sobolev spaces $H^s(-\pi,\pi)$ with $0<s<1$. The main result looks like this: if $\{e^{i\lambda_nt}\}$ is an unconditional basis in $L^2(-\pi,\pi)$, then there exist two numbers $s_0$, $s_1$ such that for $s<s_0\{e^{i\lambda_nt}\}$ forms an unconditional basis in $H^s(-\pi,\pi)$, and for $s_1<s\{e^{i\lambda_nt}\}$ forms an unconditional basis of a closed subspace in $H^s(-\pi,\pi)$ of codimension one. If $s_0\le s\le s_1$, then the family $\{e^{i\lambda_nt}\}$ is not an unconditional basis in its span in $H^s(-\pi,\pi)$.