Abstract. Motivated by the product of periodic distributions, we give a new description of the wave front and the Sobolev-type wave front of a distribution f ∈ D ′ (R d ) in terms of Fourier series coefficients.

We introduce and investigate two types of the space U * of s-ultradistributions meant as equivalence classes of suitably defined fundamental sequences of smooth functions; we prove the existence of an isomorphism between U * and the respective space D ′ * of ultradistributions: of Beurling type if * = (p! t) and of Roumieu type if * = {p! t }. We also study the spaces T * and T * of t-ultradistributions andt-ultradistributions, respectively, and show that these spaces are isomorphic with the space S ′ * of tempered ultradistributions both in the Beurling and the Roumieu case.

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