ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 6, pp. 1–5.
Allerton Press, Inc., 2018.
Original Russian Text
F.N. Garif ’yanov, E.V. Strezhneva, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 6, pp. 3–8.
Diﬀerence Equation for Functions Analytic Outside Two Squares
F. N . G a r i f ’ y a n o v
Kazan State Power Engineering University
ul. Krasnosel’skaya 51, Kazan, 420066 Russia
Kazan National Research Technical University named after A. N. Tupolev
ul. K. Marksa 10, Kazan, 420111 Russia
Received March 28, 2017
Abstract—We consider a totality of two squares built on primitive periods 1 and i and “suﬃciently
close to each other“. In a vicinity of this set we investigate four-element diﬀerence equation with
constant coeﬃcients, whose linear shifts are generating transforms of the corresponding doubly
periodic group and the inverse transforms. We seek a solution in a class of functions, which are
analytic outside this set and vanish at inﬁnity. The equation is applicable to the moments problem
for entire functions of exponential type.
Keywords: diﬀerence equations, regularization method, moments problem.
We consider two disjoint squares R
built on primitive periods 1 and i. We write generating
transformations of corresponding doubly periodic group and their inverse transformations in the form
(z)=z + i
, m = 1, 4. They map a point of interiority of each square into its complement. We
assume also that these transformations do not map points of one of the squares into closure of other one.
Let us study the following linear diﬀerence equation (l. d. e.) with constant coeﬃcients
(z)] = g
,k=1, 2, (1)
=0, k =1, 2. We seek solutions in the class of functions f, which are
holomorphic outside the set R = R
and vanish at the inﬁnity point. The boundary values f
must satisfy the H
older condition on every open side of the squares. At the vertices of the squares
the desired functions admits at most logarithmic singularities. The right-hand term is piecewise
holomorphic in R (i.e., holomorphic in every square R
), and its boundary values satisfy inclusions
(t) ∈ H(Γ
), k =1, 2, Γ
. We denote this class of solutions by B.Themost
important factor aﬀecting the solution to this problem is the incoherence of the set C \
). It consists of three connected components, and only one of them contains the point
at inﬁnity. Meanwhile Eq. (1) is deﬁned on two other components, what does not allow to apply the
classical methods of the theory of convolution operators . That is why, unlike of usual approach to
the theory of analytic solution of l. d. e., the solution and the right-hand side belong here, in general,
to distinct classes of holomorphic functions. In particular, the right-hand side must not have analytic
continuation through some segment Γ
. Even in the case of homogeneous equation
,k=1, 2, (2)