Metric unconditionality and Fourier analysis

We study several functional properties of isometric and almost isometric unconditionality and state them as a property of families of multipliers. The most general such notion is that of “metric unconditional approximation property”. We characterize this “(umap)” by a simple property of “block unconditionality” for spaces with nontrivial cotype. We focus on subspaces of Banach spaces of functions on the circle spanned by a sequence of characters e. There (umap) may be stated in terms of Fourier multipliers. We express (umap) as a simple combinatorial property of this sequence. We obtain a corresponding result for isometric and almost isometric basic sequences of characters. Our study uses the following crucial property of the L norm for even p: ∫ |f |p = ∫ |fp/2| = ∑ |f̂p/2(n)| is a polynomial expression in the Fourier coefficients of f and f̄ . As a byproduct, we get a sharp estimate of the Sidon constant of sets à la Hadamard.