Interpolation of Subspaces and Applications to Exponential Bases in Sobolev Spaces

We give precise conditions 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. We then apply this result to nonharmonic Fourier series in Sobolev spaces Hs(−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλnt) and E forms an unconditional basis in L2(−π, π). Then there exist two number s0, s1 such that E forms an unconditional basis in H for s < s0, E forms an unconditional basis in its span with codimension 1 in H for s1 < s. For s0 ≤ s ≤ s1 the exponential family is not an unconditional basis in its span.