INTERPOLATION PROBLEMS FOR FUNCTIONS WITH ZERO BALL MEANS Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 10(28), No3, 2021, pp. 129-140 DOI: 10.15393/j3.art.2021.10751

129

UDC 517.444

V. V. Volchkov, Vit. V. Volchkov

INTERPOLATION PROBLEMS FOR FUNCTIONS WITH ZERO BALL MEANS

Abstract. Let n ^ 2, Vr(Rra) be the set of functions with zero integrals over all balls in Rra of radius r. Various interpolation problems for the class Vr(Rra) are studied. In the case when the set of interpolation nodes is finite, we solve the interpolation problem under general conditions. For the problems with infinite set of nodes, some sufficient conditions of solvability are founded.

Note that an essential condition is that the definition of the class Vr(Rra) involves integration over balls. For instance, it can be shown that the analogues of our results in which the class of functions is defined using zero integrals over all shifts of a fixed parallelepiped in R™ do not hold true.

Key words: interpolation problems, spherical means, mean periodicity

2020 Mathematical Subject Classification: 44A35, 45E10,

46F10

1. Introduction. Let Rn be real Euclidean space of dimension n ^ 2 with Euclidean norm | ■ |. Assume that f E L\oc(Rra) and the equality

* |ж|^г

f (x + y)dx = 0

(1)

holds for some fixed r > 0 and all у E R™. Is it true that f = 0? This question was addressed in 1929 by the well-known Romanian mathematician D. Pompeiu, who stated that the answer is positive for n =2 (see, e. g., [15]). However, fifteen years later L. Chakalov [15] found an error in Pompeiu’s proof. Moreover, he showed that the function f (%i,x2) = sin(Axi) has zero integrals over all unit disks in R2 if A is a zero of the Bessel function J1. Later, it was found that similar examples

© Petrozavodsk State University, 2021

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V. V. Volchkov, Vit. V. Volchkov

of nonzero functions with condition (1) can be constructed by a method proposed by J. Radon as early as 1917. This method is based on the mean value theorem for eigenfunctions of the Laplacian and can be extended to an arbitrary two-point homogeneous space X (see [10, Part 2, Sect. 2.4]). Additionally, the method allows one to construct nonzero functions on X with zero integrals over all spheres of fixed radius.

Let Vr(Rra) denote the set of functions f G L\oc(Wn) satisfying (1) for all у G R™. Over the recent fifty years, this class of functions and its various analogues and generalizations have been intensively studied by F. John, J. Delsarte, J. D. Smith, L. Zalcman, C. A. Berenstein, and others (see the overviews in [1], [15], [16] and monographs [10-12], which provide extensive bibliographies). The basic directions in these studies can be listed as follows.

1. The study of zero sets and corresponding uniqueness theorems

for the class Vr (Rra) [5], [8], [10-12]. This direction goes back to the

uniqueness theorem of John [5, Chapter 6] for functions with zero spherical means.

2. The study of admissible constraints on the growth of nonzero functions of the class Vr (Rra) and its analogues on unbounded domains (theorem of the Liouville and Phragmen-Lindelof types [4-6], [8-12]).

3. The study of functions with conditions of type (1), in which r belongs to a given two-element set [1], [7], [8], [10-12], [15], [16] (two-radius theorem). The first result in this direction is Delsarte’s classical theorem on the characterization of harmonic functions by a mean-value equation satisfied by only two radii.

4. Description of functions of the class Vr (Rra) in the form of series in terms of spherical harmonics [10-13] (analogues of Taylor and Laurent expansions in the theory of analytic functions).

5. The problem of continuation [10-12] .

6. Theorems on removable singularities [10-14].

7. Integral geometry problems of reconstructing functions of specified classes from given spherical means [1], [2], [11], [12], [15], [16].

8. Approximation of functions with zero spherical means by linear combinations of special functions [10-12].

9. The study of analogues and generalizations of the class Vr(Rra) on various homogeneous spaces and groups (e.g., on Riemannian symmetric spaces) [1], [2], [7], [10-12], [14-16].

In this paper, interpolation problems for the class Vr(Rra) are studied.

Interpolation problems for functions with zero ball means

131

In the case when the set of interpolation nodes is finite, a theorem on the existence of a solution to the interpolation problem is obtained under general assumptions (see Theorem 1 below). Next, we provide some sufficient conditions for the solvability of multiple interpolation problems with an infinite number of nodes (see Theorem 2).

2. Formulations of the main results. As usual, the symbols N, Z+, and C denote the sets of positive integers, nonnegative integers, and complex numbers, respectively.

First consider the interpolation problem for the class Vr (Rra) with a finite set of interpolation nodes.

Theorem 1. Let q E N. Then, for any set of distinct points a\,..., aq in R™ and for any collection of constants bk E C, к = 1,... ,q, there exists a real analytic function f E Vr (Rra) satisfying the conditions

f (ак ) = Ьк, к =l,...,q. (2)

Note that an essential condition in Theorem 1 is that the definition of the class Vr (Rra) involves integration over balls. It can be shown that the analogue of Theorem 1, in which the class of functions is defined using zero integrals over all shifts of a fixed parallelepiped in Rn does not hold true. Indeed, any such function of the class C^(Rra) satisfies a linear difference equation relating the values of the function and its partial derivatives at the vertices of the given parallelepiped (see [11, Part 4]. Therefore, if the vertices of this parallelepiped are used as interpolation nodes, then the numbers bk in condition (2) cannot be taken as arbitrary.

Theorem 1 has the following immediate consequence, which shows that the solution of interpolation problem (2) for the class Vr (Rra) is not unique.

Corollary 1. Let q E N. Then, for any set of distinct points a\,..., aq in R™, there exists a nonzero real analytic function f E Vr (Rra) satisfying the conditions

f (ак )^ к =l,...,q.

In the general case, interpolation problems for the class Vr (Rra) with an infinite set of interpolation nodes is much more complicated than in the case of a finite set. Below, we present some sufficient conditions for the existence of a solution to the multiple interpolation problem.

We set

A = {x = (x\,... , xn) E R™ : X\ ^ 0, x2 = ... = xn = 0} .

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V. V. Volchkov, Vit. V. Volchkov

Theorem 2. Let (ak}^=1 be a sequence of distinct points in A such that

lim ak = ж. (3)

k^<X>

Then, for any sequence (mk}JJ=1 of nonnegative integers and for any collection of constants

bk,i E C (к E N, l = 0,... ,mk),

there exists a real analytic function f E Vr (Rra), such that

Шf(ak) = h'‘ (4)

for all к E N, l = 0,..., mk.

This result yields the following analogue of Corollary 1.

Corollary 1. Suppose that the sequence (ak}^ of distinct points in A satisfies condition (3). Then, for any sequence (mk}^ of nonnegative integers there exists a real analytic function f E Vr (Rra), such that

Шf (at) =0-

к E N, l

0,... ,mk,

and f Ц = 0.

3. Notation and some auxiliary statements.

For z = (zi,...,zn) E C™, C = (Ci,..., (n) E C™, we set

(z,() = Y1 zi 0.

j=i

Let t = (t1,..., tn-1) E R™ 1, x E R™, z E C,

h(x,z,t) = et(xitl+^+ж™-1*™-1) cos (xn\Jz — t\-Also, let

Sn-1 = (x E R™ : И = 1}, Ua,k = \t E R™ 1 : a < t\ + • • • + i^_1

• • • — t

2

n_ 1

< b}.

.

(5)

Interpolation problems for functions with zero ball means

133

We denote the group of rotations of R™ by SO(n). Let Jv be the Bessel function of the first kind of order v. For z > 0, we define

I, (z)

Jy (z)

z v

Note that the function I^ with v > —1 has an infinite number of positive zeros (see [3, Ch. 7, Sect. 7.9]). The following lemmas are needed in the proof of the main results.

Lemma 1. Let Wk = (wk,i,... ,Wk,n) E R™, к E {1,... ,q} and assume that

Wi,i = Wj,i for i,j E {1^.^ q}, i = j. (6)

Let 0 < а < b, z > 0. Suppose that there exist Ck E C, к = 1,... ,q, such that

g

h(wk ,z,t) = 0 (7)

k=1

for all t E Ua,b. Then ck = 0 for all k.

Proof. First consider the case n ft 3. We assume that t1 E (a, ft) for some

0 < a < ft, the numbers t2,..., tn-1 are fixed and t = fa,..., tn-1) E Ua,b.

Let - -

n—1 n—1

X = J2 Wk,i , J = Z — ^2 tf.

3=2 3=2

Consider the entire function

) = Y °ke<Wk,lC+x) cos (wk,n^p2 — (2) , C E C. (8)

k= 1

By the definition of h and relation (7), we see that <p(t1) = 0. Since t1 E (a, ft), the function <p vanishes due to the uniqueness theorem for analytic functions. Suppose now that (2 = p2 — q, where q > 0, q — and Im( —— —то. Then, in view of (6), the equality

g

Y Ckel(wk’lC+x) cos (wk,n^ft) = 0

k= 1

brings us to the conclusion that cp = 0 with wp>1 = max wk,1. Similarly

к

we obtain ck = 0 for all k, as required.

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For the case n = 2, it is sufficient to repeat the argument for the function p in (8) with A = 0, у = z. The proof of Lemma 1 is now complete. □

Lemma 2. Let t = {t\,..., tn-1) G Cn—1, z > 0 and

11 (r^z) = 0. (9)

Then the function h(x,z,t) is in the class Vr(R™).

Proof. We set (j = tj for j G {1,... ,n — 1} and

Cn

n— 1

Then ( = ((1,... ,(n) G Cn and ((,() = z. Consider the function

uc (x) = ei(x’c), x G R™.

For each у G R™, one has

|ж|^г

щ(x + y)dx = ег(у’^ ег(х,с-^dx = (2n)2 rn\« (r^z)e

i(y,0

|ж|^г

(see [3, Ch. 7, Sect. 7.12 (7)]). Together with (9), this shows that G Vr (Rra). Taking into account that

h(x,z,t) = 2 (щ (xi,... ,xn) + щ (xi,..., —xn))

we obtain the required assertion. □

4. Proof of Theorem 1. Bearing in mind that the points a1,... ,aq are pairwise different, we infer that there exists £ G Sn—1 such that

(C,ai — aj) = 0 for all i,j G {1,... ^} i = j. (10)

Since the group SO(n) acts transitively on Sra—1, one has

= (1, 0,..., 0) for some т G SO(n). (11)

We set

wk = rak = (wk,1,...,wk,n) G Rra, к G{1,...,g}. (12)

Interpolation problems for functions with zero ball means

135

Condition (10) shows that

{r^Wi - Wj ) = 0, i,j e{l,...,q}, i = j.

Together with (11), this yields

Wi,i = Wj, 1 for all i,j e{l,...,q}, i = j.

Assume now that 0 < а < b, ( > 0, and J«(r() = 0. Consider the functions

gk(t) = h{wk,(2,t), t E Uajb,

where к = l,...,q and h are defined by (5). Due to Lemma 1, the functions gk are linearly independent on Ua,b. For each m E {l,... ,q}, let Lm denote the linear subspace of L2(Ua,b) generated by the functions gk, such that к = m. Since gm E Lm, we obtain, by the Hahn-Banach theorem, that there exists a continuous linear functional Фт on L2(Ua,b), such that

Фт\Lm = 0 and Фт(дт) = l. Using the Riesz theorem, we see that Фт has the form

Ф

m\L.,

(13)

u E L2{Ua,b)

for some function <pm E L2(Ua,b). We now define the function G by the formula

G{x) = \' bm / h{x,(2, t)ipm{t)dt, x E Rn. (14)

Formula (14) guarantees that

^ ^ ЬтФт(дк) Ьк

'k

(15)

for all к E {l,..., q}. Next, for each у E R™ we have

G{x + y)dx = bm / / h{x + y,(2,t)dxipm{t)dt = 0 (16)

т=1 ^иа,ь JMyr

'm

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V. V. Volchkov, Vit. V. Volchkov

because of Lemma 2. Setting F(ж) = G(rx), we see, from (15) and (12), that

F(aк) = G(rak) = G(wk) = bk for all k.

In addition, (16) shows that F E Vr (Rra). Since the function F is real analytic, this completes the proof of Theorem 1.

5. Proof of Theorem 2. We set

= exp |afc|, к E N.

By the hypotheses on [ak}, the numbers ak are pairwise different positive numbers such that

lim ak = +ro.

k^<X>

Owing to the classical Hadamard theorem, there exists an entire function H : C ^ C satisfying the conditions

{

H(з)(ак) = 0 for j E [0,..., тк}, к E N,

H) = 0 if j = mk + 1, к E N.

Then the functions

Hk,v (ff )

н (z)

(z - ak)

к E N, v E [1,... ,mk + 1}

are entire and

{

Hkl (ak) = 0 if 0 ^ j ^ mk - v, Hkl(ak) = 0 for j = mk + 1 - у.

N7)

(18)

(19)

Let w : C ^ C be an arbitrary entire function. It is easy to see that for each l E Z+ there exist algebraic polynomials pij, j E [0,..., 1} such that

i i

(w(et)) = Уw(j)(et)pitj(et), t E C. (20)

3=0

Moreover, the polynomials pij are independent of w and pij(z) = zl. This shows that for each к E N there exist the constants ftkj E C, j E [0,... ,mk}, such that

(-)

\dtj

У fik,jPi,j(a,k) = Ькл for all l E [0,..., mk}.

3=0

(21)

Interpolation problems for functions with zero ball means

137

Using now (19), we see that there exist the constants 7^ E C (к E N, v E {1,... ,mfc + 1}), such that the functions

mk+1

Hk(z) = ^ 7k,uHk,u(z) (22)

v=1

satisfy the conditions

ЯfcJ)(«fc)= Pk,j, j E {0,...,mk}. In addition, it follows from (17), (18) and (22) that

Hk\aP ) = 0 if P E N, p= к, j E {0,..., mp}.

Assume that

and

Mk = max \Hk(z)|, Xk E N

/4

\k > 2mk + к + Mk for all к E N. We now define the function gk by the formula

(23)

(24)

(25)

9k(~) = 9k I (1 - <Г*<Л*d(, г E C,

0

where

9k =J(mk + A + 2fi, к E N.

r(mfc + 1)T(Afc + 1)

Relation (27) yields

(26)

(27)

2

Vk ^ (mk + Afc +

mfc+^fc / 1 \ \

1) E (m‘ + )

j=o v J J

^ (mk + Afc + 1)2m*+Xk.

Hence,

\9k(z)\ ^ (mk + Afc + 1)2m>+я(1 + \z\r* |z|A*+1, z E C. (28)

In addition, it follows from (26) and (27) that

9fc(1) = 1,

gfc\1) = 0 if 1 ^ s ^ mfc.

(29)

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V. V. Volchkov, Vit. V. Volchkov

Consider the function w(z)

S*(i)*

— )Hk(Z), z e C.

(30)

We claim that the series in (30) converges locally uniformly in C. Let R > 0, ak > 4R and assume that \z\ ^ R. Then

\Hk(z)| ^ max\Hk(z)| ^ max \Hk(z)| = Mk.

|^|УД \z\Zak/4

This together with (28) implies that

£ \ „ , s , , . / R \ Xk+1.

9k

(^)HtW « (m, + A, + 1)2'mfr+АУ+71+ *)”“Mt

\akl \akJ V akJ

Bearing in mind that

Xk R 1

mk <—, — ^ 7 and Mk < Xk,

2 ak 4

we obtain

< й) * <Ф Ф+0 4 D л‘/2

Since Xk > к (see (25)), this shows that the series in (30) converges uniformly in \z\ ^ R. Consequently, the series converges locally uniformly in C and the function w is entire.

Next, relations (23) and (29) show that

w(f)(ak)= pkJ, j e{0,...,mk}, к e N. Assume that Taylor’s expansion of w has the form

(31)

Then

w(z) = cpzP, z e C.

p=0

У \cp\Rp < +rc>

p=0

(32)

(33)

for each R > 0.

Interpolation problems for functions with zero ball means

139

Let v > 0, I" (rv) = 0 and p E Z+. Take ( = ((i,... ,(n) E Cn such

that (1 = — ip, (2 + ■ ■ ■ + Cn = P2 + у2 and define

fP(x) = ег(х’с\ x E Rra

The proof of Lemma 2 shows that fp E Vr (Rra). Consider the function

f(x) = ^ Cpfp(x'), x E R™

p=0

Condition (33) ensures us that the function f is real analytic and f E Vr(Rra). It follows by the definition of fp and (32) that

f (xi, 0,..., 0) = w(eX1), Ж1 E R.

Using now relations (31), (20) and (21) we see that f satisfies (4). Thus the proof of Theorem 2 is complete.

References

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DOI: https://doi.org/10.1070/SM1997v188n09ABEH000255

[14] Volchkov Vit. V., Volchkova N. P. The removability problem for functions with zero spherical means. Siberian Math. J., 2017, vol. 58, no. 3, pp. 419426. DOI: https://doi.org/10.1134/S0037446617030065

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Received May 22, 2021.

In revised form, July 17, 2021.

Accepted August 03, 2021.

Published online August 20, 2021.

Donetsk National University

24 Universitetskaya str., Donetsk 283001, Russia

E-mail:

V. V. Volchkov valeriyvolchkov@gmail.com

Vit. V. Volchkov volna936@gmail.com