Characterization of Wave Fronts of Ultradistributions Using Directional Short-Time Fourier Transform

: 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.


Introduction
The theory of directional sensitive kind of the short-time Fourier transform, in short STFT, was initially introduced and investigated in [1,2] as a blend of Radon transform and time-frequency analysis. It allows to gain information in time and frequency of a function along a certain direction or hyperplane. Following the concept of [1], in [3], the directional STFT was extended to the space of tempered distributions. Moreover, in [4], the k-directional short-time Fourier transform, in short k-DSTFT, was introduced and the results of [5] were extended to the spaces of tempered ultradistributions of Roumieu class.
Starting form [6], wave fronts have shown to be useful concepts when analyzing the propagation of different type of singularities in the theory of partial differential equations, which led to introducing various wave front sets [7][8][9][10].
Following the recent trend on studying integral transforms on the spaces of ultradistribution [11,12], authors in [4] introduce the k-directional regular sets to analyze the regularity properties of a tempered ultradistribution of Roumieu class. Furthermore, the wave front set using the k-DSTFT (k-directional wave front) of a tempered ultradistribution of Roumieu class and the partial wave front in terms of [6] are considered, and it is shown that this partial wave front is equivalent to the k-directional wave front. This paper is a continuation of our work presented in [4] for both Beurling and Roumieu cases. The main result is established in Theorem 2 where we give characterization of the Sobolev wave front of order p ∈ [1, ∞) via the k-DSTFT of tempered ultradistributions. We also consider partial wave fronts in terms of [6,13], and it is shown that these notions are equivalent with the k-directional Sobolev wave front.
The main novelty of this work is the proof of Theorem 2 where we follow the idea already proved in [4] but here with another decomposition and estimates of involved integrals. On the basis of this proof we introduce a new kind of wave front and make necessary analysis of it through our main theorem, Theorem 3.

Notation
For a given multi-index l = (l 1 , . . . , l n ) ∈ N n 0 and x = (x 1 , . . . , x n ) ∈ R n , we denote 1 · · · ∂x l n n , |l| = l 1 + · · · + l n . Points in R k are denoted byx = (x 1 , . . . , x k ). The notation Ω ⊆ R n is used for an open set and K ⊂⊂ Ω for a compact set K which is contained in Ω.
we denote the Fourier transform of a function f . The inner product of f and g in L 2 is denoted by ( f , g) and f , g means a dual paring. Thus, ( f , g) = f , g . We also use the notation Γ ξ for a cone neighborhood of ξ, L r (ξ) and B r (ξ) for an open and a closed ball with a center ξ and radius r > 0, respectively.

Ultradistribution Spaces
Let (M l ) l∈N , M 0 = 1 be a sequence of positive numbers which monotonically increases to infinity and satisfies the following: We will measure the decay properties of elements of Gelfand-Shilov spaces with respect to the Gevrey sequences M l = l! α , α > 1.
Following [11], we introduce the test spaces for spaces of Beurling and Roumieu tempered ultradistributions as a special case of ultradistributional spaces.
Let a > 0. We denoted by (S a ) α β (R n ) the Banach space of all smooth functions ϕ on R n for which The space Σ α β (R n ) (resp. S α β (R n )) is defined as a projective (resp. an inductive) limit of the space (S a ) α β (R n ): and its strong dual Σ β α (R n ) (resp. S β α (R n )) is called the space of ultradistibutions of Beurling type (resp. Roumieu type). These spaces (α + β > 1, resp. α + β ≥ 1) are closed under translation, dilation, multiplication, differentiation, and under the action of specified infinite order differential operators (see Section 1.2.1). The Roumieu type spaces are the well-known spaces of Gelfand-Shilov.

Ultradifferential Operators
It is said that P(ξ) = ∑ l∈N n 0 a l ξ l , ξ ∈ R n , is an ultrapolynomial of Beurling class (of Roumieu class), if the coefficients a l satisfy: The corresponding operator P(D) = ∑ l∈N n 0 a l D l is an ultradifferential operator of Beurling class (resp. Roumieu class). When M l = l! α it is called ultradifferential operator of class (α) (resp. class {α}). As M l satisfies (M.2), they act continuously on E (α) and D (α) (resp. E {α} and D {α} ), and the corresponding spaces of ultradistributions.

The k-DSTFT and the k-Directional Synthesis Operator
We recall some definitions and assertions from [4], where only the Roumieu case was considered. Here we state also the Beurling case, since the results of [4] also hold for the Beurling-type spaces.
and the k-directional synthesis operator of where It is shown in [4,Proposition 2.4] that for f ∈ S α β (R n ) the following reconstruction formula holds, where . Thus, the relation (7) takes the form For the sake of simplicity we transfer the STFT in direction of u k into the STFT in e k direction.
Recall the procedure (see [4]): Let A = [u i,j ] k×n be a matrix with rows u i , i = 1, . . . , k and I be the identity matrix of order n − k. Let B be an n × n matrix determined by A and I so that Bt = s, where s 1 = u 1,1 t 1 + · · · + u 1,n t n , . . . , s k = u k,1 t 1 + · · · + u k,n t n , s k+1 = t k+1 , . . . , s n = t n . The matrix B is regular, so put C = B −1 and e k = (e 1 , . . . , e k ), where e 1 , · · · , e k are unit vectors of the coordinate system of R k . If we change the variables t = Cs, and η = C T ξ, then for f ∈ L 2 (R n ), g ∈ Σ α β (R k ) (resp. g ∈ S α β (R k )), the equality (5) is transformed into: where h(s) = det(C) f (Cs) and (6) The function h(s) and it is referred to as the partial short-time Fourier transform. We have DS g,e k : ) is a continuous bilinear mapping. This is proved in Theorem 2.3 in [4] in the Roumieu case. With Theorem 2.5 and Corollary 2.7 in [4] in Roumieu case, and similarly in the Beurling case, it follows that DS * g,e k : is also continuous. This allows us to extend the definitions of the k-DSTFT and its synthesis operator to their duals (see [4] (Proposition 2.10)).

The Main Results
The STFT in the direction of u k can be used in the detection of singularities determined by the hyperplanes orthogonal to vectors u 1 , . . . , u k . For this purpose, we introduce kdirectional regular sets and wave front sets for the Beurling (resp. Roumieu)-tempered ultradistributions using the STFT in the direction of u k . To simplify our exposition we transfer the STFT in direction of u k into the STFT in e k direction by the use of (8).

Independence with Respect to a Window Function
One of the main results in [4] (Theorem 3.4) shows that the wave front set does not depend on the used window. Here, we prove the same assertion for the p-Sobolev k-directional wave fronts. The idea is similar to the one in [6,13] (see [4]) but here the decomposition of the involved integrals and the use of ultradifferential operators make the proof more complex. (12) holds for some g ∈ D (α) (R k ) (resp. g ∈ D {α} (R k )), g(0) = 0, then it holds for every h ∈ D (α) (R k ) (resp. h ∈ D {α} (R k )), h(0) = 0 supported by a ball B ρ (0), where ρ ≤ ρ 0 and ρ 0 depends on r in (12).
Since integration goes trough a subset of {ξ : |η − ξ| ≤ c|η|}, we can conclude that I 2,2 can be estimated similarly as I 1 . Additional therm κ d does not cause any problems since it belongs to E {α} .

Equivalent Definition
In this section we characterize the wave front sets given Definition 1 with the ones formulated in the next definition. We will use the Fourier transform as well as the cut-off function, and since we will show that these to definitions are equivalent but we need to distinguish them, we add the prefix "locally" in front of this notation in the Definition 2. We follow our ideas outlined in [10] and prove that both definitions determine the same sets. For the sake of completeness, we give all the details of the proof although it is the repetition of our proof of the theorem in [10] where we have considered distributions instead of ultradistributions.