Effectiveness of Cannon and Composite sets of Polynomials of two complex variables in Faber regions

The paper obtains conditions for effectiveness of Cannon and Composite sets of polynomials of two complex variables in Faber regions. It generalizes to these regions the results of Nassif on composite sets in balls of centre origin whose constituents are also Cannon sets. Mathematics Subject Classification: 30A10, 30C10, 30C20, 30D10, 32A05.


Introduction
Let C k ; k=1,2, be a Faber curve in the z k -plane and suppose that the corresponding Faber transformation is where the function φ k (t k )is regular and one-to-one for |t k | > T k . Thus the curve C k is the map in the z k -plane of the crcle |t k | > γ k by the transformation (1.1) where γ k > T k . For r > T k the map of the circle |t k | = r is the curve C k r so that C k is actually C k γ k . Following Breadze [2] the product set B r 1 ,r 2 = D(C r 1 xD(C r 2 ), r k > T k (1.2) and its closureB r 1 ,r 2 are called Faber regions in the space C 2 of the two complex variables z 1 and z 2 . For studies on Faber transformation and regions (see Newns [4],Ullman [5]).An open polycylinder in C 2 is the open connected set Γ r 1 ,r 2 = {(z 1 , z 2 ) : |z k | < r k }; r k > 0, (1.3) and its closure is denotedΓ r 1 ,r 2 . By reasoning as in [1], [2] and [4], we take as a Banach space the class of functions regular in Faber regions B r 1 ,r 2 orB r 1 ,r 2 , and let a norm defined on this space be given by for all functions F regular in B r 1 ,r 2 , ρ k < r k and M is the maximum modulus in Faber regions. Since F (z 1 , z 2 ) is regular inB R 1 ,R 2 then it admits the Faber series are the sets of Faber polynomials corresponding to the respective transformations z 1 = φ 1 (t 1 ) and z 2 = φ 2 (t 2 ). In view of the representation (1.5), the base for this space is the set {f 1 m (z 1 ).f 2 n (z 2 )}. Thus, if {P j (z 1 , z 2 )} is a sequence of basic set of polynomials of the two complex variables z 1 and z 2 , we can write the unique representation is a function regular in B r 1 ,r 2 then the basic series associated with F is given by The Cannon sum of the bassic set {P j (z 1 , z 2 )} for the Faber region B ρ 1 ,ρ 2 ,ρ k <r k is given by We can easily deduce from (1.8), (1.1) and the formula of Newns [4, p.749] that where l k is the length of the curve C k T k ; T k > T k and ∆ k (> 0) is the distance between the curves C k r k and C k T k . If the Cannon function of the same set for the same Faber region is defined as (1.14) We observe that if the series associated with the function (1.5) which leads to the basic series (1.8) is justifiable and hence the basic series associated with function (1.5) will represnt it inB ρ 1 ,ρ 2 .
2. The required condition for effectiveness of the basic set {P j (z 1 , z 2 )} in Faber regions is the following.

be a basic set for which the Cannon function is
We observe from (2.2) that r 1 r 2 = R 1 R 2 and hence Theorem 2.1 is the analogue for Faber regions of the sufficiency assertion of [1, Theorem 3.2], for polycylinders and may be derived in a manner, replacing the monomials } and, in view of (1.4) replacing the polycylinders Γ r 1 ,r 2 by the Faber regions B r 1 ,r 2 .
We, however, give another proof: Proof of Theorem 2.1. Let F (z 1 , z 2 ) be any function regular in B r 1 ,r 2 where it admits the expansion Then there are numbers ρ k > r k such that the function F (z 1 , z 2 ) is regular in B ρ 1 ,rho 2 . Hence, we can deduce that Therefore, applying (2.1), (2.2) and (2.3), we easily derive the inequality Hence the series Letting r k tend to R k in Theorems 2.1 and 2.2 and appealing to the relation (2.1) we obtain the familiar Cannon's theorem for Faber regions in the form.
3. We now consider effectiveness of composite sets of polynomials of two complex variables z 1 and z 2 in Faber regions.
Let {P 1 µ (z 1 )} and {P 2 v (z 2 )} be basic sets of polynomials. If for any mode of arrangement we write then {P j (z 1 , z 2 )} is called the composite set of polynomials whose constituents are the sets {P 1 µ (z 1 )} and {P 2 v (z 2 )}, (c.f. [3]). Following [3], suppose that the Faber polynomials {f 1 m (z 1 )} and {f 2 n (z 2 )} admit the unique representations then the product f 1 m (z 1 )f 2 n (z 2 ) will admit the unique representation where, besides ( It has been shown in [3,Theorem 5] that the composite set {P j (z 1 , z 2 )} whose constituents are Cannon sets {P 1 µ (z 1 ), P 2 v (z 2 )} will be effective in the ball S R = {(z 1 , z 2 ) : |z 1 | 2 +|z 2 | 2 ≤ R 2 }, R > 0 if and only if, each of the constituent sets is effective in the disk |z k | ≤ r for 0 < r ≤ R. To generalize this result to Faber regions, we write for the Cannon sum of the Cannon set {P 1 µ (z 1 )} in D(C r 1 ), (3.5) and in view of (3.2) the Cannon function is given by Similar notation is adopted for the Cannon set {P 2 µ (z 2 )}. With this notation our result concerning composite set is the following. Theorem 3.1. Let {P 1 µ (z 1 )} and {P 2 v (z 2 )} be Cannon sets of polynomials and suppose that {P j (z 1 , z 2 )} is their composite set. Then the Cannon function of {P j (z 1 , z 2 )} for their Faber regionB R 1 ,R 2 ; R k > T k is given by Proof. Write while (3.1) with our notation yields Introducing (3.9) into (3.8) it follows from (3.5) that Hence in view of (1.11) we obtain Let σ 1 be any finite number greater than ρ 1 and choose the number σ 2 > ρ 2 such that Then (3.12) and (3.13) together yield (3.14) From the definition (3.16) of λ ( k)(C k R k ) it follows that ω k n (C k R k ) < Kσ m k ; (m ≥ 0), K a constant, and hence (3.11) and (3.14) imply that Ω m,n (R 1 , R 2 ) < K(R 2 σ 1 ) m+n ; m, n ≥ 0.
Hence, in the limit as m + n → ∞ we have and since σ 1 can be taken arbitrarily close to λ 1 (C k R 1 ) we deduce that we can arrive at the inequality and we conclude that On the other hand, we have from the inequalities, and A combination of (3.15) and (3.16) gives the desired equality (3.7) and the theorem is established. We observe that if the set {P 1 µ (z 1 )} is effective inD(C R 1 ) and the set {P 2 v (z 2 )} is effective inD(C R 2 ) , then λ k (C k ) R k = R k and hence (3.7) implies that Moreover, if the composite set {P j (z 1 , z 2 )} is effective inB R 1 ,R 2 then the same equation (3.7) yields In a similar manner we can deduce that λ 1 (C R 2 ) = R 1 . We have, therefore proved the following result which is a generalization of the result of Nassif [3, Theorem 5] for balls: We note that if the sets {P k m (z k )} are not effective inD(C k R k ) so that λ k (C k R k ) > R k , then we would have Ω(R 1 , R 2 ) < λ 1 (C R 1 )λ 2 (C k R 2 ). We claim that the method used in establishing Theorem 3.1 for Faber regions is also valid for polycylinders Γ r 1 ,r 2 with obvious modifications (c.f. [1]). Indeed, the Cannon function λ k (R k ) of the constituent Cannon set {P k m (z k )} for the disk |z k | ≤ R k is given by where ω n (R k ) = j=0 |Π n,j |M (P k j ; R k ), R k > 0 and M denotes maximum modulus in polycylinders. The Cannon function of the composite set {P j (z 1 , z 2 )} for the closed poly-cylinderΓ R 1 ,R 2 is given by |Π m,n j |M (P j ; R 1 , R 2 ).