Symmetric representations of holomorphic functions Текст научной статьи по специальности «Математика»

124 Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 124-136

DOI: 10.15393/j3.art.2018.5390

The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.

UDC 517.53/.55

A. B. Shishkin

SYMMETRIC REPRESENTATIONS OF HOLOMORPHIC

FUNCTIONS

Abstract.

In this article a class of symmetric functions is defined and used in some special representation of holomorphic functions. This representation plays an important role in transitions from concrete problems of projective description to equivalent problems of inductive description and finds multiple applications in questions connected with spectral synthesis of differential operators.

Key words: spectral synthesis, projective description, inductive description, differential operators, symmetric functions, holomorphic functions

2010 Mathematical Subject Classification: 32A10

1. Introduction. Suppose n, q G N; G is an open set in Cn; n : G ^ Cq is a holomorphic mapping. A set g Ç G is called n-symmetric, if there exists a set V in Cq such that g = n-1(V). A function p : g ^ Cq, where g is a n-symmetric set, is called n-symmetric on g, if p = p o n, where p is some holomorphic on n(g) function. A set of all n-symmetric on g functions On (g) is a ring. This ring is a subring of the ring of all holomorphic on g functions O(g).

The class of n-symmetric functions is needed to consider some representations of holomorphic on complex domain functions. For example, consider the case of one variable. Suppose n = q =1; n is a polynomial; G is an open n-symmetric set in C. Note the following theorem: Any u G O(G) has the unique representation of the form u(z) = ^p=1 zpup(z), where up G On (G) [1]. Such presentation is called a symmetric representation of the analytic function [2]. The case n := (n1,... ,nq) : C ^ Cq,

© Petrozavodsk State University, 2018

where n1,... ,nq are polynomials, was considered in [2]. The case where G is an open set in C, (G, n, n(G)) - analytic covering, was studied in [3].

In this paper we consider a more general case. We obtain symmetric representations of some multivariate functions.

Note that the concept of symmetric representation of an analytic function plays a key role in some questions of complex analysis. For example it is used in spectral synthesis (see [3-6]).

2. Analyticity of the difference relation. 2.1. Alphabetized list of independent variables. Denote by A any

, where

,(0

p(i)

product set j z(1),... (1) } X ... x {z(n),... ,

i = 1,..., n are ordered sets of independent variables.

The set A = {zj: j = 1,... ,p} has p = p(l) x ■ ■ ■ x p(n) different elements. Any finite sequence z1,... ,zp e A is called an alphabetized list of the set A, if j < k m e [1, n) such that j1 = k1,...,jm = km, jm+1 < km+1 holds for any

{4:> ={

v(i)

(n)

.

Let z1,... ,zp be an alphabetized list of the set A. Consider a matrix

(z11) . . z(n)\

Z = Z(1) z2 . (n)

z(1) . \zp(1) . . z(n) . p(n)/

such that each j-th row equals to zj. Choose any partition of the matrix Z Z1.... ,Zp(1) where any submatrix Zj consists of p(2) • • • p(n) rows of the matrix Z. Partition of all Zk in the same way gives us submatrices Zk,i.... ,Zk,p(2). which consist of p(3) • • • p(n) rows of Z, etc. We have:

( Zi

Z

Z

Zk

k,1

Z

Zk

k,t

\Zk,p(2)j

\ZP(1)

Any submatrix of rank m has the form:

(m)

Zk ... )

Z

k,t,1

\Zk,t,p(3) /

«. . . . .j

V

(1)

(m)

(m+1) 1

z(m+1) p(m+1)

In)

(n) P(n)

Z

k

k

j

1

k

j

thus, columns 1,... ,m are equal.

Any two neighboring submatrices of rank m are called adjacent if these submatrices are in the same submatrix of rank m — 1 (note that Z has rank 0). We have:

i) any two adjacent submatrices have the same columns except exactly

one called marked;

ii) any marked column consists of equal elements.

More precisely, m-th columns of matrices Zk,...,j_1 and Zk,...,j of rank m are marked. The elements of m-th columns are equal to zjm) and zjm), respectively.

2.2. The main procedure. Let us consider the procedure used in the proof of theorem 1. Add the marked column of the upper adjacent submatrix to each submatrix Z2,..., Zp(1) of rank 1 from the right to obtain the non-rectangular matrix Z'. Each submatrix Zk,...,j of rank m of matrix Z determines the submatrix Z'k j of rank m of matrix Z' uniquely. Note that i) and ii) hold for adjacent pairs of submatrices of matrix Z'. The pair Z1 = Z1, Z2 is the only exception. The marked columns of the adjacent matrix pair Z'k_ 1 and Z'k, k = 3,... ,p(1), are the last column of Z'k_ 1 and the first column of Z'. Elements of these columns are equal to elements of z(1)2 and z(1), respectively. Add the marked column of upper adjacent submatrix to each submatrix Z3,..., Z'p^ of rank 1 from the right. We

Properties i) and

obtain a new matrix Z" and new submatrices Zk j.

ii) hold for Z

The pair Z'{ = Z1 = Z1, Z!{ = Z2 and Z!{ = Z2

Z3

are exceptions. The marked columns of the adjacent matrix pair Zk_ 1 and Z'l, k = 4,... ,p(1), are the last column of Zk_ 1 and the first column

of Z''. Elements of these columns are equal to elements of z(1)3 and respectively. Then add columns to Z4',..., Zp^, etc. Finely, p(1) — 1-th step gives us the matrix 1Z. We have:

J1)

/

1Z :=

Z

\

" k-1

" k-1

.(1)

Z1 = Z1

1 Z1

In the same way we deal with submatrices 1Zk (rank =1) of matrix 1 Z. Let us consider the first step. Properties i) and ii) are satisfied for adjacent submatrix pairs of matrix 1Z, if the submatrices have rank > 1. Then add the marked column of the upper adjacent submatrix to each

1

1

submatrix iZk,2,... ,iZk,p(2), k e 1,... ,p(1) from the right to obtain a new matrix 1Z' and new submatrices 1Zk j. Properties i) and ii) are satisfied for submatrices of rank > 2. The marked columns of the adjacent matrix pair 1Z'j_1,1Z'j, k = 1, ...,p(1), j = 3, ...,p(2), are the last column of 1Zkj_1 and the second column of 1Z'.j. Elements of these

columns are equal to elements of zj_2, zj2) respectively. Then add columns to 1Zk 3,... ,1 Z'k p(2), etc. Finally, p(2) — 1-th step gives us the matrix 2Z. We obtain:

/

2Zfc,j —

Z

z(l) zfc —1

z(D zfc —1

J1)

.(1)

Z(2)

Zj —1 Z(2)

Zj —1

(2)

(2)

2Z1,1 — Z-

1,1,

2Z1,j :

Z

1,j

\

z(2)

zj—1 z(2)

zj—1

(2)

(2)

2Zfc,1 =

Z

fc,1

\

z(1)

zfc—1 z(1)

zfc- 1

(1)

(1)

Then we change in the same way submatrices of 2Z of rank 2, etc. We have:

Z

k,

(1) k-1

.. . z

I ... |zin)1...z1n))

(if i G k,... and i — 1, then there is no i-th submatrix in the matrix nZk,...,j ). Now we stop because submatrices of nZ are rows of nZ if these submatrices have rank n.

2.3. An analytic continuation of the difference relation. Choose any m x m invertible matrix A — (akj-) and l x l nondegenerate matrix B. The matrix

/ anB ... fl1mB

Ax B —

\am1B . . . ammBy

is called the Kronecker product.

It is clear that the determinant |A x B| of the matrix A x B equals |A|'|B|m.

Suppose that a1,..., am e C. Consider the square matrix

D(a1,..., am)

1 a1 \1 am

^m— 1

1

1

1

1

1

1

1

1

1

m

Determinant of the matrix D(a1,..., am) is called the Vandermonde determinant. We have:

A(a1, ...,am)= JJ (aj — ak).

Let {z^...^}, i = 1,...,

n, be a collection of finite ordered

number sets, z1,..., zp, p = p(1) x • • • x p(n) is an alphabetized list of product set of sets jz(i1,..., j, i = 1,..., n. Consider the matrix

D = D (z™,..., 411,) x ... x D (z1n),..., z^) ; A = det(D).

It is clear that

A = AfAf ••• A"n = n n

i=1

zj — zk

where pi = p(p7y, Aj = A (z(i1,..., z^). We have:

A = A(z1,...,zp)

zp ... zp where a1,... ,ap is an alphabetized list of the set

1

1

{0,... ,p(1) — 1} x ... x {0,... ,p(n) — 1};

ak = (ak11,...,«k"1); j = (zj11)"^1' x ... x (zj"1)

("h «I"'

Let g(1),..., g(n1 be a collection of open sets in C. Take the set

,' be a

open

Iz(11,...,z(n1

lz(11 ,...,z(n

of functions that are holomorphic at the product set g(1) x ... x g(n). Let

(11 J11 \

Z1 ,...,zp(1^ ,...^z1

\Z1 ,...,zp(nl/

p

a

a i

p

be sets of independent variables where zj e g(j) . Consider the relation

where |$| = det$,

$

F = M A '

( <£>1(21) ... <£>p(z1)

\ ^1(Zp) ... <p(zp)

z1,..., zp is an alphabetized list of the product set

y(1)

J1) \

•zp(1^ x ... x i z1

(n)

»

.,zp(n)

Determinants |$| and A are holomorphic functions on g(i). From Har-togs's theorem it follows that |$| and A are holomorphic at the product

set

g = g(1) x ... x g(1) x ... x g(n) x ... x g(n).

p(1) p(n)

Then F is a holomorphic function at g \ Z(A), where Z(A) is a set of points of g and x e Z(A) ^ A(x) = 0..

Theorem 1. The function F has the unique holomorphic on G analytical continuation.

Proof. It follows from continuity of F that the analytical continuation is unique. It remains to check that it exists. Denote $ by $(Z). Consider the partition of the matrix $(Z):

$(Zk,1),..., $(Zk,p(2)).

We have:

$(Z)

i $(Z1)

\$(ZP(1))

, $(Zk)

( $(Zk,1)

\$(Zk,p(2)),

If we replace $(Zk), k = 2,... ,p(1) by

$(Z')

$(Zk) — $(Zk_1)

zw- z(1) ,

zk — zk_1

1

we obtain the matrix $(Z'). Elements of $(Z') are holomorphic functions at g. We have:

i*(z )I = (zp;i)—zp;i,_1)pi x... x (411—z(1i)pi |$(Z ')|. If we replace $(Zk), k = 3,... ,p(1) by

$(Zk) — $(Zk_ 1)

$(Z/) =

zk zk-1

we get the matrix $(Z"). Elements of $(Z") are holomorphic functions at g. We have:

l$(Z ')l= ($) - zS)-2)Pl x ... x (z31) - z|1))Pl|$(Z ")|.

If we replace $(Zk), k = 4,... ,p(1), etc, we get the matrix $(1Z). Elements of $(Z") are holomorphic functions at g. We have

|$(Z)| = Af )|. (1)

Then we replace $(1Zkj-), j = 2,... ,p(2) by

k,j ) — $ (1Zk,-

$ / Z/ N = $(1Zk,j) - $(1Zk,j-1) $ V1Zk,j/ (2) (2)

z(21 z(21 zj — zj_1

in $(1Zk) for all k G 1,... ,p(1) to obtain the matrix $(1Z'). Its elements are holomorphic functions at g. We have:

W1Z )l = ($1 — zP(1i_1)P2 x ... x (z(21 — z(21)P2 |$(1Z ')|.

If we replace $(1Zk j), j = 3,... ,p(2), etc, we get the matrix $(2Z). Its elements are holomorphic functions at g. We have:

|$(1 Z)| = AP2 |$(2Z)|. (2)

Then we get the matrix $(nZ). Its elements are holomorphic functions at g. We have

|$(n_1Z)| = A"n |$(nZ)|. (3)

It follows from (1), (2), (3) that

|$(Z)| = A |$(nZ)|.

Finally, we obtain F = |$(nZ)|. This completes the proof. □

3. The symmetric representation of an analytic function.

3.1. Analytic cover. Let A be an image of n : G ^ Cq. The mapping n is called an analytic cover of A if the following conditions hold:

1) mapping n is proper (hence, A is an n-dimensional analytic set in Cq [4, Remmert-Stein theorem]);

2) there exists an analytic subset a C A, dim a < n such that A* = A\a is an n-dimensional complex manifold in Cq;

3) the set n-1(a) is nowhere dense in G;

4) the restriction of n to G* = G\n-1(a) is a local biholomorphic p-sheeted covering on A*.

The set a is called critical. The preimage n-1(a) is an n-dimensional subset of G. Metric dimension of n-1(a) is less than or equal to 2n — 2. Hence, n-1(a) is removable. Then any bounded over n-1(a) holomorphic on G* function has the unique holomorphic on G extension. Sets n-1(A), A e A are compact analytic subsets of the set G and are called n-layers. Hence, the sets are finite [7]. Points A e A* are called ordinary, and respective n-layers are called simple. Simple n-layers consist of p different points. A single-valued mapping {1,... ,p} ^ n-1(A) is called ordering of a layer n-1(A). Ordering of a simple n-layer can be represented in the form z1,..., zp. Elements of the sequence z1,... ,zp depend on A = n(zk) e A*. Mappings

zk = (zk1)(A),...,zkn)(A)), k =1,...,p,

are holomorphic on some neighborhood of any ordinary point.

The concept of analytic cover develops the concept of local biholo-morphic k-sheeted covering. Analytical coverings appear in the Weier-strass preparation theorem. It follows from the theorem that any pure k-dimensional analytical set has some analytical cover on Ck as a local representation.

3.2. Special analytic cover. Let G(1),..., G(n) be open sets in C; n(i) : C ^ Cq(i), q(i) e N, i = 1,... ,n, be holomorphic functions; A« = n(i)(C). The mappings n(i) are analytic covers, as:

1) the mapping n(i) : C ^ Cq(i) is proper (hence, A(i) is 1-dimentional analytic set in

2) there exists the close discrete set a(j1 C A(j1 such that A(i1 = A(i1\a(i1 is a 1-dimentional complex manyfold in

3) the restriction of n(i1 to Cii1 = C\(n(i1)_1(a(i1) is a local biholo-morphic p(i)-sheeted covering on

It is clear that the product set n : C" ^ Cq, q = q(1) + ... + q(n) of mappings n(i1, i = 1,... ,n, is an analytic covering. The image n(G) = A equals to product set A(11 x . . . x

A("1

hence the image is an n-dimensional analytic set in Cq [7]. The mapping n : G ^ A is proper. Indeed, let K be a compact set in A, let K(i1 be a projection of K on

A«,

K = K(11 x ... x K(n1. It is clear that (n(i1) (K(i1) is a compact set in G(i1.

n_1(KK) = (n(11) 1 (k(11) x ... x (V"1) 1 (k("1)

n_1(KT) is a compact set in G. n_1(K) C 7f_1(KT) and n_1(K) are closed then n_1(K) is a compact set in G. This proves condition 1). Further, critical set a equals to the set

"

A \ (a!11 x... x a!"1) = U s(i1,

i=1

where

S(i1 = A(11 x ... x A(i_11 x a(i1 x A(i+11 x ... x A("1

are (n — 1)-dimensional analytic subsets in A. Indeed, a is an (n — 1)-dimensional analytic subset of A and

A! = A\a = a!11 x ... x a!"1

in an n—dimensional complex manifold in Cq. We have proved that con"

dition 2) holds. Set 7r_1(a) has the representation |J

-1 (S(i1)

, where

i=1

n_1 (S(i1) equals to

G(11 x ... x G(i_11 x _1 x G(i+11 x ... x G("1.

Since (a(i1) is a closed discrete set, it follows that

-1 (S(i1)

and

n_1(a) are (n — 1)-dimensional analytic sets. It now follows that 7r_1(a) is

a nowhere dense set in G [7]. Condition 3) holds. It is clear that condition 4) is satisfied. Note that p = p(1)... ,p(n).

A := A(1) x ... x A1n) is a topological subspace of Cq. Let A be an open subset of A, G = n-1(A). A restriction n : G ^ A of the covering map n : Cn ^ Cq to the n-symmetric set G is an analytic covering. We say that n is special. Note that any n-symmetric set is n-symmetric. In particular, any n-layer equals to the respective n-layer. 3.3. Some representation of an analytic function. Let mapping n : G ^ A be a special analytic covering; let O(A) be a ring of holomorphic on A functions; let O(A*) be a ring of locally holomorphic on A* functions; let O*(A) be a subring of the ring O(A* ) that consists of bounded on A functions. The mapping

O(A*) ^ On (G*) | 0 ^ 0 o n

is a ring isomorphism. Since the mapping n is proper and the set n-1 (a) is removable, the restriction of the mapping n to O*(A) is a ring isomorphism. It takes O*(A) on On (G*) n O(G).

Let n be a special analytic covering, z G G* and

n(z) = A = (A(1),... ,A1n)) G A*, A(i) G A*0.

Denote by A the n-layer n-1(A) . It is clear that A = A(1) x ... x A(n), where A(i) are simple n(i)-layers, i = 1,..., n. Since the n-layer A contains z = (z(1),..., z(n)), then a n(i)-layer A(i) contains the i—th coordinate z. Let z(i),..., zp)), z(i) = z(i), be an arbitrary ordering of a simple

n(i)-layer A(i); let z1,..., zp be an alphabetized list of the layer A. Consider a relation

F = M A ,

where |$| = |$|(z1,..., zp), A = A(z1,..., zp), 01,..., G O(G), define functions f : G* ^ C, f : A* ^ C, with respect to the conditions

f (z) = F (z1,..., zp), f (A) = F (z1(A),..., zp(A)).

Since the restriction of F to G\Z(A) is a symmetric function of variables {z(1),..., zP11)},..., {z1n),..., zP^} and (z1,..., zp) G G\Z(A) for

any z G G* , then the functions f, f are well defined. Indeed, the order of the set {z1,..., zp} changes but F doesn not as one changes the

order in a set {z(i),..., z^-i)}, i G 1,...,n. On the other hand, mappings zi(A),..., zp(A) are holomorphic on sufficently small neighborhoods of the ordinary points. Then f G O(A*). It follows from theorem 1 that f G O*(A). Indeed, suppose A = (A(1),..., A(n)) G a and let k(i) c A(i) be a compact neighborhood of A(i), let d(i) c A(i) be an open neighborhood of compact k(i), g(i) = (7T(i))-1(d(i)). Choose neighborhoods k(i), d(i) such that A(i) G int (k(i)) = 0, A G d(1) x ... x d(n) Ç A, A c C g(1) x ... x g(n) Ç G. Since the map n(i) is proper, K(i) = (n(i))-1(k(i)) is a compact set in g(i). It follows from theorem 1 that the function F is analytic on a set G = (g(1))p(1) x ... x (g(n) )p(n), hence, F is bounded on (K(1))p(1) x ... x (K(n))p(n). Then the function f is bounded on k1) x ... x k(n). Finally f is locally bounded on A. We have

f (z) = f (A) = (f o n)(z),

where z G G* and A = n(z).

Now note that the function f G On(G*) is locally bounded on n-1 (a). The set n-1(a) is removable. Then f has an extension that is holomorphic on G. We have

f G On (G*) n O(G). Now we can prove the following theorem. Theorem 2. For any function f G O(G) the unique representation

f = È zak f (k),f(k) G On (G*) n O(G). (4)

k = 1

is true. The restriction of f(k) to G* equals to Ak(f )/A, where Ak (f ) is the determinant obtained by replacement of k-th column by the column from f (z1),..., f (zp) in determinant A and z1,..., zp is a alphabetized list of the simple n-layer A that contains z.

Proof. It is clear that for any z G G* we have

fA = È zak Afc (f ), k=1

where f = (f (zi),...,f (zp)), zafc = (zj^,..., zp"fc). The determinant A is not equal to zero for any z G G*. Then for any z G G* we have

/ = E ^ f(k), (5)

k=1

where f(k) = Ak(f)/A G On (G*). The functions f(k) have the only extension that is in O(G). Now we have

f(k) G On (G*) n O(G),k = 1,...,p.

It is easy to see that (2) ^ (5). Uniqueness of (2) follows from the fact that we obtain f(k) from (5), using Cramer's rule. Indeed,

'f (zi) = za1 f (1)(zi) + ... + f(p) (zi)

f (zp)= za1 f P1)(zp) + ... + zpap f 1p)(zp

pp

The theorem is proved. □

References

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Received May 18, 2018. In revised form, September 3, 2018. Accepted September 3, 2018. Published online September 18, 2018.

A. B. Shishkin

Kuban State University, branch in Slavyansk-on-Kuban 200, Kubanskaya str., Slavyansk-on-Kuban, Russia E-mail: shishkin-home@mail.ru