By Tracy Kompelien

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**Sample text**

Let f ∈ L(Rn ), n 0 n f ∈ Hk,l (R ). In addition, if f (x) = u(ρ)h(x), where h ∈ Hk (Rn ) then f (x) = (2π)n/2 h(x) ik ∞ 0 tn+2k−1 u(t)I(n/2)+k−1 (t|x|)dt. 4) FOURIER TRANSFORM AND RELATED QUESTIONS 39 Proof. 29) for η = 0, we obtain that f ∈ H0k,l (Rn ). 4) can be proved in a similar way. Conversely, let f ∈ H0k,l (Rn ). Assume that g ∈ Drad (Rn ). 1). 29) for η = 0, we obtain f ∗ g ∈ Hk,l (Rn ). 3. To conclude this section we consider the Fourier transform on the space L2 (Rn ). Let f ∈ L2 (Rn ).

We say that P is elliptic on U if Char P = ∅. We require the following statement. 1. Let g ∈ D (U). Assume that f ∈ D (U) is solution of the equation P f = g. Then WFA (f ) ⊂ WFA (g) ∪ Char P. In particular, if P is elliptic on U and g ∈ RA(U) then f ∈ RA(U). 6], where a generalization of this theorem is obtained. 1. Assume that 0 ∈ U, and let f ∈ D (U) such that supp f ⊂ U ∩ x ∈ Rn : xn ε x21 + · · · + x2n−1 for some ε > 0. Assume that Pm (0, η) = 0 / supp f . for some η ∈ Rn , and let P f = 0.

Then for each x ∈ Rn there exists ∂ f (x)/∂xj and the following equality is valid ∂ f (x)/∂xj = (−iyj f (y)) (x). (2) If f ∈ L(Rn ) and g = ∂f /∂xj ∈ L(Rn ) for some j ∈ {1, . . , n} then g(x) = ixj f (x). In particular, if f ∈ D(Rn ) and α ∈ Zn+ then (∂ α f ) (x) = i|α| xα f (x). 1 follows from the deﬁnition of Fourier transform. The following statement gives the asymptotic expansion for the Fourier transforms of a broad class of functions. 1. Let 0 we set a < b, Re α > 0, Re β > 0, h ∈ C ∞ [a, b].