Elementary solutions of a homogeneous q-sided convolution equation Текст научной статьи по специальности «Математика»

Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 137-152 137

DOI: 10.15393/j3.art.2018.5410

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. A. Tatarkin, U.S. Saranchuk

ELEMENTARY SOLUTIONS OF A HOMOGENEOUS q-SIDED CONVOLUTION EQUATION

Abstract. Spectral synthesis on the complex plane related to solutions of some homogeneous equations of convolution type. There is a method to obtain solutions: we describe the elementary solutions set of the equation (spectral analysis) and prove the approximation theorem (spectral synthesis). In this paper we use the method for some homogeneous equations of convolution type, which appears from spectral synthesis problem for some differential operator.

Keywords: differential operator, holomorphic function, spectral synthesis, spectral analysis, shift operator, convolution equation

2010 Mathematical Subject Classification: 34L05

1. Introduction. Let Q0, Q be convex domains in C, e > 0, U£ = {z : |z| < e}. Suppose that Qo + U C Q ; O(Q), O(U£), O(Qo) and O(C) are spaces of holomorphic functions equipped with the topology of uniform convergence on compact sets. The operator D takes a function f to f . The operator Dq takes a function f to f(q). Consider a shift operator

Th : f (z) ^ f (z + h)

where h G Ue is the shift. The operator takes O(Q) to O(Q0) and is continuous. The shift operator Th can be identified with the infinite order differential operator exp hD. The characteristic function of the operator Th = exp hD is equal to exp hA, we have

Th(exp Az)

-= exp hA.

exp Az

@ Petrozavodsk State University, 2018

Let q be a natural number, a0,... ,aq-1 be a set of complex numbers containing a non-zero one. Denote by wq the number exp . Let A be a continuous linear operator that acts on elements of the space of entire functions O(C). Define A by the rule

q-1

g(A) ^Y, akg(^qq A),

k=0

ATh is a continuous linear operator that takes O(Q) to O(Q0) by the rule

q-1

f (z) ^Y, ak f (z + ^h).

k=0

The operator ATh : O(Q) ^ O(Q0) is called a q-sided shift operator (h € U£ is the shift). Choose a continuous linear operator S on the space O(Q0) and an arbitrary q-sided shift operator ATh : O(Q) ^ O(Q0). The equation

{S,ATh(f )> =0, f € O(Q), (1)

is called a homogeneous q-sided convolution equation. The solution space of a homogeneous q-sided convolution equation is equal to the kernel of the respective q-sided convolution operator O(Q) ^ O(Ue) | f ^ {S, ATh(f )>. This space is a closed subspace in the space O(Q) and is an invariant set for the differential operator Dq [3]. Any exponential polynomial is called an elementary solution if this exponential polynomial satisfies the condition (1).

Let q =1, f(h) = {S, exp hz> be the characteristic function of the functional S. Suppose that A is a zero of the function ip that has multiplicity n. Then exponential monomials of the form:

exp Az, z exp Az, ..., zn-1 exp Az (2)

satisfy the condition (1) and are in WS. It is well known that any exponential monomial has the form (2) if this monomial satisfies the condition (1). It follows that any elementary solution of the equation (1) is a linear combination of monomials of the form (2), where A € C is a zero of the characteristic function of the analytic functional S and n is the zero's multiplicity [1, 2].

Choose q > 1; then we obtain a more complicated description of the solution sets of homogeneous q-sided convolution equations. The spectral

synthesis problem for the differential operator Dq appears; it requires the description of the differential operator's root vectors that are in a Dq - invariant set W Ç O(Q) [3].

2. Properties of a q-sided shift operator.

Property 1. A q-sided shift operator ATh can be identified with the infinite order differential operator

~ hn

f (z) ^E bn~n\(Dnf )(z), (3)

n=0 '

where

q-1

bn ak £{0,1,...}

k=0

and the series (3) uniformly converges on each compact subset of Q0. Proof. Suppose that f £ O(Q). By definition we have:

ATh(f )(z) = Y akf (z + h) = Y ak(exp ^hD)(f )(z), k=0 k=0

where z £ Q0.

By the operator's exp hD definition we have:

q— ~ -knhn

ATh(f )(z) = £ a^ (Dnf )(z) =

k=0 n=0 ' (q-1 \ hn hn

= E E "k^) ^(Dnf )(z) = £ bn^(Dnf )(z).

n=0 \k=0 ) ' n=0 '

For any k £ {0,1,... ,q — 1} the series

~ -'lnhn

(Dnf )(z)

n=0

uniformly converges on each compact subset of to the function f (z + + -k h). Then the series (3) is uniformly convergent on compact subsets of too. □

Property 2. The characteristic function of a q-sided shift operator ATh : O(Q) ^ O(Oo) is equal to the function

9-1 fc;. h"A™

A(exp hA) := ^^ exp w^hA = ^^ bn ,

fc=0 n=0 n'

then

ATh (eXp Az) = A (exp hA). exp Az

Proof. On the one hand, by definition of the q-sided shift operator for any z € we have

9-1 9-1

ATh(exp Az) = ^^ exp A(z + w^ h) = ^^ exp w^ hA exp Az.

fc=0 fc=0

Hence,

ATh (exp Az) .

-r-= > afc exp w„hA = A(exp hA).

exp Az ^^ 9

^ fc=0

On the other hand, by Property 1

v hn ~ hn A"

ATh(exp Az) = bn—-D" exp Az = bn--— exp Az.

n' ^^ n'

n=0 n=0

Then,

ATh (exp Az) _ h"A'

exp Az " n'

K n=0

The property is proved. □

Property 3. A q-sided shift operator ATh : O(Q) ^ O(H0) is linear and continuous.

Proof. Linearity and continuity of the operator ATh : O(Q) ^ O(^0) follow from the linearity and continuity of the shift operators Tukh : O(Q) ^ O(Q0), k € {0,1,..., q — 1} and the representation

9-1

ATh(f ) = £ afc Twjh(/). (4)

fc=0

The property is proved. □

From Property 1 it follows that some of

9-1

6„ :=£ agwgfc", n €{0, 1,. ..}

g=0

are not equal to zero. Note that for any n € Z+ we have

bn+9

Hence, some of 60,... ,69-1 are not equal to zero. The determinant

(5)

A,

0,...,q-1 • —

"0 "2-1

,(q-i)(q-1)

П К -О

0^n<j^q-1

of the system of equations

'«0 + «1 + ... + 09-1 = 60 00 + W9 «1 + ... + w|-1 «9-1 = 61

,«0 + W9 «1 + ... + W9 «9-1 = 69-1

is not equal to zero. Thus, we obtain a0,..., a9-1 from 60,..., 69-1:

ao

A0,...,q-1(b°,...,bq-1)

A

0,...,q-1

(6)

where Ag ... 1(60,..., 69-1) is the determinant

0,...,q-1 (b0, . . . , bq-

"0 ■ ■ ■ b0 "0 ■ ■ ■ 61

q

0

••• 6q—

q-1

"0 "I-1

,(q-1)(q-1)

obtained by replacing the k-th column by (60,..., 69-1)T in the determinant A0,...,9-1.

Summarizing everything said above, we get the following property.

Property 4. A q-sided shift operator ATh is a differential operator of the form (3) such that:

q

q

1) bn satisfy the condition (5);

2) some of b0,..., bq-i are not equal to zero.

The coefficients a0,..., aq-i are called the determining coefficients of the q-sided shift operator ATh. The coefficients b0,..., bq-i are called the characteristic coefficients of the q-sided shift operator ATh.

Denote by nA := [n\,..., nv } any ordered set of numbers {0,..., q — 1} such that:

1) 0 < ni < ... <nv < q — 1;

2) if n G {ni,..., nv}, then bn = 0;

3) if n G {ni,..., nv}, then bn = 0.

Such a set nA := {n^ ..., nv} is called the indicator of the q-sided shift operator ATh.

3. Exponential polynomials in some Dq-invariant space. Let

W be a Dq-invariant subspace of the space O(Q). Any exponential polynomial that is in W Ç O(il) has the form

q-i j=0 k=0

where Xj G C, pjtk (z) are polynomials in z, wq := exp ^. Since the exponential polynomial is in W Ç O(Q), this polynomial is equal to the linear combination of special exponential polynomials.

Proposition 1. Any exponential polynomial from W Ç O(il) is equal to the linear combination of exponential monomials from W that have the form

q-i

£ Pk(z)exp{Lvk Xz}, (7)

k=0

where X G C, Pk(z) are polynomials in z. Proof. Suppose that a(z) G W and

q-i

a(z) := EE Pj'k (z)eXPH Xj z}, Xi = Xj, 1 = j.

j=0 k=0

Let us act by the operator — a, where a G C, on the exponential polynomial <r(z); we get

q-1

(D - a) a(z) = gj,fc(z)exp{^Az}

j=0 fc=0

where

gj,k(z) = E (q) Af Vj(z) - aj(z).

q

(z;

i=0

The following facts can be easily checked:

1) if A] = a, then for any k G {0,..., q - 1} deggj,fc(z) < degp-,fc(z);

2) if Aq = a, then for any k G {0,..., q — 1} deg(z) = deg gj,k (z) and the leading coefficient of the polynomial gj,k(z) is equal to the leading coefficient of the polynomial (z) multiplied by A] — a.

Let us act on <r(z) by the operator

(D - A?)n+1 (D - A2)n+1 • ... • (Dq - Am)n+1,

where

n = max {degpj,k(z)} , (j,k) G {0, ..., m - 1} x {0, .. ., q - 1}.

By the above relations 1) and 2), we conclude that exponential polynomial

ao(z) := (Dq - A?)n+1 (Dq - Aq)"+1 • ... • (Dq - A^)"+1 a(z) has the form

q-1

E^fc (z)exp{^k Aoz}.

k=0

The leading coefficient of the polynomial (z) is equal to that of the polynomial p0,k multiplied by

c0 := (A0 - A?)n+1 (A0 - Aq)n+1 •... • (A0 - A^)"+ = 0. Hence, the exponential polynomial

cr(z) := <r(z) - c0^0(z)

has the form

m q-1

a

(z) = ££ Pj,k (z)exp{uq X0z},

j=0 k=0

where deg po,k (z) < deg po,k (z) for any k £ {0,... ,q - 1}.

Applying the above procedure to the exponential polynomial a(z) we continue to decrease the degrees of the polynomials p0,k (z). After that, we deal with the polynomials pljk(z), etc. After a finite number of the steps we obtain:

Note that the exponential polynomials a0(z),ai(z),... ,as(z) have the required form. □

It is clear that any exponential polynomial of the form (7) has the representation

where A £ C, gk(() are holomorphic at A functions.

Proposition 2. Any exponential polynomial from W C O(il) is equal to the linear combination of exponential monomials from W that have the form (8).

Proposition 3. [3] If an exponential polynomial em(z,A) is in W C O(il) and A = 0, then the exponential polynomials

are in W.

4. Exponential polynomials and a q-sided shift operator.

Choose an arbitrary exponential polynomial

a(z) = coao(z) + c1a1(z) + ... + csas(z).

(8)

eo(z,X), ei(z,X),..., e„—i(z,X)

of the form (8). Then the q-sided shift operator

q-1

ATh : f (z) f (z + <4h)

k=Q

(where h G Ue is the shift) takes the exponential polynomial to

q-1

ATh(em(z,A)) := — ( Egk(C)ATh(exp{<kCzj)

k=Q

C=A

By Property 2 we have

ATh (exp{-q Zz}) = A(exp{wkCfe})expKfcCz} =

= E b"exp{^qfcCzL

n=0 '

where are characteristic coefficients of the q-sided shift operator ATh. Then,

ATh(em(z,A)) =

d"

dC m

q-1

00 <knz nhn E gk (C) E bn exp{<kCzj

k=Q q-1

Q

n!

C=A

^ hn am

= E bn ^ ^ E gk (0<knCn exp{<kCzj

n=0 ! Z Vk=Q

hn dm

n=Q

q-1

= E bn^ ¿^1ZnE gk (C)<kn exp{<kCzj

k=Q

C=A

C=A

Choose an arbitrary functional S G O*(^0); it takes the function ATh( em (z, A)) to

<S,ATh(em(z,A))> =

^ h" dm / q_1 \

= E b"hr dZm z" E gq(Z)-qx-qZ)

n=0 s ^ q=0 '

where ^ is a characteristic function of the functional S.

C=A

Proposition 4. We obtain the equality

(S,ATh(em(z,X))) =0 for any h G Ue iff functions

q-1

bn(Z) =: zn £ 9k(ZHnAukqC), n G nA,

k=0

are equal to zero at X with the multiplicity at least m+l, nA := {n1y... ,nv} is the indicator of the q-sided shift operator ATh.

Proof. Suppose that (S,ATh(em(z,X))) = 0 for any h G Ue. Hence, em(z,X) G WS Ç O(Q), where WS is the solution space of the homogeneous q-sided convolution equation. WS is Dq-invariant. By Proposition 3 the exponential polynomials

eo(z,X), ei(z,X),..., em-i(z,X)

are in WS. From (9) we have

(S,ATh(ep(z,X))) =

hn dp

q-1

i q-1 (zn E

k=0

for any h G U£ and p G {0, l,... ,m}. It is clear that

Y,bnn ôcpiZn£9k(zKnA-qz)

n=0 k=0

C=A

0

dp ( q 1

^ z n£ gq (z pkn^q z )

k=0

C=A

for any p G {0, l,..., m} and n G nA. Thus, the functions bn(Z), where n G nA, have a zero X with the multiplicity at least m + l. □

5. Elementary solutions of a homogeneous q-sided convolution equation. Now we obtain the description of the set consisting of exponential polynomials that are in the solution space WS Ç O(Q) of the homogeneous q-sided convolution equation (1).

0

Denote by A0,...,q-1 the Vandermonde determinant

„0 , ,o

„q „q 0i

^o . ,q-1 , ,(q-1)(q-1)

„q „ • • • „q

It is clear that

A

0,...,q-1

n („q—„j) = o.

Let c0(Z), c1(Z),..., cq-1(Z) be a set of holomorphic at A G C functions. Denote by Aq (Z, c0,..., cq-1) the determinant

Ak,...,q-^C0(Z), 1 C1(Z), . . . , Z^ICq 1 (Z)) =

C0(Z) 1 C1(Z)

V

Zq

— Cq-1(Z)

-q-1

(q-1)(q-1)

/

that is obtained by replacing the k-th column in A0,...,q-1 by

(C0 (Z ),1 C1(Z ),..., Zq-H Cq 1 (Z ))T .

Theorem 1. Any elementary solution of a homogeneous q-sided convolution equation (1) is equal to a linear combination of exponential polynomials of the form

e(z, C0, ...,Cq-1) :=

d

■q-1

V

m \ Z_/

,k=0

dZ™ ^(„q' Z)

where the holomorphic at A G C functions

Ak(C ^ . . . ,Cq-1) r q .

q - exp{„qZz}

c=a

C0 := C0(Z), . . . , Cq-1 := Cq-1(Z)

0

uj

q

q

q

q

0

0

uj

uj

q

uj

q

q

0

uj

q

q

and m G Z+ satisfy the following condition: functions

Ak (Z, CQ, .. ., Cq—1 ) Z) ,

k G {0, 1,...,q - 1}

and functions

are holomorphic at X.

Cn (Z) c

(x-nxmi ,n G nA

Proof. By Proposition 2, any exponential polynomial from a Dq-invariant WS C O(il) is equal to the linear combination of exponential polynomials from WS of the form (8). Suppose that the exponential polynomial

em(z,X) :=

dn

'q-1

dZm

J2ak(Z )exp{uk Zz}

\k=Q

Z=A

is in Ws Ç O(n). Suppose that Vn (Z ) := Znv(Z ), n G {0,1,...,q — 1}. Then for any n G {0,1,... ,q — 1} we have

q-1

q-1

bn(Z) := Zn£9k(ZK^HZ) = Y9k(Z)<Pn(ukz).

k=Q

k=Q

By Proposition 4, the functions bn(Z), n G nA have a zero X with the multiplicity at least m + 1.

Consider the system of q linear functional equations

q-1

bn(Z ) = £ 9k (Z )Vn(uk Z ), n G {0,1,...,q — 1},

k=Q

where gk(Z) are unknown holomorphic at X G C functions. The matrix of the system

( pQ(^qz) pq(^\z) ^(uQz ) MZ )

M^-1Z) \ fiH-1Z )

\Vq-1 HZ) Vq-lH Z) ■■■ <fq-l(iVqq-1Z) J

( ^°ZVK°Z) V(^1Z) ••• ^ZVK-1 Z)

vk°z ) ж vkz ) ■ ■ ■ ^q-1z vk-1z )

V „0zq-V(„0z) „¡-1Zq-V(„!Z) • • • „(q-1)(q-1)zq-V(„q-1z) J

has q rows and q columns. The determinant A0,...,q-1(Z^(Z)) of the matrix is equal to the following:

A0,...,q-1Z^(„kZ),

q=0

where A0,...,q-1 is the Vandermonde determinant

n („q—„j) = o.

Thus, the determinant A0,...,q-1(Z^(Z)) does not vanish in some neighborhood of A, except A itself. By Cramer's rule

^ A' q-1 (Zy(Z ),b0(Z ),b1(Z),..., bq-1(Z)) , fn, n

gq(Z) = ——-t-m7\\-, k G {0,1,..., q — 1},

where Aq,...,q-1(Z^(Z), b0(Z), b1(Z),..., bq-1(Z)) is the determinant

( „0ZV(„°°Z) ••• b0(Z) ••• „0ZV(„q-1Z) „0zV(„0Z) ••• b1(Z) ••• „q-1Z V(„q-1z)

V „0Zq-V(„0Z) ••• bq 1 (Z) ••• „(q-1)(q-1)Zq-V(„q-1Z)/

that we obtain by replacing the k-th column in A0,...,q-1(Z^(Z)) by

b0(Z )A(Z),..., bq-1(Z).

Then for any k G {0,1,..., q — 1} we have

AqV.,q-1(Z^(Z), b0(Z), b1(Z), . . . , bq-1 (Z)) Aq(Z, C0, . . . , Cq-1)

gk (Z)

A°,...,,-1(c^(c)) ^kz)

where

(Z) . b°(Z) (z): bq-1 (Z) c°(Z) := ^-,... ' cq-1(Z) :=-

A0,...,q-1 A0,...,q-1

Thus the exponential polynomial em(z,A) G has a representation of the form

fq—1 Afc(Zj . . . ; Cq-1) r fcA T

.Tm E--л q ex?HZz}

s 4 fc=°

Afc(Z, C°, . . . ,Cq-1)

-KZ )

C=A

and functions

Afc (Z,C°, . . . , Cq-1 ) Z )

Cn(Z )

= gk (Z )A°,...,q-1, k G{0, 1,...,q - 1},

bn(Z )

-, n G па

(Z - A)m+! (z - A)m+iAo,...,q-i are holomorphic at A.

Now consider an arbitrary exponential polynomial of the form

-q-i

e(z s

, ^ /q-1 Ak(Z,C°, . . . , Cq-1) k

e(z,c°; . . . ; Cq-1 ) .= dZ^^ --л -k^ q- eXP{^k Zz}

k=°

Z)

C=A

where the holomorphic at A G C functions

CQ := Cq(Z), . . . ,Cq-1 := Cq_l(£)

and m G Z+ satisfy the following condition: the functions

Ak (Z, C°, . . . , Cq-1 )

Z )

k G {0, 1, ...,q - 1}

and

Cn (Z)

-, n G nA

(Z - A)m+1:

are holomorphic at A. Let us show that e(z,C°,..., Cq-1) G WS. We need to prove that

(S, AThe(z,C°,... ,Cq_1)) = 0. By Property 2 we have

ATh(e(z, C°,..., Cq-1))

d m

d m / q 1

q-1

V^ Ak . . .,Cq-l)A^(^n r, ,k,

dZ m\f=Q pH Z )

ATh(eW{uk Zz})

dZ m

dm dzm

Ak (Z, CQ, . . . , Cq-1 )

pK Z )

A^xp^Tz}) exp^Tz}

E

k=Q

q-1 A / \ W ,kn/-nUn

E A E bn^ expHZz}

k=Q

P(H z )

n=Q

Z=A

Hence,

ATh(e(z,CQ,.. .,Cq-1))

E bn n ddzm (znE Ak (Z,aQ::..,Cq-1) expwq Zz}

= 0 "" v k=0 ry q Since the functional S is continuous

P(HkZ) q

C=A

(S,ATh(e(z,CQ,.. .,Cq-1 — ) =

hn dm

^brn! dZm( Z

n=Q

q-1

Ak (Z, cq,..., Cq-1)

k=Q hn dm

P(ho z)

q-1) H^KZ

C=A

AQ,...,q-^ bnn dzzm[ZnJ2

n=Q k=Q

nX^, ,kn Ak (Z,CQ,...,Cq-1)

A

Q,...,q-1

By the relations (5)

(S, ATh(e(z, cq, ..., Cq-1

q-1

_ hn dm _^

Aq.....q-^ bnn dzmi znJ2

*Q,...,q-1 7. bn'

n! dZ'

kn' Ak(Z, CQ, . . . , Cq-1 )

k=Q

A

Q,...,q-1

C=A

where n' is the remainder after division of n by q. By definition of the determinant Ak(Z, c0,..., cq-1), we obtain

q-1

E<

k=Q

kn' Ak(Z,CQ, ..., Cq-1~) _ Cn'

(Z)

A

Q,...,q-1

Zn

q

h

q

q

Then,

(S, ATh(e(z, C0,..., Cq_1))) =

A

vb hn im,Zn

0,...,q-1 bn' n| dZ^ Z n=0 ' S

Cn' (Z)

C=A

where bn = 0, if n' G nA. Note that cn (A) = 0, c^ (A) = 0,..., )(A) = = 0, if n' G nA. Hence,

<S,ATh(e(z,c0,...,Cq-1))> =0.

The theorem is proved. □

Acknowledgment. We thank the anonymous referees for valuable comments. Besides, we would like to express our acknowledgment to professor A. B. Shishkin.

References

[1] Krasichkov-Ternovskii I. F. Invariant subspaces of analytic functions. I. Spectral analysis on convex regions. Mat. Sb. (N.S.), 1972, vol. 87 (129), no. 4, pp. 459-489. DOI: https://doi.org/SM1972v017n01ABEH001488

[2] Krasichkov-Ternovskii I. F. Invariant subspaces of analytic functions. II. Spectral synthesis of convex domains. Mat. Sb. (N.S.), 1972, vol. 88, no. 1, pp. 3-30. DOI: https://doi.org/SM1972v017n01ABEH001488

[3] Shishkin A. B. Spectral synthesis for an operator generated by multiplication by a power of the independent variable. Mat. Sb., 1991, vol. 182, no. 6, pp. 828-848. DOI: https://doi.org/SM1992v073n01ABEH002542.

Received May 21, 2018. In revised form, September 3, 2018. Accepted September 3, 2018. Published online September 18, 2018.

A. A. Tatarkin Kuban State University

149 Stavropolskaya st., Krasnodar 350040, Russia E-mail: tiamatory@gmail.com;

U. S. Saranchuk Kuban State University

149 Stavropolskaya st., Krasnodar 350040, Russia E-mail: 89182859942@mail.ru