Radon’s problem for some surfaces in ${\bf R}\sp n$
Radon’s problem for some surfaces in ${\bf R}\sp n$
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${f R}sp n$ 中某些表面的氡气问题
DOI：
10.1090/s0002-9939-1987-0870790-6
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发表时间：
1987
期刊：
影响因子：
2.1
通讯作者：
A. Cormack
中科院分区：
文献类型：
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作者：
A. Cormack
R2中一族曲线的Radon问题已推广到Rn中的一族(n，1)维曲面族。该问题被假定为一组积分方程组。给出了抛物面和心形方程的解，并求出了它们的零空间和相容条件。在Rn中，设x=(X1，X2，...，xn)是一个向量，设(，q是单位向量，表示标积。设r=1x1，x=r(，p为非负实数。对于固定的p，i表示(1)r&cos{acos-L((r*7)}=pa>；？表示关于Qj对称的(n，1)维曲面，且对于该曲面，当(=Q)时，r=p。Radon的问题是确定函数f(X)，该函数给定曲面(1)上的f的积分。这是[1，2，3]中讨论的Radon问题的推广，其中(1)表示R2中的一族曲线族。在这个二维问题中，假设a是正的，这些曲线称为a-曲线，对于负的，我们定义：f=-a，并将这些曲线称为f3-曲线。由于曲线在单位圆(r，0)-t(1/r，0)内的逆成为相应的f曲线，所以a曲线和：曲线密切相关。在下文中，我们将首先考虑a的曲面(1)，即a>；0，然后在附录A中列出具有相应公式编号的f曲面的重要公式，因为处理方法是如此相似，不需要重复论证。我们首先建立了f在球谐函数中展开时的积分方程组。对于a=1/2，这些方程归结为由Wimp[13]求解的积分方程解的推广，并在附录B中讨论。它们离原点最近的点是x=pq，当p=0时，它们退化到从原点到-q方向无穷远的直线。对于a=-1/2(f=1/2)，(1)表示心形的一族。它们到原点的最大距离是x=Pr和r->；0，As(-+)。在这种情况下的解取决于Wimp的结果的进一步修正，这也在附录B中讨论。由编辑收到，1985年12月23日。1985年1月10日在新奥尔良举行的美国数学学会会议上发表的关于Radon变换和层析成像的特别会议上。1980年数学主题分类(1985年修订版)。初级44A05，45A05。关键的用法和短语。Radon变换，积分方程式。(D1987美国数学学会0002-9939/87$1.00+$0.25每页
Radon's problem for a famnily of curves in R2 has been generalized to a family of (n 1)-dimensional surfaces in Rn. The problem is posed as a set of integral equations. Solutions to these equations are given for paraboloids and cardioids, and for these cases the null spaces and consistency conditions have been found. In Rn let x = (xl, X2,.... ,xn) be a vector, let (, q be unit vectors, and let denote the scalar product. Let r = lxl, x = r(, and let p be a nonnegative real number. For a fixed p, I the expression (1) r& cos{ acos-l((r * 7)} = P a > ? represents an (n 1)-dimensional surface which is symmetrical about qj and for which r = p when ( = q. Radon's problem is to determine a function f(x) given the integrals of f over the surfaces (1). This is a generalization of Radon's problem in which (1) represented a family of curves in R2, which was discussed in [1, 2, 3]. In this two-dimensional problem a was assumed to be positive and the curves were called a-curves, and for a negative we defined:f = -a and referred to these curves as f3-curves. The a-curves and :-curves are intimately related since an acurve becomes the corrsponding f-curve under inversion in the unit circle: (r, 0) -t (1/r,0). In what follows we shall first consider the surfaces (1) for a as stated, namely, a > 0, and then in Appendix A list the important formulas for the fsurfaces with corresponding formula numbers since the treatments are so similar that the arguments need not be repeated. We first establish the integral equations for f when f is expanded in spherical harmonics. For a = 1/2 these equations reduce to an extension of an integral equation which was solved by Wimp [13], and which is discussed in Appendix B. For a = 1/2, (1) represents a family of paraboloids. Their closest point to the origin is x = pq, and for p = 0 they degenerate to the straight line from the origin to infinity in the direction -q. For a = -1/2 (f = 1/2), (1) represents a family of cardioids. Their greatest distance from the origin is x = pr and r -> 0 as ( -+ The solution in this case depends on a further modification of Wimp's result, also discussed in Appendix B. I In addition to the solutions of Radon's problem for these paraboloids and cardioids we give the null-spaces and consistency conditions for the corresponding Radon transforms. Received by the editors December 23, 1985. Presented at the American Mathematical Society meeting in New Orleans, Special Session on the Radon Transform and Tomography, January 10, 1985. 1980 Mathematic Su*ject Clasification (1985 Revisio). Primary 44A05, 45A05. Key uwods and phrases. Radon transform, integral equations. (D1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page