Bases of complex exponentials with restricted supports

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on unit interval. We analyze whether property is maintained if support restricted to possibly overlapping subsets show, example, that S1,…,SK⊂[0,1] are finite unions intervals rational endpoints cover interval, then there exists partition Z into sets Λ1,…,ΛK such ⋃k=1K{e2πiλ(⋅)χSk:λ∈Λk} Riesz L2[0,1]. Here, χS denotes characteristic function S.