New PDF release: 0-Dual Closures for Several Classes of Graphs

By Ainouche A., Schiermeyer I.

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Extra resources for 0-Dual Closures for Several Classes of Graphs

Example text

1: Source curve Γ. Proof. Write g (y, v) e−j G (y, ξ) = ξ,v dv E and rearrange the right-hand side ∞ g (y, v) e−j ξ,v dv = E f (y + tv) e−j ξ,v dtdv f (y + u) e−jr ξ,u rdudr E 0 ∞ = 0 E ∞ ∞ f (x) e−jr = 0 ξ,x dxejr ξ,y rfˆ (rξ) ejr rdr = E ξ,y dr, 0 by changing the variables v = ru, t = r−1 , x = y + u. This yields ∞ rfˆ (rξ) ejr G (y, ξ) = ξ,y dr. 0 It follows that G depends on y through the product ξ, y = ξ, x . Differentiating we obtain Gs (y, ξ) = j y ,ξ ∞ r2 fˆ (rξ) ejr ξ,x dr. 20) follows.

Proof. 19), which equals πHf (x) , where H is the Hilbert transform. The right-hand side and the function Hf are known for any t, 0 ≤ t ≤ 1, that is, for any point x in the interval [y1 , y2 ] and f is supported by this interval. 8. 11 Let Γ be a 1-smooth curve in an Euclidean space E 3 with a parameterization y = y (s), ys = 0, s ∈ [0, 1] that satisfies the completeness condition for a point x ∈ E 3 \Γ : almost any plane through x meets Γ at a point y transversely. 20) where Ω is the volume form on the unit sphere S2 and, for any ξ ∈ S2 , y is an arbitrary point in Γ such that y, ξ = x, ξ and ys , ξ = 0.

16) equals −2π 2 f (x). 5 Backprojection Filtration Method Let Γ be a connected 1-smooth curve in an Euclidean space E n of dimension n ≥ 2, f be a function on E n with compact support vanishing on Γ, and g (y, v) = Xf (y, v) be its ray transform. Let y = y (s) , a ≤ s ≤ b be a 1smooth parametrization of Γ. For a function f on E n and a point x ∈ E 3 \Γ, we take the integral along the curve arc between these some points y (s1 ) and y (s2 ) : s2 G (x, y1 , y2 ) s1 ∂ g (y (s) , x − y (s) + εy (s)) ∂ε ds.

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0-Dual Closures for Several Classes of Graphs by Ainouche A., Schiermeyer I.

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