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 define and analyse the k-directional short-time Fourier transform and its synthesis operator over Gelfand Shilov spaces S α β (R n ) and S α β (R k+n ) respectively, and their duals. Also, we investigate directional regular sets and their complementsdirectional wave fronts, for elements of S ′α α (R n ).
In this paper we give a characterization of Sobolev k-directional wave front of order p∈[1,∞) of tempered ultradistributions via the directional short-time Fourier transform.
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