Singular points for the sum of a series of exponential monomials Текст научной статьи по специальности «Математика»

72 Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 72-87

DOI: 10.15393/j3.art.2018.5310

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

UDC 517.52, 517.53

O. A. KRIYOSHEEYA, A. S. KRIYOSHEEY

SINGULAR POINTS FOR THE SUM OF A SERIES OF EXPONENTIAL MONOMIALS

Abstract.

A problem of distribution of singular points for sums of series of exponential monomials on the boundary of its convergence domain is studied. The influence of a multiple sequence A = {Ak,nk}fc=i of the series in the presence of singular points on the arc of the boundary, the ends of which are located at a certain distance R from each other, is investigated. In this regard, the condensation indices of the sequence and the relative multiplicity of its points are considered. It is proved that the finiteness of the condensation index and the zero relative multiplicity are necessary for the existence of singular points of the series sum on the R-arc. It is also proved that for one of the sequence classes A, these conditions give a criterion. Special cases of this result are the well-known results for the singular points of the sums of the Taylor and Dirichlet series, obtained by J. Hadamard, E. Fabry, G. Polya, W.H.J. Fuchs, P. Malliavin, V. Bernstein and A. F. Leont'ev, etc.

Key words: invariant subspace, series of exponential monomials, singular point, convex domain

2010 Mathematical Subject Classification: 30D10

Let A = {Ak be a sequence of different complex numbers Ak

and its multiplicities nk, |Ak+1| > |Ak| and |Ak| ^ to. We denote by n(r, A) the number of points Ak (taking into account their multiplicities) located in the disk B(0, r). The upper density and maximal density of A are the quantities

n(r, A) o^i- n(r, A) - n((1 - S)r, A) n(A)=hmsup^—-, no(A) = hmhmsup^^—---——-.

@ Petrozavodsk State University, 2018

We consider the series of exponential monomials

— 1

^ ak,nzneXkZ. (1)

k = 1,n=0

Let a = {ak,n} and D(A, a) be an open kernel of the set of all points z G C where the series (1) converges and its sum is an analytical function. We denote the sum of the series (1) by gA,a and the set of all sequences of coefficients a = {ak,n} for which D(A, a) = 0 by U(A). In this article we observe the problem of distribution of singular points for the function gA,a on a boundary dD(A,a).

Let ©(A) be the set of all partial limits of the sequence {Ak/|Ak|}^!=1. We assume

m(A) = limsup nk /|Ak |.

k^w

In [6] we showed that in the general case the set D(A,a) may not be convex and is not even connected. But if m(A) = 0 and n(A) < to then the Cauchy-Hadamard theorem for series of exponential monomials (see [6], Theorem 4.1) shows that D(A, a) is a convex domain:

D(A, a) = {z : Re(ze—i0) < h(©, a, A),ei0 G ©(A)}, (2)

%>,a, A)=inf fliminf min ln(1/|ak(j),n| A , (3)

V o^n^nfc(j)-1 |Ak(j)1 J

where the infimum is taken with respect to all subsequences {Akj)} such that Ak(j)/|Ak(j)| ^ e—Meanwhile, the series (1) diverges at every point of exterior of D(A,a) (except for the origin). Moreover, under the same conditions, by the Abel theorem (see [6], Theorem 3.1) for the series of the kind, the expansion (1) converges absolutely and uniformly on every compact subset of D(A,a).

The problem of describing the set of singular points for gA,a on the boundary dD(A,a) counts a long history. It originates in the investigation (started as early as the 19th century) of domains of existence for functions representable by power series. In this regard, we note the works of J. Hadamard [4] and E. Fabry [2]. In the works of G. Polya [17], W. Fuchs [3] and P. Malliavin [16] the following result was obtained. The necessary and sufficient condition of existence for each sum of the Taylor series, converging in the unit disk, of a singular point on any arc of the

boundary of this disk of length 2nr is t = n0 (A) (A is a sequence indexes of Taylor series with nonzero coefficients). G. Polya (see [18], [19]), Carlson and Landau (see [18], Chapter II, §5.2) and V. Bernstein [1] extended this result to the case of Dirichlet series. In work [7] the result is obtained for these series whose special cases are all specified results for Taylor and Dirichlet series. It was proved that each sum of the Dirichlet series has a singular point on the segment of length 2nT lying on the convergence line, then and only then, when t = n0(A) and Sa = 0 (we will define the index of condensation SA below). The singular points for the general series (1) and the series of exponents are studied in [8], [15].

This paper studies singular points for the sum of series (1) on arcs of the boundary dD(A,a) of the following form. Let 7 be the arc of the boundary connecting points z1 and z2. The arc 7 will be called an R-arc if |z2 - z11 = R.

Let A be a regular sequence, i.e., it is a part of a regularly distributed set (see [15], Chapter II). It follows from Theorem 4.1 [8] that in this case, under certain restrictions on D(A,a), each gA,a has a singular point on any R-arc if and only if Sa =0. A is regular if and only if the maximal density n°(A,-0,^>) (in the angle bounded by the rays re®^, re®v) does not exceed the length of the corresponding arc of the boundary of a convex compact set (see [13], Theorem 1). Thus, sufficient conditions for existence of singular points for gA,a on R-arcs are the special boundedness of maximal density n°(A,-0,^>) and the equality SA = 0. In this research it is shown that in the general case these conditions are also necessary for the existence of singular points for gA,a on R-arcs. Moreover, for one class of the sequences A (which are concentrated along some ray re®v) it is proved that these conditions give a criterion for the existence of singular points for gA,a on R-arcs. All the results on Taylor and Dirichlet series, which are marked above, and the previously mentioned result from [7] are particular cases of this statement.

First of all, we study the influence of some characteristics of A on the presence of singular points for gA,a on R-arcs. We assume (see [9], [11])

S° = liminf ^. Sa = lim Sa(S), Sa(S) = liminf -ln (Ak,S)|

S^0 S S^o k^TO |Ak |

<z,S) = n (3-m

XmEB(Xk ,S|Afc |),m=k 1 m|

Let nA(z,S) be the number of points Ak G B(z,S|z|) with their multiplicities nk taken into account.

Theorem 1. Let A = {Ak,nk}, m(A) = 0 and SA = -to. Then for each R > 0 there exists a sequence a G U(A) such that function gA,a has no singular points on some R-arc of boundary dD(A,a).

Proof. R > 0 and number S0 G (0,1/15) satisfies the condition 0 G (0, n/4) if 0 G (0, n/2) and |e^ -1| < 2So(1 - So)-1. Since SA = -to, according to the definition of SA we find S G (0, S0) and the sequence {Ak(p)} such that

ln |qA(p)(Afc(p),S)| < -^|Afc(p)|, p > 1, p = 12RS. (4)

Passing to the subsequence, we can assume that

Afc(p)/|Afc(p)| ^ e-^ P ^ TO ^fctp+i)! > ^^(p)^ P > 1 (5) The function gA,a is found as the sum of the series

w

g(z) = 53 cpgp(z). (6)

p=i

To do this, we construct an auxiliary sequence A2 C A, A2 = Up^1A2,p.

Let Bp(a) = B(Ak(p), aS|Ak(p)|). By (5) and inequality S0 < 1/15, disks

Bp(1), p > 1, do not intersect in pairs. We fix p > 1. The set A2,p is formed from multiple points Ak G Bp(1). If

nA(Afc(p),S) - nfc(p) < P|Afc(p)| + 1, (7)

then, A2,p pair Ak(p),1 is taken as and all pairs Ak,nk such that k = k(p) and Ak G Bp. In this case we assume mp = nA(Ak(p) ,S) - nk(p) + 1. Let now

nA (Ak(p),S) - nk(p) > P|Afc(p)| + 1. (8)

Then we reduce the multiplicity Ak(p) to 1 and from the disk Bp(1) we withdraw as many points Ak without taking Ak(p), or we reduce their multiplicities nk (without changing their designations) that inequalities

P|Afc(p)| < mp - 1 < Afc(p)| + 1, (9)

are satisfied, where mp is the number of remaining points Ak with taking into account their multiplicities. In this case, as A2,p all the remaining

pairs Ak,nk are taken. Thus the sequence A2 c A is constructed. We will show that n(A2) < to. Let r > 0 and p(r) be the maximal number of disk Bp(1) having a non-empty intersection with B(0, r). Then r ^ ^ |Ak(p)|(1 — S) and by (7), (9) we obtain:

n(r, A) < n(|Ak(p(r))|(1 + S) A) < y^ mp <

r < |Ak(p(r))|(1 — S) < -= |Ak(p(r))|(1 — S) <

< 2-p ^k^ + 1 + 1 < 2-p + 2 A

^ -= |Ak(p(r))! ^ -= ^(-(r^ v |Ak(p) U Using inequality (5), we get:

nM) < Cyl^kwl < cpr^T- < 2C.

r -= |Ak(-(r))| -= 2-(r)—-

It follows that n(A2) < to. We show now that inequalities (4) are not violated if A is replaced by A2. If inequality (7) is true then by construction

qA(-) (Ak(-),S) = qA(2-)(Ak(-),S).

Suppose now that inequality (8) is true. Then inequality (9) is true, and according to the definition of qA2, we have:

ln |qA2-)(Ak(-),S)| = E nm ln |Akg.. Am| <

vm |

1( v -k(-^ ^ 3S|Amr

Am£Bp(1),m=k 1 m|

| A k ( p ) |

< E nm ln3|Am[ <

AmeBp(1),m=k 1 m|

< — ln(3(1 — S))(m- — 1) < m- — 1 < —S|Ak(-)|, P > 1.

Thus,

ln |qA(-)(Ak(-),S)| < —^|Ak(-)|. (10)

Let us now define the function gp, p > 1, by the formula

1 f eAz dA

g-(z) = f

dBp(5) T-(A — Ak(-))qA2-)(A,S)'

where we define numbers tp > 1 below. We get estimates from above on gp. We have:

|A - Afc|

3J|Afc

> 1, A £BP(5), Afc e Bp(1).

Since tp > 1, we obtain:

|gp(z)| < sup |eAz| < exp(Re(Afc(p)z)+5^|Afc(p)||z|), z e C. (11)

AedBp(5)

Let us now define the coefficients of cp. Let

Cp = exp(-^|Afc(p) |), p > 1. (12)

We will show that series (6) converges on compact sets K in the domain D = B(0, 2R) n {z : Re(ze-i^) < 2RJ} uniformly. We take e > 0 such that

Re(ze-iv) < RJ - 2e, z e K. By (5) there is a number p0 such that

Re(Afc(p)z) < (Re(ze-i^) + e)|Afe(p)|, z e K, p > po. Then, we get for z e K by (11) and (12):

ww w

E |cpgp(z)| < E Cp exp ((RJ - e + 5JR)|Afc(p) |) < E e-6^1.

p=po p=po p=po

Since n(A2) < to then the last series converges. Thus, the function g is analytical in the domain D for any tp ^ 1.

Since J e (0,1/15) and the points Ak e A2 belong to the disks Bp(1), then by (5) there is 0 e (0,n/2) such that - 1| < 2J(1 - J)-1 and ©(A2) C {eie, 0 e (^ - 0,^ + 0)} holds true. We will show that for any tp > 1 the function g is represented by series (5) which converges uniformly on compact sets in the angle

r = {z : Re(ze-i(v+^) < 0} n {z : Re(ze-i(v-^) < 0} .

esidue ca every p ^ 1 we have

Using the residue calculus and the definition of the function qA(p), for

b nfc-11

gp(z) = eak(p)z + EE z"eAfc z,

Tp Afc,nfceA2,PIfc=fc(p) n=0 Tp

where &fc(p),o = (qA(2P)(Afc(p), • Let bfc(p),n = 0, n = 1,nfc(p) - 1. We define the coefficients {ak,n} :

flfc,n = 0, Afc,nfc€ A2, afc,„ = p fc,n , Afc,nfc € A2,p, p > 1.

Since the disks Bp(1), p > 1 do not intersect in pairs, the definition is well-defined. Let us find the numbers tp, p > 1. By (10) and (12) we have:

max ln |cp6fc,„| > ln |cp6fe(p),o| > ln cp + ^|Afc(p)| = 0.

Afc ,nfc£A2,p

We choose the numbers tp > 1 such that

max ln |ak,n| = max ln |cp6k,„| — ln ap = 0. (13)

We find the convergence domain of series (1) with the coefficients ak,n defined above. Since n(A2) < to, m(A) = m(A2) = 0 then the Cauchy-Hadamard and Abel theorems (see [6]) show that series (1) converges uniformly on compact sets in the convex domain D(A,a), defined by formulas (2) and (3), and diverges at each point of its exterior (except for the origin of coordinates). By (13) and (3)

uto awi- ■ f • ln(1/K,n|) h(0, a, A) > limmf mm —

k^w O^n^nfc — 1 |Ak |

= limsup max —^ k,w| = 0

for all

©(A2). In addition, there are numbers s(p), n(p) such that the pair As(p), ns(p) is an element of the sequence A2 and

ln |as(p),„(p)| =0, p > 1. (14)

We assume that As(p)/|As(p)| ^ e-ip, p ^ to. Then eip € ©(A2). And we can assume p = ^ + a, where a € (—By (14) we have:

w , • f ln(1/|as(p),n(p)|)

+ a, a, A) < liminf-,— = 0.

|As(p)|

p

Thus, series (1) diverges at each point of n = {z : Re(ze-i(v+a)) > 0}. In addition, it converges uniformly on compact sets in the angle r C D0(a,A), along with series (6). Therefore, the function gA,a = g is analytical in the domain G = D U r.

Let a > 0 and w e dB(0,2R) n (L0 = {z : Re(ze-i^) = 0}), Im(we-iV) > 0 (the case a < 0 is similar). Straight lines that are perpendicular L3 = {z : Re(ze-i(v+^)) = 0} and pass through the points 0 and w, are denoted by L1 and L2. Since 0 e (0, n/4), the distance between these lines is strictly greater than R. Let Q be the area bounded by the lines L0, L1, L2. It lies in the domain G. Also, some neighborhood V of the interval (0,w) C L0 n dQ lies in the domain G. Thus, the function gAj0 is analytical in the domain Q U V.

Since 0 e (0, n/4) then the half-string which is limited by lines L1, L2, lies in the angle r C D(A,a). Series (1) diverges in n. Therefore, there exists an R-arc which lies in the intersection of the domain D U V and the half-plane {z : Re(ze-i(v+a)) < 0}. By construction, gA a has no singular points on this arc. □

Example 1. Let A = {Ak,nk}, nk = 1 and A2k = k, A2k-1 = k — e efc, k > 1, where e > 0. Then m(A) =0. If 5 G (0,1/3), then

|qAk (A2fc,¿)| <

Therefore,

A2fc — A2fc-1 3^|A2fc-i|

e-£fc

<

3J(k — e-efc)'

SA = lim liminf —)| < lim liminf k 1 ln ( -——-r- | = — e.

fc^w |A; | fc^w \(35(k — e-efc)J

By the last inequality it follows that = —to. We also note that the work [20], Chapter II, §4, has a nontrivial example of a sequence A for which SA = 0 and = —to.

Theorem 2. Let A = {Ak, nk} and m(A) = 0. Then for every R > 0 there exists a sequence a G U(A) such that the function gA,a has no singular points on any R-arc of the boundary dD(A,a).

Proof. Let R > 0, a G (0,1] and

D = {z : Rez < a}n{z : |Imz| < aR}, = {z : Rez = a, |Imz| < aR}.

The segment lies on the boundary of the half-string . We choose a G (0,1) such that (1 - a)2 + aR2 < 1 and a < (2V3R)-1. Then Tj lies in disk B(1,1), the domain G1 = \ {z : Rez < — 2-1} lies in the disk B(0,1), and \ G1 lies in the truncated angle

r = {Re(zein/6) < 0} n {Re(ze-in/6) < 0} n {z : Rez < —2-1}.

Since Tj is a compact set, then there exists an e G (0,1) such that the rectangle Tj (e) =DCT n {z : Rez > a — e} lies in the disk B(1,r0) for any ro G (e-1,1).

According to the condition m(A) = 0 there exists a sequence {Ak(p)} such that Afc(p)/|Afc(p)| ^ e-iV and ank(p)/|Ak(p)| > t > 0. We suppose Mp = eiVa-1Afc(p). Then Mp/M ^ 1 and nfc(p)/|^p| > t, p > 1. We assume that |mp+1| ^ 2|mp|, p > 1. Let 0 <7 < min{4-1e, 2-1t, 8-1}. We can also assume that

yM < m(p) < min{4-1e|^p|,nfc(p)}, p > 1, (15)

where m(p) are some positive natural numbers. By construction

Re(Mpz) < |MP|(Rez — 2-1ylnr0), z G B(0,t), p > pt. (16)

We suppose cp = exp( —(a + 4 1ylnr0)|^p|), p > 1 and consider the series

(z)=£ Cp(z - 1)m(p)eMpZ, (17)

P=1

We show that it converges uniformly on compact sets in the domain . Considering the embedding T(e) c B(1,ro), by (15) and (16) we have:

|cp(z - 1)m(p)eMpZ | < exp(7|Mp | ln ro - (a - Rez + 4-137 ln ro)M)) <

< exp(4-1YlnroiMpi), z e T(e), p > pi. (18)

Since Gi c B(0,1), ro > e-1 then by (15), (16) and the definition of T(e) we have:

|cp(z - 1)m(p)eMpZ| < exp(m(p) ln 2-(a - Rez + 4-137lnro)|Mp|)) < exp((-e/2 + 37/4) |Mp|) <

< exp(-5e/16|Mp|), z e G1 \ T(e), p > p1. (19)

Considering the embedding DCT \ G1 C r, we get:

|cp (z - l)m(p)empz | < exp(m(p)ln(1 + |z|) - (a - Rez + 4-137 ln r0 )M)) <

< exp(|^p||z|/4 + (Rez + 3/32)^)) < exp((|Rez|/2 + Rez + 3/32)^|) <

< exp(-5/32|Mp|), z e (Dff \ G1) n B(0,t), p > pt.

By the last inequality and (18), (19) it follows that series (17) converges uniformly on the compact sets of the domain DCT. Therefore, gCT e H(DCT). Opening brackets in (17), we obtain:

w,m(p) 1

E = E

p=1,n=0 k = 1,n=0

gff (z)= E = E , (20)

where z = ae ®vw, ak,n = 0, if k = k(p) or k = k(p) and n > m(p). Since r0 < 1 then G0 = {z : Rez > a + (7lnr0)/8} n DCT = 0. Similarly to (16), we obtain:

Re(^z) > |^p|(Rez + (7lnr0)/8), z e G0, p > p0.

It is easy to notice that |bp,0| = cp. Therefore, taking into account the definition of cp we have:

|bp,0||eMpZ| > 1, z e G0, p > p0.

Thus, the first series in (20) diverges at every point z e G0. Let

r = {Re(zein/6) < 0} n {Re(ze-in/6) < 0} n {z : Rez < -3},

and z e r0. Coefficients 6p,„ are estimated above by 2m(p). Therefore, taking into account (15), (16) and definitions cp, r0, e, 7 we have:

|6Pj„zne^z| < exp(m(p) + m(p) ln(1 + |z|) + (Rez - 4-137lnr0)|^|) <

< exp((e/4 + e/4|z| + 37/4 + Rez)|^|) < exp((1/2 + Rez/2)|^|) < .

It means that the first series in (20) converges uniformly on r0.

It follows from the above that on a certain aR-arc, lying in the string {z : |Imz| < aR} and on the boundary of the convergence domain of this series, its sum gCT has no singular points. Then the sequence of coefficients of the second series in (20) is the required one. □

Example 2. Let A = {Ak,—k}, Ak = 2k and —k = 2k-1, k > 1. Then —(A) < 1 and m(A) = 1/2.

By theorems 1,2 conditions > -to, m(A) = 0 are necessary for the existence of singular points for all functions #A,a, a e U(A), on every R-arc of boundaries of the convergence domains corresponding to series (1). We show that these conditions are sufficient for one of the classes of sequences as well.

Lemma 1. Let A = {Ak, —k} such that m(A) =0, > -to, and Ak/|Ak| ^ e-iv, k ^ to. Then —o(A) < to.

Proof. Let S e (0,1). As in Theorem 1, we obtain:

ln |qk(Afc,S)| < - ln(3(1 - S))(nA(Afc,S) - 1). Therefore, taking into account equality m(A) =0 we obtain:

=liminfliminfln |q|(Ak,S)| < 5^o fc^TO o|Afc |

< liminfliminf - ln(3(1 - S))—A(Ak,S) = 5^o fc^TO o|Afc |

—A(Afc,S)

- limsupln(3(1 - S)) limsup ■

5—► o fc^oo S | Afc

S|Afc|

= - limsuplimsup A(^k, ). (21)

fc^to S|Afc|

Let —o(A) > 0. We choose a sequence rp,5, p > 1 such that n(r,A) - n((1 - S)r,A)

lim sup

Sr

—(rp 5 ,A) — —((1 — S)rp 5 ,A)

lim ( p 5, )--) p,A, ) > 0, (22)

p^TO Srp,5

S e (0,So). We assume that any ring B(0,rp,5) \ B(0, (1 - S)rp,5) contains at least one Ak. Let Up,5 be the group of all points Ak, belonging to it and Afc(p,5) e Up,5. According to the condition Up,5 c B(Afc(p,5),4S|Afc(p,5)|), p > p(S). Therefore, taking into account (22) and (21) we obtain:

-A(Afc(p,5),4S) . —A (Ak(p,5),4S) _

—o(A) ^ limsuplimsup- ' )- ^ limsuplimsup

5^o p^TO Srp(S) 5^o p^TO S|Ak(p,5)|

= 4 lim sup limsup ) ^ 4 limsup limsup A(^k, ) ^ -■4S°.

5^0 p^TO S|Ak(p'5^ 5^0 k^to S|Ak|

□

Let A = {Ak,nk}, D be a convex domain, W(A,D) be a closure in H(D) (in the topology of uniform convergence on compact sets) of linear span £ (A) = {z"eAfc z}. We need the criterion of fundamental principle from [11], Theorem 3.2. Let us express it in the particular case. We assume

L(y,D) = dD n {w : Re(we-iv) = HD(y)}, HD(0) = sup Re(ze-i^)

is a supported function of the domain D, t(y, D) is the length of L(y, D) (possibly equal to zero).

Lemma 2. Let A = {Ak, nk} such that Ak/|Ak| ^ e-iV, and D be a bounded convex domain. If W (A, D) is non-trivial (i. e., the system £ (A) is not complete in H(D)), then the following statements are equivalent:

1) £ (A) is a basis in W (A, D);

2) Sa = 0 and n0(A) < t(y,D)/2n.

We note that the system £(A) is not complete in H(D) when and only when there exists an entire function f of the exponential type, the shift of the conjugate diagram (see [15], Chapter I, §5) of which lies in D.

Theorem 3. Let A = {Ak, nk} such that Ak/|Ak| ^ e-iV, and R > 0; the following statements are equivalent:

1) Each function gA,a on every R-arc 7 C dD(A, a) has a singular point.

2) SA = 0 and n0(A) < R/2n.

Proof. Let us assume that statement 2) holds true and a e U(A). It is easy to notice that from inequality n0(A) < to relations m(A) = 0 and n(A) < to follow. Then, as in Theorem 1, domain D(A, a) is defined by formula (2). According to condition ©(A) = {eiV}. Hence, D(A, a) = = {z : Re(ze-iV) < c}, and any R-arc of the boundary dD(A, a) is a segment [z1,z2] C {z : Re(ze-iV) = c} of length R. Let's assume that gA,a has no singular points on [z1,z2].

Then there exists S > 0 such that the function gA,a is analytical in the domain Q = K + B(0,3S), where K is a square with the side [z1, z2],

lying in the closure D(A, a). By Lemma 2.1 from work [12] A is a part of the sequence A1 = {^p,mp} having density —(A1) = —o(A). It can be assumed that ^p/|^p | ^ e-iV (arguments do not affect the density A1). We get

f (z)= ,n 0 - (¿2

The function f is entire and has the exponential type, its conjugate diagram coincides with the segment [-a, a], where a = n—o(A)ei(v+n/2) (see [13], Chapter 2, §1, Theorem 2). The shift of this segment lies in the domain G = K + B(0,S) - Seiv c D(A, a). Hence, E(A) is not complete in H(G).

Since gA,a is represented by the series (1) in the domain D(A, a) then gA,a e W(A, G). In addition, gA,a is analytical in the domain G + B(0,S) c 0. Thus, the conditions of Theorem 12.1 from [5] on the continuation of the spectral synthesis are fulfilled. Therefore, according to it, gA,a e W(A, G + B(0,S)).

As we have shown above, E(A) is not complete in H(G + B(0, S)). According to statement 2) and taking into account the construction we have: SA = 0 and —o(A) < R/2n < t(y>, G + B(0, S))/2n. Then, by Lemma 2, the function gA,a is represented by the series (1) in the domain G+B(0,S). Thus, we have two representations of the function gA a by the series (1) in the domain G c D(A,a) n (G + B(0,S)). Since E(A) is not complete in H(G), then (see [15], Chapter 2, § 6, Theorem 6.2) these representations are the same. According to the definition of D(A, a) it means that there exists an embedding G + B(0, S) c D(A, a). By construction, however, this embedding is incorrect. We have a contradiction. Hence, gA,a has at least one singular point on the segment [z1, z2].

Let us assume that statement 1) holds true. Then by Theorem 1 inequality > -to holds true. Referring to the definition of the quantities and SA we get SA = 0. Theorem 2 implies that the equality m(A) = 0 is also true. Then, by Lemma 1 we get —o(A) < to. It remains to prove that —o(A) < R/2n. Let us assume the opposite: p = —o(A) > R/2n. We choose £ > 0 such that p - e > R/2n.

We suppose L1 = {z e C : Rez =1}, L2 = {z e C : Rez = -1}. Let (L4) be a straight line passing through the points with coordinates (0, ¿np) and (1,«n(p - e)) ((0, -¿np) and (1, -¿n(p - e))). We suppose D = eiVDo, where Do is the domain bounded by straight lines L1, L2, L4. It is an isosceles trapezoid. One of its bases of length 2n(p - e)

lies on the line Li and the other base of length > 2np lies on the line L2 •

The vertical segment with length 2np lies in D0 by construction. Therefore, the domain D contains a shift of a conjugate diagram of the function f • It means that E(A) is not complete in H(D). We consider the subspace W (A, D). We have n0(A) = p > p - e = t (y, D)/2n. Then there exists a function g G W(A,D) by Lemma 2 which is not represented by the series (1) uniformly convergent on a compact set in the domain D.

Let us consider the domain D1 = eiVD0,1, where D0,1 = D0 n {z G C : Rez < 0}. It is an isosceles trapezoid, one of its bases coincides with the corresponding base of the trapezoid D, and the other base coincides with the segment eiV [—inp, inp]. The domain D1 contains a shift of a conjugate diagram of the function f by construction. Therefore, E(A) is not complete in H(D1). We have Sa = 0 and n0(A) = p = t(y,D1)/2n. Then the function g is represented by the series (1) in the domain D1 by Lemma 2.

Since n0(A) < to, this series converges in some half-plane {z : Re(ze-iV) < c} (c > 0) and diverges at each point of its exterior, as in the beginning of the proof. The inequality c < 1 holds true. Indeed, otherwise the function g may be represented by the series (1) which converges in the domain D, which is impossible owing to choosing g.

By construction the function g is analytical in the domain D which crosses the line {z : Re(ze-iV) = c} at intervals of length strictly greater than 2n(p — e). Thus, given the choice of the number e > 0 we have the sum of the series (1), which has no singular points on a certain R-arc (a segment with length R) of the boundary of its convergence domain. We come to a contradiction with 1). Therefore, the assumption that n0(A) > R/2n is incorrect. □

Example 3. Let A = {Ak,nk}, Ak = kh and nk = 1, k ^ 1. Then Sa = 0 (see [12], §2) and n0(A) = h.

The particular cases of Theorem 3 are all the above mentioned results for Dirichlet and Taylor series.

Acknowledgment. The work of the first author was supported by a grant of the Russian Science Foundation (project 18-11-00002).

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Received May 11, 2018. In revised form,, August 29, 2018. Accepted August 31, 2018. Published online September 6, 2018.

O. A. Krivosheeva Bashkir State University 32 Z. Validi, Ufa 450076, Russia E-mail: [email protected]

A. S. Krivosheev Institute of Mathematics

with Computing Centre — Subdivision of the Ufa Federal Research Centre of the Russian Academy of Science 112 Chernyshevsky str., Ufa 450008, Russia E-mail: [email protected]