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2URAL MATHEMATICAL JOURNAL, Vol. 9, No. 1, 2023, pp. 187-200
DOI: 10.15826/umj.2023.1.017
ON ONE ZALCMAN PROBLEM FOR THE MEAN VALUE OPERATOR
Natalia P. Volchkova
Donetsk National Technical University, 58 Artioma str., Donetsk, 283000, Russian Federation volchkova.n.p@gmail.com
Vitaliy V. Volchkov
Donetsk State University, 24 Universitetskaya str., Donetsk, 283001, Russian Federation volna936@gmail.com
Abstract: Let D'(Rn) and £'(Rn) be the spaces of distributions and compactly supported distributions on Rn, n > 2, respectively, let E'(Rn) be the space of all radial (invariant under rotations of the space Rn)
distributions in £'(Rn), let T be the spherical transform (Fourier—Bessel transform) of a distribution T € £'(Rn),
and let Z+(T) be the set of all zeros of an even entire function T lying in the half-plane Re z > 0 and not belonging to the negative part of the imaginary axis. Let ar be the surface delta function concentrated on the sphere Sr = {x € Rn : |x| = r}. The problem of L. Zalcman on reconstructing a distribution f € D'(Rn) from known convolutions f * ari and f * o>2 is studied. This problem is correctly posed only under the condition ri/r2 / Mn, where Mn is the set of all possible ratios of positive zeros of the Bessel function Jn/2—1. The paper shows that if ri/r2 / Mn, then an arbitrary distribution f € D'(Rn) can be expanded into an unconditionally convergent series
_ _ tXil - ,-
Aez+(ori) Mez+(iv2) (A2 - ri(A)Qr2M
/= E E —-, - /-(P^(A)((/*^)*n*J-Pri(A)((/*arJ*n£))
in the space D'(Rn), where A is the Laplace operator in Rn, Pr is an explicitly given polynomial of degree [(n + 5)/4], and Q r and Qr are explicitly constructed radial distributions supported in the ball |x| ^ r. The proof uses the methods of harmonic analysis, as well as the theory of entire and special functions. By a similar technique, it is possible to obtain inversion formulas for other convolution operators with radial distributions.
Keywords: Compactly supported distributions, Fourier—Bessel transform, Two-radii theorem, Inversion formulas.
1. Introduction
The study of functions f € C(R2) with zero integrals over all sets congruent to a given compact set of positive Lebesgue measure (for example, with zero integrals over all discs of a fixed radius in R2) goes back to Pompeiu [17, 18]. Motivated by the works of Pompeiu, Nicolesco in his paper [16] presents the following erroneous statement concerning integrals over circles of a fixed radius: if a real-valued function u(x,y) belongs to the class Cs(R2) for some s € Z+, r is a fixed positive number, and the function
r 2n
vs(x,y,r)= u(x + r cos d,y + r sin 9)e%sd dd 0
does not depend on (x,y), then u(x,y) is a solution to the equation
( d .d V , .
In particular, if u € C(R2) and u has constant integrals over all circles of fixed radius, then u = const. The impossibility of such a result is shown by the following proposition from a paper by Radon published back in 1917 (see [19, Sect. C]).
Proposition 1. Let r > 0 be fixed, and let Xr be an arbitrary positive zero of the Bessel function J0 ■ Then, for any k € Z, the function
Ik(z) = Jk(Xp)e%kip (p and p are the polar coordinates of z)
has zero integrals over all circles of radius r ■
Similar examples related to the zeros of the Bessel function Jn/2-\ can also be constructed for spherical means in Rn for n > 2. This shows that knowing the averages of a function f over all spheres of the same radius is insufficient to reconstruct f uniquely. Subsequently, the class of functions f € C(Rn) that have zero integrals over all spheres of fixed radius in Rn was studied by many authors see [2, 23, 25, 27, 35, 36], and the references therein). A well-known result in this direction is the following analog of Delsarte's famous two-radius theorem [6] for harmonic functions.
Theorem 1 [7, 33]. Let r1,r2 € (0, let Yn = {71,72,•••} be the sequence of all positive
zeros of the function Jn/2-1 numbered in ascending order, and let Mn be the set of numbers of the form a/0, where a, ft € Yn.
(1) If ri/r2 / Mn, f € C(Rn), and
f f (x)da(x) = Î f (x)da(x) =0, y € Rn, (1.1)
■ 'lx-yl=r1 ~/|x-y|=r-2
(da is the area element), then f = 0.
(2) If r1/r2 € Mn, then there exists a nonzero real analytic function f : Rn ^ C satisfying the relations in (1.1).
In terms of convolutions (see formula (2.2) below), Theorem 1 means that the operator
Pf = (f * ,f * °T2), f € C(Rn) (1.2)
is injective if and only if r\/r2 € Mn. Hereinafter, ar is a surface delta function concentrated on the sphere
Sr = {x € Rn : |x| = r},
that is,
(ar,<p) = J <p(x)da(x), <p € C(Rn).
In this regard, Zalcman [34, Sect. 8] posed the problem of finding an explicit inversion formula for the operator P under the condition r\/r2 € Mn (see also [19, Sect. C]). A similar question for ball means values was studied by Berenstein, Yger, Taylor, and others (see [1, 3, 4]). Note that their methods are also applicable in the case of spherical means. In particular, the following local result is valid (see the proof of Theorem 9 in [1]).
Theorem 2. Let
Ti/V2 /Mn, R>ri + T2, Br = {x € Rn : |x| < R}, and let {£k}£=1 be a .strictly increasing sequence of positive numbers with limit
R/(ri + r2) - 1, Rk = (ri + r2)(1+ £k), Ro = 0.
Then, for all r > 0, r € [Rk-1, Rk), and every spherical harmonic Y of degree m on the unit sphere Sn-1, one can explicitly construct two sequences Cl and Dl of compactly supported distributions in BR-ri and BR-r2, respectively, such that the following estimate holds for l ^ cm2 and every function f / C~(Br):
/ f (ra)Y(a)da - (Ci, f * ari) - (Di, f * ar2)
IS"-1
< l(R-r)-Nr-{n-3)/2 max I H^N
|x| <R'
dlal
№
dx
, (1.3)
k
where
N = [(n + 13)/2] + 1, R'k = (2R + Rk )/3, and y and c are positive constants depending on rl, r2, R, n, and £1.
Here it is appropriate to make a few remarks. The distributions Cl and Dl have a very complex form and are constructed as inverse Fourier-Bessel transforms to some linear combinations of products of rational and Bessel functions (see the proof of Proposition 8 and Theorem 9 in [1]). Further, every function f / CBR) can be represented as a Fourier series
ro dm
f (x) = E Efmj(r)Y(m)(*)> x = m, a € (1.4)
m=0j=1
converging in the space Cro(BR), where {Yjm) j l is a fixed orthonormal basis in the space of spherical harmonics of degree m on Sn-1,
fmj(r)= [ f(ra)Y}m\a)da
J S"-1
(see, for example, [10, Ch. 1, Sect. 2, Proposition 2.7], [24, Sect. 1]). Therefore, estimate (1.3) as l ^ to and expansion (1.4) imply the reconstruction of a function f / Cro(BR) from its spherical means f *ari and f *ar2 in the ball BR. The transition to the class C(BR) can be done by smoothing f by convolutions of the form f * <p£, where <p£ € Cro(Rn), supp<p£ c B£ (see [1, Sect. 3]).
The above remarks and Theorem 2 for R = to give a procedure for finding a function from its two spherical means. However, "explicit" inversion formulas for the operator (1.2) were unknown. This work aims to solve this problem.
2. Statement of the main result
In what follows, as usual, Cn is an n-dimensional complex space with the Hermitian scalar product
n
(C, 0 = E 0^ C = (Ci,•• •, Cn), = (ft, • • •, •?«), j=i
D'(Rn) and E'(Rn) are the spaces of distributions and compactly supported distributions on Rn, respectively.
The Fourier-Laplace transform of a distribution T € E'(Rn) is the entire function
T(Z) = <T(x), e-i(z'x)), Z € Cn. In this case, T grows on Rn not faster than a polynomial and
<T,i>) = <T,$), ^ €S(Rn), (2.1)
where S(Rn) is the Schwartz space of rapidly decreasing functions from C^(Rn) (see [13, Ch. 7]). If Ti,T2 € D'(Rn) and at least one of these distributions has compact support, then their convolution Ti * T2 is a distribution in D'(Rn) acting according to the rule
<Ti * T2,p) = <T2(y), <Ti(x),f(x + y))>, f € D(Rn), (2.2)
where D(Rn) is the space of finite infinitely differentiable functions on Rn. For T1,T2 € E'(Rn), the Borel formula
ff*T2 = Ti T2 (2.3)
is valid.
Let EjJ(Rn) be the space of radial (invariant under rotations of the space Rn) distributions in E'(Rn), n > 2. The simplest example of distribution in the class E^(Rn) is the Dirac delta function 5 with support at zero. We set
The spherical transform T of a distribution T € E^(Rn) is defined as
T(z) = <T,fz), z € C, (2.4)
where fz is a spherical function on Rn, i.e.,
<p2(x) = 2»/2-ir (2) ln/2-i{z\x\), X €
(see [9, Ch. 4]). The function fz is uniquely determined by the following conditions:
(1) fz is radial and fz (0) = 1;
(2) fz satisfies the Helmholtz differential equation
A(fz)+ z2fz = 0. (2.5)
We note that T is an even entire function of exponential type and the Fourier transform T is expressed in terms of T as
TXZ)= Z2 +... + zn), Z €Cn. (2.6)
The set of all zeros of the function T that lie in the half-plane Re z > 0 and do not belong to the negative part of the imaginary axis will be denoted by Z+(T). For T = CTr, we have (see [27, Part 2, Ch. 3, formula (3.90)])
3v (z) = (2n)n/2rn-1In/2-i(rz). (2.7)
Hence, by the formula
IV (z) = -zIv+i(z) (2.8)
(see [12, Ch. 7, Sect. 7.2.8, formula (51)]), we find
Q (z) = — (2n)n/2 rn+izIn/2(rz). (2.9)
Using the well-known properties of zeros of Bessel functions (see, for example, [12, Ch. 7, Sect. 7.9]), one can obtain the corresponding information about the set Z+(<rr). In particular, all zeros of <rr are simple, belong to R\{0}, and
Z+(3FP) = {^,^>...}. (2.10)
In addition, since the functions Jn/2-i and Jn/2 do not have common zeros on R\{0}, the function
is well defined, where %r is the indicator of the ball Br Let
m
Pr (z) = fl Z -
I, rn
j=1
n + 5
For A € Z+ (Qr), we set if A € Z+(3y) and if Pr (—A2) = 0, where
(2.11)
Qr = Pr (A)CTr. (2.12)
Then, by the formula
p(A)T(z) = p(—z2)T(z) (p is an algebraic polynomial), (2.13)
we have
Q r (z) = Pr (—z2)Q (z), (2.14)
/'I
and all zeros of Qr are simple. Besides,
z+(ari) nz+(arn) = 0 —£Mn. (2.16)
r2
= Pr (A)cta (2.17)
Q A = Qr,A(A)ar (2.18)
QrM = (2-19)
The main result of this work is the following theorem.
4
Theorem 3. Let
^<£Mn, /eD'(Kn), n> 2.
Then
f= E E (p.mf*«,)*<)
Aez+(nn) Mez+(nr2) (A V )"ri(^)"r2(V) (2.20)
—Pn(A)((f * ari) * ^)), where the series (2.20) converges unconditionally in the space D'(Rn).
Equality (2.20) reconstruct a distribution f / D'(Rn) from its known convolutions f * ari and f * ar2 (see (2.11), (2.14), (2.15), and (2.17)-(2.19)). Thus, Theorem 3 gives a solution to the Zalcman problem formulated above. Note that there is great arbitrariness in the choice of polynomials Pri and Pr2 in formula (2.20) (see the proof of Corollary 1 and Lemma 5 in Section 3). In particular, they can be defined fully explicitly without using the zeros of the function Jn/2-i. For other results related to the inversion of the spherical mean operator, see [5, 8, 11, 20, 21, 26, 28-32].
3. Auxiliary statements
Let us first describe the properties of the functions Iv, which we will need later.
Lemma 1. (1) The following inequality holds for v > -1/2 and z € C:
e|Im z|
(2) If v € R, then
1 e|Im z|
Im^oo. (3.2)
(3) Let v > —1 and let {yvj }°=i be the sequence of all positive zeros of the function Iv numbered in ascending order. Then
In addition,
lim ^Y+3/2\Iu+1(^)\ = J-. (3.4)
j ^ro v n
Proof. (1) By the Poisson integral representation [12, Ch. 7, Sect. 7.12, formula (8)], we have
i
2i-v r
M-) = + 1/2) J cos(uz)(1 -u2)v~1/2d,u.
0
Hence,
i
2i-v f
<
2
1 — V
^-B f-,v + - ) e'Im~' =
3|Im z|
Vvrr(z/+1/2) 2 V2' 2/ 2T(i/ + l)'
which is required.
(2) The asymptotic expansion of Bessel functions [12, Ch. 7, Sect. 7.13.1, formula (3)] implies the equality
Iv(z) =
e
|Im z|
1 — ( COS (z — — — -j) + Oi ,,
7T \ V 2 4/ I |z|
z —y to, —n < arg z < n. (3.5)
Considering that
e|Im w|
I cos wI ~—-—, Im«i —> oo,
by (3.5), we obtain (3.2).
(3) The asymptotic behavior (3.3) for the zeros of Iv is well known (see, for example, [25, Ch. 7, formula (7.9)]). Then
cos Yvj
2 4
)=cos(vr,-|+oQ))
= O
j — to.
It follows that
lim
nv n
sm I 7^ - Y ~ 4
= 1.
Using this relation and the equality
W;) = /»U ft - ^ - OÍ'
24
|z|
z — to, —n < arg z < n,
(see (3.5)), we arrive at (3.4).
□
Corollary 1. For all r > 0,
£
a e (n r )
Proof. Using (2.14) and (2.9), we find
|Qr (A)|
< +to.
(3.6)
Qr'(A)=Pr(—A2)Q(A) - 2AP;(-A2)àr(A) = -(2n)n/2rn+1APr(-A2)In/2(rA) - 2AP;(-A2)Q(A). Now, from (2.10) and (2.15), we have
y. 1 1 1 y._1_
This series is comparable with the convergent series
j=i J
1
;2m— (n— 1)/2
(see (2.11), (3.3), and (3.4)). Hence, we obtain the required assertion.
□
1
J
1
Lemma 2. Let g : C ^ C be an even entire function, and let g(X) = 0 for some X € C. Then
Xg{z)
z2- X2
< max |g(Z)|, z € C;
|C-z|<2
(3.7)
the left-hand side in (3.7) for z = ±X is extended by continuity. Proof. We have
2Xg(z) g{z) g{z) < g{z) + g{z)
z2 — X2 z — X z + X z-X z + A
Let us estimate the first term on the right-hand side of (3.8). If |z - X| > 1, then
g(z)
zX
< |g(z)| < max |g(Z)|. - iyv 71 - |z—z|<2
(3.8)
(3.9)
Assume that |z — X| < 1. Then, applying the maximum-modulus principle to the entire function g(Z)/(Z — X), we obtain
g(z)
zX
< max K—A|<1
g(Z)
Z — X
= max |g(Z)|. |z—A| = 1 1 V 71
Considering that the circle |Z — X| = 1 is contained in the disc |Z — z| < 2, we arrive at the estimate
g(z)
zX
< max \q(Z)|,
which is valid for all z € C (see (3.9)). Similarly,
' g(z)
z + X
< max |g(Z)|, z € C,
because g(—X) = 0. From (3.10), (3.11), and (3.8) the required assertion follows.
(3.10)
(3.11) □
Lemma 3. The function satisfies the equation
A(a$) + X2a$ = — ar, X € Z+(S>). (3.12)
Proof. For every function p € D(Rn), we have
<A(a$) + X2a$, p) = (a$, (A + X2)p)
1 f In/2—l(X|x|) 1 f In/2—i(X|x|) , , ,
=--To / —i—7T \ Aip{x)dx - - / —-—— r (p(x)dx.
rX2 7|x|<r In/2(Xr) r 7|x|<r In/2(Xr)
We apply Green's formula
f (vAu — uAv)dx = f (v^- — da
Jo Jdo\ dn dnj
to the former integral (see, for example, [22, Ch. 5, Sect. 21.2]). Since X € Z+ (<rr), we have
,1 i , ,d i In/2—l(X|x|) \ 1 f In/2—l(X|x|) . ..
1 ^ { I„/2(Ar) ) ^ - - i,s, I„/2(Ar) ^
Hence, by (2.5), we obtain
r) + A ^' ^ = ^ JSr { Ira/2(Ar) J da(x)■
Now, using the formula
d x
-№l)) = №|>, n = -,
and relation (2.8), we find
(A(^) + AV, <p) = -~ [ <p(x) M I;/2(.AJ'Y') da(x) = - [ ^(x)da(x) = ~(ar,
r JSr In/2(Ar) Jsr
This proves equality (3.12). □
Remark 1. From (2.13) and the injectivity of the spherical transform, it follows that, for distributions U,T € S'(Rn) and A € Z+(T),
A U + A 2U = -T U (z) = (3-13)
z2 — A2
Therefore, relation (3.12) implies the equality
= A GZ+(3?P). (3.14)
Lemma 4. Let A € Z+ ( Qr). Then
Proof. Formula (3.15) easily follows from (2.13) and Remark 1. Indeed, if A € Z+ (ar), then, by (2.17), (2.13), (3.14), and (2.14), we have
W*) = Pr(~Z2№) = Pri/Jll{Z) =
Similarly, if Pr(—A2) = 0, then
W*) = QrA-^Mz) = {Uz)
Z2 - A2 Z2 - A2
(see (2.18), (2.19), (2.13), and (2.14)). □
Lemma 5. Let
=
r - .
Qr (A)
2A ~
—Qr> AGZ+(flr). (3.16)
Then
r
E *A = 5, (3.17)
Ae z+(nr)
where the series in (3.17) converges unconditionally in the space D'(Rn).
Proof. For an arbitrary function p € D(Rn), we define a function ^ € S(Rn) as follows:
Mv) = 7A1 / <p{x)Sx^dx, ye R'\ (2n) it"
Then (see (2.1), (2.6), and (3.15))
= = [ tp(x)^(\x\)dx = [ tP(x)^X\ldx.
Jt" Qr (X) Jt" |x|2 — X2
Using this representation and Lemma 2, we get
\№,<P)\ < /2 ,/ 1^)1, max \ar(Q\dx. |Qr (X)\Jt" K—MI<2'
From (2.14), (2.7), and (3.1), we obtain
max |Qr(Z)\ = (2n)n/2rn—1 max \Pr(—(2)| |In/2— 1(rZ)| K—MI^1 rVVl v ; K—MI^1 n s 711 n/2 1 V|
2n'a/2 rn—1 2nn/2 rn—1p2r
< £E_1_ max IP (_Z2) I .er ImC < —_-_— max IP f-Z2)l
" r(n/2) |CHf<2|n( U| 6 " r(n/2) |CHi<2|n(
Therefore,
4_n/2rn— 1e2r r
№,<p) <-FT—r / №0*01, max \pr(-(2)\dx, (3.18)
r (n/2) |Qr (X^Jt" |z—|x||<2'
This inequality and Corollary 1 show that the series in (3.17) converges unconditionally in the space to some distribution / supported in Br. By Lemma 4, the spherical transform of this
distribution satisfies the equality
f(z)= £ *}{z)= y: (3-19)
~ ~ Q (X) z2 — X2
Aez+ (nr) Aez+ (nr) 1 r (X)
In this case, if p € Z+ ( Qr), then
m = J?- lim = (3-20)
Qr (p) z2 — P2
Further, since f(z) — 1 and Qr(z) are even entire functions of exponential type, by (3.20) and the simplicity of the zeros of Qr, their ratio
„.(;) = m^i
Q r(z)
is an entire function of at most first order (see [15, Ch. 1, Sect. 9, Corollary of Theorem 12]). For
Im z = ±Re z, z = 0, it is estimated as follows:
< m + 1
£ 1
Q '(A) Vz - A z + A
A eZ+(Qr) 12 r (A)
|Qr (z)| |Q r (z)|
1
+
<
(2n)n/2 rn-1|Pr (-z2 )I„/2-l(rz)| 1 1 1 1
^ I^ ( Ir - XI + Ir -U XI ) +
, , ,n , ,Q '(A) I V |z - A| |z + A^ (2n)n/2rn-1|Pr(-z2)I„/2-i(rz)| aez+(ur) ' ' '
2y/2 1 1
E +
|z| Iq' (A) I (2n)n/2 rn-1|Pr (-z2)I„/2-i(rz)r
A € Z+(Ur ) I ' I
It can be seen from this estimate and relations (3.6) and (3.2) that
lim h(z) = 0. (3.21)
Im z=±Re z
Then, according to the Phragmen-Lindelof principle, h is bounded on C. Now it follows from (3.21) and Liouville's theorem that h = 0. Hence, f = 1, i.e., f = 5. Thus, Lemma 5 is proved. □
Lemma 6. Let A € Z+( Qri), ^ € Z+( Qr2). Then
(A2 - At2)1/A * = „ , 4Xil, (ar2 * fi* - Qri * Q?) . (3.22)
Q ri (A)Q r2 7
Proof. By (3.15), (3.13), and (3.16), we have
(A + A2) = —-J-j——ilri, (3.23)
V J Q n (A) 2^
(A + (i2){^) = -^—ilr2. (3.24)
Qr2 M
From (3.23), (3.16) and the permutation of the differentiation operator with convolution, we obtain
—4A^
(A + A2) * = —-àB.-Qri * qm .
Qri (A)Qr2 M
Similarly, it follows from (3.24) that
4A^
-(A + ¿t2) (V * = —-^r— ar2 * fi*.
V 1 V Q f\\Q f,,\ 1
Qri (A)Qr2 M
Adding the last two equalities, we arrive at relation (3.22). □
4. Proof of Theorem 3
By Lemma 5, we obtain
E < = *, E =s- (4.1)
Aez+(n r1) ^ez+(n r2)
We claim that
E E * = I (4.2)
Aez+(nri) ^ez+(n r2)
where the series in (4.2) converges unconditionally in the space D'(tn). Let p € D(tn), — € S(tn),
and let p = For X € Z+( Qri) and p € Z+( Qr2), we have (see (2.3) and the proof of estimate (3.18))
* ,p)
= |( = K ^n ^ A) | =
-(x)M (|x|)^2 (|x|)dx
<
|Qri (X)Qr2(p)
167Tra(rir2)ra-1e2(ri+r2) \a'(\)arl(ri\T2(n/2) JR"
ltn |x|2 — X2 |x|2 — p2
/ |-(x)Lma^ |pri(—C2)| max |pr2(—C2)|dx
Jt" |z—|x||<2 IC-MK2
This and (3.6) imply that
E ( E K^A *,p>|) <
Aez+(nri) ^ez+(nr2)
Therefore (see, for example, [14, Ch. 1, Theorem 1.24]), the series in (4.2) converges unconditionally in the space D'(tn). In addition (see (2.2) and (4.1)),
E E «* = E ( E <*£(*),(^Ai(x),p(x+y))]
Aez+(nri) (nr2) Aez+(nri) ^ez+(nr2)
= E <*Ai (x),p(x)) = p(0),
AeZ+(Q ri)
which proves (4.2).
Convolving both parts of (4.2) with f and taking into account the separate continuity of the convolution of f € D'(tn) with g € E(tn), (3.22) and (2.16), we find
4Xp
/= E E —
Aez+(nri) (nr2) (X2 — p2)Qri(X)Qr2(p)
— (f * (1 r2 * 1 rAi) — f * (11 ri * 1 £)). (4.3)
Finally, using (4.3), (2.12), and the commutativity of the convolution operator with the differentiation operator, we arrive at formula (2.20). Thus, Theorem 3 is proved. □
t
"
4
5. Conclusion
The proof of Theorem 3 shows that the key role in formula (2.20) is played by the expansion of the delta function into a series of distributions »A, A € Z+( Qr) (see Lemma 5). This system of distributions is biorthogonal to the system of spherical functions ^ € Z+( Qr), i.e.,
(»A,.,,> = (° f " = A'
11 if ^ = A
(see (2.4), (3.15) and (3.16)). Using similar expansions, it is possible to obtain inversion formulas for other convolution operators with radial distributions.
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