A Characterization of Some Sets of Spectral Synthesis

Abstract

I will describe joint work with the late Bob Warner. We knew that $$H^1(R)$$ gave better results for singular integrals than $$L^1(R)$$ ; our question was: Would the same be true for spectral synthesis?. We extend the Beurling–Pollard argument to give sufficient conditions for spectral synthesis in $$H^1(R)$$ . We motivate and construct a class of Q-scets which satisfy the boundary and union property of synthesis, and give examples of Q-sets. To some extent the technical parts of the argument extend to d-dimensional Euclidean spaces for $$d \ge 1$$ .