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

Authors: V. V. Andrievskii; H.-P. Blatt; R. K. Kovacheva
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The distribution of zeros and poles of best rational approximants is well understood for the functionsf(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.


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## Body

## 1.  Introduction

LetB be a subset of ℂ; we denote by (1.1)m1(B):=inf⁡⁡∑ν‍|Uν|
the m1-measure ofB, where the infimum is taken over all coverings {Uν} of B by disks Uν and |Uν| is the radius of the disk Uν.LetD be a region in ℂ and φ a function defined in D with values in ℂ¯. A sequence {φn}n∈ℕ of meromorphic functions in D is said to converge to a functionφwith respect to them1-measure  insideD if for every ɛ>0 and any compact set K⊂D we have
(1.2)m1({z∈K:|(φ-φn)(z)|≥ɛ})⟶0asn⟶∞
(cf. Gončar [1]).The sequence{φn}n∈ℕ is said to converge to φm1-almost geometrically insideD if for any ɛ>0 there exists a set Ω(ɛ) in ℂ with m1(Ω(ɛ))<ɛ such that
(1.3)limsup⁡n→∞‖φ-φn‖K∖Ω(ɛ)1/n<1
for any compact set K⊂D. We note that ∥·∥B is the supremum norm on a subset B of ℂ.Forn∈ℕ0=ℕ∪{0}, we denote by 𝒫n the collection of all polynomials of degree at most n, and let
(1.4)Rn,m∶={r=pq:p∈Pn,q∈Pm,q≢0}.
In [2], sequences {rn}n∈ℕ, rn∈ℛn,n, on a region D were investigated if the number of poles of rn in D is bounded. It turns out that the geometric convergence of {rn}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, letB⊂ℂ and let ℳm(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.
LetS be a continuum in ℂ and D a region with S⊂D. Let {rn}n∈ℕ, rn∈ℛn,n, be a sequence of rational functions converging geometrically to a function f on S, that is,
(1.5)limsup⁡n→∞‖f-rn‖S1/n<1,
and assume that f≢0 on S. If there exists a fixed integer m∈ℕ such that rn∈ℳm(D) for all n and
(1.6)N0(rn,K)=o(n)asn⟶∞
for each compact set K⊂D, then the sequence {rn}n∈ℕ converges m1-almost geometrically inside D to a meromorphic function f∈ℳm(D).Here, the numberN0(rn,K) denotes the number of zeros of rn 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
(1.7)Eρ∶={z∈C:G(z,∞)<log⁡ρ},ρ>1,
be the Green domain to the parameter ρ, that is, ℰρ is the open Joukowski-ellipse with foci at +1 and -1 and major axis ρ+1/ρ.Letf∈C[-1,1] be real-valued on [-1,1]. For abbreviation, we will write ∥·∥ for ∥·∥[-1,1]. Given n,m∈ℕ0, let rn,m*=rn,m*(f)∈ℛn,m denote the real rational function of best uniform approximation to f∈C[-1,1] with respect to ℛn,m, that is,
(1.8)En,m(f)∶=‖f-rn,m*‖=inf⁡⁡{‖f-r‖:r∈Rn,m,rreal-valuedon⁡R}.
Moreover, let {mn}n∈ℕ be a sequence in ℕ with
(1.9)lim⁡n→∞mn=∞,mn=o(nlog⁡n)asn⟶∞,
and let us consider a function f∈C[-1,1] that can be continued meromorphically into ℰρ for some ρ>1. Then the sequence {rn,mn*}n∈ℕ converges m1-almost geometrically inside ℰρ to f [3]. Using Theorem A, we obtain results about the distribution of the a-values in the neighborhood of a point z0∈∂Eρ. For a∈ℂ¯ and B⊂ℂ, we denote by
(1.10)Na(r,B)∶=#{z∈B:r(z)=a}
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,
(1.11)limsup⁡n→∞Na(rn,mn*,U)=∞.
Particulary, such a point z0 is either an accumulation point of zeros or of poles of rn,mn*.On the other hand, iff is not holomorphic on [-1,1], so far results about the distribution of the zeros of rn,mn*(f) are only known in the case that mn=0 for all n∈ℕ (polynomial approximation) or in the case that mn=m∈ℕ is fixed (rational approximation with a bounded number of free poles). In the polynomial case, the normalized zero counting measures of rn,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 ofrn,mn*, the behavior of the alternation points of f-rn,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

Letf 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
(2.1)f(z)=∑n=1∞‍Anz-αn,
where the points αn are situated in ℂ∖[-1,1] such that any point of [-1,1] is a limit point of the sequence {αn} and the coefficients An converge to zero sufficiently fast. Hence, it is possible that there exists a sequence {rn}n∈ℕ, rn∈ℛn,n, such that
(2.2)limsup⁡n→∞‖f-rn‖1/n<1,
and f is continuous on [-1,1], but nowhere holomorphic on [-1,1].But it turns out that in this case TheoremA immediately yields the following.Theorem 2.1.
Letf∈C[-1,1] be not holomorphic on [-1,1], and let {rn}n∈ℕ, rn∈ℛn,n, be a sequence such that
(2.3)limsup⁡n→∞‖f-rn‖1/n<1.
Then for any non holomorphic point z0∈[-1,1] of f any neighborhood U of z0 either
(2.4)limsup⁡n→∞N∞(rn,U)=∞
or
(2.5)limsup⁡n→∞Na(rn,U)n>0
for all a∈ℂ.In the following we consider functionsf∈C[-1,1] that are always real-valued on [-1,1]. Then the case that
(2.6)limsup⁡n→∞En,n1/n(f)=1
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{mn}n∈ℕ is a sequence with
(2.7)mn≤n,mn≤mn+1≤mn+1.
For abbreviation, let
(2.8)En∶=En,mn(f),rn*∶=rn,mn*(f)=pn*qn*,
where pn*∈𝒫n and qn*∈𝒫mn have no common factor. We define
(2.9)δn∶=min⁡(n-deg⁡pn*,mn-deg⁡qn*)
as the defect of rn* and dn∶=n+mn+1-δn. According to the alternation theorem of Chebyshev (cf. Meinardus [17], Theorem 98) there exist dn+1 points xk(n),
(2.10)-1≤x0(n)<x1(n)<⋯<xdn(n)≤1,
which satisfy
(2.11)λn(-1)k(f-rn*)(xk(n))=‖f-rn*‖[-1,1],0≤k≤dn,
where λn=+1 or λn=-1 is fixed. For each pair (n,mn) let
(2.12)An=An(f)∶={xk(n)}k=0dn
denote an arbitrary, but fixed alternation set for the best approximation rn*∈ℛn,mn, and let νn denote the normalized counting measure of An, that is,
(2.13)νn([α,β])∶=#{xk(n):α≤xk(n)≤β}dn+1
for any interval [α,β]⊂[-1,1]. Since νn is a probability measure on [-1,1], there exists a subsequence Λ⊂ℕ such that
(2.14)νn→*νasn⟶∞,n∈Λ,
in the weak*-topology and ν is again a probability measure on [-1,1].Theorem 2.2.
Letf∈C[-1,1] be real-valued, and let (2.6) hold. Moreover, let f be approximated with respect to ℛn,mn, where the sequence {mn}n∈ℕ satisfies (2.7). Then there exists a subsequence Λ⊂ℕ with the following properties:  (i)
(2.15)νn→*νasn⟶∞,n∈Λ,(ii)
letz0∈supp(ν), a∈ℂ, and let U be a neighborhood of z0 with f(z)≢a on U∩[-1,1]; then
(2.16)eitherlimsupn∈Λ,n→∞N∞(rn*,U)=∞orlimsup⁡n∈Λ,n→∞Na(rn*,U)n>0.Applying to the approximation in the upper half of the Walsh table, we obtain the following.Corollary 2.3.
Letf∈C[-1,1] with (2.6) and let the subsequence {mn}n∈ℕ satisfy
(2.17)mn≤cnwith0≤c<1,mn≤mn+1≤mn+1.
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 Theorem2.2 is the interaction between alternation points and poles of best rational approximants.Letτn denote the normalized counting measure of the poles of rn*, counted with their multiplicities, and let us denote by τ^n the balayage measure of τn onto [-1,1]. Then the following distribution results hold for the interaction between the alternation points of An and the poles of rn* and rn+1*.Theorem B (See [11]).
Letf be not a rational function, and let {mn}n∈ℕ satisfy (2.7). Then there exists a subsequence Λ⊂ℕ such that
(3.1)νn-αn(τ^n+τ^n+1)-(1-αn)μ→*0asn⟶∞,n∈Λ,
where
(3.2)αn=deg⁡qn*+deg⁡qn+1*dn+1
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
(3.3)Λ∶={n∈N:En+En+1En-En+1≤n2}.
Inspecting the proof of (3.1) in [11], it turns out that we can modify the definition of Λ by
(3.4)Λ∶={n∈N:En+1≤(1-1n2)En}.
The existence of such sequences Λ is based on the divergence of the infinite product
(3.5)∏n=0∞‍En+1En=∏n=0∞‍(1-En-En+1En)
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 approximationspn* to f,
(3.6)pn*(z)=anzn+⋅⋅⋅,
the asymptotic behavior of the highest coefficient an plays an essential role, namely,
(3.7)limsup⁡n→∞|an|1/n=1cap([-1,1])limsup⁡n→∞⁡en1/n,
where
(3.8)en=‖f-pn*‖=inf⁡pn∈Pn‖f-pn‖
and cap([-1,1])=1/2 is the logarithmic capacity of [-1,1].Iff∈C[-1,1] is not holomorphic on [-1,1], then limsupn→∞en1/n=1 and we can choose a subsequence Λ⊂ℕ such that
(3.9)lim⁡n∈Λ,n→∞en1/n=1
and moreover,
(3.10)lim⁡n→Λ,n→∞|an|1/n=2.
If en≠en+1, then the polynomial
(3.11)pn(z)∶=pn*(z)an
is monic and satisfies
(3.12)‖pn‖≤(12-ɛ)n
for all n∈Λ which are sufficiently large, where ɛ>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-pn* 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 subsequenceΛ⊂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.
Letf∈C[-1,1] with (2.6). Then there exists a subsequence Λ⊂ℕ such that
(3.13)En+1≤(1-1n2)Enforn∈Λ,lim⁡n∈Λ,n→∞En1/n=1.
Moreover, let {ξn}n∈Λ be a sequence in [-1,1] with |(f-rn*)(ξn)|=∥f-rn*∥; then
(3.14)lim⁡n∈Λ,n→∞|(rn*-rn+1*)(ξn)|1/n=1.Proof.
Using the above arguments of the beginning of this section, there exists a subsequenceΛ1⊂ℕ such that
(3.15)En+1≤(1-1n2)Enforn∈Λ1.
First, we show that there exists Λ⊂ℕ such that (3.13) holds.
For proving this, we define(3.16)Λ~∶={n∈N:En+1≤(1-1n2)En}.
Since Λ1⊂Λ~, Λ~≠∅, and Λ~ is not finite, hence the complement(3.17)Λ~c∶=N∖Λ~
of Λ~ in ℕ has the property that
(3.18)En+1>(1-1n2)Enforn∈Λ~c.
If Λ~c is a finite set, then there exists m∈ℕ such that
(3.19)Λ∶={n∈N:n≥m}
satisfies property (3.13).
IfΛ~c is an infinite set, then observing that Λ~ is not a finite set, we can define subsequences {mj}j∈ℕ and {nj}j∈ℕ of ℕ such that
(3.20)nj-1<mj≤nj<mj+1,Λ~={n∈N:mj≤n≤nj,j≥1}.
Next, we consider a fixed integer m≥m1. If
(3.21)nj-1<m<mj,j≥2,
then m∉Λ~ and we deduce
(3.22)Emj>(1-1(mj-1)2)Emj-1>(1-1(mj-1)2)(1-1(mj-2)2)Emj-2>⋯>∏k=0mj-m-1‍(1-1(m+k)2)Em.
Since the infinite product
(3.23)S=∏n=2∞‍(1-1n2)
converges, there exists a constant β, 0<β<1, such that all partial products
(3.24)Sν,μ∶=∏n=νμ‍(1-1n2),2≤ν<μ,
of S are bounded by β from below, that is, Sν,μ≥β.
By (3.22), Emj>βEm and
(3.25)Emj1/mj≥Emj1/m>β1/mEm1/mforEmj≤1.
Let us define for m≥m1(3.26)ν(m)∶={m,ifm∈Λ~mj,ifnj-1<m<mj.
Next, we choose a subsequenceΛ2={kj}j∈ℕ of ℕ such that k1≥m1 and
(3.27)lim⁡j→∞Ekj1/kj=1.
If Λ2⊂Λ~, then we are done. As for the general case, let us define
(3.28)Λ∶=⋃j=1∞‍{ν(kj)};
then Λ⊂Λ~ and (3.25)–(3.27) imply
(3.29)lim⁡j→∞Eν(kj)1/ν(kj)=1.
Hence, (3.13) is proved.
Moreover, forn∈Λ,
(3.30)|(rn*-rn+1*)(ξn)|≥|(f-rn*)(ξn)|-|(f-rn+1*)(ξn)|≥En-En+1≥En-(1-1n2)En=1n2En,1≥limsup⁡n→∞|(rn*-rn+1*)(ξn)|1/n≥limsup⁡n∈Λ,n→∞|(rn*-rn+1*)(ξn)|1/n≥limsup⁡n∈Λ,n→∞((1n2)1/nEn1/n)=lim⁡n∈Λ,n→∞En1/n=1.
Hence,
(3.31)lim⁡n∈Λ,n→∞|(rn*-rn+1*)(ξn)|1/n=1,
and (3.14) is proved.


## 4.   Proofs

Proof of Theorem2.2.
First we will prove the theorem fora=0.
According to the lemma in Section3, 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 νn are probability measures on [-1,1], we may assume that
(4.1)νn→*νasn⟶∞,n∈Λ.
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 Theorem2.2 does not hold. Hence, there exists m∈ℕ(4.2)N∞(rn*,U)≤m∀n∈N,(4.3)N0(rn*,U)=o(n)asn→∞.
Of course, we may assume that U is a bounded symmetric region with respect to the real axis. Let ln be the number of poles ξn,i of rn* in U counted with their multiplicities. Then we define
(4.4)qn(z)∶={∏i=1ln‍(z-ξn,i),ln≥1,1,ln=0.
Because qn,qn+1∈𝒫m, there exists a subsequence Λ1⊂Λ and q~0,q~1∈𝒫m such that
(4.5)lim⁡n∈Λ1,n→∞qn+i=q~ifori=0,1.
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
(4.6)|q~i(x)|≥κforx∈[α,β],i=0,1,(4.7)|f(x)|≥κforx∈[α,β].
Let kn be the number of zeros (with multiplicities) of rn* in U. If kn≥1, let ηn,i,  1≤i≤kn, be the zeros of rn* in U and let
(4.8)πn(z)∶={∏i=1kn‍(z-ηn,i),kn>01,kn=0.
Because of (4.3), kn=o(n) as n→∞ and we obtain
(4.9)limsup⁡n→∞‖πn‖K1/n≤1
for any compact set K in ℂ. Now, let us define
(4.10)hn(z)∶=1nlog⁡|Φn(z)|
with
(4.11)Φn(z)∶=πn(z)rn*(z)qn(z).
Then Φn is holomorphic in U and hn harmonic in U.
Considerz∈[α,β] and Λ1 as before. Then by (4.5)–(4.7) there exists n~∈ℕ such that
(4.12)|rn+i*(z)|≥κ2,|qn+i(z)|≥κ2
for z∈[α,β], i=0,1, and n∈Λ1, n≥n~. Then for i=0,1(4.13)‖πn+irn+i*qn+i‖[α,β]≤4(d+1)kn+iκ2,
where
(4.14)d=sup⁡z∈U|z|.
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
(4.15)‖πn+irn+i*qn+i‖K≤λn+i‖πn+irn+i*qn+i‖[α,β]
for i=0,1. For example, λ([α,β],U,K) can be chosen as
(4.16)λ([α,β],U,K)∶=max⁡z∈Ksup⁡t∈C¯∖Uexp⁡(G[α,β](z,t)),
where G[α,β](z,t) is the Green function of ℂ¯∖[α,β] with pole at t.
Next, we choose a regionW⊂U, W symmetric to the real axis, with z0∈W, W¯⊂U and [α,β]⊂W, then
(4.17)hn+i(z)≤λ([α,β],U,W¯)+1n+ilog⁡4κ2+kn+in+ilog⁡(1+d)
for i=0,1. Hence for i=0,1, the sequences {hn+i}n∈Λ1 are uniformly bounded in W from above as n→∞, n∈Λ1, i=0,1. By Harnack’s theorem, either
(4.18)hn(z)⟶-∞locallyuniformlyinWasn⟶∞,n∈Λ1,
or there exists a subsequence Λ2⊂Λ1 such that {hn}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: ifC>0 is such that
(4.19)max⁡z∈[α,β]hn(z)≤-C
for n∈Λ1 and n sufficiently large, then
(4.20)n⋅max⁡z∈[α,β]|hn(z)|≤max⁡z∈[α,β]log⁡|πn(z)rn*(z)qn(z)|≤-nC.
Hence, by (4.5)–(4.7) there exists a constant c1>0 such that
(4.21)max⁡z∈[α,β]|πn(z)|≤c1e-nC.
Since πn∈𝒫kn is a monic polynomial and kn=o(n) as n→∞, this is a contradiction to
(4.22)‖πn‖[α,β]≥2((β-α)4)kn.
Next, we consider (4.17) for i=1. Again by Harnack’s theorem, either
(4.23)hn+1(z)⟶-∞locallyuniformlyinWasn⟶∞,n∈Λ2,
or there exists a subsequence Λ3⊂Λ2 such that {hn+1}n∈Λ3 converges locally uniformly to a function h1 in W and h1 is harmonic in W.
As above for{hn}n∈Λ1, the first situation cannot occur. Consequently,
(4.24)max⁡z∈[α,β]hi(z)≥0fori=0,1.
On the other hand, using (4.13) we deduce for i=0,1 that
(4.25)limsup⁡n∈Λ1,n→∞max⁡z∈[α,β]hn+i(z)≤0.
Summarized, we have for i=0,1 that
(4.26)hi(z)≡0forz∈[α,β].
By definition, the regions U,W are symmetric to ℝ as well as the functions
(4.27)|rn+i*(z)|,|πn+i(z)|,|qn+i(z)|
for i=0,1. This symmetry, together with (4.26), implies that
(4.28)hi(z)≡0∀z∈W
for i=0,1. Hence,
(4.29)lim⁡n∈Λ3,n→∞‖rn+i*qn+i‖K1/n≤1
for all compact sets K in W,  i=0,1.
Combining (4.29) for i=0,1, we obtain
(4.30)lim⁡n∈Λ3,n→∞‖(rn*-rn+1*)qnqn+1‖K1/n≤1
for all compact sets K⊂W. Hence, the function V(z)≡0 is a harmonic majorant for the sequence {Fn}n∈Λ3 of subharmonic functions in W, where
(4.31)Fn(z)∶=1nlog⁡|(rn*-rn+1*)(z)qn(z)qn+1(z)|,n∈N.
Next, we want to show that V(z)≡0 is an exact harmonic majorant for {Fn}n∈Λ3 and also for any {Fn}n∈Λ4 for any subsequence Λ4⊂Λ3.
Let us assume that this assertion would be false: then there exists a subsequenceΛ4⊂Λ3⊂Λ (Λ as in the Corollary of Section 3) and a continuum K⊂W such that
(4.32)limsup⁡n∈Λ4,n→∞max⁡z∈KFn(z)<0.
Since V(z)≡0 is a harmonic majorant for {Fn}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 byδz the Dirac measure of the point z∈ℂ¯, and let δ^z be the associated balayage measure of δz 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
(4.33)ddxδ^z(x)=∂∂n+G(x,z)+∂∂n-G(x,z),
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](4.34)0≤δ^z([α,β])≤1,lim⁡z→η⁡η∈[-1,1]∖[α,β]⁡δ^z([α,β])=0.
Let z∈ℂ¯∖[-1,1], ξ∈U∩[-1,1], and ɛ>0; then
(4.35)lim⁡ɛ→0δ^z([ξ-ɛ,ξ+ɛ])=0.
Consider the exterior of the ɛ-neighborhood of [-1,1]; that is, let
(4.36)Wɛ∶={z∈C¯:dist⁡(z,[-1,1])≥ɛ}.
Then we can obtain a sharpening of (4.35), namely,
(4.37)lim⁡ɛ→0max⁡z∈Wɛ∖Uδ^z([ξ-ɛ,ξ+ɛ])=0.
Since ξ∈U∩[-1,1] and (4.2) holds, (4.34)–(4.37) imply
(4.38)lim⁡ɛ→0lim⁡n→∞τ~n([ξ-ɛ,ξ+ɛ])=0.
Because (3.1) and (4.1) hold for n∈Λ, ξ cannot be an isolated point of supp(ν).
Consequently, sincez0∈supp(ν) there exists a sequence {ξk}k∈ℕ in U,  ξk∈supp(ν), such that
(4.39)z0=lim⁡k→∞ξk
and each ξk is not an isolated point of supp(ν). Hence, for any k∈ℕ and any open interval (α,β) with ξk∈(α,β) we have ν((α,β))>0. Taking into account (4.39) and the fact that the zero set
(4.40)Z∶={z∈C:p~0(z)=0orp~i(z)=0}
of the polynomials p~0, p~1 in (4.5) is finite, there exists an interval [α~,β~]⊂U∩[-1,1], α~<β~, with
(4.41)ν([α~,β~])>0,[α~,β~]∩Z=∅.
Using (4.5) we conclude that there exists n1∈ℕ and a constant κ~>0 such that
(4.42)|qn+i(z)|≥κ~forz∈[α~,β~],
where n∈Λ1, n≥n1, and i=0,1.
Let us choose forK in (4.32) the interval [α~,β~]. Then there exists, by definition of Fn(z) in (4.31), a constant δ, 0<δ<1, and n2∈ℕ, n2≥n1, such that
(4.43)max⁡z∈[α~,β~]|(rn*-rn+1*)(z)qn(z)qn+1(z)|≤δn.
for all n∈Λ4, n≥n2. By (4.42) we obtain
(4.44)max⁡z∈[α~,β~]|(rn*-rn+1*)(z)|≤δnκ~2,(4.45)limsupn∈Λ4,n→∞∥rn*-rn+1*∥[α~,β~]1/n≤δ<1
contradicting the property (3.14) and ν([α~,β~])>0.
Hence,V(z)≡0 is an exact harmonic majorant for {Fn}n∈Λ3 and for any subsequence {Fn}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(4.46){(rn*-rn+1*)qnqn+1}n∈Λ3
of holomorphic functions in W can be applied (Walsh [21], Theorem 16, page 221): for every compact set K in W we have
(4.47)N0((rn*-rn+1*)qnqn+1,K)=o(n)asn∈Λ3,n⟶∞.
Choosing for K the interval [α~,β~], then the number of alternations of f-rn* in [α~,β~] is a lower bound for the number
(4.48)N0((rn*-rn+1*)qnqn+1,[α~,β~])
of zeros of (rn*-rn+1*)qnqn+1 in [α~,β~]. Because of (4.1) and ν([α~,β~])>0,
(4.49)lim⁡n∈Λ3,n→∞νn([α~,β~])=ν([α~,β~])>0
which contradicts (4.47).
Hence, the theorem is proved fora=0. The case a≠0 can be reduced to a=0 by defining
(4.50)f~(z)∶=f(z)+a,z∈[-1,1],(4.51)r~(z)∶=r(z)+aforr∈ℛn,n,z∈ℂ.
If a∈ℂ, we note that the inequality (4.30) is equivalent to
(4.52)lim⁡n∈Λ3,n→∞‖(r~n*-r~n+1*)qnqn+1‖K1/n≤1
and (3.14) is equivalent to
(4.53)lim⁡n∈Λ,n→∞|(r~n*-r~n+1*)(ξn)|1/n=1,
where {ξn}n∈Λ, ξn∈[-1,1], and |(f~-r~n*)(ξn)|=∥f~-r~n*∥. Therefore, all arguments for the sequence {Fn} are invariant by replacing in definition (4.10) the functions rn*,rn+1* by r~n*,r~n+1*. Hence, Theorem 2.2 is true for all a∈ℂ.Proof of the Corollary.
In the proof of Theorem2.2, the subsequence Λ was chosen such that
(4.54)νn-αn(τ^n+τ^n+1)-(1-αn)μ→*0asn⟶∞,n∈Λ,
where
(4.55)αn=deg⁡qn*+deg⁡qn+1*dn+1.
Since {mn} fulfills (2.17), we obtain
(4.56)αn=deg⁡qn*+deg⁡qn+1*n+mn+1-δn≤deg⁡qn*+deg⁡qn+1*n+mn+1-(mn-deg⁡q*n)=1-n+1-deg⁡qn+1*n+1+deg⁡qn*<1-n+1-c(n+1)n+1+c(n+1)=1-1-c1+c.
Hence, by (3.1)
(4.57)νn([α,β])≥1-c1+cμ([α,β])
for any interval [α,β]⊂C[-1,1]. Therefore, property (i) of Theorem 2.2 implies that
(4.58)νn→*ν,supp⁡(ν)=[-1,1]
and Theorem 2.2 holds for all z0∈[-1,1].


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

Theorem2.2 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
(5.1)(n(s),m(s))∈N0×N0
depend on parameters s∈ℕ. For abbreviation, let
(5.2)Es∶=En(s),m(s)(f),rs*=rn(s),m(s)*(f)=ps*qs*,
where ps* and qs* have no common factor. As above, let
(5.3)δs∶=min⁡(n(s)-deg⁡ps*,m(s)-deg⁡ps*)
be the defect of rs*, and let As=As(f)={xk(s)}k=0d(s) be an alternation point set to f-rs*, where
(5.4)ds=n(s)+m(s)+1-δs.
We denote by νs the normalized counting measure of As. 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
(5.5)n(s)≤n(s+1)≤n(s)+1,m(s)≤m(s+1)≤m(s)+1,
and let us approximate f∈C[-1,1], with respect to ℛn(s),m(s), where
(5.6)m(s)≤n(s)+κ(s),s∈ℕ,κ(s)=o(slog⁡s)ass⟶∞.
If f∈C[-1,1]  satisfies (2.6), then there exists a subset Λ⊂ℕ with the following properties: (i)
νs→*ν as  s→∞, s∈Λ.(ii)
leta∈ℂ; then for any z0∈supp(ν) and any neighborhood U of z0 with f(z)≢a on U∩[-1,1] either
(5.7)limsup⁡s→∞N∞(rs*,U)=∞or
(5.8)limsup⁡s→∞Na(rs*,U)s>0.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
(5.9)νs-αs(τ^s+τ^s+1)-(1-αs)μ→*0ass⟶∞,s∈Λ.
Again, we use in (5.9) the balayage measures of the normalized pole counting measures τs and τs+1 of rs*, respectively,  rs+1*, onto [-1,1] and
(5.10)αs∶=deg⁡qs*+deg⁡qs+1*ds+1.
Then the proof of (5.7) and (5.8) follows the same lines as the proof of Theorem 2.2 if
(5.11)limsup⁡s→∞Es1/s=1.
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
(5.12)limsup⁡s→∞Es1/s=limsup⁡s→∞En(s),m(s)1/s≥limsup⁡s→∞(EM(s),M(s)1/M(s))M(s)/s=1,
since
(5.13)s≥M(s)-M(1).


## 6. Remarks

For the functionf(x)=|x|α, α>0, the distribution of alternation points of the optimal error curves, as well as the zeros and poles of rn,m* is very well investigated [7].Letα∈ℝ+∖2ℕ, and let (n,mn)∈ℕ×ℕ with
(6.1)lim⁡n→∞mnn=c≤1,n≥mn+2[α2].
Since all best approximants of f(x)=|x|α are even functions, we can assume that n,mn∈ℕ are even. Moreover, the error function f-rn,mn* has always exactly n+mn+3 points [7]. By νAn=νn we denote the normalized alternation counting measure and νPn denotes the normalized pole counting measure of rn,mn* and νZn the normalized zero counting measure of rn,mn*. Then
(6.2)νAn→n→∞*2c1+cδ0+1-c1+cμ,(6.3)νPn→*n→∞δ0,(6.4)νZn→*n→∞cδ0+(1-c)μ
(cf. Theorems 1.6 and 1.7 in [7]).Forc<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 rn,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.Forc=1, νAn→*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 rn,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 off(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.



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*Source: 961209-2012-07-05.xml*