ON RATIONAL APPROXIMATION IN A BALL IN C

We study rational approximations of elements of a special class of meromorphic functions which are characterized by their holomorphic behavior near the origin in balls in CN by means of their rational approximants. We examine two modes of convergence for this class: almost uniform-type convergence analogous to Montessus-type convergence andweaker formof convergence using capacity based on the classical Tchebychev constant. These methods enable us to generalize and extend key results of Pommeranke and Gonchar.


Introduction.
This paper is an attempt to extend the theoretical basis of rational approximation by means of rational approximants in C N to elements of a certain class of meromorphic functions on the ball B N ρ := {z ∈ C N : N k=1 |z k | 2 < ρ}, that are holomorphic at the origin.This investigation of rational approximants in several complex variables, which began in the early seventies, offers new insights into the problem of analytic continuation from the local neighborhoods of holomorphy into the open connected regions of meromorphy.The local holomorphic expansions from which one traditionally extracted rational approximants, largely involved polynomial expansions of multiple degrees and power series expansions not in terms of homogeneous polynomials.This approach, although it gave rise to some interesting results (see [8,9,10]) lacked the flexibility of the formulation discussed in this paper.Part of the advantage gained in the latter formulation, is that one sets up initial definitions in a relatively simple, almost Padé-like fashion using slice functions.A useful consequence of this is that any investigation of the vertical and diagonal sequences of a (µ, ν)-rational approximant table, analogous to the Padé table, is easily accessible.
There are two main types of convergence behavior of interest in this paper.The first is the almost uniform-type (see [8]) associated with vertical sequences analogous to Montessus-type convergence.The second convergence behavior is the weaker of the two, and it is given in terms of convergence in capacity.The methods of investigations of the main diagonal sequences in C N , chiefly use capacity based on the classical Tchebychev constant (see [1,2,12]).Our main result associated with the Tchebychev constant (transfinite diameter) generalizes a result of Pommeranke [11] and also extends a result of Gonchar [5].
We now give a brief description of the contents of this paper.Section 2 introduces and develops most of the required preliminaries by way of definitions, lemmas, and propositions.Section 3, considers Montessus-type convergence analogous to that given in [8] for the polydisc.In Section 4, we discuss convergence in capacity which gives a considerably sharp version of a result of Gonchar (see [5]), proved using R 2N -Lebesgue measure.
From the contents of [3,4,6,7], the authors take a different tack on Padé approximants and the problems of convergence of de Montessus de Ballore type and hence prove totally different theorems than our own, represented in this paper.
2. Some preliminaries.We begin this section by developing index sets most suited for handling expansions expressed in terms of homogeneous polynomials.
Let I := {0, 1, 2,...,} and We introduce a partial order in I N as follows: for each pair α, Here it should be noted that λ, µ ∈ I but not in I N whereas α, β ∈ I N .We now introduce the notion of an index set E N µν for µ, ν ∈ I. where Definition 2.2.E N µν is called maximal if its cardinality satisfies Remark 2.3.The concept of maximality, as will be determined later, becomes central in dealing with the question of normality for rational approximants.
Let ᏻ B N ρ be the ring of holomorphic functions on B N ρ and ᏹer B N ρ the ring of meromorphic functions on B N ρ .In particular, let ᏹer 1 B N ρ be the subring of ᏹer B N ρ characterised by the following properties: (P.I) ∀f ∈ ᏹer 1 B N ρ , there exists a neighborhood of the origin in B N ρ , where f is holomorphic.
(P.II) For each f ∈ ᏹer 1 B N ρ , there is a nonhomogeneous normalized polynomial q(z) of minimal degree such that in B N ρ , the zero set of q(z) denoted by ᐆ(q) coincides with the polar set ᐆ(f −1 ) of f , that is (2.3) (P.III) For each f ∈ ᏹer 1 B N ρ and its corresponding minimal polynomial q as in (P.II), f q ∈ ᏻ B N ρ .(P.IV) For f and q as in (P.III) ᐆ(f q) ∩ ᐆ(q) ∩ B N ρ = ∅ except possibly at the points of indeterminacy of f on B N ρ .We shall now introduce the slice function on B N ρ .For any g ∈ ᏻ B N ρ and for each z ∈ ∂B N ρ , let L z denote the complex line through the origin zero and z.The slice function of g is determined from so that (t, z) tz g(tz) and we have ) In the rest of this paper, we shall apply the slice definitions to rational as well as polynomial functions.
For each µ, ν ∈ I, we let µν be the class of rational functions of the form P µ (z)/Q ν (z), where P µ (z) and Q ν (z) are nonhomogeneous polynomials expended in terms of homogeneous polynomials up to degrees µ and ν, respectively.Furthermore, Q ν (0) ≠ 0; P µ (z) and Q ν (z) are relatively prime in C N except at the points of indeterminacy of ) where (2.9) Proof.The result follows from comparing the coefficients of t k in the equal but separate Taylor expansions of f z (t) in 1 and f (tz) in U.

6) holds if and only if
(2.10) (ii) Equation (2.7) holds if and only if (2.11) Proof.The results follow from Proposition 2.6, using the linear independence of monomial vectors , in the right-hand side of the equation (2.8) that generate the homogeneous subspace {z α : |α| = k} ⊂ C[z 1 ,...,z N ], the latter being the algebra of polynomials in C N .Here the two cases are covered as follows: When E N µν is maximal, the system of equations produced by (2.11) with the normalization Q ν (0) = 1, gives rise to a linear system of maximal rank N+ν N − 1.The latter equals the number of unknown coefficients of Q ν (z).The solution of the linear system of equations with maximal rank leads to the uniqueness of the resulting (µ, ν)-rational approximant with respect to E N µν maximal.We call such (µ, ν)-rational approximants each with a normalized denominator polynomial, (µ, ν)-unisolvent rational approximants (in short URA) and denote it by π µν .
The array or table of {π µν } µν of uniquely determined entries is called normal.The fact that there are many E N µν that are maximal, suggests the following conjecture: "There are as many maximal E N µν index interpolation sets as there are normal tables (all analogs of a normal Padé-table) associated with a given function holomorphic at zero." This situation is very different from the one variable case where there is one and only one maximal Padé index set giving rise to a single normal Padé table .Lemma 2.8.Let f ∈ ᏹer 1 B N ρ and let q ω (z) be its corresponding nonhomogeneous polynomial of minimal fixed degree ω ∈ I. Suppose π µνz (t (2.12) (2.14) The result then follows from (2.13) and (2.14).

Montessus-type convergence.
One of the key results in this section is Lemma 3.4 which establishes an inequality that is central to all the proofs of convergence.
As in the hypothesis of Lemma 2.8, we shall assume f ∈ ᏹer 1 B N ρ and let q ω (z) be its corresponding nonhomogeneous normalized polynomial of minimal degree ω, for which the characterizing properties (P.I)-(P.IV) given in Section 2 hold.
To prove Theorem 3.
, where τ ∈ 1 , Cauchy's integral formula yields where z ∈ B N ρ .But 1/(1 − t) = ∞ k=0 1/t k is absolutely and uniformly convergent in ∂ 1+ , so (3.3) gives This series is compactly convergent in B N ρ .Using the expression (3.1) for H µνω (z), it is immediate that where we have used Lemma 2.8 to yield The proof then follows from Claim 3.3.
Proof.This is immediate from direct computation, using Leibniz rule as follows: This completes the proof of the claim.
The proof of Lemma 3.2 is thus an immediate consequence of this claim together with (3.5b).
Then with > 0 as in the proof of Lemma 3.2, where K := sup K | | and

.10)
Proof.Following the set up of the preceding lemma, with z ∈ B N ρ , and using Cauchy's estimate we obtain (3.11)where M is as stated in the lemma.Hence for z ∈ K the desired inequality follows.
Proof of Theorem 3.1.This is obtained immediately from the inequality (3.9) on letting µ → ∞ while remains positive.Theorem 3.5 (Montessus-type).Let ν ∈ I be fixed.Suppose f ∈ ᏹer 1 B N ρ , i.e., f is characterized by the four properties (P.I)-(P.IV).Suppose π µν (z) µ is a "column" sequence of a (µ, ν)-URA table to f at the origin z = 0, with its polar set on B N ρ determined by ᐆ(π −1 µν )∩B N ρ which is closed in B N ρ .Then as µ → ∞, (modulo the sets of indeterminacy of f and π µν ) Proof.Recall that π µν (z) = P µν (z)\Q µν (z).Without loss of generality, we shall assume that both Q µν and q ν have been normalized in the same manner.
To prove (i), let K be any compact subset of B N ρ such that there is a ρ , satisfying ρ .Now since Q µν 's are normalized, {Q µν } µ is a uniformly bounded sequence in K and therefore, it contains a subsequence {Q µjν } j that converges uniformly to, say, S ν , in K. From Theorem 3.1, we know that H µν (z) = Q µν (z)f (z)q ν (z)−P µν (z)q ν (z) → 0 uniformly on K, and so the uniform convergence of the subsequence {Q µjν (z)} j induces the uniform convergence of a similar subsequence {P µjν (z)} j of {P µν (z)} µ to a limit, say, T (z) on K. Thus from the limit of H µν (z) → 0, we get S ν (z)f (z)q ν (z) = T (z)q ν (z).
Proof.Take any point b ∈ ᐆ(q ν ) ∩ B N ρ with b not in the set of points of indeterminacy of f .Then q ν (b) = 0 makes T (b)q ν (b) = 0 and from (3.12) we deduce that S ν (b)f (b)q ν (b) = 0.But from property (P.IV) of f ∈ ᏹer 1 B N ρ , we have ρ , with b not in the set of points of indeterminacy of f , then S ν (b) = 0, and hence S ν (b)f (b)q ν (b) = 0. Once again from (3.12), we deduce that T (b)q ν (b) = 0. From the relative primeness condition of P µν and Q µν for all µ,ν ∈ I we get ᐆ(Q µjν )∩ᐆ(P µjν )∩B N ρ = ∅, except at points of indeterminacy of π µjν for all j with ν ∈ I fixed.Thus we must have ᐆ(S ν ) ∩ ᐆ(T ) ∩ B N ρ = ∅, except at points of indeterminacy of f .This implies that T (b) ≠ 0. Therefore, q ν (b) = 0, i.e., b ∈ ᐆ(q ν ) ∩ B N ρ modulo the points of indeterminacy of f and so ᐆ(S ν ) ∩ B N ρ ⊂ ᐆ(q ν ) ∩ B N ρ .This completes the proof of the claim.
Proof of Theorem 3.5 continued.Now every subsequence {Q µjν (z)} j is constrained by (3.12) to converge to S ν .Hence the sequence {Q µν (z)} µ converges uniformly in B N ρ to S ν .This implies that By the claim, the desired result follows for (i).

Convergence in capacity.
In this final section of the paper, we consider diagonal sequences {π µµ } µ of (µ, µ)-sequences of URA's to f at zero for f ∈ ᏹer 1 B N ρ .Here convergence is considered in terms of capacity and not in terms of uniform convergence.This is done because there is a significant growth in the polar set of π µµ for sufficiently large values of µ, which tend to thwart uniform convergence on compact subsets of B N ρ , except in a small neighborhood of the origin zero.For any integer d ≥ 1, let ᏼ d (C N ) be the class of polynomials P d (z) = |α|≤d a α z α , with normalization max |α|≤d {|a α |} = 1, so that on the unit ball B N ρ , Pd BN ρ ≥ 1, where Pd is a homogeneous polynomial of degree d.Let K ⊂ B N ρ with ρ > 1, be compact.Then we can find a ρ such that 1 < ρ < ρ and K ⊂ B N ρ B N ρ .The Tchebychev constant for a compact set K may be defined from (see [1,2]) as a capacity of K by Proof.From [12] we know that for each d ≥ 1, there is a Tchebychev polynomial With respect to σ > 0, σ ∈ I, following [12], we can find numbers τ and r with 0 ≤ r < τ, σ satisfying σ = kτ +r , so that g σ (z) = z r (p * τ (z)) k .Thus Now let L = sup (z 1 ,...,z N )∈K |z r | > 0, and take c 1 = max(1/L, 1).Then Hence from the definition of T (K), the desired result follows.