Diagonal Pade Sequences for Functions Meromorphic in the Unit Ball Approximate Well Near 0

D. S. Lubinsky

Abstract. Let f be analytic at 0 and meromorphic in the unit ball. It is well known that its diagonal Pade sequence {[n/n]}_{n=1}^{∞} need not converge in measure, even in any arbitrarily small ball. Neverthless, it has recently been shown that this sequence provides good uniform approximation on almost \frac{1}{8} of the circles centre 0. Here we show that in a neighbourhood of 0, we have good approximation most of the time, at least for a subsequence. If errors of approximation behave regularly, this is true for the full sequence. Estimates of the number of poles near 0 are derived as an auxiliary result.

§1. Introduction

Let f be a function analytic at 0, and hence possessing a Maclaurin series there. Recall that if m, n ≥ 0, the (m, n) Padé approximant to f is a rational function

[m/n] (z) = (p/q) (z),

where p, q are polynomials of degree ≤ m, n respectively, with q not identically zero, and

(fq − p)(z) = O (z^{m+n+1}) .

The order relation indicates that the coefficients of 1, z, z^{2}, … z^{m+n} in the Maclaurin series of the left-hand side vanish. For an introduction to the subject, see [1].

The convergence theory of Padé approximation is intricate and complicated. It is known that if f is analytic at 0, and meromorphic in the whole plane, then {[n/n]}_{n=1}^{∞} converges in measure – the Nuttall-Pommerenke Theorem [11,12]. More precisely, given r, ε > 0,

m_{2} {z : |z| ≤ r and |f − [n/n]| (z) > ε^{n}} → 0, n → ∞.

Trends in Approximation Theory 1 Kirill Kopotun, Tom Lyche, and Mike Neamtu (eds.), pp. 1–7. Copyright o^{c} 2001 by Vanderbilt University Press, Nashville, TN. ISBN 0-8265-xxxx-x. All rights of reproduction in any form reserved.

Here m_{2} denotes planar measure, and it may be replaced by capacity. There are many generalisations [4], and deeper analogues for functions with branchpoints [15,16].

One feature of the theorem is that it requires f to be meromorphic in C. There are functions analytic in the unit ball for which {[n/n]}_{n=1}^{∞} does not converge in capacity or measure in any ball, no matter how small, contained in the unit ball [5,6,14]. Such counterexamples suggest that there is nothing positive that can be said about full diagonal Padé sequences for functions analytic in the unit ball. For subsequences, there is the still unresolved Baker-Gammel-Wills Conjecture, and its cousins – see [1,8,17].

In a recent paper [10], the author showed that nevertheless, there is something positive (and we believe significant) that can be said for full diagonal sequences, for functions with finite radius of meromorphy:

Theorem 1. Let f be analytic at 0 and meromorphic in {z : |z| < 1}. Then given 0 < δ < 1, there exists 0 < ε < 1 and n_{0} such that for n ≥ n_{0},

m_{1} \{r ∈ [0, 1] : max_{|z|=r} |f − [n/n]| (z) ≤ ε^{n}\} ≥ \frac{1}{8} − δ.

Here m_{1} denotes linear Lebesgue measure.

Thus \{[n/n]\}_{n=1}^{∞} provides good approximation on almost \frac{1}{8} of the circles centre 0, within the unit ball. This may be viewed as a much sharper form of the results of [9], where it was shown that [n/n] provides approximation on a set of positive proportion, independently of n. The main idea of [10] was the use of potential theory to derive a sharp lower bound for

m_{1} (\{r ∈ [0, 1] : r^{an} \begin{matrix}max & |t|=1 & |P (t)| \\ min & |t|=r & |P (t)| \end{matrix} < ε^{n}\}) ,

where P is an arbitrary polynomial of degree n, and a ≥ 0, ε > 0.

In this paper, we shall use these results to estimate the density of poles of [n/n] near 0, for large n, and then prove that in a neighbourhood of 0, we obtain approximation on most circles centre 0 within that neighbourhood (rather than just \frac{1}{8} of the circles). We need some notation to formulate our results. Let

π_{n}(r) := \frac{1}{n} × ( Number of poles of [n/n] in |z| ≤ r) ,

counting multiplicity, and the upper and lower densities of poles,

π^{+}(r) := lim sup_{n→∞} π_{n} (r) ;
π^{−}(r) := lim inf_{n→∞} π_{n} (r) .

Our first result is:

Theorem 2. Assume that f is analytic at 0, and meromorphic in the unit ball. There exists C > 0 independent of f and r such that

π^{−}(r) ≤ \frac{C}{|log r|}, r ∈ (0, 1) .  (1.1)

In particular, the lower density has limit 0 at 0. To bound the upper density, we need errors of rational interpolation and their asymptotic behaviour: the error of approximation of f by rational functions of type (n, n) on |z| ≤ r is

R_{n} (f; r) := inf \{∥f − \frac{p}{q}∥_{L_{∞}(|z|≤r)} : deg (p) , deg (q) ≤ n, q ≢ 0\} .

Moreover, we set

ρ^{+}(r) := lim sup_{n→∞} R_{n} (f; r)^{1/n};
ρ^{−}(r) := lim inf_{n→∞} R_{n} (f; r)^{1/n} .

Of course ρ^{±}(r) are increasing functions of r, and it is known [13,18] that

ρ^{−}(r) ≤ r^{2}; ρ^{+}(r) ≤ r.  (1.2)

If ρ^{+}(r) = 0 for one r, then f belongs to the Goncar-Walsh class and ρ^{+}(r) = 0 for all r [2]. It follows from (2.9) below that if ρ^{−} (r) = 0 for one r, then ρ^{−}(r) = 0 for all r. If f belongs to the Gonchar-Walsh class, then {[n/n]}_{n=1}^{∞} converges in capacity in compact subsets of the domain of meromorphy, and so much stronger results are possible. Therefore, we can assume

ρ^{+}(r) > 0, r ∈ (0, 1).  (1.3)

Theorem 3. Assume that f is analytic at 0, and meromorphic in the unit ball and that (1.3) holds. There exists C > 0 such that

π^{+}(r) ≤ \frac{C}{|log r|} (1 + |\frac{log ρ^{+}(r)}{log ρ^{−}(r)}|) + |1 − \frac{log ρ^{+}(r)}{log ρ^{−} (r)}| , r ∈ (0, 1). (1.4)

In particular,

lim_{r→0+} \frac{log ρ^{+}(r)}{log ρ^{−}(r)} = 1 ⇒ lim_{r→0+} π^{+} (r) = 0.  (1.5)

For functions for which R_{n}(f; r) decays regularly with n, so that ρ^{+}(r) = ρ^{−}(r) for all r, we then see that π^{+}(r) admits the same estimate (1.1) as π^{−}(r).

From Theorems 2 and 3, we deduce that a subsequence of {[n/n]}_{n=1}^{∞} approximates well on most circles in a neighbourhood of 0:

Theorem 4. Assume that f is analytic at 0, and meromorphic in the unit ball. Let 0 < η < 1. Then there exists δ, ε ∈ (0, 1) and an infinite sequence J of positive integers such that for n ∈ J,

\frac{1}{δ}m_{1} \{r ∈ [0, δ] : max_{|z|=r} |f − [n/n]| (z) ≤ ε^{n}\} ≥ 1 − η.  (1.6)

If (1.5) holds, then instead this holds for all sufficiently large n.

Thus in a neighbourhood of 0, [n/n] approximates well on a proportion of at least 1−η of the circles centre 0 in that neighbourhood. The proof shows that one may take ε = δ^{1/4}. Theorem 4 is closely related to the main result of [7].

§2. Proofs

Throughout, we assume the hypotheses of Theorem 2 on f. We shall also for a given s ∈ (0, 1), assume that f has poles of total multiplicity ℓ = ℓ (s) in |z| ≤ s, and S is a polynomial of degree ℓ such that fS is analytic in |z| ≤ s. We shall denote absolute (numerical) positive constants by A, and positive constants independent of n, r, s by C. The same symbol does not necessarily denote the same constant in different occurrences. Our basic tool is:

Lemma 2.1. For s ∈ (0, \frac{1}{2}),

R_{n(1−π_{n}(s))+ℓ(2s)} (f; s) ≤ CA^{4n}R_{n−ℓ(2s)} (f; 2s)  (2.1)

where A is an absolute constant, and C is independent of n, s.

Proof: Choose a rational function p_{n}^{∗}/q_{n}^{∗} of type (n, n) such that

max_{|z|≤s} |f − p_{n}^{∗}/q_{n}^{∗}| (z) = R_{n} (f; s) .  (2.2)

This is possible for n ≥ ℓ := ℓ (s). Write [n/n] = p_{n}/q_{n}. We use the error formula [4]

(fq_{n} − p_{n}) (z) = \frac{1}{2πi} ∫_{|t|=s} \frac{(fq_{n−ℓ}^{∗} − p_{n−ℓ}^{∗}) (t) (Sq_{n}) (t)}{(t − z) (Sq_{n−ℓ}^{∗}) (z)} (\frac{z}{t} )^{2n+1} dt,

|z| < s. This yields for r < s, the estimate

max_{|z|=r} |f − [n/n]| (z) ≤ \frac{Cs}{s − r} (\frac{r}{s})^{2n} \frac{max_{|t|=s} |Sq_{n−ℓ}^{∗}q_{n}| (t)}{min_{|z|=r} |Sq_{n−ℓ}^{∗}q_{n}| (z)}R_{n−ℓ} (f; s) .

Now it follows from Theorem 2.3 in [10] that there is an absolute constant A > 0 such that

m_{1} \{r ∈ [0, 1] : \begin{matrix}max & |t|=1 & |P (t)| \\ min & |z|=r & |P (z)| \end{matrix} ≤ A^{m}\} ≥ \frac{3}{4},  (2.3)

for every m ≥ 1 and polynomial P of degree ≤ m. (In the notation of [10], A := ε = exp (−G_{0} (3/4)) and a = 0). Let us apply this with m = 2n and to P (t) := (Sq_{n−ℓ}^{∗}q_{n}) (t/s). This gives

max_{|z|=r} |f − [n/n]| (z) ≤ \frac{Cs}{s − r} (A\frac{r}{s})^{2n} R_{n−ℓ} (f; s) ,

for r/s ∈ S_{n}, where m_{1} (S_{n}) ≥ \frac{3}{4}. Next, we need a result of Gonchar-Grigorjan [3]: let g be analytic in |z| ≤ r, except for poles of total multiplicity m say. Let g_{a} be the analytic part of g, that is g_{a} equals g less the sum of its principal parts. (The latter is a rational function of type (m − 1, m)). Then

max_{|z|≤r} |g_{a}(z)| ≤ 7m^{2} max_{|z|=r} |g(z)| .  (2.4)

Applying this to g := f − [n/n], which has poles of total multiplicity at most ℓ + nπ_{n}(r) in |z| ≤ r, and noting that [n/n]_{a} is a rational function of type (n (1 − π_{n}(r)) , n (1 − π_{n}(r))), we deduce that for at least one r ∈ [\frac{s}{2}, \frac{7s}{8}],

R_{n(1−π_{n}(r))+ℓ} (f; r) ≤ max_{|z|≤r} |f_{a} − [n/n]_{a}| (z) ≤ C7n^{2}A^{2n}R_{n−ℓ} (f; s) .

Since π_{n}(r) ≥ π_{n} (s/2) and R_{n} (f; r) increases in r, we deduce that

R_{n(1−π_{n}(s/2))+ℓ} (f; s/2) ≤ CA^{4n}R_{n−ℓ} (f; s) .

Now change s to 2s.

Corollary 2.2. For s ∈ (0, \frac{1}{2}),

ρ^{+} (s)^{1−π^{−}(s)} ≤ A^{4}ρ^{+} (2s) ;  (2.5)
ρ^{−} (s)^{1−π^{−}(s)} ≤ A^{4}ρ^{−} (2s) ;  (2.6)
ρ^{−} (s)^{1−π^{+}(s)} ≤ A^{4}ρ^{+} (2s) .  (2.7)

Proof: We prove first (2.5). Let δ > 0. For large enough n,

π_{n} (s) ≥ π^{−} (s) − δ,

so if [x] denotes integer part, Lemma 2.1 gives for large enough n,

R_{[n(1−π^{−}(s)+δ)]+ℓ(2s)} (f; s) ≤ CA^{4n} (ρ^{+} (2s) + δ)^{n−ℓ(2s)} .  (2.8)

Now choose an infinite sequence of positive integers T such that

lim_{n→∞,n∈T} R_{[n(1−π^{−}(s)+δ)]+ℓ(2s)} (f; s)^{1/([n(1−π^{−}(s)+δ)]+ℓ(2s))} = ρ^{+} (s) .

To see that it is possible to choose such a T , one uses the fact that for given α, β > 0, every positive integer ≥ α + β is at most a distance α + 1 away from an integer of the form [αm + β]. Moreover, one needs the fact that R_{m} (f; s) is decreasing as m increases. To finish, take nth roots in (2.8), let n → ∞, and then let δ → 0+.

For the proof of (2.6), one chooses T to give ρ^{−} (2s) from the right-hand side of (2.1); and for (2.7), one chooses T such that as n → ∞ through T , π_{n} (s) → π^{+} (s).

Next, we compare R_{n} (f; 2s) and R_{n} (f; s):

Lemma 2.3. For s ∈ (0, \frac{1}{6}) and n ≥ n_{0} (s), and for some C independent of f, n, s,

R_{n} (f; 2s) ≤ R_{n−ℓ(4s)} (f; s)^{1+ \frac{C}{log s}} .  (2.9)

Proof: Let p_{n}^{∗}/q_{n}^{∗} satisfy (2.2). Choose for large n, m = m (n) such that

(5s)^{m+1} < R_{n} (f; s) ≤ (5s)^{m} .

Since (1.2) implies that for large m,

R_{m} (f; 4s) ≤ (5s)^{m+1}

we obtain for large n, that

R_{m} (f; 4s) ≤ R_{n} (f; s) ≤ (5s)^{m} .  (2.10)

Here we see that

n ≤ m = m (n) ≤ \frac{log R_{n} (f; s)}{log (5s)}.  (2.11)

Next if p_{m}^{#}/q_{m}^{#} satisfies

max_{|z|≤4s} |f − p_{m}^{#}/q_{m}^{#}| (z) = R_{m} (f; 4s) ,

we obtain

max_{|z|≤s} |p_{m}^{#}/q_{m}^{#} − p_{n}^{∗}/q_{n}^{∗}| (z) ≤ 2R_{n} (f; s) .

Normalizing

max_{|z|=4s} |q_{m}^{#}q_{n}^{∗}| (z) = 1,

gives

max_{|z|≤s} |p_{m}^{#}q_{n}^{∗} − p_{n}^{∗}q_{m}^{#}| (z) ≤ 2R_{n} (f; s) ,

and the Bernstein inequality for growth of polynomials yields

max_{|z|≤4s} |p_{m}^{#}q_{n}^{∗} − p_{n}^{∗}q_{m}^{#}| (z) ≤ 2 · 4^{m+n}R_{n} (f; s) .

Hence for r ≤ 4s,

max_{|z|=r} |p_{m}^{#}/q_{m}^{#} − p_{n}^{∗}/q_{n}^{∗}| (z) ≤ 4^{m+n+1}R_{n} (f; s) \frac{max_{|t|=4s} |q_{m}^{#}q_{n}^{∗}| (t)}{min_{|z|=r} |q_{m}^{#}q_{n}^{∗}| (z)}
≤ 4^{m+n+1}R_{n} (f; s) A^{m+n},

by (2.3), for r/ (4s) ∈ S_{n}, where m_{1} (S_{n}) ≥ \frac{3}{4}. Then our choice of p_{m}^{#}/q_{m}^{#} and (2.10) give for at least one r ≥ 2s,

max_{|z|=r} |f − p_{n}^{∗}/q_{n}^{∗}| (z) ≤ R_{n} (f; s) (1 + 4^{m+n+1}A^{m+n}) .

Applying the Gonchar-Grigorjan inequality (2.4) much as in the proof of Lemma 2.1 gives for large n,

\begin{aligned}R_{n+ℓ(4s)} (f; 2s) & ≤ R_{n+ℓ(4s)} (f; r) ≤ R_{n} (f; s) 8^{m+n}A^{m+n} \\ & ≤ R_{n} (f; s) (8A)^{2m} ≤ R_{n} (f; s)^{1+2 \begin{matrix}log & (8A) \\ log & (5s) \end{matrix} ,} \end{aligned}

by (2.11). Replacing n by n − ℓ (4s) gives (2.9).

The above lemma gives

ρ^{±} (2s) ≤ ρ^{±} (s)^{1+ \frac{C}{log s}} , s ∈ (0, \frac{1}{6}) .  (2.12)

Proof of Theorem 2: From (2.5) and (2.12),

(1 − π^{−} (s)) log ρ^{+} (s) ≤ 4 log A + (1 + \frac{C}{log s}) log ρ^{+} (s)

⇒ π^{−} (s) ≤ \frac{4 log A}{|log ρ^{+} (s)|} + \frac{C}{|log s|}.

Since ρ^{+} (s) ≤ s, we obtain (1.1) for r ∈ (0, \frac{1}{6}), with C independent of f, r. Since π^{−} (r) ≤ 1, (1.1) follows also for r ∈ [\frac{1}{6}, 1) with a suitable C.

Proof of Theorem 3: This follows similarly from (2.7) and (2.12).

Proof of Theorem 4: Let 0 < η < 1. It follows easily from Theorem 2.3 of [10] that there exists a_{0} such that for a ≥ a_{0}, and polynomials P of degree ≤ m,

m_{1} \{r ∈ [0, 1] : r^{am} \begin{matrix}max & |t|=1 & |P (t)| \\ min & |z|=r & |P (z)| \end{matrix} ≤ 1\} ≥ 1 − η.  (2.13)

Fix such an a > 4. Now choose δ ∈ (0, \frac{1}{4}) and an infinite sequence of positive integers T such that for n ∈ T , [n/n] = p_{n}/q_{n} has at most n/a^{2} poles in |z| ≤ δ. (This is possible by Theorem 2). Let S be a polynomial of degree ℓ say such that fS is analytic in |z| ≤ δ. We use the contour integral error formula

(S (fq_{n} − p_{n})) (z) = \frac{1}{2πi} ∫_{|t|=δ} \frac{(Sfq_{n}) (t)}{t − z} (\frac{z}{t} )^{2n+1} dt, |z| < δ

to deduce that for r ≤ δ^{2}

max_{|z|=r} |f − [n/n]| (z) ≤ C (\frac{r}{δ} )^{2n} \frac{max_{|t|=δ} |Sq_{n}| (t)}{min_{|z|=r} |Sq_{n}| (z)},  (2.14)

where C is independent of n, r. Now split

Sq_{n} = u_{n}v_{n},

where u_{n} has zeros in |z| ≥ δ and v_{n} has zeros in |z| < δ. Then v_{n} has degree ≤ ℓ + n/a^{2} for n ∈ T . We have that for r ≤ δ^{4},

(\frac{r}{δ} )^{n} \frac{max_{|t|=δ} |u_{n}| (t)}{min_{|z|=r} |u_{n}| (z)} ≤ δ^{3n} (\frac{2}{1 − δ^{3}} )^{n} < δ^{n}.  (2.15)

Also if m = m (n) is the degree of v_{n}, then if n is large enough, m ≤ 2n/a^{2}, and we may use Bernstein’s inequality for growth of polynomials to deduce that

(\frac{r}{δ} )^{n} \frac{max_{|t|=δ} |v_{n}| (t)}{min_{|z|=r} |v_{n}| (z)} ≤ (\frac{r}{δ} )^{a^{2}m/2} δ^{−3m} \frac{max_{|t|=δ^{4}} |v_{n}| (t)}{min_{|z|=r} |v_{n}| (z)}

≤ (\frac{r}{δ} )^{a^{2}m/2} δ^{−3m} (\frac{r}{δ^{4}} )^{−am} ,

for r/δ^{4} ∈ S_{m}, where m_{1} (S_{m}) ≥ 1 − η, by (2.13). Now since a^{2}/2 ≥ 2a, we may continue this as

≤ (\frac{r}{δ} )^{am} δ^{−3m+3am} ≤ 1.

Putting this, and (2.14), (2.15) together gives

max_{|z|=r} |f − [n/n]| (z) ≤ Cδ^{n} for r/δ^{4} ∈ S_{m}.

First setting ε = δ^{1/2}, and then replacing δ^{4} by δ gives (1.6) for large enough n. Finally, if the conclusion of Theorem 2 holds, we can take T to be the set of all large enough positive integers.

Remark. The above proof works more generally for any sequence of rational functions r_{n} = p_{n}/q_{n}, n ≥ 1, such that

(i) (fq_{n} − p_{n}) (z) = O (z^{n+m_{n}}), with m_{n} → ∞, n → ∞; (ii) lim_{s→0+} π^{−} (s) = 0, if we define π^{−} in the same way as for {[n/n]}_{n=1}^{∞}.

§3. References

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4. Karlsson, J., Rational Interpolation and Best Rational Approximation, J. Math. Anal. Applns. 53 (1976), 38–52.

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7. Lubinsky, D. S., Convergence of Diagonal Padé Approximants for Functions Analytic Near 0, Trans. Amer. Math. Soc. 723 (1995), 3149–3157.

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9. Lubinsky, D. S., On the Maximum and Minimum Modulus of Rational Functions, Canad. J. Math. 52 (2000), 815–832.

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The John Knopfmacher Centre for Applicable Analysis and Number Theory Department of Mathematics, Witwatersrand University Wits 2050 South Africa.

036dsl@cosmos.wits.ac.za