Interpolation of Subspaces and Applications to Exponential Bases

Precise conditions are given under which the real interpolation space [Y0, X1]θ,p coincides with a closed subspace of [X0,X1]θ,p when Y0 is a closed subspace of codimension one. This result is applied to the study of nonharmonic Fourier series in the Sobolev spaces Hs(−π, π) with 0 < s < 1. The main result looks like this: if {eiλnt} is an unconditional basis in L2(−π, π), then there exist two numbers s0, s1 such that for s < s0 the family {eiλnt} forms an unconditional basis in Hs(−π, π), and for s1 < s this family forms an unconditional basis of a closed subspace in Hs(−π, π) of codimension one. If s0 ≤ s ≤ s1, then the family {eiλnt} is not an unconditional basis in its span in Hs(−π, π). §