# Generalized Functions - Vol 5: Integral Geometry and by I M Gel'fand (or Gelfand), M I Graev, N Ya Vilenkin, Eugene

By I M Gel'fand (or Gelfand), M I Graev, N Ya Vilenkin, Eugene Saletan

The 1st systematic concept of generalized features (also referred to as distributions) was once created within the early Nineteen Fifties, even though a few features have been constructed a lot past, such a lot significantly within the definition of the Green's functionality in arithmetic and within the paintings of Paul Dirac on quantum electrodynamics in physics. The six-volume assortment, Generalized capabilities, written by means of I. M. Gelfand and co-authors and released in Russian among 1958 and 1966, supplies an creation to generalized features and provides a variety of purposes to research, PDE, stochastic procedures, and illustration idea. the most aim of quantity four is to improve the sensible research setup for the universe of generalized services. the most idea brought during this quantity is the thought of rigged Hilbert house (also often called the built Hilbert house, or Gelfand triple). Such house is, actually, a triple of topological vector areas \$E \subset H \subset E'\$, the place \$H\$ is a Hilbert area, \$E'\$ is twin to \$E\$, and inclusions \$E\subset H\$ and \$H\subset E'\$ are nuclear operators. The ebook is dedicated to numerous functions of this concept, akin to the idea of optimistic certain generalized features, the idea of generalized stochastic techniques, and the examine of measures on linear topological areas.

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Example text

These c^ are constants, or rather are independent of the intersecting plane. To see this, let φ^ξ, p) be the Radon transform of the charac­ teristic function of one of the octants. Now let us perform a dilation in one of the coordinate directions, say the x1 direction: χ ι = ^ι^ > 0)» x 2 = x 2 > x'z = xs · This changes φ 1 (^ 1 , ξ2 , £8 ; p) into λ-1φ1(λ-1ξ1, ξ2, ξζ ; p) [see Eq. 3]. Since, however, the octant is invariant under this trans­ formation, the generalized areas of the intersections must remain invariant.

58 RADON TRANSFORM OF TEST FUNCTIONS Ch. I Let us now use the method of analytic continuation in the coefficients of P (as described in the appendix to this section). On so doing, we arrive at the following result. Assume the nondegenerate quadratic form P(x) to have k positive and / negative coefficients in the canonical form. Then the Radon transform of (P + ί0)λ is i Γ ( - λ ) | Δ |* * V | p |2A+n-l(Q _ fO)-A-in (7) where Q = Q(£) is the quadratic form dual to P, and Δ is the discriminant of P.

Xn). It is easily shown that the 30 RADON TRANSFORM OF TEST FUNCTIONS Ch. I Radon transform of ax (x; X) is then obtained by replacing ξ1 in our previous formulas by the inner product (£, x0). Thus the Radon transform of the generalized function defined by (5) is # ( £ , *o);1-A + />-(£. , (6) and />*(£, ^); 1 "* + P-{L x«)-1-* - ( - ^ 8 w [(f, * „ ) ] / In I * I for λ = * = 0, 1, ... , xj, both concentrated on the entire xx axis. , * Λ ) is l/> ΙΛI ^ |-X~A, for ΧΦ2Η (A = 0 , 1 , . . ) (7) (here | £x I-1-2*1 is an associated homogeneous generalized function).

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