By V.V. Volchkov
Integral geometry bargains with the matter of deciding on services by means of their integrals over given households of units. those integrals de?ne the corresponding integraltransformandoneofthemainquestionsinintegralgeometryaskswhen this remodel is injective. however, once we paintings with advanced measures or types, operators look whose kernels are non-trivial yet which describe very important periods of capabilities. lots of the questions bobbing up right here relate, in a single manner or one other, to the convolution equations. many of the renowned guides during this ?eld comprise the works by way of J. Radon, F. John, J. Delsarte, L. Zalcman, C. A. Berenstein, M. L. Agranovsky and up to date monographs via L. H¨ ormander and S. Helgason. till lately learn during this zone was once conducted ordinarily utilizing the means of the Fourier rework and corresponding equipment of advanced research. in recent times the current writer has labored out an basically di?erent method in response to the outline of assorted functionality areas when it comes to - pansions in detailed services, which has enabled him to set up very best ends up in a number of renowned problems.
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Example text
3, we obtain that βk,l,η = 0 for all k, l, η. 6) holds and the proof is complete. 5). 5) is called the Helmholtz equation. Let U ⊂ Rn be a non-empty open set. 9) Br for all x ∈ U, r ∈ (0, dist(x, ∂U)). In particular, for λ = 0 we have the classical mean value theorem for harmonic functions. Furthermore, let ϕ ∈ Erad (Rn ), and assume that there exists a non-empty open set U1 ⊂ Rn such that U1 − supp ϕ ⊂ U. 8). 5) is called a polyharmonic equation. Γ ν + n2 + 1 Br ν=0 for all x ∈ U and r ∈ (0, dist(x, ∂U)).
45)) we obtain assertion (1). 8. Assertion (2) can be proved in a similar way. 9. Let U = BR and let A∞ ϕ (U) = {0}. Then Aϕ (U) contains a radial function distinct from zero. 5 Proof. By the hypothesis there exists a function f ∈ A∞ ϕ (U) distinct from zero. 9) belongs to A∞ ϕ (U). Let k be the smallest non-negative integer such that the set A∞ ϕ (U) contains the function (k) of the form g(ρ)Yl (σ) distinct from zero for some l ∈ {1, . . , dk } (there exists such a k because f = 0 in BR ).
Let (y, ξ) ∈ (N ∗ S(x0 , r0 )) \ {0}, and assume that f is real analytic in a neighborhood of the Tx0 -mirrored point to y. Then (y, ξ) ∈ / WFA (f ). 1. Some general results Let U be a non-empty open subset of Rn . 1) |α| m where cα ∈ RA(U). For (x, ξ) ∈ U × Rn we set Pm (x, ξ) = |α|=m cα (x)ξ α . We denote Char P = {(x, ξ) ∈ U × (Rn \{0}) : Pm (x, ξ) = 0}. 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.