CC BY
11
УДК 517.55
Power Series Nonextendable Across the Boundary of their Convergence Domain
Aleksandr D. Mkrtchyan*
Faculty of Mathematics and Mechanics, Yerevan State University, Alex Manoogian, 1, Yerevan, 375010
Armenia
Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041
Russia
Received 10.03.2013, received in revised form 14.04.2013, accepted 20.05.2013 In the article we construct a new power series in a single variable nonextendable through the boundary circle of the convergence disk. This series refines the known Fredholm‘s example.
Using this series we construct a double power series that does not admit an analytic continuation across the boundary of its convergence domain.
Keywords: power series, analitic continuation, infinitely differentiate, Dirichlet series.
The problem of describing the relations between singularities power series in one variable and their coefficients attracted mathematicians’ attention already at the end of 19th century. Remarkable results were obtained in the first half of the 20th century which allowed thinking that the development in this direction was almost completed. Many obtained results touch upon the question about series non extendable analytically across the boundary of their convergence domain and these results are connected with the names of famous Hungarian mathematicians Sego and Polya (see, for example, articles [1,2] and also the list of their articles in the book of L.Bieberbach [3]). Examples of series that are non extendable analytically across the boundary of their convergence domain we can find in the text-books about theory functions of complex variables. These examples deal with the so-called "strong lacunar" series, in other words, having "many" monomials with zero coefficients. Such series, for instance, are
OO OO
n=0 n=0
In 1891, Fredholm [4] gave examples of "moderate lacunar" non extendable series, moreover, these series represented infinitely differentiable function in the closure of the convergence disk. These series depend on a parameter a, and they have the following form
O
0 < a < 1.
n=0
Here n2 has a power order 2 respective to the summation index n, therefore we say that Fred-holm’s series have the lacunarity order 2.
*[email protected] © Siberian Federal University. All rights reserved
n=0
n
A more general result on a non extendable series in terms of lacunarity belongs to Fabry (see [3] or [5]). It claims that, if the sequence of natural numbers mn increases faster than n (i.e. n = o(mn)), then there is series
tt
E«nzmn,
n=0
converging in the unit disk and not extending across his boundary.
The purpose of this work is to construct a lacunar scale of power series oinone variable that are not extendable across the convergence boundary and represent infinitely differentiable functions in the closed disk. Furthermore, they should include Fredholm‘s series. Besides, we construct examples of double power series that converge in bidisk and do not extend across its boundary.
One of the main results is given by Theorem 1.1. It demnstrates that Fredholm‘s example may be refined to the power order of lacunarity from 2 to 1+£. The precise formulation is the following: if the increasing sequence of natural numbers nk satisfies the inequality nk > const * k1+e with £ > 0, then the power series
tt
y^qfcznk, 0 < |a| < 1
k=0
is not extending across the unit circle boundary and represents infinitely differentiable function in the closed disk.
Also, we give examples of double power series that are not extendable outside of the unit bidisk U2 = {(z1, z2) : |z11<1, |z21<1} and represent infinitely differentiable function in U2 \T2 where T2 = {(zi, z2) : |zi| = 1, |z21 = 1}.
1. Generalization of the Fredholm result
Theorem 1.1. If the increasing sequence of natural numbers nk satisfies the inequality nk ^ const * k1+e with £ > 0, then the power series
tt
Yak znk, 0 < |a| < 1 (1)
k=0
are not extendable across the boundary circle and represents infinitely differentiable function in the closed disk.
Proof. Consider the following series
tt
tf(t, u) = Y enkt+ku, where t,u e C. (2)
k=0
Its terms exponentially decrease in the product n x n of subspaces n = {u : Re u < 0} and n = {t : Re t ^ 0}. These series converge uniformaly on compact subsets of n x n, and therefore
¥>(t, u) is holomorphic in the product n x n of open subspaces. This property is preserved for
all derivatives with respect to the variable t of these series. Consequently the function y>(t, u) is infinitely differentiable in n for each fixed u e n.
Introduce the following notation
tt tt
F(-t) = Y eku0 e-nk(-t) = E enkt+ku° = ^(t,uo) (3)
k=0 k=0
for t e n and for each fixed u0 e n. Here, the function F(— t) is represented by a Dirichlet series
a, e-^fct
tt
] ak e
k=0
with exponential indexes Ak = nk and coefficiens ak = eku°.
Let us compute the value
l = urn ^.
k—►tt Ak
We obtain
-— ln k -— ln k -— ln k
L = lim —— = lim ----------- ^ lim 1 — = 0.
k——tt Ak k—-tt nk k——tt k1—£
Therefore, the abscissa of convergence for the series (3) we can be found as follows
-— ln |eku° | -— ln ekReu° -— k Re u0 -— kRe u0
r = lim -----------= lim --------------= lim ------------ ^ lim — = 0.
k—-tt nk k—-tt nk k—-tt nk k—-tt k1—£
Now we demonstrate that for the function F(—t) is perform the conditions of Polya‘s theorem [6]:
k
If 0 < nk | to, lim — = 0, nk+1 — nk ^ h > 0 and series
k —tt nk
tt
F(z) = £ ake-nkz k=1
has a finite abscissa of convergence —to < r < +to, then the line of convergence Re z = r is the natural boundary for the function F(z).
Indeed,
kk
0 < nk | to, lim — ~ lim -----------------► 0
k—-tt nk k—-tt k1—£
and
nk + 1 — nk ~ (k +1)1—£ —k1—£ = k1—^ ^ (\ + — ^ = k1—^ ^(1 + £) — + o( W
Consequently, the function F(—t) is not analitically extendable. Then, denoting a = eu (fixed) and z = et, from (2) we get (1), as desired. □
Theorem 1.2. For an arbitrary pair of natural numbers p > q, the series
tt
f (z) = ^ avq zvP, 0 < a < 1, (4)
v=0
is not expendable across of the unit disk boundary |z| < 1 and represents an infinitely differentiable function in the closed disk.
Proof. We can prove this theorem directly, without referring to the Polya‘s theorem.
We consider the following series
tt
^(t,u) = £ evPt+v’u, where t,u e C. (5)
Its terms are exponentially decreasing in the product n x n of subspaces n = {u : Re u < 0}
and n = {t : Re t ^ 0}. The series converges uniformly on the compact subsets of n x II,
therefore f (t, u) is holomorphic in the product of open subspaces n x n. Besides, the function f holomorphic in u G n for any fixed to G n.
We consider the Taylor expansion of f
(t ) ^ (t ) (u - uo)k (6)
f(t,u) = 1^ duk(t,u0)--------k----, (6)
f=0 '
with the centre uo G n, regarding t G n as a parameter. In view of (5), we have
g (t,u) = 2>q )f e'-+'-
V = 0
duf
Substituting this expression in (6), we obtain
°° I °° \ / \ f
I ^ (V9)fcevpt+vq«0 j (u - uo) . (7)
f=0 \v=0 / *
We demonstrate that the series (7) has a finite convergence radius for any fixed t0 from boundary d n (i.e. Re t0 =0).
The series (5) diverges if Re u > 0 and Re t0 = 0, because its general term
|evPt0+vq «| = |evPto ||evq «| = (eRe«)vq
does not tend to 0. Besides, the series (5) can be considered as a power series in the variable w = e«. Using these facts, we obtain that function f(t0,u) has a singularity point u such that Re u = 0. Hence, the series (7) has a finite convergence circle.
By using these facts and the Cauchy-Hadamar formula, we obtain that there is a sequence k, with the following property
!>q)
Vq)ki evpto + vq«0
v=0
k,*
—f with k, ^ to, (8)
Pf
where p is the convergence radius of the series (5), which depends on the choice of points u0 £ n and t0 G n.
Assume that the function f(t,u0) extends analytically with respect to t from n across some boundary point t0 £ d n for some fixed u0 G n. We denote by f(t, u0) the analytic continuation of the function f(t,u0). Its Taylor series is the following:
~(t ) dff(t )(t - t0)f f (t ) (t - t0)f
f(t,u0) = atr(t0,u0)—k— = ^ atf (t0,u0)—k—'
f=0 ■ f=0
Taking into account (5), we have
df ((,») = P)f e^"-
v=0
Substituting this expression in (9), we obtain
co
oo
f(t,u0) = ^ ]T(v p)f e^t0+^«0
f=0 V=0
(t — -0)S k!
oo
f=0
(vq)pfevPt0+vq“0j (t kt0)f . (10)
We investigate the convergence radius of this series by the Cauchy-Hadamard theorem. In the sequence
v=0
we consider the subsequence taking kq = qk,:
qfci
(qk,)!
^ ' (Vq)pki evpt0+vq«0
Using the estimate (8), we obtain
(qk,)! ppfl
(qk,)!
By Stirling’s formula
qfcl. I (pk,)pfl- 2 e-pfl pp ^ k,
, 1 p q ~ -t--------► to with p > q.
y (qk,)qfl-2e-qfl k, fi^°
Thus, the series (9) has empty convergence domain.
It follows that the series (5) does not continue analitically with respect to t across the point t0 G d II.
Denoting a = e« (fixed) and z = e4, from (5) we obtain (4). The theorem was proved. □
2. Double power series not extendable across the unit bidisk
Theorem 2.1. If the support A of the double power series
E zifl Z2 f 2
(k12A
is of type
A = {(k1,k2) G Z+2 : k2 > k11+e}u{(k1,k2) G Z+2 : k1 > k21+e} with e> 0,
then the double series (11) is not extendable across of the boundary of bidisk
U2 = {(z1, Z2) : |z 11 < 1, |z21 < 1} and represents an infinitely differentiable function in U2 \ T2 where
T2 = {(z1, Z2) : |z11 = 1, |z21 = 1}.
(11)
o
f
1
P
P
q
Proof. We can present the power series (11) by the sum of two series:
oo
z1
fl=0 &2=0 fi=0 f2=0
Y Y z1f1 Z2f2 + [f11 + E] + Y Y z1f1 + [f21 + elz2f2 =
o o o o
E Z2f^ Z1f1 Z2 11 + £] + £ Z1 Z2f2 Z1lf21 + el
f2=0 f1=0 f1=0 f2 = 0
oo 1 f- n ■ 1 f2 ~ «fc2
E Z1f1 Z2”fc1 + E Z2f2 Z1nfc2 .
1 - z2 1 2 1 - z1
2 f1=0 1 f2=0
Here [kj1+e] means the integer part of the number kj1+e. According to the Theorem 1.1 the
j series
Ez1f1 Z2nfc1, (12)
f1=0
considered respective the variable z2, converges in the unit disk and is not extending across the boundary circle, when 0 < |z1| < 1.
Using the change of variables e« = z1 and e4 = z2, we rewrite (12) as an exponential series
o
ef1«enk14 that represents an infinitely differentiable function in {(u, t) : Re u ^ 0, Re t ^ 0}\
f1=0
{(u,t) : Re u = 0, Re t = 0}. Consequently, the series (13) represent an infinitely differentiable function in U2 \ T2.
Similar properties one gets for the series
E Z2f2 Z1”fc2
f2=0
converges in the unit disk, does not continue with respect to the variable z1 with 0 < |z2| < 1 and represent infinitely differentiable function in the U2 \ T2. Therefore, we obtain the require statement for series (11). □
Proposition 1. Let K be the sector with integer generate vectors m1 = (m11,m12) and m2 = (m21,m22). Then the series
f (z)= Z1f1 Z2f2 (13)
few 2nK
represents the rational function
f (z) = Ti----------------------7 with P (z) = 1 + za (14)
V 7 (1 - z1m11 z2m12 )(1 - z1m21 z2m22 ) W ^ V 7
V 1 2 A 1 2 1 ae(N2n int D)
where intD is the interior of parallelogram D with verteces (0, 0), m1, m2 and m1 + m2.
Proof. We can cover the all integer points of K by semigroup L = {11m1 +12 m2, G i =
1, 2} and its shifts Lj = aj + L where aj runs over N2 n intD,. Thus, we have
E z1f1 z2f2 = E z1f1 z2f2 + E z1f1 z2f2 + ... + E z1f1 z2f2 .
few 2nK feL feL1 feLp
where p is the cardinality of N2 n int D
53z1f1 z2f2 = Y, z,1m1+,2m2 = Y (zm1 ),1 (zm2)^ = (15)
feL ,1l,2>0 ,1,,2>0
o
- (1 - zm1 )(1 - zm1 ) _ (1 - z1m11 z2 m12 )(1 - z1m21 z2m22 )’
and
Y z1f1 z2f2 = z1°j1 z2°j2 Y z1f1 z2f2, aj = (aj1,aj2).
f e L j f e L
Therefore we obtain
V' f1 f2 1 + z1a11 z2°12 + ... + z1°p1 z2ap2
\ 1 zo 2 ---- --------------------------------------
1 2 (1 - z1m11 z2m12 )(1 - z1m21 z2m22 )
few 2nK v 1 2 A 1 2 1
as desired. □
In conclusion, we are interested the following question. Consider the series (13) for general (not necessary integer) generators m1 and m2. Is it true, that this series either is not extandable across of the boundary of converge domain, or represents a rational function of type
f (z) = 7i-----------------------------------“St-t , (16)
W (1 - z1m11 z2m12 )(1 - z1m21 z2m22 )’ V '
where P(z) is polynomial?
We can interpret this fact as a two-dimentional analogue of Sego‘s theorem [1] (see also [3]) on series with the finite number of different Taylor coefficients. In the multivariate case, the interest of studing series of form (13) arises in the thermodynamics of several hamiltonians [7].
1
1
References
[1] G.Szego, C/ber Potenzreihen mit endlich vielen verschiedenen Koeffizienten, Sitzgsber. pruefi. Akad. Wiss., Math.-phys. K1., (1922), 88-91.
[2] G.Polya, C/ber Potenzreihen mit ganzzalhigen koeffizienten, Math. Ann., 77(1916), 497-513.
[3] L.Bieberbach, Analytische Fortsetzung, Berlin, Springer-Verlag, 1955.
[4] G.Mittag-Leffeler, Sur une transcendente remarquable trouvee par M. Fredholm. Extrait d‘une letter de M. Mittag-Leffler a M. Poincare, Acta mathematica, 15 Imprime le 21, 1891.
[5] E.Fabry, Sur les points singuliers d’une fonction donnee par son developpement de Taylor, Ann. ec. norm. sup., 13(1896), 367-399.
[6] A.F.Leont’ev, Entire functions, exponential series, Nauka, Moscow, 1983 (in Russian).
[7] M.Passare, D.Pochekutov, A.Tsikh, Amoebas of complex hypersurfaces in statistical thermodynamics. Math. Phys., Analysis and Geometry, 16(2013), no. 3, 89-108.
О степенных рядах, непродолжимых через границу области сходимости
Александр Д. Мкртчян
Мы построили пример степенных рядов, которые не продолжаются через границу своей области сходимости и представляют бесконечно дифференцируемую функцию в замыкании круга. Он усиливает известный пример Фредгольма. Используя полученный результат, мы строим двумерные степенные ряды, которые не продолжаются через единичный бикруг.
Ключевые слова: степенные ряды, аналитическое продолжение, бесконечная дифференцируемость, ряды Дирихле.
