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We study certain ideals in some spaces of analytic functionals with unbounded carriers introduced by T. Kawaï, M. Morimoto and J. W. de Roever. Using Banach algebra methods, we show an example of space without spectral synthesis. Using Hörmander’s      L 2 {L^2}       estimates, we prove a spectral synthesis theorem for mean periodic functions.

Analytic functionals with unbounded carriers and mean periodic functions