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UDC 513.88+517.948
MATHEMATICS
G. I. Shlakhsin
ON THE EXPRESSION OF A FUNCTION THROUGH ITS INTEGRALS OVER ELLIPSOIDS
(Presented by Academician I. G. Petrovskii, May 22, 1965)
Let
\[ P=P(x)=\sum_{s,t=1}^{n} a_{st}x_sx_t \]
be a positive definite quadratic form in Euclidean space of odd dimension \(n\), and let \(\delta^{(p)}(c^2-P)\) be a functional on the space \(K\) of finite infinitely differentiable functions. To a function \(f(x)\) from \(K\) we associate its integrals over ellipsoids
\[ \check f_p(x)=\int f(x+y)\delta^{(p)}(c^2-P(y))\,dy \]
for fixed \(p\), where \(dy=dy_1\ldots dy_n\); the integration is carried out over all values of the variables (which is also understood for the other integrals). It is required, knowing \(\check f_p(x)\), to reconstruct the function \(f(x)\).
The following results are obtained in the paper:
- If \(f(x)\) is concentrated inside the domain \(P(x)\leq c^2\), then
\[ f(x)=(-1)^{(p-q)/2}\frac{2c^2\Delta}{\pi^{\,n-1}} \int \check f_p(x+y)\delta^{(q)}(c^2-P(y))\,dy, \tag{1} \]
where \(\Delta=\det\|a_{st}\|\); \(q\) is determined from the condition \(p+q=n-1\) (this equality is assumed to hold also below); \(p=0,1,\ldots,n-1\), and also
\[ f(x)=-(-1)^{(p-q)/2}\frac{\Delta}{2\pi^{\,n-1}} \int L_p\check f_{p-1}(x+y)\delta^{(q-1)}(c^2-P(y))\,dy, \tag{2} \]
where
\[ L_p=\sum_{s,t=1}^{n} a^{st}\frac{\partial^2}{\partial x_s\partial x_t}; \]
\(a^{st}\) are the coefficients of the matrix inverse to \(\|a_{st}\|\); \(p=1,2,\ldots,n-2\).
- If \(\check f_p(x)\) and \(\check f_{p-1}(x)\) are known, then
\[ f(x)=\frac{(-1)^{(p-q)/2}\Delta}{4\pi^{\,n-1}} \left[ 4c^2\int \check f_p(x+y)\delta^{(q)}(c^2-P(y))\,dy -\int L_p\check f_{p-1}(x+y)\delta^{(q-1)}(c^2-P(y))\,dy \right], \]
where \(f(x)\) is an arbitrary function from \(K\), \(p=1,2,\ldots,n-2\).
This establishes the equivalence of (1) and (2) for those values of \(p\) for which (2) is defined.
- For an arbitrary function from \(K\),
\[ f(x)=\frac{2\Delta c^2}{\pi^{\,n-1}} \sum_{k=0}^{\infty}(-1)^k(2k+1) \int \check f_{(n-1)/2}(x+y)\delta^{((n-1)/2)}(c^2(2k+1)^2-P(y))\,dy, \]
\[ f(x)=-\frac{\Delta}{2\pi^{\,n-1}} \sum_{k=0}^{\infty}\int L_P\check f_{(n-3)/2}(x+y) \delta^{((n-3)/2)}(c^2(2k+1)^2-P(y))\,dy, \tag{3} \]
where the sums terminate for sufficiently large \(k\).
Relation (1) for \(p=0\) in the case when \(P\) is a sum of squares was obtained earlier by another method by N. Ya. Vilenkin. For \(n=3\), (3) is John’s formula for expressing a function through its spherical means over spheres of fixed radius in an affinely invariant notation \((({}^{1}),\) p. 109).
We next prove some relations for delta-functions, from which these results follow directly.
Denote by
\[ Q=Q(\sigma)=\sum_{s,t=1}^{n} a^{st}\sigma_s\sigma_t \]
the quadratic form conjugate to \(P\). As is known from \(({}^{2})\), the following formulas hold:
\[ F\left[\delta^{(p-1)}(c^2-P)\right] = \frac{\pi^{n/2}}{\sqrt{\Delta}} \left(\frac{2c}{Q^{1/2}}\right)^{n/2-p} J_{n/2-p}(cQ^{1/2}), \tag{4} \]
\[ F[L_P\delta(x)]=-Q, \tag{5} \]
where \(F\) denotes the Fourier transform; \(p\ge 1\) is an integer; \(c>0\), \(\delta(x)=\delta(x_1,\ldots,x_n)\) is the delta-function; \(J_{n/2-p}\) is a Bessel function. The left-hand sides of (4) and (5) are understood as functionals on the space \(Z\), dual to \(K\) with respect to the Fourier transform.
Consider the generalized function \(\delta^{(p)}(c^2-P)*\delta^{(q)}(c^2-P)\), defined by means of convolution. Since the inverse Fourier transform \(F^{-1}\) of the function \(1\) is \(\delta(x)\), using the properties of the Fourier transform of a convolution (see \(({}^{3})\)) and (4), and introducing the notation \(z=cQ^{1/2}\), \(m=(p-q)/2\) (\(m\) an integer), we shall have
\[ \delta(x)=(-1)^m \frac{2c^2\Delta}{\pi^{n-1}} \delta^{(p)}(c^2-P)*\delta^{(q)}(c^2-P)+A_m(c,P), \tag{6} \]
where
\[ A_m(c,P)=F^{-1}\left[1-(-1)^m\pi z J_{-m-1/2}(z)J_{m-1/2}(z)\right], \qquad p=0,1,\ldots,n-1. \]
Similarly, taking into account (5), the differentiation rule for convolution, and the equality
\[ J_\nu(z)J_{-\nu+1}(z)+J_{-\nu}(z)J_{\nu-1}(z)=\frac{2\sin \nu\pi}{\pi z}, \]
one can obtain
\[ \delta(x)=-(-1)^m \frac{\Delta}{2\pi^{n-1}} \delta^{(p-1)}(c^2-P)*L_P\delta^{(q-1)}(c^2-P)-A_m(c,P). \]
We shall show that the functional \(A_m(c,P)\) is concentrated outside the domain \(P(x)<(2c)^2\).
For the proof, using the representation of Bessel functions with half-integral index in terms of Lommel polynomials \(R_{m,\nu}(z)\) \((({}^{4}),\) p. 326), we transform the expression for \(A_m(c,P)\):
\[ F[A_m]=U_m(z)\cos 2z+V_m(z)\sin 2z, \]
where
\[ U_m(z)=R_{m-1,\,3/2}(z)R_{m-1,\,1/2}(z) + R_{m,\,1/2}(z)R_{m-2,\,3/2}(z), \]
\[ V_m(z)=R_{m-1,\,3/2}(z)R_{m-2,\,3/2}(z) - R_{m,\,1/2}(z)R_{m-1,\,1/2}(z). \]
Here we have used Krell’s formula for Lommel polynomials \((({}^{4}),\) p. 328). The functions \(U_m(z)\) and \(V_m(z)\) are polynomials in negative powers of \(z\), and the validity of the assertion follows from the following lemma.
Lemma. The functional \(F^{-1}[\cos z/z^{2k}]\) for integer \(k<n/2\) and the functional \(F^{-1}[\sin z/z^{2k+1}]\) for integer \(k<(n-1)/2\) are concentrated outside the domain \(P(x)<c^2\).
For the proof of the lemma it suffices, using formulas 8.411,7; 6.699,1 and 2 from \(({}^{5})\), to establish two relations with a hypergeometric function. We give one of them:
\[ z^{\lambda-n}\cos ze^{-i(x,\sigma)}\,d\sigma = \frac{2\sqrt{\Delta}\,\pi^{n/2}}{c^n\Gamma(n/2)} \Gamma(\lambda)\cos\frac{\pi\lambda}{2} F\left( \frac{\lambda}{2},\, \frac{\lambda+1}{2},\, \frac{n}{2};\, \frac{P}{c^2} \right). \]
Here \(\lambda\) is a complex parameter, \(d\sigma=d\sigma_1\ldots d\sigma_n\), \(P(x)<c^2\). For \(0<\operatorname{Re}\lambda<(n+1)/2\) the integral converges directly; for other values of \(\operatorname{Re}\lambda\) it is understood in the regularized sense, by virtue of the uniqueness of analytic continuation.
Let \(k\) be an integer. For the case \(m=0\), analogously to (6), we have
\[ F\left[2\Delta c^2 \delta^{((n-1)/2)}(c^2-P)*\delta^{((n-1)/2)}(c^2(2k+1)^2-P)\right] = \]
\[ = \pi^{n-1}\bigl(\cos 2kz+\cos 2(k+1)z\bigr), \]
and further
\[ \delta(x)=\frac{2\Delta c^2}{\pi^{n-1}}\delta^{((n-1)/2)}(c^2-P)* \]
\[ * \sum_{k=0}^{r-1}(-1)^k(2k+1)\delta^{((n-1)/2)}\bigl(c^2(2k+1)^2-P\bigr)+(-1)^r B_n(cr,P), \]
where \(r\ge 1\) is an arbitrary integer,
\[ B_n(c,P)=\frac{2c\sqrt{\Delta}}{\pi^{(n-1)/2}}\delta^{((n-1)/2)}(4c^2-P). \]
In the same way one easily obtains
\[ \delta(x)=-\frac{\Delta}{2\pi^{n-1}}\delta^{((n-3)/2)}(c^2-P)*L_P \sum_{k=0}^{r-1}\delta^{((n-3)/2)}\bigl(c^2(2k+1)^2-P\bigr)+B_n(cr,P). \]
In conclusion I express my gratitude to M. I. Graev and N. Ya. Vilenkin for discussion and useful advice.
Received
8 IV 1965
CITED LITERATURE
- F. John, Plane Waves and Spherical Means, IL, 1958.
- I. M. Gel'fand, G. E. Shilov, Generalized Functions and Operations on Them, Moscow, 1959.
- I. M. Gel'fand, G. E. Shilov, Spaces of Basic and Generalized Functions, Moscow, 1958.
- G. N. Watson, A Treatise on the Theory of Bessel Functions, Part I, IL, 1949.
- I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Sums, Series, and Products, Moscow, 1962.