Unconditional Schauder Frames of Exponentials and of Uniformly Bounded Functions in L p Spaces

It is known that there is no unconditional basis of exponentials in the space Lᵖ(Ω ), p ≠ 2, for any set Ω ⊂ Rᵈ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space Lᵖ(Ω ) does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word “basis” is replaced with “Schauder frame”. On the other hand, we prove that if Ω has nonempty interior then there are no unconditional Schauder frames of exponentials in the space Lᵖ(Ω ), p ≠ 2.