On the Distribution of Zeros and Poles of Rational Approximants on Intervals

Abstract

The distribution of zeros and poles of best rational approximants is well understood for the functions 𝑓(𝑥)=|𝑥|^𝛼, 𝛼>0. If 𝑓∈𝐶[−1,1] is not holomorphic on [−1,1], the distribution of the zeros of best rational approximants is governed by the equilibrium measure of [−1,1] under the additional assumption that the rational approximants are restricted to a bounded degree of the denominator. This phenomenon was discovered first for polynomial approximation. In this paper, we investigate the asymptotic distribution of zeros, respectively, 𝑎-values, and poles of best real rational approximants of degree at most 𝑛 to a function 𝑓∈𝐶[−1,1] that is real-valued, but not holomorphic on [−1,1]. Generalizations to the lower half of the Walsh table are indicated.

1. Introduction

Let 𝐵 be a subset of ℂ; we denote by 𝑚₁(𝐵)∶=inf_𝜈𝑈_𝜈 (1.1) the 𝑚₁-measure of 𝐵, where the infimum is taken over all coverings {𝑈_𝜈} of 𝐵 by disks 𝑈_𝜈 and |𝑈_𝜈| is the radius of the disk 𝑈_𝜈.

Let 𝐷 be a region in ℂ and 𝜑 a function defined in 𝐷 with values in ℂ. A sequence {𝜑_𝑛}_𝑛∈ℕ of meromorphic functions in 𝐷 is said to converge to a function 𝜑with respect to the 𝑚₁-measure inside 𝐷 if for every 𝜀>0 and any compact set 𝐾⊂𝐷 we have 𝑚₁𝑧∈𝐾∶𝜑−𝜑_𝑛(𝑧)≥𝜀⟶0as𝑛⟶∞ (1.2) (cf. Gončar [1]).

The sequence {𝜑_𝑛}_𝑛∈ℕ is said to converge to 𝜑𝑚₁-almost geometrically inside 𝐷 if for any 𝜀>0 there exists a set Ω(𝜀) in ℂ with 𝑚₁(Ω(𝜀))<𝜀 such that limsup_𝑛→∞𝜑−𝜑_𝑛^1/𝑛_{𝐾⧵Ω(𝜀)}<1 (1.3) for any compact set 𝐾⊂𝐷. We note that ‖⋅‖_𝐵 is the supremum norm on a subset 𝐵 of ℂ.

For 𝑛∈ℕ₀=ℕ∪{0}, we denote by 𝒫_𝑛 the collection of all polynomials of degree at most 𝑛, and let ℛ_𝑛,𝑚𝑝∶=𝑟=𝑞∶𝑝∈𝒫_𝑛,𝑞∈𝒫_𝑚,𝑞≢0. (1.4) In [2], sequences {𝑟_𝑛}_𝑛∈ℕ, 𝑟_𝑛∈ℛ_𝑛,𝑛, on a region 𝐷 were investigated if the number of poles of 𝑟_𝑛 in 𝐷 is bounded. It turns out that the geometric convergence of {𝑟_𝑛}_𝑛∈ℕ on a continuum 𝑆⊂𝐷 implies that the sequence converges 𝑚₁-almost geometrically inside 𝐷 to a meromorphic function 𝑓 in 𝐷 with at most a finite number of poles in 𝐷.

To be precise, let 𝐵⊂ℂ and let ℳ_𝑚(𝐵) denote the subset of meromorphic functions in 𝐵 with at most 𝑚 poles in 𝐵, each pole counted with its multiplicity. The main result of [2] can be stated as follows.

Theorem A. Let 𝑆 be a continuum in ℂ and 𝐷 a region with 𝑆⊂𝐷. Let {𝑟_𝑛}_𝑛∈ℕ, 𝑟_𝑛∈ℛ_𝑛,𝑛, be a sequence of rational functions converging geometrically to a function 𝑓 on 𝑆, that is, limsup_𝑛→∞𝑓−𝑟_𝑛𝑆^1/𝑛<1, (1.5) and assume that 𝑓≢0 on 𝑆. If there exists a fixed integer 𝑚∈ℕ such that 𝑟_𝑛∈ℳ_𝑚(𝐷) for all 𝑛 and 𝑁₀𝑟_𝑛,𝐾=𝑜(𝑛)as𝑛⟶∞ (1.6) for each compact set 𝐾⊂𝐷, then the sequence {𝑟_𝑛}_𝑛∈ℕ converges 𝑚₁-almost geometrically inside 𝐷 to a meromorphic function 𝑓∈ℳ_𝑚(𝐷).

Here, the number 𝑁₀(𝑟_𝑛,𝐾) denotes the number of zeros of 𝑟_𝑛 in 𝐾, each zero counted with its multiplicity.

The above result was applied in [2] to Chebyshev approximation on [−1,1]. Let 𝐺(𝑧,∞) be the Green function of Ω=ℂ⧵[−1,1] with pole at ∞, and let ℰ_𝜌∶=𝑧∈ℂ∶𝐺(𝑧,∞)<log𝜌,𝜌>1, (1.7) be the Green domain to the parameter 𝜌, that is, ℰ_𝜌 is the open Joukowski-ellipse with foci at +1 and −1 and major axis 𝜌+1/𝜌.

Let 𝑓∈𝐶[−1,1] be real-valued on [−1,1]. For abbreviation, we will write ‖⋅‖ for ‖⋅‖_[−1,1]. Given 𝑛,𝑚∈ℕ₀, let 𝑟^∗_𝑛,𝑚=𝑟^∗_𝑛,𝑚(𝑓)∈ℛ_𝑛,𝑚 denote the real rational function of best uniform approximation to 𝑓∈𝐶[−1,1] with respect to ℛ_𝑛,𝑚, that is, 𝐸_𝑛,𝑚𝑓∶=𝑓−𝑟^∗_𝑛,𝑚=inf𝑓−𝑟∶𝑟∈ℛ_𝑛,𝑚,𝑟real-valuedonℝ. (1.8) Moreover, let {𝑚_𝑛}_𝑛∈ℕ be a sequence in ℕ with lim_𝑛→∞𝑚_𝑛=∞,𝑚_𝑛𝑛=𝑜log𝑛as𝑛⟶∞, (1.9) and let us consider a function 𝑓∈𝐶[−1,1] that can be continued meromorphically into ℰ_𝜌 for some 𝜌>1. Then the sequence {𝑟^∗_{𝑛,𝑚𝑛}}_𝑛∈ℕ converges 𝑚₁-almost geometrically inside ℰ_𝜌 to 𝑓 [3]. Using Theorem A, we obtain results about the distribution of the 𝑎-values in the neighborhood of a point 𝑧₀∈𝜕𝐸_𝜌. For 𝑎∈ℂ and 𝐵⊂ℂ, we denote by 𝑁_𝑎(𝑟,𝐵)∶=#{𝑧∈𝐵∶𝑟(z)=𝑎} (1.10) the number of 𝑎-values of the rational function 𝑟 in 𝐵 and each 𝑎-value is counted with its multiplicity. If 𝑓 cannot be continued meromorphically to 𝑧₀, then for any neighborhood 𝑈 of 𝑧₀ and any 𝑎∈ℂ, with at most one exception, limsup_𝑛→∞𝑁_𝑎𝑟^∗_{𝑛,𝑚𝑛},𝑈=∞. (1.11) Particulary, such a point 𝑧₀ is either an accumulation point of zeros or of poles of 𝑟^∗_{𝑛,𝑚𝑛}.

On the other hand, if 𝑓 is not holomorphic on [−1,1], so far results about the distribution of the zeros of 𝑟^∗_{𝑛,𝑚𝑛}(𝑓) are only known in the case that 𝑚_𝑛=0 for all 𝑛∈ℕ (polynomial approximation) or in the case that 𝑚_𝑛=𝑚∈ℕ is fixed (rational approximation with a bounded number of free poles). In the polynomial case, the normalized zero counting measures of 𝑟^∗_𝑛,0(𝑓) converge in the weak*-sense to the equilibrium measure of [−1,1], at least for a subsequence 𝑛∈Λ⊂ℕ [4]. This result was generalized to rational approximation with a bounded number of poles (cf. [5, Theorem 4.1]). Moreover, Stahl [6] and Saff and Stahl [7] have investigated for the function 𝑓(𝑥)=|𝑥|^𝛼, 𝛼>0, the distribution of zeros and poles of rational approximants, as well as the alternation points of the optimal error function.

In contrast to the distribution of zeros of 𝑟^∗_{𝑛,𝑚𝑛}, the behavior of the alternation points of 𝑓−𝑟^∗_{𝑛,𝑚𝑛} for 𝑓∈𝐶[−1,1] is well understood, not only in the polynomial case (cf. [8, 9]), but also for rational approximations (cf. [10–14]). The aim of the present paper is to investigate the distribution of the zeros of the rational approximants via the distribution of the alternation points.

2. Main Results

Let 𝑓 be continuous on [−1,1], possibly complex-valued. It is well known that the rate of approximation by rational functions does not guarantee the holomorphy of the function 𝑓. Gončar ([15], p. 101) pointed out the example 𝑓(𝑧)=^∞_𝑛=1𝐴_𝑛𝑧−𝛼_𝑛, (2.1) where the points 𝛼_𝑛 are situated in ℂ⧵[−1,1] such that any point of [−1,1] is a limit point of the sequence {𝛼_𝑛} and the coefficients 𝐴_𝑛 converge to zero sufficiently fast. Hence, it is possible that there exists a sequence {𝑟_𝑛}_𝑛∈ℕ, 𝑟_𝑛∈ℛ_𝑛,𝑛, such that limsup_𝑛→∞𝑓−𝑟_𝑛^1/𝑛<1, (2.2) and 𝑓 is continuous on [−1,1], but nowhere holomorphic on [−1,1].

But it turns out that in this case Theorem A immediately yields the following.

Theorem 2.1. Let 𝑓∈𝐶[−1,1] be not holomorphic on [−1,1], and let {𝑟_𝑛}_𝑛∈ℕ, 𝑟_𝑛∈ℛ_𝑛,𝑛, be a sequence such that limsup_𝑛→∞𝑓−𝑟_𝑛^1/𝑛<1. (2.3) Then for any non holomorphic point 𝑧₀∈[−1,1] of 𝑓 any neighborhood 𝑈 of 𝑧₀ either limsup_𝑛→∞𝑁_∞𝑟_𝑛,𝑈=∞ (2.4) or limsup_𝑛→∞𝑁_𝑎𝑟_𝑛,𝑈𝑛>0 (2.5) for all 𝑎∈ℂ.

In the following we consider functions 𝑓∈𝐶[−1,1] that are always real-valued on [−1,1]. Then the case that limsup_𝑛→∞𝐸^1/𝑛_𝑛,𝑛𝑓=1 (2.6) is not covered by Theorem 2.1. By Bernstein’s theorem, condition (2.6) implies that 𝑓∈𝐶[−1,1] is not holomorphic on [−1,1]. Examples for (2.6) are functions which are piecewise analytic on [−1,1] (Newman [16], Gončar [15]).

In the following, we assume that {𝑚_𝑛}_𝑛∈ℕ is a sequence with 𝑚_𝑛≤𝑛,𝑚_𝑛≤𝑚_𝑛+1≤𝑚_𝑛+1. (2.7) For abbreviation, let 𝐸_𝑛∶=𝐸_{𝑛,𝑚𝑛}𝑓,𝑟^∗_𝑛∶=𝑟^∗_{𝑛,𝑚𝑛}𝑓=𝑝^∗_𝑛𝑞^∗_𝑛, (2.8) where 𝑝^∗_𝑛∈𝒫_𝑛 and 𝑞^∗_𝑛∈𝒫_𝑚𝑛 have no common factor. We define 𝛿_𝑛∶=min𝑛−deg𝑝^∗_𝑛,𝑚_𝑛−deg𝑞^∗_𝑛 (2.9) as the defect of 𝑟^∗_𝑛 and 𝑑_𝑛∶=𝑛+𝑚_𝑛+1−𝛿_𝑛. According to the alternation theorem of Chebyshev (cf. Meinardus [17], Theorem 98) there exist 𝑑_𝑛+1 points 𝑥_𝑘^(𝑛), −1≤𝑥₀^(𝑛)<𝑥₁^(𝑛)<⋯<𝑥_𝑑^(𝑛)_𝑛≤1, (2.10) which satisfy 𝜆_𝑛(−1)^𝑘𝑓−𝑟^∗_𝑛𝑥_𝑘^(𝑛)=𝑓−𝑟^∗_𝑛[−1,1],0≤𝑘≤𝑑_𝑛, (2.11) where 𝜆_𝑛=+1 or 𝜆_𝑛=−1 is fixed. For each pair (𝑛,𝑚_𝑛) let 𝐴_𝑛=𝐴_𝑛𝑓∶=𝑥_𝑘^{(𝑛)𝑑𝑛}_𝑘=0 (2.12) denote an arbitrary, but fixed alternation set for the best approximation 𝑟^∗_𝑛∈ℛ_{𝑛,𝑚𝑛}, and let 𝜈_𝑛 denote the normalized counting measure of 𝐴_𝑛, that is, 𝜈_𝑛#𝑥𝛼,𝛽∶=^𝑘(𝑛)∶𝛼≤𝑥_𝑘^(𝑛)≤𝛽𝑑_𝑛+1 (2.13) for any interval [𝛼,𝛽]⊂[−1,1]. Since 𝜈_𝑛 is a probability measure on [−1,1], there exists a subsequence Λ⊂ℕ such that 𝜈−→_𝑛^∗𝜈as𝑛⟶∞,𝑛∈Λ, (2.14) in the weak*-topology and 𝜈 is again a probability measure on [−1,1].

Theorem 2.2. Let 𝑓∈𝐶[−1,1] be real-valued, and let (2.6) hold. Moreover, let 𝑓 be approximated with respect to ℛ_{𝑛,𝑚𝑛}, where the sequence {𝑚_𝑛}_𝑛∈ℕ satisfies (2.7). Then there exists a subsequence Λ⊂ℕ with the following properties: (i) 𝜈−→_𝑛^∗𝜈as𝑛⟶∞,𝑛∈Λ, (2.15)(ii) let 𝑧₀∈supp(𝜈), 𝑎∈ℂ, and let 𝑈 be a neighborhood of 𝑧₀ with 𝑓(𝑧)≢𝑎 on 𝑈∩[−1,1]; then eitherlimsup_{𝑛∈Λ,𝑛→∞}𝑁_∞𝑟^∗_𝑛,𝑈=∞orlimsup_{𝑛∈Λ,𝑛→∞}𝑁_𝑎𝑟^∗_𝑛,𝑈𝑛>0. (2.16)

Applying to the approximation in the upper half of the Walsh table, we obtain the following.

Corollary 2.3. Let 𝑓∈𝐶[−1,1] with (2.6) and let the subsequence {𝑚_𝑛}_𝑛∈ℕ satisfy 𝑚_𝑛≤𝑐𝑛with0≤𝑐<1,𝑚_𝑛≤𝑚_𝑛+1≤𝑚_𝑛+1. (2.17) Then there exists a subsequence Λ⊂ℕ with the following property: Let 𝑎∈ℂ, 𝑧₀∈[−1,1], and let 𝑈 be a neighborhood of 𝑧₀ with 𝑓(𝑧)≢𝑎 on 𝑈∩[−1,1]; then either (i) or (ii) holds.

3. Auxiliary Tools

One of the essential tools for proving Theorem 2.2 is the interaction between alternation points and poles of best rational approximants.

Let 𝜏_𝑛 denote the normalized counting measure of the poles of 𝑟^∗_𝑛, counted with their multiplicities, and let us denote by 𝜏_𝑛 the balayage measure of 𝜏_𝑛 onto [−1,1]. Then the following distribution results hold for the interaction between the alternation points of 𝐴_𝑛 and the poles of 𝑟^∗_𝑛 and 𝑟^∗_𝑛+1.

Theorem B (See [11]). Let 𝑓 be not a rational function, and let {𝑚_𝑛}_𝑛∈ℕ satisfy (2.7). Then there exists a subsequence Λ⊂ℕ such that 𝜈_𝑛−𝛼_𝑛𝜏_𝑛+𝜏_𝑛+1−1−𝛼_𝑛𝜇−−^∗→0as𝑛⟶∞,𝑛∈Λ, (3.1) where 𝛼_𝑛=deg𝑞^∗_𝑛+deg𝑞^∗_𝑛+1𝑑_𝑛+1 (3.2) and 𝜇 is the equilibrium distribution of [−1,1].

We remark that in the proof of Theorem B in [11], the subsequence Λ⊂ℕ is defined by 𝐸Λ∶=𝑛∈ℕ∶^𝑛+𝐸_𝑛+1𝐸_𝑛−𝐸_𝑛+1≤𝑛². (3.3) Inspecting the proof of (3.1) in [11], it turns out that we can modify the definition of Λ by Λ∶=𝑛∈ℕ∶𝐸_𝑛+1≤11−𝑛²𝐸_𝑛. (3.4) The existence of such sequences Λ is based on the divergence of the infinite product ^∞_𝑛=0𝐸_𝑛+1𝐸_𝑛=^∞_𝑛=0𝐸1−^𝑛−𝐸_𝑛+1𝐸_𝑛 (3.5) to 0 if 𝑓 is not a rational function. This argument has already been used by Kadec [9] in his proof for the distribution of the alternation points in polynomial approximation.

Concerning the distribution of the zeros of best polynomial approximations 𝑝^∗_𝑛 to 𝑓, 𝑝^∗_𝑛(𝑧)=𝑎_𝑛𝑧^𝑛+⋅⋅⋅, (3.6) the asymptotic behavior of the highest coefficient 𝑎_𝑛 plays an essential role, namely, limsup_𝑛→∞𝑎_𝑛^1/𝑛=1cap([−1,1])limsup_𝑛→∞𝑒_𝑛^1/𝑛, (3.7) where 𝑒_𝑛=𝑓−𝑝^∗_𝑛=inf_{𝑝𝑛∈𝒫𝑛}𝑓−𝑝_𝑛 (3.8) and cap([−1,1])=1/2 is the logarithmic capacity of [−1,1].

If 𝑓∈𝐶[−1,1] is not holomorphic on [−1,1], then limsup_𝑛→∞𝑒_𝑛^1/𝑛=1 and we can choose a subsequence Λ⊂ℕ such that lim_{𝑛∈Λ,𝑛→∞}𝑒_𝑛^1/𝑛=1 (3.9) and moreover, lim_{𝑛→Λ,𝑛→∞}𝑎_𝑛^1/𝑛=2. (3.10) If 𝑒_𝑛≠𝑒_𝑛+1, then the polynomial 𝑝_𝑛(𝑧)𝑝∶=^∗𝑛(𝑧)𝑎_𝑛 (3.11) is monic and satisfies 𝑝_𝑛≤12−𝜀^𝑛 (3.12) for all 𝑛∈Λ which are sufficiently large, where 𝜀>0 can be chosen arbitrarily. Then the Erdős-Turán Theorem [18] (cf. [19]) implies a weak*-version of Kadec’s result, namely, the weak*-convergence of the normalized counting measures of alternation sets of 𝑓−𝑝^∗_𝑛 to the equilibrium measure 𝜇 of [−1,1], at least for a subsequence Λ, 𝑛∈Λ.

The objective of this section is to show that there exists a subsequence Λ⊂ℕ such that (3.4) and the analogue of (3.9) for rational approximation hold simultaneously with consequences for the behavior of the difference of two consecutive best approximants.

Lemma 3.1. Let 𝑓∈𝐶[−1,1] with (2.6). Then there exists a subsequence Λ⊂ℕ such that 𝐸_𝑛+1≤11−𝑛²𝐸_𝑛for𝑛∈Λ,lim_{𝑛∈Λ,𝑛→∞}𝐸_𝑛^1/𝑛=1. (3.13) Moreover, let {𝜉_𝑛}_𝑛∈Λ be a sequence in [−1,1] with |(𝑓−𝑟^∗_𝑛)(𝜉_𝑛)|=‖𝑓−𝑟^∗_𝑛‖; then lim_{𝑛∈Λ,𝑛→∞}𝑟^∗_𝑛−𝑟^∗_𝑛+1𝜉_𝑛^1/𝑛=1. (3.14)

Proof. Using the above arguments of the beginning of this section, there exists a subsequence Λ₁⊂ℕ such that 𝐸_𝑛+1≤11−𝑛²𝐸_𝑛for𝑛∈Λ₁. (3.15) First, we show that there exists Λ⊂ℕ such that (3.13) holds.
For proving this, we define Λ∶=𝑛∈ℕ∶𝐸_𝑛+1≤11−𝑛²𝐸_𝑛. (3.16) Since Λ₁⊂Λ, Λ≠∅, and Λ is not finite, hence the complementΛ^𝑐∶=ℕ⧵Λ (3.17) of Λ in ℕ has the property that 𝐸_𝑛+1>11−𝑛²𝐸_𝑛for𝑛∈Λ^𝑐. (3.18) If Λ^𝑐 is a finite set, then there exists 𝑚∈ℕ such that Λ∶={𝑛∈ℕ∶𝑛≥𝑚} (3.19) satisfies property (3.13).
If Λ^𝑐 is an infinite set, then observing that Λ is not a finite set, we can define subsequences {𝑚_𝑗}_𝑗∈ℕ and {𝑛_𝑗}_𝑗∈ℕ of ℕ such that 𝑛_𝑗−1<𝑚_𝑗≤𝑛_𝑗<𝑚_𝑗+1,Λ=𝑛∈ℕ∶𝑚_𝑗≤𝑛≤𝑛_𝑗,𝑗≥1. (3.20) Next, we consider a fixed integer 𝑚≥𝑚₁. If 𝑛_𝑗−1<𝑚<𝑚_𝑗,𝑗≥2, (3.21) then 𝑚∉Λ and we deduce 𝐸_𝑚𝑗>11−𝑚_𝑗−1²𝐸_𝑚𝑗−1>11−𝑚_𝑗−1²11−𝑚_𝑗−2²𝐸_𝑚𝑗−2>⋯>^{𝑚𝑗−𝑚−1}_𝑘=011−(𝑚+𝑘)²𝐸_𝑚. (3.22) Since the infinite product 𝑆=^∞_𝑛=211−𝑛² (3.23) converges, there exists a constant 𝛽, 0<𝛽<1, such that all partial products 𝑆_𝜈,𝜇∶=^𝜇_𝑛=𝜈11−𝑛²,2≤𝜈<𝜇, (3.24) of 𝑆 are bounded by 𝛽 from below, that is, 𝑆_𝜈,𝜇≥𝛽.
By (3.22), 𝐸_𝑚𝑗>𝛽𝐸_𝑚 and 𝐸^1/𝑚𝑗_𝑚𝑗≥𝐸_𝑚^1/𝑚_𝑗>𝛽^1/𝑚𝐸_𝑚^1/𝑚for𝐸_𝑚𝑗≤1. (3.25) Let us define for 𝑚≥𝑚₁𝜈(𝑚)Λ𝑚∶=𝑚,if𝑚∈_𝑗,if𝑛_𝑗−1<𝑚<𝑚_𝑗. (3.26)
Next, we choose a subsequence Λ₂={𝑘_𝑗}_𝑗∈ℕ of ℕ such that 𝑘₁≥𝑚₁ and lim_𝑗→∞𝐸^1/𝑘𝑗_𝑘𝑗=1. (3.27) If Λ₂⊂Λ, then we are done. As for the general case, let us define Λ∶=^∞_𝑗=1𝜈𝑘_𝑗; (3.28) then Λ⊂Λ and (3.25)–(3.27) imply lim_𝑗→∞𝐸^{1/𝜈(𝑘𝑗)}_{𝜈(𝑘𝑗)}=1. (3.29) Hence, (3.13) is proved.
Moreover, for 𝑛∈Λ, 𝑟^∗_𝑛−𝑟^∗_𝑛+1𝜉_𝑛≥𝑓−𝑟^∗_𝑛𝜉_𝑛−𝑓−𝑟^∗_𝑛+1𝜉_𝑛≥𝐸_𝑛−𝐸_𝑛+1≥𝐸_𝑛−11−𝑛²𝐸_𝑛=1𝑛²𝐸_𝑛,1≥limsup_𝑛→∞𝑟^∗_𝑛−𝑟^∗_𝑛+1𝜉_𝑛^1/𝑛≥limsup_{𝑛∈Λ,𝑛→∞}𝑟^∗_𝑛−𝑟^∗_𝑛+1𝜉_𝑛^1/𝑛≥limsup_{𝑛∈Λ,𝑛→∞}1𝑛^21/𝑛𝐸_𝑛^1/𝑛=lim_{𝑛∈Λ,𝑛→∞}𝐸_𝑛^1/𝑛=1. (3.30) Hence, lim_{𝑛∈Λ,𝑛→∞}𝑟^∗_𝑛−𝑟^∗_𝑛+1𝜉_𝑛^1/𝑛=1, (3.31) and (3.14) is proved.

4. Proofs

Proof of Theorem 2.2. First we will prove the theorem for 𝑎=0.
According to the lemma in Section 3, there exists a subsequence Λ⊂ℕ such that (3.13)–(3.14) hold. Then Theorem B applies and (3.1) holds for 𝑛∈Λ. Because 𝜈_𝑛 are probability measures on [−1,1], we may assume that 𝜈−−_𝑛^∗→𝜈as𝑛⟶∞,𝑛∈Λ. (4.1) Let 𝑧₀∈supp(𝜈) and 𝑈 a neighborhood of 𝑧₀ such that 𝑓(𝑧)≢0 on 𝑈∩[−1,1].
Let us assume that (ii) of Theorem 2.2 does not hold. Hence, there exists 𝑚∈ℕ𝑁_∞𝑟^∗_𝑛𝑁,𝑈≤𝑚∀𝑛∈ℕ,₀𝑟^∗_𝑛,𝑈=𝑜(𝑛)as𝑛→∞. (4.2)(4.3) Of course, we may assume that 𝑈 is a bounded symmetric region with respect to the real axis. Let 𝑙_𝑛 be the number of poles 𝜉_𝑛,𝑖 of 𝑟^∗_𝑛 in 𝑈 counted with their multiplicities. Then we define 𝑞_𝑛(𝑧)∶=^𝑙𝑛∏_𝑖=1𝑧−𝜉_𝑛,𝑖,𝑙_𝑛≥1,1,𝑙_𝑛=0. (4.4) Because 𝑞_𝑛,𝑞_𝑛+1∈𝒫_𝑚, there exists a subsequence Λ₁⊂Λ and 𝑞₀,𝑞₁∈𝒫_𝑚 such that lim_{𝑛∈Λ1,𝑛→∞}𝑞_𝑛+𝑖=𝑞_𝑖for𝑖=0,1. (4.5) Together with 𝑓(𝑧)≢0 for 𝑧∈𝑈∩[−1,1], this implies that there exists an interval [𝛼,𝛽]⊂𝑈∩[−1,1],𝛼≠𝛽, and a constant 𝜅>0 such that 𝑞_𝑖(𝑥𝑓(𝑥.)≥𝜅for𝑥∈𝛼,𝛽,𝑖=0,1,)≥𝜅for𝑥∈𝛼,𝛽 (4.6)(4.7) Let 𝑘_𝑛 be the number of zeros (with multiplicities) of 𝑟^∗_𝑛 in 𝑈. If 𝑘_𝑛≥1, let 𝜂_𝑛,𝑖, 1≤𝑖≤𝑘_𝑛, be the zeros of 𝑟^∗_𝑛 in 𝑈 and let 𝜋_𝑛(𝑧)∶=^𝑘𝑛∏_𝑖=1𝑧−𝜂_𝑛,𝑖,𝑘_𝑛>01,𝑘_𝑛=0. (4.8) Because of (4.3), 𝑘_𝑛=𝑜(𝑛) as 𝑛→∞ and we obtain limsup_𝑛→∞𝜋_𝑛𝐾^1/𝑛≤1 (4.9) for any compact set 𝐾 in ℂ. Now, let us define ℎ_𝑛(𝑧)1∶=𝑛logΦ_𝑛(𝑧) (4.10) with Φ_𝑛(𝑧)𝜋∶=^𝑛(𝑧)𝑟^∗_𝑛(𝑧)𝑞_𝑛(𝑧). (4.11) Then Φ_𝑛 is holomorphic in 𝑈 and ℎ_𝑛 harmonic in 𝑈.
Consider 𝑧∈[𝛼,𝛽] and Λ₁ as before. Then by (4.5)–(4.7) there exists 𝑛∈ℕ such that 𝑟^∗_𝑛+𝑖(𝑧≥𝜅)2,𝑞_𝑛+𝑖(𝑧≥𝜅)2 (4.12) for 𝑧∈[𝛼,𝛽], 𝑖=0,1, and 𝑛∈Λ₁, 𝑛≥𝑛. Then for 𝑖=0,1𝜋_𝑛+𝑖𝑟^∗_𝑛+𝑖𝑞_{𝑛+𝑖[𝛼,𝛽]}≤4(𝑑+1)^{𝑘𝑛+𝑖}𝜅², (4.13) where 𝑑=sup_𝑧∈𝑈|𝑧|. (4.14) According to a Lemma of Gončar [20, Lemma 1, page 153], for any compact set 𝐾⊂𝑈 there exists a constant 𝜆=𝜆([𝛼,𝛽],𝑈,𝐾)>1 such that 𝜋_𝑛+𝑖𝑟^∗_𝑛+𝑖𝑞_{𝑛+𝑖𝐾}≤𝜆^𝑛+𝑖𝜋_𝑛+𝑖𝑟^∗_𝑛+𝑖𝑞_{𝑛+𝑖[𝛼,𝛽]} (4.15) for 𝑖=0,1. For example, 𝜆([𝛼,𝛽],𝑈,𝐾) can be chosen as 𝜆𝛼,𝛽,𝑈,𝐾∶=max_𝑧∈𝐾sup_{𝑡∈ℂ⧵𝑈}exp𝐺_[𝛼,𝛽](𝑧,𝑡), (4.16) where 𝐺_[𝛼,𝛽](𝑧,𝑡) is the Green function of ℂ⧵[𝛼,𝛽] with pole at 𝑡.
Next, we choose a region 𝑊⊂𝑈, 𝑊 symmetric to the real axis, with 𝑧₀∈𝑊, 𝑊⊂𝑈 and [𝛼,𝛽]⊂𝑊, then ℎ_𝑛+𝑖(𝑧)≤𝜆𝛼,𝛽,𝑈,𝑊+14𝑛+𝑖log𝜅²+𝑘_𝑛+𝑖()𝑛+𝑖log1+𝑑 (4.17) for 𝑖=0,1. Hence for 𝑖=0,1, the sequences {ℎ_𝑛+𝑖}_𝑛∈Λ1 are uniformly bounded in 𝑊 from above as 𝑛→∞, 𝑛∈Λ₁, 𝑖=0,1. By Harnack’s theorem, either ℎ_𝑛(𝑧)⟶−∞locallyuniformlyin𝑊as𝑛⟶∞,𝑛∈Λ₁, (4.18) or there exists a subsequence Λ₂⊂Λ₁ such that {ℎ_𝑛}_𝑛∈Λ2 converges locally uniformly to ℎ₀ as 𝑛→∞, 𝑛∈Λ₂, in the region 𝑊 and the function ℎ₀ is harmonic in 𝑊.
Next, let us show that the first situation cannot occur: if 𝐶>0 is such that max_{𝑧∈[𝛼,𝛽]}ℎ_𝑛(𝑧)≤−𝐶 (4.19) for 𝑛∈Λ₁ and 𝑛 sufficiently large, then 𝑛⋅max_{𝑧∈[𝛼,𝛽]}ℎ_𝑛(𝑧)≤max_{𝑧∈[𝛼,𝛽]}𝜋log^𝑛(𝑧)𝑟^∗_𝑛(𝑧)𝑞_𝑛(𝑧)≤−𝑛𝐶. (4.20) Hence, by (4.5)–(4.7) there exists a constant 𝑐₁>0 such that max_{𝑧∈[𝛼,𝛽]}𝜋_𝑛(𝑧)≤𝑐₁𝑒^−𝑛𝐶. (4.21) Since 𝜋_𝑛∈𝒫_𝑘𝑛 is a monic polynomial and 𝑘_𝑛=𝑜(𝑛) as 𝑛→∞, this is a contradiction to 𝜋_{𝑛[𝛼,𝛽]}≥2𝛽−𝛼4^𝑘𝑛. (4.22) Next, we consider (4.17) for 𝑖=1. Again by Harnack’s theorem, either ℎ_𝑛+1(𝑧)⟶−∞locallyuniformlyin𝑊as𝑛⟶∞,𝑛∈Λ₂, (4.23) or there exists a subsequence Λ₃⊂Λ₂ such that {ℎ_𝑛+1}_𝑛∈Λ3 converges locally uniformly to a function ℎ₁ in 𝑊 and ℎ₁ is harmonic in 𝑊.
As above for {ℎ_𝑛}_𝑛∈Λ1, the first situation cannot occur. Consequently, max_{𝑧∈𝛼,𝛽}ℎ_𝑖(𝑧)≥0for𝑖=0,1. (4.24) On the other hand, using (4.13) we deduce for 𝑖=0,1 that limsup_{𝑛∈Λ1,𝑛→∞}max_{𝑧∈[𝛼,𝛽]}ℎ_𝑛+𝑖(𝑧)≤0. (4.25) Summarized, we have for 𝑖=0,1 that ℎ_𝑖(𝑧)≡0for𝑧∈𝛼,𝛽. (4.26) By definition, the regions 𝑈,𝑊 are symmetric to ℝ as well as the functions 𝑟^∗_𝑛+𝑖(𝑧),𝜋_𝑛+𝑖(𝑧),𝑞_𝑛+𝑖(𝑧) (4.27) for 𝑖=0,1. This symmetry, together with (4.26), implies that ℎ_𝑖(𝑧)≡0∀𝑧∈𝑊 (4.28) for 𝑖=0,1. Hence, lim_{𝑛∈Λ3,𝑛→∞}𝑟^∗_𝑛+𝑖𝑞_{𝑛+𝑖𝐾}^1/𝑛≤1 (4.29) for all compact sets 𝐾 in 𝑊, 𝑖=0,1.
Combining (4.29) for 𝑖=0,1, we obtain lim_{𝑛∈Λ3,𝑛→∞}𝑟^∗_𝑛−𝑟^∗_𝑛+1𝑞_𝑛𝑞_𝑛+1𝐾^1/𝑛≤1 (4.30) for all compact sets 𝐾⊂𝑊. Hence, the function 𝑉(𝑧)≡0 is a harmonic majorant for the sequence {𝐹_𝑛}_𝑛∈Λ3 of subharmonic functions in 𝑊, where 𝐹_𝑛(𝑧)1∶=𝑛log𝑟^∗_𝑛−𝑟^∗_𝑛+1(𝑧)𝑞_𝑛(𝑧)𝑞_𝑛+1(𝑧),𝑛∈ℕ. (4.31) Next, we want to show that 𝑉(𝑧)≡0 is an exact harmonic majorant for {𝐹_𝑛}_𝑛∈Λ3 and also for any {𝐹_𝑛}_𝑛∈Λ4 for any subsequence Λ₄⊂Λ₃.
Let us assume that this assertion would be false: then there exists a subsequence Λ₄⊂Λ₃⊂Λ (Λ as in the Corollary of Section 3) and a continuum 𝐾⊂𝑊 such that limsup_{𝑛∈Λ4,𝑛→∞}max_𝑧∈𝐾𝐹_𝑛(𝑧)<0. (4.32) Since 𝑉(𝑧)≡0 is a harmonic majorant for {𝐹_𝑛}_𝑛∈Λ4 in 𝑊, then (4.32) implies that the inequality (4.32) holds for any continuum 𝐾⊂𝑊.
First, let us note that under the condition (4.2) a point 𝜉∈𝑈∩[−1,1] cannot be an isolated point of supp(𝜈).
To prove this, let us denote by 𝛿_𝑧 the Dirac measure of the point 𝑧∈ℂ, and let 𝛿_𝑧 be the associated balayage measure of 𝛿_𝑧 to the interval [−1,1]. For 𝑧∉[−1,1] the density of the balayage measure 𝛿_𝑧 at the point 𝑥∈(−1,1) is given by 𝑑𝛿𝑑𝑥^𝑧(𝑥)=𝜕𝜕_𝑛+𝐺()+𝜕𝑥,𝑧𝜕_𝑛−𝐺(𝑥,𝑧), (4.33) where 𝑛₊ (resp., 𝑛₋) denotes the normal at the point 𝑥 to the upper half (resp., lower half) plane and 𝐺(𝜉,𝑧) is the Green function for 𝜉∈ℂ⧵[−1,1] with pole at 𝑧, continuously extended by 𝐺(𝑥,𝑧)=0 to 𝜉=𝑥∈[−1,1].
Then for any interval [𝛼,𝛽]⊂[−1,1]0≤𝛿_𝑧𝛼,𝛽≤1,lim_{𝑧→𝜂𝜂∈[−1,1]⧵[𝛼,𝛽]}𝛿_𝑧𝛼,𝛽=0. (4.34) Let 𝑧∈ℂ⧵[−1,1], 𝜉∈𝑈∩[−1,1], and 𝜀>0; then lim_𝜀→0𝛿_𝑧𝜉−𝜀,𝜉+𝜀=0. (4.35) Consider the exterior of the 𝜀-neighborhood of [−1,1]; that is, let 𝑊_𝜀∶=𝑧∈ℂ∶dist(𝑧,[−1,1])≥𝜀. (4.36) Then we can obtain a sharpening of (4.35), namely, lim_𝜀→0max_{𝑧∈𝑊𝜀⧵𝑈}𝛿_𝑧𝜉−𝜀,𝜉+𝜀=0. (4.37) Since 𝜉∈𝑈∩[−1,1] and (4.2) holds, (4.34)–(4.37) imply lim_𝜀→0lim_𝑛→∞𝜏_𝑛𝜉−𝜀,𝜉+𝜀=0. (4.38) Because (3.1) and (4.1) hold for 𝑛∈Λ, 𝜉 cannot be an isolated point of supp(𝜈).
Consequently, since 𝑧₀∈supp(𝜈) there exists a sequence {𝜉_𝑘}_𝑘∈ℕ in 𝑈, 𝜉_𝑘∈supp(𝜈), such that 𝑧₀=lim_𝑘→∞𝜉_𝑘 (4.39) and each 𝜉_𝑘 is not an isolated point of supp(𝜈). Hence, for any 𝑘∈ℕ and any open interval (𝛼,𝛽) with 𝜉_𝑘∈(𝛼,𝛽) we have 𝜈((𝛼,𝛽))>0. Taking into account (4.39) and the fact that the zero set 𝑍∶=𝑧∈ℂ∶𝑝₀(𝑧)=0or𝑝_𝑖(𝑧)=0 (4.40) of the polynomials 𝑝₀, 𝑝₁ in (4.5) is finite, there exists an interval [𝛼,𝛽]⊂𝑈∩[−1,1], 𝛼<𝛽, with 𝜈𝛼,𝛽>0,𝛼,𝛽∩𝑍=∅. (4.41) Using (4.5) we conclude that there exists 𝑛₁∈ℕ and a constant 𝜅>0 such that 𝑞_𝑛+𝑖(𝑧)≥𝜅for𝑧∈𝛼,𝛽, (4.42) where 𝑛∈Λ₁, 𝑛≥𝑛₁, and 𝑖=0,1.
Let us choose for 𝐾 in (4.32) the interval [𝛼,𝛽]. Then there exists, by definition of 𝐹_𝑛(𝑧) in (4.31), a constant 𝛿, 0<𝛿<1, and 𝑛₂∈ℕ, 𝑛₂≥𝑛₁, such that max_𝑧∈[𝛼_,𝛽_]𝑟^∗_𝑛−𝑟^∗_𝑛+1(𝑧)𝑞_𝑛(𝑧)𝑞_𝑛+1(𝑧)≤𝛿^𝑛. (4.43) for all 𝑛∈Λ₄, 𝑛≥𝑛₂. By (4.42) we obtain max_𝑧∈[𝛼_,𝛽_]𝑟^∗_𝑛−𝑟^∗_𝑛+1(𝑧≤𝛿)^𝑛𝜅²,limsup_{𝑛∈Λ4,𝑛→∞}𝑟^∗_𝑛−𝑟^∗_𝑛+1[^1/𝑛𝛼_,𝛽_]≤𝛿<1 (4.44)(4.45) contradicting the property (3.14) and 𝜈([𝛼,𝛽])>0.
Hence, 𝑉(𝑧)≡0 is an exact harmonic majorant for {𝐹_𝑛}_𝑛∈Λ3 and for any subsequence {𝐹_𝑛}_𝑛∈Λ4, Λ₄⊂Λ₃, in the region 𝑊.
This is now the situation that a distribution result of Walsh about the zeros of the sequence 𝑟^∗_𝑛−𝑟^∗_𝑛+1𝑞_𝑛𝑞_{𝑛+1𝑛∈Λ3} (4.46) of holomorphic functions in 𝑊 can be applied (Walsh [21], Theorem 16, page 221): for every compact set 𝐾 in 𝑊 we have 𝑁₀𝑟^∗_𝑛−𝑟^∗_𝑛+1𝑞_𝑛𝑞_𝑛+1,𝐾=𝑜(𝑛)as𝑛∈Λ₃,𝑛⟶∞. (4.47) Choosing for 𝐾 the interval [𝛼,𝛽], then the number of alternations of 𝑓−𝑟^∗_𝑛 in [𝛼,𝛽] is a lower bound for the number 𝑁₀𝑟^∗_𝑛−𝑟^∗_𝑛+1𝑞_𝑛𝑞_𝑛+1,𝛼,𝛽 (4.48) of zeros of (𝑟^∗_𝑛−𝑟^∗_𝑛+1)𝑞_𝑛𝑞_𝑛+1 in [𝛼,𝛽]. Because of (4.1) and 𝜈([𝛼,𝛽])>0, lim_{𝑛∈Λ3,𝑛→∞}𝜈_𝑛𝛼,𝛽=𝜈𝛼,𝛽>0 (4.49) which contradicts (4.47).
Hence, the theorem is proved for 𝑎=0. The case 𝑎≠0 can be reduced to 𝑎=0 by defining 𝑓(𝑧)(𝑧)[],(𝑧)(𝑧)∶=𝑓+𝑎,𝑧∈−1,1𝑟̃∶=𝑟+𝑎for𝑟∈ℛ_𝑛,𝑛,𝑧∈ℂ. (4.50)(4.51) If 𝑎∈ℂ, we note that the inequality (4.30) is equivalent to lim_{𝑛∈Λ3,𝑛→∞}(𝑟^∗_𝑛̃−𝑟̃^∗_𝑛+1)𝑞_𝑛𝑞_𝑛+1𝐾^1/𝑛≤1 (4.52) and (3.14) is equivalent to lim_{𝑛∈Λ,𝑛→∞}𝑟^∗_𝑛̃−𝑟̃^∗_𝑛+1𝜉_𝑛^1/𝑛=1, (4.53) where {𝜉_𝑛}_𝑛∈Λ, 𝜉_𝑛∈[−1,1], and |(𝑓−𝑟^∗_𝑛̃)(𝜉_𝑛)|=‖𝑓−𝑟^∗_𝑛̃‖. Therefore, all arguments for the sequence {𝐹_𝑛} are invariant by replacing in definition (4.10) the functions 𝑟^∗_𝑛,𝑟^∗_𝑛+1 by 𝑟^∗_𝑛̃,𝑟̃^∗_𝑛+1. Hence, Theorem 2.2 is true for all 𝑎∈ℂ.

Proof of the Corollary. In the proof of Theorem 2.2, the subsequence Λ was chosen such that 𝜈_𝑛−𝛼_𝑛𝜏_𝑛+𝜏_𝑛+1−1−𝛼_𝑛𝜇−→^∗0as𝑛⟶∞,𝑛∈Λ, (4.54) where 𝛼_𝑛=deg𝑞^∗_𝑛+deg𝑞^∗_𝑛+1𝑑_𝑛.+1 (4.55) Since {𝑚_𝑛} fulfills (2.17), we obtain 𝛼_𝑛=deg𝑞^∗_𝑛+deg𝑞^∗_𝑛+1𝑛+𝑚_𝑛+1−𝛿_𝑛≤deg𝑞^∗_𝑛+deg𝑞^∗_𝑛+1𝑛+𝑚_𝑛+1−𝑚_𝑛−deg𝑞∗_𝑛=1−𝑛+1−deg𝑞^∗_𝑛+1𝑛+1+deg𝑞^∗_𝑛()<1−𝑛+1−𝑐𝑛+1𝑛+1+𝑐(𝑛+1)=1−1−𝑐.1+𝑐 (4.56) Hence, by (3.1) 𝜈_𝑛𝛼,𝛽≥1−𝑐𝜇1+𝑐𝛼,𝛽 (4.57) for any interval [𝛼,𝛽]⊂𝐶[−1,1]. Therefore, property (i) of Theorem 2.2 implies that 𝜈−−_𝑛^∗→𝜈,supp(𝜈)=[−1,1] (4.58) and Theorem 2.2 holds for all 𝑧₀∈[−1,1].

5. Generalization to the Lower-Half of the Walsh Table

Theorem 2.2 restricts the approximation to the upper half of the Walsh table. In the following, we also want to allow approximations in the lower half of the Walsh table. We assume that the pairs (𝑛(𝑠),𝑚(𝑠))∈ℕ₀×ℕ₀ (5.1) depend on parameters 𝑠∈ℕ. For abbreviation, let 𝐸_𝑠∶=𝐸_{𝑛(𝑠),𝑚(𝑠)}𝑓,𝑟^∗_𝑠=𝑟^∗_{𝑛(𝑠)(𝑠),𝑚}𝑓=𝑝^∗_𝑠𝑞^∗_𝑠, (5.2) where 𝑝^∗_𝑠 and 𝑞^∗_𝑠 have no common factor. As above, let 𝛿_𝑠∶=min𝑛(𝑠)−deg𝑝^∗_𝑠,𝑚(𝑠)−deg𝑝^∗_𝑠 (5.3) be the defect of 𝑟^∗_𝑠, and let 𝐴_𝑠=𝐴_𝑠(𝑓)={𝑥_𝑘^(𝑠)}^𝑑(𝑠)_𝑘=0 be an alternation point set to 𝑓−𝑟^∗_𝑠, where 𝑑_𝑠=𝑛(𝑠)+𝑚(𝑠)+1−𝛿_𝑠. (5.4) We denote by 𝜈_𝑠 the normalized counting measure of 𝐴_𝑠. Then Theorem 2.2 can be generalized in the following way.

Theorem 5.1. Let (𝑛(𝑠),𝑚(𝑠)), 𝑠∈ℕ, be a strictly increasing subsequence of ℕ₀×ℕ₀ with 𝑛(𝑠)≤𝑛(𝑠+1)≤𝑛(𝑠)+1,𝑚(𝑠)≤𝑚(𝑠+1)≤𝑚(𝑠)+1, (5.5) and let us approximate 𝑓∈𝐶[−1,1], with respect to ℛ_{𝑛(𝑠),𝑚(𝑠)}, where 𝑚(𝑠)(𝑠)(𝑠)𝜅(𝑠)𝑠≤𝑛+𝜅,𝑠∈ℕ,=𝑜log𝑠as𝑠⟶∞. (5.6) If 𝑓∈𝐶[−1,1] satisfies (2.6), then there exists a subset Λ⊂ℕ with the following properties: (i)𝜈−→_𝑠^∗𝜈 as 𝑠→∞, 𝑠∈Λ. (ii)let a∈ℂ; then for any 𝑧₀∈supp(𝜈) and any neighborhood 𝑈 of 𝑧₀ with 𝑓(𝑧)≢𝑎 on 𝑈∩[−1,1] either limsup_𝑠→∞𝑁_∞𝑟^∗_𝑠,𝑈=∞ (5.7)or limsup_𝑠→∞𝑁_𝑎𝑟^∗_𝑠,𝑈𝑠>0. (5.8)

For the proof, we use a generalization of Theorem B to the previous situation (see [10]): if (5.5) and (5.6) hold, then there exists a subsequence Λ⊂ℕ such that 𝜈_𝑠−𝛼_𝑠𝜏_𝑠+𝜏_𝑠+1−1−𝛼_𝑠𝜇−−^∗→0as𝑠⟶∞,𝑠∈Λ. (5.9) Again, we use in (5.9) the balayage measures of the normalized pole counting measures 𝜏_𝑠 and 𝜏_𝑠+1 of 𝑟^∗_𝑠, respectively, 𝑟^∗_𝑠+1, onto [−1,1] and 𝛼_𝑠∶=deg𝑞^∗_𝑠+deg𝑞^∗_𝑠+1𝑑_𝑠.+1 (5.10) Then the proof of (5.7) and (5.8) follows the same lines as the proof of Theorem 2.2 if limsup_𝑠→∞𝐸_𝑠^1/𝑠=1. (5.11) Because of (5.5), the index 𝑛(𝑠) runs from 𝑛(1) to ∞. Moreover, let 𝑀(𝑠)∶=max(𝑛(𝑠),𝑚(𝑠)), 𝑠∈ℕ; then 𝑀(𝑠) runs from 𝑀(1) to ∞ and limsup_𝑠→∞𝐸_𝑠^1/𝑠=limsup_𝑠→∞𝐸_{𝑛(𝑠)(𝑠)}^1/𝑠_,𝑚≥limsup_𝑠→∞𝐸^1/𝑀(𝑠)_{𝑀(𝑠),𝑀(𝑠)}^{𝑀(𝑠)/𝑠}=1, (5.12) since 𝑠≥𝑀(𝑠)−𝑀(1). (5.13)

6. Remarks

For the function 𝑓(𝑥)=|𝑥|^𝛼, 𝛼>0, the distribution of alternation points of the optimal error curves, as well as the zeros and poles of 𝑟^∗_𝑛,𝑚 is very well investigated [7].

Let 𝛼∈ℝ₊⧵2ℕ, and let (𝑛,𝑚_𝑛)∈ℕ×ℕ with lim_𝑛→∞𝑚_𝑛𝑛=𝑐≤1,𝑛≥𝑚_𝑛𝛼+22. (6.1) Since all best approximants of 𝑓(𝑥)=|𝑥|^𝛼 are even functions, we can assume that 𝑛,𝑚_𝑛∈ℕ are even. Moreover, the error function 𝑓−𝑟^∗_{𝑛,𝑚𝑛} has always exactly 𝑛+𝑚_𝑛+3 points [7]. By 𝜈_𝐴𝑛=𝜈_𝑛 we denote the normalized alternation counting measure and 𝜈_𝑃𝑛 denotes the normalized pole counting measure of 𝑟^∗_{𝑛,𝑚𝑛} and 𝜈_𝑍𝑛 the normalized zero counting measure of 𝑟^∗_{𝑛,𝑚𝑛}. Then 𝜈→_𝐴𝑛^∗_𝑛→∞2𝑐𝛿1+𝑐⁰+1−𝑐𝜈1+𝑐𝜇,_𝑃𝑛∗−−→_𝑛→∞𝛿₀,𝜈−−_𝑍𝑛^∗→_𝑛→∞𝑐𝛿₀+(1−𝑐)𝜇 (6.2)(6.3)(6.4) (cf. Theorems 1.6 and 1.7 in [7]).

For 𝑐<1, we would obtain by (3.1) and by the corollary of Theorem 2.2 that any point of [−1,1] is either a limit point of poles or of 𝑎-values of 𝑟^∗_{𝑛,𝑚𝑛}, 𝑎∈ℂ, as 𝑛→∞. Since by (6.3) the normalized pole counting measures converge to the Dirac measure at 0, any point of [−1,1], with 0 as only possible exception, is a limit point of 𝑎-values.

For 𝑐=1, 𝜈−→_𝐴𝑛^∗𝛿₀. Hence Theorem 2.2 can only tell us that the point 0 is either a limit point of poles or of 𝑎-values, 𝑎∈ℂ. But (6.3) and (6.4) show that 0 is as well a limit point of zeros as of poles of 𝑟^∗_{𝑛,𝑚𝑛}. Hence, the investigations in [7] for the special functions 𝑓(𝑥)=|𝑥|^𝛼 lead to deeper results for the zeros and poles of the best approximants.

But the example of 𝑓(𝑥)=|𝑥|^𝛼 shows an interesting area for further investigations, namely, a weak*-type analogue of relation (3.1) for the distribution of zeros, respectively, 𝑎-values, and poles of rational approximation would be desirable. Moreover, the approximation problem should be moved from the interval [−1,1] to more general compact sets 𝐸 in ℂ.