The method of spherical harmonics for integral transforms on a sphere Текст научной статьи по специальности «Математика»

UDC 517.983.2, 517.444

THE METHOD OF SPHERICAL HARMONICS FOR INTEGRAL TRANSFORMS ON A SPHERE

V.N. Stepanov

Associate Professor, Ph.D. (Math), e-mail: [email protected]

Omsk State Technical University

Abstract. The integral convolution-type equations of the first kind on the sphere are important for the geometric tomography. They have been studied by many researchers. In this paper, we consider the uniqueness and stability of solutions of such equations. We prove the uniqueness of the solution for the equation with the kernel of convolution type and obtain a formula for the average value of a function on a subsphere. The latter is used for the deriving of the inversion formula of Radon spherical transformation on sphere. For the Blaschke-Levy equation and for the convolution type singular integral equations of the linear transfer theory, the uniqueness theorems are proved and find estimates for the stability of solutions are found. In all the cases we use the expansion of a function into series of spherical harmonics.

Keywords: geometric tomography, convolution-type equations of the first kind, uniqueness, stability, Radon spherical transform, inversion formula, Blaschke-Levy equation, equation with a singular kernel.

1. Introduction and preliminaries

Let Sn-1 be the unit sphere in Rn, |Sn-1| be its volume, u,v e Sn-1, u± be the orthogonal complementary subspace for u, Sn-2= Sn-1 n u± be the big subsphere, (,) be the inner product and an-1 be the spherical (n — 1)-dimensional Lebesgue measure on Sn-1. In the geometric tomography, equations of the first kind with the following convolution type integral operators on a sphere are important:

In the cases K((u,v)) = 8((u,v)), where 8(t) is the delta-function at zero,

K((u,v)) = \(u,v)\, K((u,v)) = \(u,v)\a, K((u,v)) = x((u,v)), where x(t) is

Heaviside's function, the integral transforms are well-known. They are called Radon spherical transform Rz, cosine-transform Cz, a-cosine-transform Caz, and hemispherical transform Hz, respectively.

The transforms above have the following geometrical meaning: if (n — 1)z(v) = = pn-1(v), where p(v), is the radial function of a star body B, then the Radon transform (Rz)(u) is the volume of the intersection of B with the subspace u±. If

z(v) is the product of principal curvature radii of the closed smooth convex surface SB in a point with normal v, then 2(Cz)(u) is the (n — 1)-dimensional volume of the projection of the convex body B onto the subspace u±, and (Hz)(u) is the area of the "lit" part of the surface SB. Thus the inversion formulas for the integral transforms and the uniqueness and stability of the solution of the integral equations are of great interest.

Integral transforms with the kernels |(u,v)|, |(u,v)|a,x((u,v)) are closely related to the Radon spherical transform. For example, the cosine transformation C is connected with the Radon transform R via the Laplace-Beltrami operator As on Sn-1 due to the identity DC = R, where □ = (As + n — 1)/(2 ■ |Sn-2|) [10,11]. The expansion of functions into series of spherical harmonics is a powerful tool in the investigation of the convolution-type equations (see, for example, [18]). Thus the following theorem is useful.

Theorem 1 (Funk-Hecke theorem, [3]). Let K(t) be a bounded measurable function on [—1,1] and Yk (u) be a spherical harmonic of order k. Then

i K((u,v))Yk(vK-i(dv) = XkYk(u), u e Sn-1, (1)

Jsn-1

where

Ak =|S+1) /; K (t) cr-1. (1 — t2) - dt (2)

and ckn/2-1)(t) are the Gegenbauer polynomials.

The Funk-Hecke formula can be extended onto the case of the kernel K(t) e L;[—1,1] (see [3]).

In this paper, we apply the method of spherical harmonics to the problem of uniqueness for the solutions of convolution-type equations of the first kind. We got the inversion formula for the spherical Radon transform on S3 and proved estimates of the solutions stability of a-cosine transform and equation with singular kernel.

2. Uniqueness of the solution of the equation for measure

We consider the equation of the first kind

f (u)= / K (<u,v))j(dv), (3)

JS"-1

where ^ is the unknown signed measure on Sn-1. Let {Yk(u)} be a complete system of spherical functions on Sn-1 and M be Banach space of signed measures (charges) on Sn-1. Let us introduce moments of the signed measure with respect to the system {Yk (u)}:

^k = Yk (u)ju(du). (4)

J S"-1

Lemma 1. The signed measure ¡j, on Sn-1 is uniquely defined by its moments ¡j,k, k = 0,1,2,..., with respect to system {Yk(u)}.

Proof. Let and p,2 be two measures whose moments coincide. Set p, = p1 — p2. It is sufficient to prove that the equalities

/ Yk (u)p(du) = 0, k = 0,1, 2,...,

imply p = 0. Any polynomial is a linear combination of spherical harmonics on Sn-1. It follows from Weierstrass's Theorem that the linear span of spherical harmonics is dense in the Banach space of all continuous functions on the sphere

/ ip(u)p(du) = 0 Jsn-1

for any continuous function p(u) e C(Sn-1). Thus p = 0, since the space of measures is dual to the space of continuous functions. ■

Theorem 2. If K((u,v))

e L1 [— 1, 1] and the system of its eigenfunctions is complete in L1 (Sn-1), then the equation (3) admits at most one solution in M.

Proof. Multiplying equation (3) by Yk(u), integrating it over the Lebesgue measure an-1, and applying Fubini's theorem, we get:

f (u)Yk(u)an-1 (du) = Yk(u) / K({u,v))p(dv) ) an-i(du)

J Sn-1 \Jsn-1

an-i(du) K ({u,v))Yk (u)ß(dv) =

J sn-1

K({u,v))Yk(u)an-i(du)) ß(dv).

n

fSn-1 JSn-1

J sn—i \JSn-1 J

According to Funk-Hecke formula (1),

I K({u,v))Yk(v)an-i(dv) = XkYk(u), k = 0,1, 2,..., Jsn-i

where k same for any spherical harmonics of order k, k = 0,1,2,.... This equality together with (4) imply

fk = / f (u)Yk (u)an-i(du) = Xk ßk, k = 0,1, 2,... .

S

'n — 1

Since the system of eigenfunctions is complete, Xk = 0. Hence the Fourier coefficients fk of the function f (u) uniquely define the moments of p by the equality pk = fk/Xk, k = 0,1,2,... . Applying Lemma 1, we conclude that the measure p is uniquely determined by its moments. ■

n1

S

3. The average of a function over subspheres and spherical transform of harmonics

Let u be the pole of Sn-1, v e Sn-1 and 7 = (u/v). Point v e Sn-1 can be presented as v = (v' sin 7, cos 7), where v' e S^-2 = Sn-1 n u±.

The Lebesgue measure an-1 on sphere Sn 1 and Lebesgue measure an_27 on subsphere SY^-2 = {v e Sn-1 : (u,v) = cos y} are subject to the equality an-1(dv) = sinn-27d7an-2,7(dv'),0 ^ 7 ^ n/2 [12]. For a function f(v) on Sn-1 we define its average value on the subsphere S^-2 as

/(Y,u) = Tô1^ / f (v) an-2,7(dv).

|sn-2|

-7 I ^ sy

Since |Sn 2| = sinn 2 y|Sn 2| and an-2,Y(dv)=an-2,Y(sinYdv')=sinn 2Yan-2(dv'), if Y = const, v' e Sn-2, we have

f(Y, u) = T^1^ I f (v) ^n-2,7(dv) =

f (v' sin 7, cos 7) sinn-2 Yan-2(dv') = (5)

sinn-2 y|Sn-2Us;-2

= , Cn_2] f (v' sin Y, COs Y) ^n-2(dv').

|Sn 2Usu-2

We use the ¿-function to obtain the formula for the average. Let us consider the integral

/ ¿(M — 1) ■ ^((x/|x|,y) — Cos Y)f^^

J Rn

where ¿(|y| — 1) ■ ¿((x/|x|,y) — cos y) is the direct product of the delta functions and f(y) is the smooth finite continued of f(v),v e Sn-1, in the area y e Rn : 0 < r < |y| < R, 0 < r < 1 < R. According to the formula [5],

I ¿(P(x))f(x)dx- ^ f(x)dax

lp(x)=0 |DP(x)|

where P(x) = {p1(x),p2(x),... ,pk(x)} = {0,0,..., 0} is the (n — k)-surface in Rn, |DP| = ^det(Vpj, Vp) is the square root of the Gram determinant of the vectors {Vp1(x),Vp2(x),...,Vpk(x)}, da is the element of surface area P(x) = 0, for

P1(y) = ¿(|y| — 1), P2(y) = ¿((x/|x|,y) — cos y) we get:

¿(M - 1) ■ ^(<x/|x|,y) - cos Y)f (y)dy =

f (y)* n—2,7 (dy)

'y| = 1,(x/|x|,y>=cos Y A /1 —

V M%|

|x|2|y|2

f (y) *n—2,y (dy)

'|y| = 1,(x/|x|,y>=cos Y Sin Y

n

Therefore,

/<Y'U) _ sin- y|sJiv)_. y f (V) _

_ 1 f f (v) ^n-2,7 <dv) _

sinn-3 y|sn-2 | </<u,v>=cos Y Sin Y

-—n-3 ^ 2, / - 1) ■ ^<(x/|x|,y)- cos Y)f <y)dy

sinn-3 y|Sn-21 .

■ n-3 1 I Cn_ 2, / ^<(x/|x|,y)-cos Y)f(y)^n-1 <dy) sinn 3 Y|Sn 21 ,/sn-i

(6)

To find out the average value of spherical harmonics Yk(u), let us use the Funk-Hecke formula and (6). We have:

Yk(Y,u) _ . n-3 1 I Cn_2, / ¿((x/|x|,v) - COS Y)Yfc<v)a„-i(dv) Sinn 3 Y|Sn 2| Jsn-1

Yk(u) |Sn-2|r(n - 2)r(k + 1) f1 ^(n/2-iw, ^n-3

-^—öt ■-^77-- ■ / - cos y) Ck ' ;<i)<1 - t2) 2 dt

sinn-3 y|Sn-2| r(k + n - 2) " k y

_ jn-3-r<k+ + -)'2) ■ Ci"/2-1)(cos Y)s,n-3YYk<u) _ _r(n<- ^ ■Cr/2-1,(cos Y)Yk(u).

Thus, the average value of the spherical harmonic Yk(u) over the subsphere

S™-2 _ {v G Sn-1 : (u,v) _ cos y} is equal to

r(n - 2)r(k + 1)^(n/2-1) I r(k + n - 2)

W \ r(n - 2)r(k + 1)^»(n/2-1) / /-x

Yfc(y,u) _ -—— C^ (cos Y)Yfc(u). (7)

For the spherical Radon transform of the spherical harmonics Y2k(u), formula (7) means that

(RY2k)(U) = ^ ^(«M») = r(r-2+rn- ^ C^1'(0)K2k(«).

* (8)

4. The inversion formula for Radon spherical transform on S3

The results concerning the Radon transform on Rn at Riemannian manifolds of negative and positive curvature are given in the monographs [6], [9], [14] and in the article of [17] B. Rubin. The central point of the Radon transform is the reconstruction of a function by its integrals on submanifolds. Starting with the classical results by Funk [13], Radon [16], a lot of different variants of inversion formulas for Radon spherical transform and the related integral transforms were

obtained. The history of the inversion formulas can be found in the article [17] by B. Rubin. In the case n ^ 3, the inversion formula for the spherical Radon transform R was obtained by Helgason in 1959, but it was not published until 1990. It is presented in [14], Theorem 3.13, p. 54. Note that the Radon transform is used in the solution of the inverse problems of the scattering theory [15]. The problem is to restore the shape, size, and the electromagnetic parameters of the scattering body. Our goal is to derive the inversion formula for the spherical Radon on S3 by the method of spherical harmonics.

Theorem 3. If z(u) is a sufficiently smooth even function on S3, then the inversion formula for the spherical Radon transform on S3

1

(Rz)(u) = — z(v) a2(dv)

J s3nu±

has the following form:

1 d2 f

z(u)= TT la (Rz)(v)l(u,v)l °3(dv)

2n dt2 J(u,v)2>t

t=0

u

_L

Proof. The average of the spherical harmonic Y2k(u) on the subsphere SU = S3 n u (i.e., the spherical Radon transform on S3) is given by the formula (8). If

n = 4,C^(0) = (-1)k, hence

(RY2k)(u) = Y2k(n/2, u) = r(l)r(,2k + ^ c2)}(0)Y2k(u) = (u).

We claim that

r(2k + 2)

1 d2 C

Ö-Ä* (RY2k)(v)Ku,v)l as(dv)

2n dt* J(u,v)2 >t

2k + 1

(9)

t=0

t=0

Y2k(v)l(u,v)l a3(dv)

coincides with Y2k(u).

We transform the integral and use the formulas (5) and (7):

1 d2 C

(RY2k )(v)\(u,v}\03(dv)

2^ at2 j(u,v)2>t

_ 2(-1)k r = 2n(2k + 1) dt2 JM>Vt

= (-1)k o2 r n(2k + 1) dt2 J{Utv)>Vt

/"arccos y/t

' dt?Jo

4(—1)k d2 farccos \/t

= 2k + 1 dt2 Jo

4(—1)k d2 f arccos \/t

t=0 2

Y2k(v' sin y, cos j) sin2 Y cos j a2(dv')dj

t=0

1

sin j cos j ( -¡— Y2k(v' sin j, cos j) <J2(dv') I dj

J sir2

4n(-1)k

nökTTj' = (10)

t=0

(2k + 1)2 dt2 J0

Y2k (j,u) sin2 j cos jdj C2k (cos j) sin2 j cos jdj

t=0

Y2k (u).

t=0

Let I(t) denote the second derivative of the integral. We find it by differentiating with the respect to the parameter t.

d2 |>arccos Vt d

1 (t) _ di? J0 C2k)(cos Y) sin2 Y cos Yd Y _ dt

' c2k)(Vt)(1 - t)vt'

2\fi\f1-t

1 r „—_ „(D„ ^c2k)(Vt) VT-1 dc2k)(Vt)

C2k)^v/t)

2

c2k)(Vt) dc2k)(Vt)

4v/T—i 2 di

V1 - i d\ft '

For the Gegenbauer polynomials there is the following differentiation formula [3]:

(1 - i2)dCT _ (2k + 2)iC2k)(i) - (2k + 1)C2kU).

Applying it, we get

1 (i) _ cvrvl - [(2k + 2) V^V) - (2k + 1)c2k++1(Vi)

c2k)(Vi) (2k + 2)c2k)(Vi) + (2k + 1)c2kUVi) _

V1 - i V1 - i 4\ft\fY-t

_ (2k + 1)c2k)(Vi) + (2k + 1)c2k++1 (Vi)

4 V1 - i 4^71-1 .

It follows from the equality [3]

k

(-1)m(1)2k+1-m(2s)2k+1-2m (1)2k+1(2s)2k+1

C(1) (s) _ V^ ^ _ _ vs^^t^__| +

+ ¿H m!(2k + 1 - 2m)! (2k + 1)! +... +

+ (-1)k-1(1)k+2(2s)3 + (-1)k (1)k+1(2s)1 + 3!(k -1)! + 1!(k)! ,

where (1)p _ r(p + 1)/r(1) _ p! that

C2k)+1^v/i)

Vi

_2(-1)k (k + 1). Thus, we get 1 (0):

_ 2(-1)k(1)k+1 _ 2(-1)kr(k + 2) _ 2(-1)k(k + 1)! 1!k! r(1)k! k!

t=o

(2k + 1) -C2k)(0) + 2(-1)k(k + 1) because C2k)(0) _ (-1)k

(-1)k (2k + 1)2

1 (0) _-^-4-^ = 4 (11)

According to formula (9), (10), (11)

1 d2 C

— (RY2fc)(v)|<u,v>| ^(dv)

2n dt2 J(u,v)2>t

= Y2k (u), k = 0,1, 2,... .

4(-1)k (-1)k (2k + 1)2

t=0

(2k + 1)2

4

Y2k (u)

This proves the inversion formula for the spherical harmonics Y2k (u), k = 0,1, 2,... .

Let z(u) be a sufficiently smooth even function on S3 . Let us expand it into series of spherical harmonics of even order:

<x

z(u) = Y%k(u).

k=0

Its Radon transform is

(Rz)(u) = ¿(RY* )(u).

k=0

Hence

1 d2

— — (Rz)(v)|<u,v>| a3(dv)

2n dt2 J (u,v)2 >t 1 d2

t=0

, „ (RY2k )(v)|<u,v>| aa(dv) = £ Y2k (u) = z(u)

k=0

' (u,v)2>t

t=0 k=0

and we get the inversion formula for the spherical Radon transform of an even smooth function on S3. ■

5. The estimate of the stability of the equation Blaschke-Levy solution

Let us consider the Blaschke-Levy equation (a-cosine transform)

f (u) = ^/ |(u,v)|az(v)a2(dv), u G S2, (12)

4tT Js2

where a2(■) is the Lebesgue measure on S2, z(u) is an even function on S2, a > -1, a = 0, 2,4,..., 2m,..., m G N. We shall formulate the restrictions on f (u) later assuming now that f (u) is a sufficiently smooth even function. Eigenvalues of the operator (12) can be found by the Funk-Hecke formula (2):

i f i

Ak = ^J jrcki/2)(t)di.

The Gegenbauer polynomials Cki/2)(t) coincide with the Legendre polynomials Pk(t), which satisfy the condition Pk(—t) = (—1)kPk(t). Consequently, eigenfunc-tions of the equation (12) are spherical harmonics Y2k(u) of even order and the

corresponding eigenvalues A2k are determined by the integral

A2fc = i (t)dt.

'0

This integral converges at a > —1 and its value is given in [4]:

a(a — 2)(a — 4) ■ ... ■ (a — 2k + 2)(a — 2k + 4)

A

2fc

(a + 1)(a + 3) ■ ... ■ (a + 2k — 1)(a + 2k + 1) ' Expanding the function f(u) over the system of spherical functions {Y2k(u)}

f (u) = £ Y2k(u), fc=0

where, by the Laplace formula [20],

4k + 1

Y2k (u) = 4-^/ P2k (<u,v))f (v^dv).

4n ./S2

we get the solution of the equation (5.1) in a form of series:

z(u) = ^ a2fc Yifc (u), fc=0

where

1 (a + 1)(a + 3) ■ ... ■ (a + 2k — 1)(a + 2k + 1)

a2fc

A2fc a(a — 2)(a — 4) ■ ... ■ (a — 2k + 2)(a — 2k + 4)'

If a > —1 is not an even nonnegative number, then the equation (5.1) has not more than one solution.

Let us estimate the stability. Since the Legendre polynomials P2k(<u,v)) are spherical functions, we have

p2k (<"■"»= [2k(2k+1)]*-AS p2k (<u-v)). The surface Laplace operator S is self-adjoint. Hence

Y2k (u) = /,2 «». ^ f (<0*2«fc).

Together with the Cauchy-Schwartz inequality and the inequality ||P2k(<u,v))||L2[_i,i] ^ Cifc"1/2, Ci = const [20], this implies

(u)iil2(s2) ^ 4n[2(4k2+ +)1)], lAf (u)||l2(s2) ■ |P2k (<u, v)) ||l2[_i,i] ^ ^ C2 ■ (2fc)_2i+1/2||ASf(u)|L2(S2), C2 = const.

It follows from the above inequality that

IIY*(U)IIL2(S2) ^ C3 ■ k1-4^ ||ASf (u)||L2(S2), C3 = const. Let us estimate the norm of the solution z(u) of the equation (12) in L2(S2).

Iz(u)|k(s2 ) =

\

s2 \fc=Q

Y2fc(un a2(du) =

\

2

5>2fcl|Y2k(u)H2(s2) ^

k=Q

^ C

\

Ë «2k k1-" y AS f (u)|t2(s,), C = const.

k=Q

To analyze the series a2kk1 on the convergence we use the Gaussian criterion. The ratio

_ «2k (k + 1)4^-1 _ (1 - a/2k)2(1 + 1/k)4^-1

^2fc+2 «2fc+2 k4^-1 [1 + (a + 3)/2k]2

is equivalent to

= 1 + 4i - 2a - 4 + vfc(i,a)

^2fc+2 k k2 where (i, a) is bounded. By the Gaussian criterion, the series converges if p = 4/ — 2a — 4 > 1, i.e. if i > (2a + 5)/4. Thus we obtain the estimates for the solution of (12) depending on a.

Theorem 4. If f (u) e C2^(Sn-1), then for any a > -1,a = 0, 2,4,..., 2m, m e N, there exists the unique solution of class L2(S2) for the equation (12). Moreover, if -1 < a < -0.5, then ||z(u)||l2(s?) ^ cJAf (u)||l2(S2); if (4k - 1)/2 ^ a < (4k + 3)/2, then ||z(u)||l2(s2) ^ Q||ASf(u)||l2(*>), where i = k + 2, k = 0,1, 2,... .

6. Equations with singular kernels on Sn-1

The analysis of some kinetic equations in the neutron transport theory [7] lead to the problem of the determination of the scattering indicatrix z(:,t) from the equation

f (u,t) = i K((u,:))z(:,t) a„-1(dv), u e Sn-1, t ^ 0.

Jsn-1

In particular, the following equation with singular kernel was obtained in [1]:

t, s f z(v) a„-1(dv)

f (u) = -7-r".

Jsn-1 1 -(u,:)

We consider a more general equation

f(u) = / f0 7-'(;;:). (13)

Jsn-1 [1 -(U,:)]

2

If a < (n - 1)/2, then the kernel K(t) = (1 - t)-a satisfies condition [2]

f (1 ~ ^^n-3Q)/2dt = J1 (1 - t)(n-2a-3)/2(1 + t)(n-3)/2dt =

= 2"-1-«B f n-i ,n-2a - 1 ^ <

where B(x,y) is the beta-function. Thus we may use formula (2) to find the eigenvalues [19, p. 431]:

Afc =|S n-2Jr(n+-2)r(k+1) / (1 - t)(n-2a-3)/2(1+t)(n-3)/2ckn/2-i)(t)dt.

We use the following asymptotic formula for the Gegenbauer polynomials Ckra/2-1)(t) as k —y to, which was found in the paper [8]:

Cn/2-1)(t) 2n/2-1r((n - 1)/2) (k + n - 3)! (1 t2)-n/4+1/2 C (t) vn(n - 3)! k!kn/2-1 (1 -1) ■

■ cos [(k + n/2 - 1) arccos t + (2n - nn)/4], -1 < t < 1, n ^ 3. It follows that the eigenvalues are subject to the following asymptotic formula:

2«/2n(n-2)/2

A

rs^

k n/2-1

■J (1 - t)n/4-a-1(1 + t)n/4-1 cos [(k + n/2 - 1) arccos t + (2n - nn)/4] dt

as k — to. The integral converges if a < n/4.

Due to the fast oscillation of the integrand, the integral is small for large k. The stationary phase method shows that the integral is o(1) (see [8]), whence Ak = o(k-n/2+1) as k — to. Moreover, there are constant a > 0 and b > 0 such that ak-n/2+1 ^ |Ak| ^ bk-n/2+1,k — to. Thus, the eigenfunctions of the integral operator in the right-hand side of (13) are the spherical harmonics Yk (u) and Ak are its eigenvalues.

Let f (u) = E^o Yfc(u), where (see [20, p. 489]

(n + 2k - 2)r(n/2 - 1) f (n/2-1),, , ,, ,

Yfc(u) =-4^-J ' )((u,v))f(vK-1(dv)

by the Laplace formula, be the expansion of the function f (u) into the series of spherical harmonics. Then

1

fc=0

z(u) = £- Yk (u) (14)

— Afc

is the solution to equation (13). If f (u) e C2^(Sra-1), then

)*(n + 2k - 2)r(n/2 - 1) r 4nn/2[k(n + k - 2)]* Jsn-i

(-1)*(n + 2k - 2)r(n/2 - 1) f ^(„/2-1),,

Applying the L2(Sn-1) - estimates ||Cf/2-1)((u,v))||L2(Sn-i) ^ c(n)kn/2-2 [20] for the Gegenbauer polynomials, the Cauchy-Schwartz inequality, and the asymptotics of eigenvalues we get:

inM ^ c(n)k-^lAS/^(S-1) < C(„)k —211^„Ll(Sn-i), £ S 1,

| Afc I I Afc I

c(n) = const.

Consequently, £ ^ 1 implies the uniform convergence of the series (14) and the continuity of the sum.

Thus, we obtain the following result.

r n — 11

Theorem 5. If /(u) e C2i(S--1) and £ ^ —-— + 1, then there exists the

unique smooth solution of the equation (13). Moreover, it admits the following estimate:

||z(u)|C(Sn-1) ^ C(n)|A|f (u)|L2(Sn-1).

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МЕТОД СФЕРИЧЕСКИХ ГАРМОНИК ДЛЯ ИНТЕГРАЛЬНЫХ ПРЕОБРАЗОВАНИЙ НА СФЕРЕ

В.Н. Степанов

доцент, к.ф.-м.н., e-mail: [email protected]

Омский государственный технический университет

Аннотация. Интегральные уравнения первого рода типа свёртки на сфере Sn-1 имеют важное значение в геометрической томографии и исследовались многими авторами. В представленной работе рассматриваются вопросы единственности и устойчивости решений таких уравнений. Доказана единственность решения относительно меры уравнения с ядром K((u,v)) класса Li[—1,1]. Получена формула для среднего значения функции на подсферах, которая затем используется для вывода формулы обращения сферического преобразования Радона на сфере S3. Для уравнения Бляшке-Леви и для уравнения типа свёртки с сингулярным ядром, встречающимся в линейной теории переноса, доказаны теоремы единственности и даны оценки устойчивости решений. Во всех случаях используется метод разложения функций по полной системе сферических гармоник.

Ключевые слова: геометрическая томография, уравнение первого рода типа свёртки, единственность, устойчивость, сферическое преобразование Радона, формула обращения, уравнение Бляшке-Леви, уравнение с сингулярным ядром.

Дата поступления в редакцию: 06.12.2016