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EDN: MNYKKX УДК 517.55
On the Blaschke Factors in Polydisk
Matvey E. Durakov*
Siberian Federal University Krasnoyarsk, Russian Federation
Received 10.07.2022, received in revised form 15.09.2022, accepted 20.10.2022 Abstract. The purpose of this work is to construct a multidimensional analogue of the Blaschke factors. The relevance of the construction of this analogue was prompted by a recent joint article by Alpay and Yger devoted to the multidimensional interpolation theory for functional spaces in special Weyl polyhedra. By such a factor we understand a set of special inner rational functions in a unit polydisk. We construct inner rational functions for the case of three complex variables, in particular, using the Lee-Yang polynomial from the theory of phase transitions in statistical mechanics.
Keywords: Blaschke product, Lee-Yang polynomial.
Citation: M.E. Durakov, On the Blaschke Factors in Polydisk, J. Sib. Fed. Univ. Math. Phys., 2023, 16(2), 245-252 . EDN: MNYKKX.
Introduction
In 1915 Wilhelm Blaschke introduced a very important class of functions of one complex variable, which allowed solving important problems of interpolation theory in a unit disk.
Definition 0.1 ([1]). The one-dimensional Blaschke product is a function of the form:
z - Zk
1 - ZkZ '
bw = IIT—Z (1)
k>1
where {z1, z2,..., zn,... } is a sequence of points in the unit disk D C C.
In the case of a finite number of points {zk} from the disk, no restrictions are imposed on them, however, when moving to a countable set of points, the so-called Blaschke condition is added for the correctness of the definition:
]T(1 -\zk |) < œ. (2)
k = 1
This concept made it possible to solve important problems of interpolation theory in a single disk. For example, Blaschke's theorem states that a sequence {zk} in a disk is a zero set for a holomorphic function bounded in D if and only if the series in (2) converges.
Except for the case of bounded functions, similar descriptions have been obtained for functions from Hardy classes.
*[email protected] © Siberian Federal University. All rights reserved
Before proceeding to an analogue of the Blaschke factors in the multidimensional case, let us take a closer look at the Blaschke factors in the one-dimensional case. Note that each factor
bk = Zk _ Z of the product (1) is a fractional rational function of the form:
1 - Zk z
b = pz) = zq(1/z) k q(z) q(z) '
In the case when q has real coefficients, the functions bk can be represented as:
bk=Zq^/zi
k = z q(z) •
The idea of our generalization of the Blaschke factors is to construct such 'elementary' factors for several complex variables. We would like to note that such a construction was carried out under the influence of the results of Alpay and Yger [2].
1. Multidimensional analogue of the Blaschke factor in the space C3
By the analogue of the Blaschke factor in C3 we shall understand the triple of special inner rational functions in the unit polydisk of C3. We will construct inner rational functions using the Lee-Yang polynomials (see [3]). In order to do this, we fix an arbitrary symmetric n x n matrix (ajk) with real coefficients satisfying the condition 0 < \a,jk\ < 1 The corresponding Lee-Yang polynomial is constructed according to the given matrix as follows:
f (zi, ( zj 11 ajk) , J jeJ V k/j )
where J runs over the set of all subsets of {1, 2,... ,n}.
Let us present some important properties of this polynomial. Recall that the amoeba Af
of the polynomial f is defined as the image Log V of the zero set V = {z e (C\0)n : f (z) = 0}
under the map Log : (z1,..., zn) ^ (ln \z1\,..., ln \zn\) (see [4, 5, 6]). Taking into account the
following expression:
f(zi,z2, •• .,zn) = ziz2 • • •znf(1/zi, 1/z2,..., 1/zn)
the amoeba of the polynomial f is symmetric with respect to the origin. Moreover, the following theorem is valid.
Theorem 1.1 (M. Passare, A. Tsikh [7]). Let A be the amoeba of the Lee-Yang polynomial, then the closed positive and negative orthants ±R+ intersect the amoeba A only at the origin:
R+ n A = -R+ n A = {0} Consider the Lee-Yang polynomial in three variables associated with the matrix
a11 a b
(ajk) = I a a22 c
b c a33
where {a11, a22,a33, a, b, c} e (-1,1)\{0}. The corresponding Lee-Yang polynomial is f = (z1z2 z3 + bcz1z2 + abz2z3 + acz1z3) + (abz1 + acz2 + bcz3 + We introduce the following
Notation. Split the polynomial into two parts and denote
• f-i = Z1Z2Z3 + bcziz^ + abz2z3 + acziz^,
• f2 = abz- + acz2 + bcz3 + 1.
Next, we fix an arbitrary (z0, z0, z0) from the distinguished boundary
A = {\zj\ = 1,j = 1,2, 3}
of the polydisk D3 c C3 and consider the following sequence of functions:
Pi = fi(z0, z2,z0), p2 = f1(z1,zi0,z3), p3 = fi(z-,z2,z0), q1 = f2(z 0,z2,z3), q2 = f2(z i,z0,z3), q3 = f2(z-,z2,z0).
Definition 1.1. We call the map (—,— , — ) a three-dimensional analogue of the Blaschke
q q2 q3
factor if the zeros of the polynomial f2 = abz 1 + acz2 + bcz3 + 1 do not intersect the open unit polydisk D3 (in this case, qi will be called valid).
Recall an important definition.
Definition 1.2 (see [8]). A function g € H(Dn) is called inner if its radial boundary values g*(s) satisfy the condition \g* (w)| = 1 almost everywhere on Tn.
In fact, the Blaschke product is an inner function. Our theorem below shows that the definition we introduced corresponds to this property.
Theorem 1.2. The functions pj/qj in Definition 1.1 of the Blaschke factors are inner functions in the polydisk D3.
Proof. Since q 1 = f2(z0, z2,z3), the zeros of the denominator q1 on the unit distinguished boundary are also zeros of the polynomial f2. It can be noted that for \ab\ + \ac\ + \bc\ < 1, the polynomial f2 has no zeros in the closure of the unit polydisk D3, but then the denominators qj have no zeros in the same closure. For \ab\ + \ac\ + \bc\ = 1, the polynomial f2 has a single zero on the distinguished boundary (z1, z2,z3) and has no zeros inside the polydisk. In this case, the denominator qi has a single zero on the distinguished boundary if zi = z0; otherwise, qi does not vanish in the closure of a single polydisk. If the inequality \ab\ + \ac\ + \bc\ > 1 is satisfied, then the polynomial f2 has zeros inside the polydisk, so the corresponding denominators qj are not valid. Thus, the permissible denominators vanish at no more than one point from the distinguished boundary. Therefore, almost everywhere on
T3
we have:
p 1 = f 1 (z0, z2,z3) = z0z2z3 + bcz0z2 + abz2z3 + acz0z3
0 bc ab ac 0 1 1 1
= z 0z2 z3[— + — + — + 1 = z 0z2z3f2[—,—,— , \z3 z0 z2 ) \z0 z2 z3)
P2 = f1 (z 1, z0, z3) = z 1z0z3 + bcz 1z0 + abz0z3 + acz 1z3 =
0 bc ab ac 0 1 1 1
= z 1z2 z3[--1---+ — + 1 = z 1z2 z3f2 —, —, — h
2 z3 z z20 2 z z20 z3
P3 = f1 (z 1, z2,z0) = z 1z2z0 + bczz + abz2z0 + acz 1z0
0 bc ab ac 0 1 1 1
z1z2z°0{ z0 + z; + v2 +l)= z1z2z3f2{ ,v2,4)
From these equalities we obtain the following chains of equalities for modules of functions
—, —, —, which are valid almost everywhere on the distinguished boundary: qi q2 q3
Pi Z0Z2Z3/2(, , ) = |z0 z2z3| f2( z° , Z2 , Z3 )
qi f2 (z2,z2,z3) /2(z2,Z2,Z3)
f2( ,1,1)
/2 (^1,^2,^3)
f2(z0,z2,z3)
/2(^1,^2,^3)
/2 (z1, Z2 , Z3)
/2 (z1, Z2 , Z3)
P2 ziz2°z3/2(z,, Z0, z3) = |ziz2z3| f2(z,, z0, z3)
q2 f2 (zi, z2 , z3) f2 (zi ,z2,z3)
f2 (i,, z0, L)
f2 (zi,z2,z3)
f2(zi,z^,z3)
f2(zi,z2,z3)
f2(zi,z0,z3)
f2(zi,z0,z3)
P3 ziz2z3f2 ( z, i i) z2 , z0 ) = |z1z2z0| f2( z1! , z2 , 4 )
q3 /2(zi, z2 z30) f2 (zi ,z2,z°)
f2( z! , z2 , z° )
f2 (zi,z2,z0)
f2(zi,z2 ,z3)
f2(zi,z2 ,z3)
f2(zi,z2,z3)
f2(zi,z2,z3)
1.
That is, moduli of the radial boundary values are almost everywhere on the distinguished boundary equal to 1, so functions
Pi P2 P3
qi' q2' q3
are inner by definition. □
To describe the valid denominators of q«, we need the following
Definition 1.3 (see [9], Sec. 14, p. 125). Let C be a nonempty convex set. Then the closed convex set
C = {x|Vx* e C, (x,x*) < 1},
is called the polar of the set C.
If the set C itself is closed and contains the origin, then it coincides with the polar of its polar
set
C = C.
For more information about convex sets and other properties of the polar, see [9].
Let us find the polar of the cube K = [—1,1]3. For this we need to find all points that satisfy the equation
xx* + yy* + zz* < 1 Vx* e [—1,1], y* e [—1,1], z* e [—1,1].
Since we compare everything with one in this equation, there is no point in checking the fulfillment of this inequality for intermediate values. Because if the inequality holds for boundary values, then it holds automatically for values inside the segment. Therefore, we have the system:
xx* + yy* + zz* < 1 Vx*,y*,z* e { — 1, 1},
which can be written as a single inequality x + y + Izl < 1 in the conjugate space. We need the constructed polar to describe the valid denominators, namely
1
1
Theorem 1.3. The denominators qi are valid if and only if the pairwise products (ab, ac, bc) = (x,y,z) lie in the polar and satisfy the system of inequalities:
1 > yz >0
x
1 > xz >0
—
y
1> xy >0
z
Proof. If the denominators qj are valid, then the polynomial f2 = abzi + acz2 + bcz3 + 1 has no zeros inside the unit polydisk. And this is possible, as we have shown above, if and only if \ab\ + \ac\ + \bc\ < 1, that is, when the pairwise products of (ab,ac,bc) lie in the polar. The system of inequalities arises from the following reasoning:
a G (-1, 1)\{0} b G (-1, 1)\{0} c G (-1, 1)\{0}
0 <a2 < 1 0 <b2 < 1 0 < c2 < 1
0< 0< 0<
bc-a• bc ac-b• ac ab-c ab
<1 <1 <1
|0 < ab-ac bc < 1 |0 < xy z <
~ < 0 < ab-bc ac < 1 ~ < 0 < xz y <
1» < ac-bc ab < 1 N < yz_ x <
□
Let us visualize the resulting set. In Fig. 1, 2, we can see the points from the boundary of the polar satisfying the resulting system of inequalities. In the first figure, the edges of the polar set of the cube (of the regular octahedron) are highlighted in red, and the intersection is green.
Fig. 1. Visual representation of the intersection of solutions of the system of inequalities and the boundary of the polar
10 0.5 oo -0.5 -1.0
"10 "O S 0.0 0.5
Fig. 2. Computer representation of the intersection of solutions of the system of inequalities and the boundary of the polar
2. The second approach to constructing an analogue of the Blaschke factors
In this section, when constructing a generalization of the Blaschke factors, we start from the
zk — z
1 — zkz
form of this factor in the one-dimensional case. As we noted earlier, each factor —k- of the
product (1) is a rational function of the form:
b = Pf) = zq(1/z ) k q(z) q(z) '
where q(z) = 1 — zkz is a polynomial of the first degree that has no zeros inside the unit disk D. In this case, when switching to the multidimensional case, we can take as such a polynomial the following one
q(zi, . . . , zn ) = 1 + Cizi +... + Zn zn ,
where (Z^ ... ,(n) e {(z4, ..., zn) e Dn : |z4| + ... + ^^ < 1}. With such restrictions on the coefficients of a given linear polynomial, it will not have zeros inside the unit polydisk Dn. Note that these constraints are consistent with the one-dimensional case. As a numerator p(z4, ..., zn) we can take the following polynomial, which also agrees with the case of one complex variable:
p(zi, ...,zn) = zi ■ ... ■ zn ■ q(1/z i,..., 1/zn). Now fix an arbitrary point (z2,... ,zn) from the distinguished boundary
A = {z | = 1,j = 1,...,n} of the polydisk Dn and consider the following set of functions:
Pi = P(z0,...,zn), Pn = p(zi,...,zn),
qi = q(z0,...,zn), qn = q(zi,...,zn).
(Pi Pn ^
Definition 2.1. The map —,...,— ) is called the multidimensional analogue of the Blaschke
V qi qn J
factor.
Theorem 2.1. The functions Pj/qj in the definition of the analogue of the Blaschke factor are inner rational functions in the polydisk Dn.
Proof. Since qi = q(z°,... ,zn), the zeros of the denominator qi on the unit distinguished boundary are also zeros of the polynomial q. It can be noted that for ^ | +... + |Zn | < 1 the polynomial q has no zeros in the closure of the unit polydisk Dn, but then the denominators qj have no zeros in the same closure. For |Z4| + ... + |Zn| = 1, the polynomial q has a single zero on the distinguished boundary (z4, z2, i3) and has no zeros inside the polydisk. In this case, the denominator q« has a single zero on the distinguished boundary if z« = z°; otherwise, q« does not vanish in the closure of a single polydisk. Thus, q« vanish at no more than one point from the distinguished boundary. Therefore, almost everywhere on Tn we have
p1 = p(z0, Z2,. .., Zn) = Z0Z2 • . . . • zn + Ciz2 ■■■zn + . . . + Cnz? Z2 . . . Zn-1
= z0 • • z (+ ... + C^ + Z0 • ... • Zn • q (i
z zn z
(q?'...'zn).
From this equality we obtain the following chain of equalities for the module of the function
P1
—, which holds almost everywhere on the distinguished boundary:
qi
p 1 z i •• • zn • q( i? ,•••, in ) = \z0 • • • • zn\ 1( i? . • •' in )
qi q(z0, • • • ,zn) q(z 0 ,• • , zn)
i? ~tz)
q(z 0, •••,zn)
q(4,
q(z0,
,Zn)
,Zn)
That is, the module of the radial boundary values are almost everywhere on the distinguished
P
boundary equal to 1, so the function — is inner by definition. Having carried out similar
qi
reasoning for the remaining rational functions
P2_ q2'
Pn ' qn'
we obtain the statement of the theorem. □
The investigation was supported by the Russian Science Foundation, grant no. 20-11-20117.
1
References
[1] W.Blaschke, Eine Erweiterung des Satzes von Vitali über Folgen analytischer Funktionen, Berichte Math.-Phys. Kl., Sachs. Gesell. der Wiss. Leipzig, 67(1915), 194-200.
[2] D.Alpay, A.Yger, Cauchy-Weil formula, Schur-Agler type classes, new Hardy spaces of the polydisk and interpolation problems, Journal of Mathematical Analysis and Applications, 504(2021), no. 2, 125437. DOI: 10.1016/j.jmaa.2021.125437
[3] Tsung Dao Lee, Chen Ning Yang, Statistical theory of equations of state and phase transitions, Physical Rev., 87(1952), 404-419.
[4] I.Gel'fand, A.Zelevinskii, M.Kapranov, Hypergeometric functions and toral manifolds, Funct. Anal. Its Appl., 23(1989), 94-106. DOI: 10.1007/BF01078777
[5] M.Forsberg, M.Passare, A.Tsikh, Laurent Determinants and Arrangements of Hyperplane Amoebas. Advances in Mathematics 151(2000), 45-70. D0I:10.1006/AIMA.1999.1856
[6] G.Mikhalkin, Real algebraic curves, the moment map and amoebas, Annals of Mathematics, 151(2000), 309-326.
[7] M.Passare, A.Tsikh, Amoebas: their spines and their contours, Contemporary Mathematics, 377(2005), 275-288.
[8] W.Rudin, Function theory in polydiscs, New York: W. A. Benjamin, 1969.
[9] R.Rockafellar, Convex Analysis, Princeton Landmarks in Mathematics and Physics, 1970.
О множителях Бляшке в полидиске
Матвей Е. Дураков
Сибирский федеральный университет Красноярск, Российская Федерация
Аннотация. Цель настоящей работы состоит в построении многомерного аналога множителя Бляшке. На актуальность построения данного аналога нас натолкнула недавняя совместная статья Алпая и Ижера, которая посвящена многомерной интерполяционной теории для функциональных пространств в специальных полиэдрах Вейля. Под таким множителем мы будем понимать набор специальных внутренних рациональных функций в единичном поликруге. Построение внутренних рациональных функций для случая трех комплексных переменных произведем, в частности, с помощью многочлена Ли-Янга из теории фазовых переходов статистической механики.
Ключевые слова: произведение Бляшке, многочлен Ли-Янга.
