PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 31–39 DOI: 10.2298/PIM1410031B

CONVERGENCE IN CAPACITY OF RATIONAL APPROXIMANTS OF MEROMORPHIC FUNCTIONS

Hans-Peter Blatt

Abstract. Let f be meromorphic on the compact set E ⊂ ℂ with maximal Green domain of meromorphy E_{ρ(f)} , ρ(f) < ∞. We investigate rational approximants with numerator degree ⩽ n and denominator degree ⩽ m_{n} for f. We show that the geometric convergence rate on E implies convergence in capacity outside E if m_{n} = o(n) as n → ∞. Further, we show that the condition is sharp and that the convergence in capacity is uniform for a subsequence Λ ⊂ ℕ.

1. Introduction

Let E be compact in ℂ with connected complement Ω = \overline{ℂ} ∖ E. The set Ω is called regular if there exists a Green function G(z) = G(z, ∞) on Ω with pole at ∞ satisfying G(z) → 0 as z → ∂Ω. Note that lim_{z→∞} (G(z) − log|z|) = − logcapE. Here, capE is the logarithmic capacity and capE > 0 if Ω is regular (cf. Tsuji [5]). Moreover, we define the Green domains E_{ρ} by

E_{ρ} := {z ∈ Ω : G(z) < logρ} ∪ E, ρ > 1

and E_{1} := E^{∘}, where E^{∘} is the set of interior points of E.

For B ⊂ ℂ, we denote by C(B) the class of continuous functions on B, and M(B) represents the class of functions f that are meromorphic in some open neighborhood of B.

If f ∈ M(E), then there exists a maximal ρ(f) > 1 such that f ∈ M(E_{ρ_{(f)}} ). ρ(f) = ∞ if and only if f is meromorphic on ℂ.

Given n, m ∈ ℕ_{0}, ℕ_{0} := ℕ ∪ {0}, let R_{n,m} be the collection of all rational functions,

R_{n,m} := {r = p/q : p ∈ P_{n}, q ∈ P_{m}, q ≢ 0},

where P_{n} (resp. P_{m}) denotes the collection of all algebraic polynomials with degree at most n (resp. m).

2010 Mathematics Subject Classification: 41A20, 41A25, 30E10.

Key words and phrases: rational approximation, convergence in capacity.

Dedicated to Giuseppe Mastroianni.

Let r_{n,m}^{∗} := r_{n,m}^{∗} (f) ∈ R_{n,m} denote a rational function of best uniform approximation to f on E, i.e.,

e_{n,m} (f) := inf_{r∈R_{n,m}} ∥f − r∥_{E} = ∥f − r_{n,m}^{∗} ∥_{E},

where we use ∥ · ∥_{B} for the supremum norm on B ⊂ ℂ.

By Walsh’s theorem (cf. Walsh [6]), we know that

lim sup_{n→∞}∥f − r_{n,m_{n}}^{∗} ∥_{E}^{1/n} ⩽ \frac{1}{ρ(f)} ,

if lim_{n→∞} m_{n} = ∞. Now, the starting point in [1] was a sequence of rational approximants {r_{n,m_{n}} }_{n∈ℕ} such that

(1.1)  lim sup_{n→∞}∥f − r_{n,m_{n}} ∥_{∂E}^{1/n} ⩽ \frac{1}{τ} < 1.

In [1] the problem was considered whether the convergence (1.1) can be transferred to domains E_{σ}, σ > 1. Concerning the convergence of {r_{n,m_{n}} }_{n∈ℕ} , the m_{1}-measure was used: Let e be subset of ℂ, and set m_{1}(e) := inf{∑ |U_{ν}|}, where the infimum is taken over all coverings {U_{ν}} of e by disks U_{ν}, and |U_{ν}| is the radius of the disk U_{ν}.

Let D be domain in ℂ and φ a function defined in D with values in \overline{ℂ}. A sequence of functions {φ_{n}}, meromorphic in D, is said to converge to a function φ m_{1}-almost uniformly inside D if for any compact set K ⊂ D and any ε > 0 there exists a set K_{ε} such m_{1}(K_{ε}) < ε and {φ_{n}} converges uniformly to φ on K ∖ K_{ε}.

Theorem 1.1. [1, Theorem 1] Let E be compact in ℂ with regular, connected complement Ω = \overline{ℂ} ∖ E, \{m_{n}\}_{n=1}^{∞} a sequence in ℕ_{0} with m_{n} = o(n/ logn) as n → ∞, {r_{n,m_{n}} }_{n∈ℕ} a sequence of rational functions, r_{n,m_{n}} ∈ R_{n,m_{n}} , such that for f ∈ M(E)

lim sup_{n→∞}∥f − r_{n,m_{n}} ∥_{∂E}^{1/n} ⩽ \frac{1}{τ} < 1.

Then there exists an extension f̃ of f to E_{τ} with the following property:

For any ε > 0 there exists a subset Ω(ε) ⊂ ℂ with m_{1}(Ω(E)) < ε such that f̃ is a continuous function on E_{τ} ∖ Ω(ε) with

lim sup_{n→∞}∥f̃− r_{n,m_{n}} ∥_{\overline{E}_{σ} ∖Ω(ε)}^{1/n} ⩽ \frac{σ}{τ}

for any σ with 1 < σ < τ and {r_{n,m_{n}} }_{n∈ℕ} converges m_{1}-almost uniformly to f̃ inside E_{τ}.

In [1] it was noted that it is not known for f ∈ M(E_{ρ}), ρ > 1, whether the continuous extension f̃ of Theorem 1.1 is m_{1}-equivalent to f on E_{τ} ∩ E_{ρ} if lim_{n→∞} m_{n} = ∞, m_{n} = o(n/ logn) as n → ∞.

The main result of this paper is to show that this is true even if m_{n} = o(n) as n → ∞. Moreover, we can show not only convergence in m_{1}-measure, but more stronger, convergence in capacity and even uniform convergence in capacity at least for a subsequence of {r_{n,m_{n}} }_{n∈ℕ} .

2. Convergence in capacity

Let D be a domain in ℂ, φ a function in ℂ with values in \overline{ℂ}. A sequence φ_{n} : D → \overline{ℂ}, n ∈ ℕ, converges in capacity inside D if for any compact set K ⊂ D and any ε > 0 one has cap({z ∈ K : |(φ − φ_{n})(z)| ⩾ ε} → 0 as n → ∞. Moreover, {φ_{n}}_{n∈ℕ} converges uniformly in capacity inside D to φ if for any compact set K ⊂ D and any ε > 0 there exists a set K_{ε} ⊂ K such that capK_{ε} < ε and {φ_{n}}_{n∈ℕ} converges uniformly to φ on K ∖ K_{ε} (cf. Gonchar [2]).

Our main theorems for convergence in capacity can be formulated as follows.

Theorem 2.1. Let E be compact in ℂ with regular connected complement, {m_{n}}_{n∈ℕ} a sequence in ℕ with

(2.1)  m_{n} = o(n) as n → ∞ and lim_{n→∞} m_{n} = ∞.

Let f ∈ M(E) and let {r_{n,m_{n}} }_{n∈ℕ} be a sequence of rational functions, r_{n,m_{n}} ∈ R_{n,m_{n}} , such that

lim sup_{n→∞}∥f − r_{n,m_{n}} ∥_{∂E}^{1/n} ⩽ \frac{1}{τ} < 1.

Then the sequence {r_{n,m_{n}} }_{n∈ℕ} converges in capacity to f inside E_{min (τ,ρ(f))} .

Theorem 2.1 can be proved by using the methods of the proof of the following Theorem 2.2. For the statement of Theorem 2.2 we choose a parameter d > 1 such that diameter(E_{ρ(f)} ) < d, if ρ(f) < ∞. The parameter d results from the subadditivity theorem for the capacity, due to Nevanlinna [3, p. 217] (cf. Pommerenke [4]).

Theorem 2.2. Let f ∈ M(E) with ρ(f) < ∞, {m_{n}}_{n∈ℕ} with (2.1), {r_{n,m_{n}} }_{n∈ℕ}

a sequence of rational functions, r_{n,m_{n}} ∈ R_{n,m_{n}} , such that

(2.2)  lim sup_{n→∞}∥f − r_{n,m_{n}} ∥_{∂E}^{1/n} ⩽ \frac{1}{ρ(f)} .

Let σ, 1 < σ < ρ(f), and 1 < θ < ρ(f)/σ. Then there exists n_{0} = n_{0}(σ, θ) and compact sets Ω_{n}(σ, θ) ⊂ \overline{E}_{σ} such that for all n ⩾ n_{0}(σ, θ)

capΩ_{n}(σ, θ) ⩽ d^{1/2} ( 1 − \frac{θ − 1}{1 + 3θ} )^{n/2m_{n}} , (2.3)

∥f − r_{n,m_{n}} ∥_{\overline{E}_{σ} ∖Ω_{n} (σ,θ)} ⩽ ( \frac{θσ}{ρ(f)} )^{n}. (2.4)

Concerning uniform convergence inside E_{ρ(f)} the following theorem holds.

Theorem 2.3. Let f, {m_{n}}_{n∈ℕ} and {r_{n,m_{n}} }_{n∈ℕ} be as in Theorem 2.2 and let (2.2) hold. Then there exists a subset {n_{k}}_{k∈ℕ} of ℕ such that the subsequence {r_{n_{k} ,m_{n_{k}}} }_{k∈ℕ} converges uniformly in capacity to f inside E_{ρ(f)} .

Such a type of geometric convergence in capacity was proved by Gonchar [2] for the Padé approximation. In [1] geometric uniform convergence in m_{1}-measure of real rational approximants to real functions was proved only for Chebyshev approximation on an interval. So far Theorem 2.3 seems to be the first result for uniform convergence in capacity.

3. Proofs

As already mentioned, we may restrict ourselves to the proof of Theorem 2.2.

Proof of Theorem 2.2. For abbreviation, we write ρ = ρ(f). Let ε := (θ − 1)/4; then we get

ε = \frac{θ − 1}{4} < \frac{ρ/σ − 1}{4} = \frac{1}{4} \frac{ρ − σ}{σ} < ρ − σ.

We choose τ such that ρ − ε < τ < ρ, and we denote by h^{τ} the monic polynomial whose zeros are the poles of f in \overline{E}_{τ}, counted with their multiplicities. Then

(fh^{τ})(z) = f(z)h^{τ}(z)

is holomorphic in \overline{E}_{τ}. Let us denote by p_{n}^{τ} ∈ P_{n} the best uniform approximation of fh^{τ} on E. Then there exists n_{1} = n_{1}(σ, ε) such that for n ⩾ n_{1}(σ, ε)

(3.1)  ∥fh^{τ} − r_{n,m_{n}} h^{τ}∥_{∂E} ⩽ \frac{1}{2}( \frac{1}{ρ − ε} )^{n},
(3.2)  ∥fh^{τ} − p_{n}^{τ}∥_{E} ⩽ \frac{1}{2}( \frac{1}{ρ − ε} )^{n},
(3.3)  ∥fh^{τ} − p_{n}^{τ}∥_{\overline{E}_{σ}} ⩽ \frac{1}{2}( \frac{σ}{ρ − ε} )^{n},
(3.4)  degree(h^{τ}) ⩽ m_{n}.

For (3.1) we have used (2.2), the theorem of Bernstein–Walsh for (3.2) and (3.3), (3.4) follows from (2.1).

Combining (3.1) and (3.2),

(3.5)  ∥r_{n,m_{n}} h^{τ} − p_{n}^{τ}∥_{∂E} ⩽ ( \frac{1}{ρ − ε} )^{n}, n ⩾ n_{1}(σ, ε).

Let r_{n,m_{n}} (z) = p_{n}(z)/q_{m_{n}} (z), normalized by

q_{m_{n}} (z) := q_{m_{n}}^{∗} (z) ∏_{ξ_{n,i} ∉E_{ρ}} ( 1 − \frac{z}{ξ_{n,i}} ) and q_{m_{n}}^{∗} (z) := ∏_{ξ_{n,i} ∈E_{ρ}} (z − ξ_{n,i} )

where ξ_{n,i} denote the poles of r_{n,m_{n}} . Then for any compact set K ⊂ ℂ

lim sup_{n→∞}∥q_{m_{n}} ∥_{K}^{1/n} ⩽ 1.

Because of (3.5) and the normalization of q_{m,n} , there exists a constant c > 0 such that for z ∈ E

|p_{n}(z)h^{τ}(z) − p_{n}^{τ}(z)q_{m_{n}} (z)| ⩽ c^{m_{n}} ( \frac{1}{ρ − ε} )^{n}.

We apply the lemma of Bernstein–Walsh to the polynomial

w(z) = p_{n}(z)h^{τ}(z) − p_{n}^{τ}(z)q_{m_{n}} (z) ∈ P_{n+m_{n}}

and obtain |w(z)| ⩽ (cσ)^{m_{n}} ( \frac{σ}{ρ−ε} )^{n} for z ∈ \overline{E}_{σ}. Consequently, for z ∈ \overline{E}_{σ}, where q_{m_{n}} (z) ≠ 0, we get

|r_{n,m_{n}} (z)h^{τ}(z) − p_{n}^{τ}(z)| = | \frac{w(z)}{q_{m_{n}} (z)}| ⩽ (cσ)^{m_{n}} ( \frac{σ}{ρ − ε} )^{n} \frac{1}{|q_{m_{n}} (z)|}.

CONVERGENCE IN CAPACITY OF RATIONAL APPROXIMANTS 35

Hence, there exists n_{2} = n_{2}(σ, ε), n_{2} ⩾ n_{1}, such that

|r_{n,m_{n}} (z)h^{τ}(z) − p_{n}^{τ}(z)| ⩽ \frac{1}{2}( \frac{(1 + ε)σ}{ρ − ε} )^{n} \frac{1}{|q_{m_{n}}^{∗} (z)|}

for all z ∈ \overline{E}_{σ} with q_{m_{n}}^{∗} (z) ≠ 0 and all n ⩾ n_{2}. Let us consider the set

S_{n}(σ, ε) := \{z ∈ \overline{E}_{σ} : |r_{n,m_{n}} (z)h^{τ}(z) − p_{n}^{τ}(z)| ⩾ \frac{1}{2}( \frac{(1 + 2ε)σ}{ρ − ε} )^{n}\};

then

S_{n}(σ, ε) ⊂ e_{n} = e_{n}(σ, ε) := \{z ∈ \overline{E}_{σ} : |q_{m_{n}}^{∗} (z)| ⩽ ( \frac{1 + ε}{1 + 2ε} )^{n}\}.

Since q_{m_{n}}^{∗} is monic and degree (q_{m_{n}}^{∗} ) ⩽ m_{n}, we obtain

(3.6)  cape_{n} ⩽ ( \frac{1 + ε}{1 + 2ε} ) ^{\frac{n}{degree (q{∗ \atop mn} )}} ⩽ ( \frac{1 + ε}{1 + 2ε} ) ^{\frac{n}{mn}} .

Therefore, we have shown that for z ∈ \overline{E}_{σ} ∖ e_{n} and n ⩾ n_{2} = n_{2}(σ, ε)

(3.7)  |r_{n,m_{n}} (z)h^{τ}(z) − p_{n}^{τ}(z)| ⩽ \frac{1}{2}( \frac{(1 + 2ε)σ}{ρ − ε} )^{n}.

By (3.3) and (3.7), we have for z ∈ \overline{E}_{σ} ∖ e_{n} and n ⩾ n_{2}

|f(z)h^{τ}(z) − r_{n,m_{n}} (z)h^{τ}(z)| ⩽ ( \frac{(1 + 2ε)σ}{ρ − ε} )^{n}

or

|f(z) − r_{n,m_{n}} (z)| ⩽ ( \frac{(1 + 2ε)σ}{ρ − ε} )^{n} \frac{1}{|h^{τ}(z)|},

when h^{τ}(z) ≠ 0. Let us consider

S̃_{n} (σ, ε) := \{z ∈ \overline{E}_{σ} : |f(z) − r_{n,m_{n}} (z)| ⩾ ( \frac{(1 + 3ε)σ}{ρ − ε} )^{n}\};

then

S̃_{n} (σ, ε) ⊂ ẽ_{n} = ẽ_{n}(σ, ε) := \{z ∈ \overline{E}_{σ} : |h^{τ}(z)| ⩽ ( \frac{1 + 2ε}{1 + 3ε} )^{n}\}

and by (3.4)

(3.8)  cap ẽ_{n} ⩽ ( \frac{1 + 2ε}{1 + 3ε} )^{n/m_{n}} .

Summarizing, we have obtained for z ∈ \overline{E}_{σ} ∖ (e_{n} ∪ ẽ_{n}) and n ⩾ n_{2}

(3.9)  |f(z) − r_{n,m_{n}} (z)| ⩽ ( \frac{(1 + 3ε)σ}{ρ − ε} )^{n}.

Because of the subadditivity of the capacity (Nevanlinna [3], Pommerenke [4]) 1/ log \frac{d}{cap(e_{n} ∪ ẽ_{n})} ⩽ 1/ log \frac{d}{cape_{n}} + 1/ log \frac{d}{cap ẽ_{n}} , where d is greater than the diameter of e_{n} ∪ ẽ_{n} and d > 1. The parameter d of Theorem 2.2 fulfills these conditions. Using (3.7) and (3.9), we get

1/ log \frac{d}{cap(e_{n} ∪ ẽ_{n})} ⩽ 1/ log[d( \frac{1 + 2ε}{1 + ε} )^{n/m_{n}} ] + 1/ log[d( \frac{1 + 3ε}{1 + 2ε} )^{n/m_{n}} ]

⩽ 2/ log[d( \frac{1 + 3ε}{1 + 2ε} )^{n/m_{n}} ]

or

log \frac{d}{cap(e_{n} ∪ ẽ_{n})} ⩾ \frac{1}{2} log[d( \frac{1 + 3ε}{1 + 2ε} )^{n/m_{n}} ]

and finally

(3.10)  cap(e_{n} ∪ ẽ_{n}) ⩽ d^{1/2} ( \frac{1 + 2ε}{1 + 3ε} )^{n/2m_{n}} .

Since ε = (θ − 1)/4, we obtain

\frac{1 + 2ε}{1 + 3ε} = \frac{2 + 2θ}{1 + 3θ} = 1 − \frac{θ − 1}{1 + 3θ} < 1

and some calculations show that

\frac{1 + 3ε}{ρ − ε} < \frac{θ}{ρ}.

Inserting these inequalities into (3.9) and (3.10), and define the compact sets Ω_{n}(σ, θ) := e_{n}(σ, θ) ∪ ẽ_{n}(σ, θ). Then we have proved the inequalities (2.3) and (2.4) of Theorem 2.2. □

We remark that Theorem 2.1 follows directly from Theorem 2.2 if we choose θ so small that θ < ρ(f)/σ and keeping in mind that

lim_{n→∞} cap(e_{n}(σ, ε) ∪ ẽ_{n}(σ, ε)) = 0

with ε = (θ − 1)/4. Moreover, for τ < ρ(f) the same method of proof leads to the result of Theorem 2.1 under the condition

(3.11)  lim sup_{n→∞}∥f − r_{n,m_{n}} ∥_{∂E}^{1/n} ⩽ \frac{1}{τ} < 1.

Then the technique of the proof leads immediately to the following Corollary of Theorem 2.2.

Corollary 3.1. Let f ∈ M(E) with ρ(f) < ∞, {m_{n}}_{n∈ℕ} a sequence with

(2.1), {r_{n}, m_{n}}_{n∈ℕ} , r_{n,m_{n}} ∈ R_{n,m_{n}} , a sequence such that (3.11) holds. Then there exists for any σ, 1 < σ < min(τ, ρ(f)) and arbitrary θ, 1 < θ < min(τ, ρ(f))/σ, a natural number n_{0} = n_{0}(σ, θ) and set Ω_{n}(σ, θ) ⊂ \overline{E}_{σ} such that (2.3) and (2.4) hold for n ⩾ n_{0}(σ, θ).

Proof of Theorem 2.3. We consider a monotonically increasing sequence {σ_{i}} such that lim_{i→∞} σ_{i} = ρ(f) and a monotonically decreasing sequence {θ_{i}}, 1 < θ_{i} < ρ(f)/σ_{i}, such that lim_{i→∞} θ_{i} = 1. Let Ω_{n}(σ, θ) and n_{0}(σ, θ) be defined as in Theorem 2.2, i.e., for n ⩾ n_{0}(σ, θ)

|f(z) − r_{n,m_{n}} (z)| ⩽ ( \frac{θσ}{ρ(f)} )^{n} for z ∈ \overline{E}_{σ} ∖ Ω_{n}(σ, θ)

and capΩ_{n}(σ, θ) ⩽ d^{1/2} γ^{n/2m_{n}} , where γ = 1 − \frac{θ−1}{1+3θ} . Replacing (σ, θ) by (σ_{1}, θ_{1}),

we can find, by using m_{n} = o(n) as n → ∞, a subsequence Λ_{1} = {n_{j}^{(1)}}_{j=1}^{∞} of ℕ

such that n_{j}^{(1)} ⩾ n_{0}(σ_{1}, θ_{1}) and

\frac{m_{n_{j}}^{(1)}}{n_{j}^{(1)}} / log \frac{1}{γ_{1}} ⩽ \frac{2}{(j + 1)^{2}}

for j = 1, 2, …, where

γ_{1} = 1 − \frac{θ_{1} − 1}{1 + 3θ_{1}} .

Recursively, we can define subsequences Λ_{k} = {n_{j}^{(k)}}_{j=1}^{∞} ⊂ Λ_{k−1} (k = 2, 3, …)

such that n_{j}^{(k)} ⩾ n_{0}(σ_{k}, θ_{k}) and

\frac{m_{n_{j}}^{(k)}}{n_{j}^{(k)}} / log \frac{1}{γ_{k}} ⩽ \frac{2}{(k + j)^{2}}

for j = 1, 2, . . .. We define Λ := {n_{1}^{(k)}}_{k=1}^{∞} and we have to show that Λ fulfills the assertions of our theorem.

Let K be compact in E_{ρ(f)} and ε > 0. For ε we can find an index i^{∗} > 1 such that

(3.12)  2 ∑_{j=1}^{∞} \frac{1}{(i^{∗} + j)^{2}} < 1/ log \frac{d}{ε}.

Then we define

k^{∗} := max(i^{∗}, min\{i : K ⊂ E_{σ_{i}} \}) and K_{ε} := \overline{⋃_{j=1}^{∞} Ω_{n_{j}^{(k^{∗} )}} (σ_{k^{∗}} , θ_{k^{∗}} )} .

We know that

(3.13)  |f(z) − r_{n,m_{n}} (z)| ⩽ ( \frac{θ_{k^{∗}} σ_{k^{∗}}}{ρ(f)} )^{n}

for z ∈ \overline{E}_{σ_{k^{∗}}} ∖ Ω_{n_{j}^{(k^{∗} )}} (σ_{k^{∗}} , θ_{k^{∗}} ) and

(3.14)  capΩ_{n_{j}^{(k^{∗} )}} (σ_{k^{∗}} , θ_{k^{∗}} ) ⩽ d^{1/2} (γ_{k}^{∗})^{n_{j}^{(k^{∗} )} /2m_{n{(k^{∗} ) \atop j}}}

for j = 1, 2, . . .. The subadditivity of the capacity (Nevanlinna) yields with (3.12), (3.14)

1/ log \frac{d}{capK_{ε}} ⩽ ∑_{j=1}^{∞} 1/ log \frac{d}{capΩ_{n_{j}^{(k^{∗} )}} (σ_{k^{∗}} , θ_{k^{∗}} )}

⩽ ∑_{j=1}^{∞} ( logd − \frac{1}{2} logd − \frac{n_{j}^{(k^{∗})}}{2m_{n_{j}^{(k^{∗} )}}} logγ_{k}^{∗}) ^{−1}

⩽ 2 ∑_{j=1}^{∞} \frac{m_{n_{j}^{(k^{∗} )}}}{n_{j}^{(k^{∗})}} / log \frac{1}{γ_{k}^{∗}} ⩽ 2 ∑_{j=1}^{∞} \frac{1}{(k^{∗} + j)^{2}} < 1/ log \frac{d}{ε},

and consequently capK_{ε} < ε. Since {n ∈ Λ : n ⩾ n_{1}^{(k^{∗})}} ⊂ Λ_{k^{∗}} we obtain by (3.13)

|f(z) − r_{n,m_{n}} (z)| ⩽ ( \frac{θ_{k^{∗}} σ_{k^{∗}}}{ρ(f)} )^{n}, z ∈ K ∖ K_{ε}.

for all n ∈ Λ, n ⩾ n_{1}^{(k^{∗})}. Hence the uniform convergence in capacity of {r_{n,m_{n}} }_{n∈Λ} to f inside E_{ρ(f)} is proven. □

4. Sharpness of the theorems

The result in Theorem 2.1 is sharp in the sense that in (2.1) the condition m_{n} = o(n) as n → ∞ is essential. To verify this we consider the following example: Let E be compact with regular connected complement, f ∈ M(E) with ρ(f) < ∞. If {m_{n}}_{n∈ℕ} is a sequence in ℕ with (2.1), then Walsh’s theorem implies that there exist best uniform rational approximants r_{n,m_{n}}^{∗} ∈ R_{n,m_{n}} to f on E such that

lim sup_{n→∞}∥f − r_{n,m_{n}}^{∗} ∥_{E}^{1/n} ⩽ \frac{1}{ρ(f)} .

According to Theorem 2.1 the sequence {r_{n,m_{n}}^{∗} }_{n∈ℕ} converges in capacity to f inside E_{ρ(f)} .

Furthermore, let {m̃_{n} }_{n∈ℕ} be a sequence in ℕ with

(4.1)  m̃_{n} ⩾ m_{n} and lim_{n→∞} \frac{m̃_{n}}{n} > 0.

We choose a point ξ ∈ E_{ρ(f)} ∖ E, hence α := dist(ξ, E) > 0. Then we define the sequence

r_{n,m̃_{n}} := r_{n,m_{n}}^{∗} + R_{n}(z) ∈ R_{n,m̃_{n}} , where R_{n}(z) = \frac{α^{m̃_{n} −m_{n}}}{ρ(f)^{n}} \frac{1}{(z − ξ)^{m̃_{n} −m_{n}}} .

Then

∥f − r_{n,m̃_{n}} ∥_{E} ⩽ ∥f − r_{n,m_{n}}^{∗} ∥_{E} + \frac{1}{ρ(f)^{n}} ,

lim sup_{n→∞}∥f − r_{n,m̃_{n}} ∥_{E}^{1/n} ⩽ \frac{1}{ρ(f)} .

Consider the disks

D_{n} := \{z ∈ ℂ : |z − ξ| ⩽ \frac{α}{ρ(f)^{n/(m̃_{n} −m_{n} )}} \}.

Using (2.1) and (4.1), we conclude that there exists a number κ > 0 and n_{0} ∈ ℕ such that

\frac{n}{m̃_{n} − m_{n}} ⩽ κ for all n ⩾ n_{0}.

Hence, with r := α/ρ(f)^{κ} we get D_{n} ⊃ K := K_{r}(ξ) = {z ∈ ℂ : |z − ξ| ⩽ r} for all n ⩾ n_{0}. Moreover, we can choose κ big enough such that K ⊂ E_{ρ(f)} .

Now, fix ε > 0, 0 < ε < 1, and consider the sets

S_{n}(ε) := {z ∈ K : |(f − r_{n,m_{n}}^{∗} )(z)| ⩾ ε}.

By Theorem 2.1, we know that {r_{n,m_{n}}^{∗} }_{n∈ℕ} converges in capacity to f inside E_{ρ(f)} . Therefore

(4.2)  lim_{n→∞} capS_{n}(ε) = 0.

By definition of r_{n,m̃_{n}} , we have

|(f − r_{n,m̃_{n}} )(z)| ⩾ |R_{n}(z)| − |(f − r_{n,m_{n}}^{∗} )(z)|.

Consequently,

|(f − r_{n,m̃_{n}} )(z)| ⩾ 1 − ε for all z ∈ K ∖ S_{n}(ε).

Since capK = r > 0, we obtain by Nevanlinna’s inequality, together with (4.2), that

lim inf_{n→∞} cap{z ∈ K : |(f − r_{n,m̃_{n}} )(z)| ⩾ 1 − ε} > 0.

Hence, {r_{n,m̃_{n}} }_{n∈ℕ} does not converge in capacity to f inside E_{ρ(f)} , and the condition “m_{n} = o(n) as n → ∞" is essential in Theorem 2.1.

References

1. H.-P. Blatt, R. K. Kovacheva, Growth behavior and zero distribution of rational approximants, Constr. Approx. 34(3) (2011), 393–420.

2. A. A. Gončar, On the convergence of generalized Padé approximants of meromorphic functions, Mat. Sb. 98 (140) (1975), 564–577; English translation in Math. USSR Sb. 27(4) (1975), 503– 514.

3. R. Nevanlinna, Eindeutige analytische Funktionen, Springer-Verlag, Berlin, 1974.

4. Ch. Pommerenke, Univalent Functions, Vandenhoek and Ruprecht, Göttingen, 1975.

5. M. Tsuji, Potential Theory in Modern Function Theory, Maruzen, Tokyo, 1959.

6. J. L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Colloq. Publ., Am. Math. Soc. 20, Providence, Rhode Island, 1969.

Mathematisch-Geographische Fakultät

Katholische Universität Eichstätt-Ingolstadt

Eichstätt Germany

    hans.blatt@ku.de