Abstract Generated abstract
This note studies the real Euclidean analogue of reconstructing a function in three-dimensional affine space from its integrals over straight lines belonging to a three-parameter complex. The authors formulate the space of lines as a pseudo-Riemannian manifold, identify the relevant Laplace-Beltrami operator, and relate the line-integral transform to harmonic functions on this manifold, proving continuity, inversion, projection, and compatibility with the ordinary Laplacian. They then reinterpret reconstruction from restricted families of lines as a boundary-value problem for the Laplace equation on the manifold of lines. A Green-type formula using currents is proposed, showing that in general both boundary values and normal derivatives are required, with special hypersurfaces allowing recovery from boundary values alone.
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UDC 517.948.5
MATHEMATICS
V. I. SEMYANISTYI, Z. F. SHIBASOVA
INTEGRAL GEOMETRY IN EUCLIDEAN SPACE AND ITS CONNECTION WITH BOUNDARY-VALUE PROBLEMS
(Presented by Academician I. G. Petrovskii, 19 III 1966)
In the present note we consider the real analogue of the problem of reconstructing a function, defined in three-dimensional affine space, from its known integrals over lines belonging to a certain complex (a three-parameter family), solved for the case of complex affine space by I. M. Gelfand and M. I. Graev (\(^1,{}^2\)). A connection is established between this problem and boundary-value problems for a differential equation.
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Let \(R_3\) denote real Euclidean space, and let \(\Pi_4\) be the manifold of its lines. We shall agree to specify each point in \(\Pi_4\), i.e. a line in \(R_3\), by a direction vector \(l \ne 0\) and a point \(a\) of \(R_3\) lying on this line. Thus we define a mapping
\[ p:\ (R_3 \setminus \{0\}) \times R_3 = A_6 \to \Pi_4 . \]
We shall agree, for brevity, to call any function or differential form on \(A_6\) that coincides with the image of some function or differential form on \(\Pi_4\) under the mapping \(p^*\) a function or differential form defined on \(\Pi_4\). Let \((x,y)\) denote the scalar product in \(R_3\), and let \(\partial_x\) be the operator which assigns to a function of \(x\) its gradient. It is easy to see that, in order that a function or differential form on \(A_6\) be defined on \(\Pi_4\), it is necessary and sufficient that it be invariant with respect to the transformations
\[ l \to tl,\qquad a \to a + sl, \tag{1} \]
where \(t \ne 0\) and \(s\) are arbitrary real parameters. For differentiable functions this condition is equivalent to the following:
\[ (l,\partial_l)h \equiv l_1\frac{\partial h}{\partial l_1} + l_2\frac{\partial h}{\partial l_2} + l_3\frac{\partial h}{\partial l_3}=0,\qquad (l,\partial_a)h \equiv l_1\frac{\partial h}{\partial a_1} + l_2\frac{\partial h}{\partial a_2} + l_3\frac{\partial h}{\partial a_3}=0 . \tag{2} \] -
The manifold \(\Pi_4\) is naturally a pseudo-Riemannian space with metric
\[ ds^2 = (l,dl,da)/(l,l) \tag{3} \]
(the numerator here is the mixed product in \(R_3\)). This metric corresponds to the volume element
\[ \omega = \frac{(l_1dl_2\wedge dl_3 + l_2dl_3\wedge dl_1 + l_3dl_1\wedge dl_2)\wedge (l_1da_2\wedge da_3 + l_2da_3\wedge da_1 + l_3da_1\wedge da_2)} {(l,l)^2}, \tag{4} \]
the Laplace–Beltrami operator
\[ \square = (l,\partial_l,\partial_a) \equiv l_1\left(\frac{\partial^2}{\partial l_2\partial a_3} -\frac{\partial^2}{\partial l_3\partial a_2}\right) + l_2\left(\frac{\partial^2}{\partial l_3\partial a_1} -\frac{\partial^2}{\partial l_1\partial a_3}\right) + l_3\left(\frac{\partial^2}{\partial l_1\partial a_2} -\frac{\partial^2}{\partial l_2\partial a_1}\right), \tag{5} \]
and the codifferentiation operator \(\delta\) (divergence), whose explicit form we shall not write down, but only note that if, as usual, we denote \(h\omega = *h\), then
\[ \delta * h = \frac{(l_1dl_2\wedge dl_3 + l_2dl_3\wedge dl_1 + l_3dl_1\wedge dl_2)\wedge(\partial_a,\partial_l)h} {(l,l)} +\alpha, \tag{6} \]
where \(a\) is a certain closed form. In this case, as is not hard to notice,
\(d\delta * h = * \Box h\).
Let us note that on \(\Pi_4\), besides \(\Box\), there also exist other invariant differential operators of second order. We shall encounter, for example, the operator
\[ \hat{\Delta}=(\partial_a,\partial_a)=\partial^2/\partial a_1^2+\partial^2/\partial a_2^2+\partial^2/\partial a_3^2. \tag{7} \]
We shall call a function \(h(l,a)\), defined on \(\Pi_4\), rapidly decreasing if it itself and all its products by arbitrary polynomials on \(\Pi_4\) in the coordinates of the vector \(a\) are integrable over the whole space \(\Pi_4\). The set of all infinitely differentiable rapidly decreasing functions on \(\Pi_4\) will be denoted by \(\hat S\). By \(\hat K\) we shall denote the set of all infinitely differentiable functions on \(\Pi_4\) with compact support. In the linear space \(\hat S\) one introduces, as is customarily done in the theory of generalized functions, the topology of uniform convergence of all derivatives on compact sets, and in \(\hat K\) the topology of the inductive limit of subspaces in \(\hat S\) consisting of functions whose supports lie in one and the same compact set (cf., for example, \((^3)\)). We shall call a function from \(\hat S\) or \(\hat K\) harmonic if it is taken to zero by the operator \(\Box\). The subspaces of harmonic functions in \(\hat S\) or \(\hat K\) will be denoted by \(H\hat S\) or \(H\hat K\), respectively.
- Let \(S\) and \(K\) denote, respectively, the spaces of infinitely differentiable and finite functions on \(R_3\) (see \((^3)\)). To each function \(f(x)\in S\) we assign its integrals over all possible straight lines in \(R_3\):
\[ \hat f(l,a)=|l|\int_{-\infty}^{\infty} f(a+lt)\,dt. \tag{8} \]
Theorem 1. The operation \(f\to \hat f\) is a one-to-one and continuous mapping of the spaces \(S\) and \(K\) onto \(H\hat S\) and \(H\hat K\), respectively. Moreover,
\[ (\Delta f)^{\hat{\ }}=\hat{\Delta}\hat f * . \tag{9} \]
Let us note that the essential part of this theorem is due to F. John, who in 1938 established that integrals over straight lines satisfy a system of equations equivalent to (2), (6), and also that every function on \(\Pi_4\) satisfying this system is representable in the form (8) (see \((^4)\)).
- If \(x\) is an arbitrary point in \(R_3\), and \(p(l,a)\) is a point in \(\Pi_4\), i.e. a straight line in \(R_3\), then the Euclidean distance between them is equal to
\[ \rho(x;l,a)=\frac{|[l,a-x]|**}{|l|}. \tag{10} \]
For each function \(h(l,a)\in\hat S\) put
\[ \check h(x)=-\frac{1}{4\pi^3}\int_{\Pi_4}\rho^\lambda(x;l,a)h(l,a)\omega\big|_{\lambda=-3}. \tag{11} \]
The right-hand side here must be understood in the following way. The integral here converges and defines an analytic function of \(\lambda\) for \(\operatorname{Re}\lambda>-2\). The right-hand side of (11) is the value at \(\lambda=-3\) of the analytic continuation of this integral.
Theorem 2. The operation \(h\to \check h\) is a continuous mapping of the spaces \(\hat S\) and \(\hat K\) onto \(S\) and \(K\), respectively. The kernel of this mapping is the orthogonal complement in the spaces \(\hat S\) and \(\hat K\) of the subspaces \(H\hat S\) and \(H\hat K\) in the sense of the scalar product
\[ (h_1,h_2)=\int_{\Pi_4}\bar h_1h_2\omega. \tag{12} \]
\[ \text{* } \Delta \text{ denotes the Laplace operator in } R_3. \]
\[ \text{** } [a,b] \text{ denotes the vector product in } R_3. \]
In this case
\[ (\Delta \hat h)^\vee=\Delta \check h. \tag{13} \]
Theorem 3. The operation \(h\to \hat h\) is a one-to-one and continuous mapping of the spaces \(H\hat S\) and \(H\hat K\) onto \(S\) and \(K\), respectively.
In the case \(h\in H\hat S\), formula (11) is equivalent to the following:
\[ \check h(x)=-\frac{1}{2\pi^2} \int_{[m[l,a-x]]=0} p^\lambda(x;l,a)h(l,a)\,\theta\big|_{\lambda=-2}, \tag{14} \]
where \(m\) is an arbitrary vector from \(R_3\setminus\{0\}\),
\[ \theta=(m,l,dl)\wedge(m,l,da)/(m,m)|l|^3. \tag{15} \]
We note that the integration in (14) is carried out over the set of lines lying in a plane passing through \(x\); the independence of the integral (14) from the choice of such a plane is a necessary and sufficient condition for the harmonicity of the function \(h(l,a)\).
Theorem 4. For every function \(f\in S\) the inversion formula
\[ f=(\hat f)^\vee, \tag{16} \]
holds, and for every function \(h\) from \(\hat S\) the mapping \(h\to(\check h)^\wedge\) is the orthogonal projection onto \(H\hat S\).
- We now turn to the problem of reconstructing a function \(f\in K\) (or \(S\)) from its known integrals over lines belonging to some complex. Formulas (16) and (14) make it possible to solve it for the special case of a complex that is a one-parameter family of plane fields. For example, for the complex of lines intersecting a fixed line, the reconstruction problem is solved completely in this way, while for the complex of lines tangent to a cone, the value of the function \(f(x)\) is reconstructed only at points lying outside this cone.
Of course, it is very interesting to find an inversion formula for a broader class of complexes. Instead of this problem we shall further consider the equivalent boundary-value problem for the Laplace equation in \(\Pi_4\): knowing the values of the function \(h\in H\hat K\) on some hypersurface \(D\) in \(\Pi_4\), find this function (the equivalence of these problems follows directly from Theorems 3 and 4). It turns out, however, that the stated problem is solvable only for a comparatively narrow class of hypersurfaces. In the general case, in order to find a function \(h\in H\hat K\), one must know not only its values on \(D\), but also the values of its normal derivatives (in the sense of the metric (3)) on this hypersurface.
- To solve the boundary-value problem just formulated, we shall now propose a certain analogue of Green’s formula. Let us begin with some preliminaries.
Let \(p(m,b)\) be a fixed point of \(\Pi_4\), and let \(k\) be some vector from \(R_3\setminus\{0\}\). Consider a current of degree zero (a generalized function) assigning to each compactly supported 4-form \(\varphi\) the number
\[ \langle \varphi,T^\lambda(k;m,b;l,a)\rangle = \int_{\Pi_4} \left\{ \frac{(l,m,a-b)\operatorname{sgn}\{(l,k)(m,k)\}+i0}{|l|\,|m|} \right\}^{\lambda} \varphi. \tag{17} \]
The integral here converges for \(\operatorname{Re}\lambda>0\); for the remaining \(\lambda\) it must be understood in the regularized sense, i.e. as an analytic continuation in \(\lambda\). Let \(C\) denote some 4-chain, which, as usual (see (5)), we identify with the current
\[ \langle \alpha,C\rangle=\int_C \alpha. \]
Represent this chain in the form
\[ C=\sum_{i\in I} t_i C_i, \]
where \(C_i\) are elements of the chain, and assign to each \(i\in I\) a vector \(k_i\in R_3\setminus\{0\}\) such that for all points \(p(l,a)\) belonging to the set \(|C_i|\) of points of the element \(C_i\), \((l,k_i)\ne 0\). The existence of the vector \(k_i\) follows from the fact that the set \(|C_i|\) is situated in a single coordinate neighborhood. Put now
\[ T_i^\lambda(m,b;l,a)=t_i T^\lambda(k_i;m,b;l,a). \tag{18} \]
We shall say that the point \(p(m,b)\) is situated regularly with respect to the chain \(C\) if, in a neighborhood of any point of intersection of the isotropic cone with vertex \(p(m,b)\)* and the boundary \(\partial C_i\) of an arbitrary element \(C_i\) of the chain \(C\), the cone and the boundary \(\partial C_i\) can serve as a pair of coordinate neighborhoods in some system of local coordinates. If this condition is fulfilled, then the products of currents
\[ \sum_{i\in I} T_i^\lambda \partial C_i,\qquad \sum_{i\in I} \delta * T_i^\lambda \wedge \partial C_i \tag{19} \]
are defined.
It is clear that the common support of the currents (19) is the hypersurface \(D\) in \(\Pi_4\) lying on the union of all \(|\partial C_i|\); generally speaking, this surface depends on the representation of the chain \(C\) and on the choice of the vectors \(k_i\); it may be larger than \(|\partial C|\).
Theorem 5. If the point \(p(m,b)\) is situated regularly with respect to the chain \(C=\sum_{i\in I}t_i C_i\), and if it lies inside the set \(|C|\), then for any function \(h\in \hat H\hat K\)
\[ h(m,b)=\frac{1}{\pi^2}\sum_{i\in I} \{\langle h,\delta * T_i^\lambda \wedge \partial C_i\rangle -\langle \delta * h,T_i^\lambda\partial C_i\rangle\}_{\lambda=-1}. \tag{20} \]
Let us note that, in order to compute the first term on the right, it is necessary to know the values of \(h\) on \(D\), while in order to compute the second term, the values on \(D\) of the normal derivatives of this function are needed. However, if \(D\) is given by the equation \(F(l,a)=0\), with \(\delta * F=0\) on \(D\), then these normal derivatives can be found from the values of \(h\) on \(D\).
The authors express their gratitude to I. M. Gel'fand, M. I. Graev, and Z. Ya. Shapiro for discussion of the present work.
Kolomna Pedagogical Institute
Received
16 III 1966
CITED LITERATURE
- I. M. Gel'fand, M. I. Graev, DAN, 138, No. 6, 1266 (1961).
- I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions, Vol. 5, Moscow, 1962.
- I. M. Gel'fand, G. E. Shilov, Generalized Functions, Vol. 1, Moscow, 1959.
- F. John, Duke Math. J., No. 4, 300 (1938).
- G. de Rham, Differentiable Manifolds, IL, 1956.
* That is, the cone consisting of all points at zero distance from \(p(m,b)\).