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

and Applied Analysis 3 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 Eρ : { z ∈ C : G z,∞ < log ρ, ρ > 1, 1.7 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. 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 ∈ Rn,m denote the real rational function of best uniform approximation to f ∈ C −1, 1 with respect to Rn,m, that is, En,m ( f ) : ∥f − r∗ n,m ∥∥ inf ∥f − r∥ : r ∈ Rn,m, r real-valued onR } . 1.8 Moreover, let {mn}n∈N be a sequence in N with lim n→∞ mn ∞, mn o ( n logn ) as n −→ ∞, 1.9 and let us consider a function f ∈ C −1, 1 that can be continued meromorphically into Eρ for some ρ > 1. Then the sequence {r∗ n,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 Na 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, lim sup n→∞ Na ( r∗ 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 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 r∗ n,0 f converge in the weak ∗-sense to the equilibriummeasure 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. 4 Abstract and Applied Analysis 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 1 An z − αn , 2.1 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 lim sup n→∞ ∥f − rn ∥∥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 {rn}n∈N, rn ∈ Rn,n, be a sequence such that lim sup n→∞ ∥f − rn ∥∥1/n < 1. 2.3 Then for any non holomorphic point z0 ∈ −1, 1 of f any neighborhood U of z0 either lim sup n→∞ N∞ rn,U ∞ 2.4


Introduction
Let B be a subset of C; we denote by the m 1 -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 ϕ with 2 Abstract and Applied Analysis respect to the m 1 -measure inside D if for every ε > 0 and any compact set K ⊂ D we have The sequence {ϕ n } n∈N is said to converge to ϕ m 1 -almost geometrically inside D if for any ε > 0 there exists a set Ω ε in C with m 1 Ω ε < ε such that lim sup for any compact set K ⊂ D. We note that • B is the supremum norm on a subset B of C.
For n ∈ N 0 N ∪ {0}, we denote by P n the collection of all polynomials of degree at most n, and let R n,m : r p 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 m 1 -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, 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 for each compact set K ⊂ D, then the sequence {r n } n∈N converges m 1 -almost geometrically inside D to a meromorphic function f ∈ M m D .
Here, the number N 0 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 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 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 z 0 , then for any neighborhood U of z 0 and any a ∈ C, with at most one exception, lim sup Particulary, such a point z 0 is either an accumulation point of zeros or of poles of r * n,m n .On the other hand, if f is not holomorphic on −1, 1 , so far results about the distribution of the zeros of r * n,m n 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,m n , the behavior of the alternation points of f − r * n,m n 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.

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 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 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.

2.3
Then for any non holomorphic point 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 lim sup In the following, we assume that {m n } n∈N is a sequence with For abbreviation, let where p * n ∈ P n and q * n ∈ P m n have no common factor.We define 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 which satisfy where λ n 1 or λ n −1 is fixed.For each pair n, m n let 12 denote an arbitrary, but fixed alternation set for the best approximation r * n ∈ R n,m n , and let ν n denote the normalized counting measure of A n , that is, 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,m n , where the sequence {m n } n∈N satisfies 2.7 .Then there exists a subsequence Λ ⊂ N with the following properties: ii let z 0 ∈ supp ν , a ∈ C, and let U be a neighborhood of z 0 with f z / ≡ a on U ∩ −1, 1 ; then

2.16
Applying to the approximation in the upper half of the Walsh table, we obtain the following.

2.17
Then there exists a subsequence Λ ⊂ N with the following property: Let a ∈ C, z 0 ∈ −1, 1 , and let U be a neighborhood of z 0 with f z / ≡ a on U ∩ −1, 1 ; then either i or ii holds.

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 where and μ is the equilibrium distribution of −1, 1 .

Abstract and Applied Analysis 7
We remark that in the proof of Theorem B in 11 , the subsequence Λ ⊂ N is defined by Inspecting the proof of 3.1 in 11 , it turns out that we can modify the definition of Λ by The existence of such sequences Λ is based on the divergence of the infinite product 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, the asymptotic behavior of the highest coefficient a n plays an essential role, namely, lim sup where 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

3.14
Proof .Using the above arguments of the beginning of this section, there exists a subsequence

3.15
First, we show that there exists Λ ⊂ N such that 3.13 holds.
For proving this, we define

3.16
Since Λ 1 ⊂ Λ, Λ / ∅, and Λ is not finite, hence the complement of Λ in N has the property that

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

3.20
Next, we consider a fixed integer m ≥ m 1 .If then m / ∈ Λ and we deduce

3.22
Since the infinite product converges, there exists a constant β, 0 < β < 1, such that all partial products S ν,μ : of S are bounded by β from below, that is, S ν,μ ≥ β.By 3.22 , E m j > βE m and

3.26
Next, we choose a subsequence Λ 2 {k j } j∈N of N such that k 1 ≥ m 1 and lim

3.27
If Λ 2 ⊂ Λ, then we are done.As for the general case, let us define then Λ ⊂ Λ and 3.25 -3.27 imply

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 Let z 0 ∈ supp ν and U a neighborhood of z 0 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 ∈ N 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 : 1, l n 0.

4.4
Because q n , q n 1 ∈ P m , there exists a subsequence Λ 1 ⊂ Λ and q 0 , q 1 ∈ P m such that lim n∈Λ 1 ,n → ∞ q n i q i for i 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 x ≥ κ for x ∈ α, β , i 0, 1, 4.6 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

4.11
Then Φ n is holomorphic in U and h n harmonic in U.
Consider z ∈ α, β and Λ 1 as before.Then by 4.5 -4.7 there exists n ∈ N such that for z ∈ α, β , i 0, 1, and n ∈ Λ 1 , n ≥ n.Then for i 0, 1 where 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 for i 0, 1.For example, λ α, β , U, K can be chosen as 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 z 0 ∈ W, W ⊂ U and α, β ⊂ W, then for i 0, 1. Hence for i 0, 1, the sequences {h n i } n∈Λ 1 are uniformly bounded in W from above as n → ∞, n ∈ Λ 1 , i 0, 1.By Harnack's theorem, either or there exists a subsequence Λ 2 ⊂ Λ 1 such that {h n } n∈Λ 2 converges locally uniformly to h 0 as n → ∞, n ∈ Λ 2 , in the region W and the function h 0 is harmonic in W.

4.22
Next, we consider 4.17 for i 1. Again by Harnack's theorem, either or there exists a subsequence Λ 3 ⊂ Λ 2 such that {h n 1 } n∈Λ 3 converges locally uniformly to a function h 1 in W and h 1 is harmonic in W.
As above for {h n } n∈Λ 1 , the first situation cannot occur.Consequently, max z∈ α,β h i z ≥ 0 for i 0, 1.

4.24
On the other hand, using 4.13 we deduce for i 0, 1 that lim sup Summarized, we have for i 0, 1 that By definition, the regions U, W are symmetric to R as well as the functions for i 0, 1.This symmetry, together with 4.26 , implies that for all compact sets K in W, i 0, 1. Combining 4.29 for i 0, 1, we obtain lim for all compact sets K ⊂ W. Hence, the function V z ≡ 0 is a harmonic majorant for the sequence {F n } n∈Λ 3 of subharmonic functions in W, where Next, we want to show that V z ≡ 0 is an exact harmonic majorant for {F n } n∈Λ 3 and also for any {F n } n∈Λ 4 for any subsequence Λ 4 ⊂ Λ 3 .
Let us assume that this assertion would be false: then there exists a subsequence Λ 4 ⊂ Λ 3 ⊂ Λ Λ as in the Corollary of Section 3 and a continuum K ⊂ W such that lim sup Since V z ≡ 0 is a harmonic majorant for {F n } n∈Λ 4 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 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 .

Abstract and Applied Analysis 15
Then for any interval α, β ⊂ −1, Consider the exterior of the ε-neighborhood of −1, 1 ; that is, let Hence, V z ≡ 0 is an exact harmonic majorant for {F n } n∈Λ 3 and for any subsequence This is now the situation that a distribution result of Walsh about the zeros of the sequence r * n − r * n 1 q n q n 1 n∈Λ 3 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

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 Hence, the theorem is proved for a 0. The case a / 0 can be reduced to a 0 by defining If a ∈ C, we note that the inequality 4.30 is equivalent to Proof of the Corollary.In the proof of Theorem 2.2, the subsequence Λ was chosen such that where Since {m n } fulfills 2.17 , we obtain

4.56
Hence, by 3.1   We denote by ν s the normalized counting measure of A s .Then Theorem 2.2 can be generalized in the following way.

. 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,m n } n∈N converges m 1 -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 z 0 ∈ ∂E ρ .For a ∈ C and B ⊂ C, we denote by N a r, B : #{z ∈ B : r z a} 1.10

5 . 3 be
δ s : min n s − deg p * s , m s − deg p * s the defect of r * s , and let A s A s f {x an alternation point set to f − r * s , where d s n s m s 1 − δ s .5.4

Theorem 5 . 1 .ForBecause of 5 . 5 ,s
Let n s , m s , s ∈ N, be a strictly increasing subsequence of N 0 × N 0 withn s ≤ n s 1 ≤ n s 1, m s ≤ m s 1 ≤ m s 1, 5.5and let us approximate f ∈ C −1, 1 , with respect to R n s ,m s , wherem s ≤ n s κ s , s ∈ N, 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.Abstract and Applied Analysis 19ii let a ∈ C; then for any z 0 ∈ supp ν and any neighborhood U of z 0 with f z / ≡ a on U ∩ −1, 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 thatν s − α s τ s τ s 1 − 1 − α s μ * → 0 as s −→ ∞, s ∈ Λ.5.9Again, we use in 5.9 the balayage measures of the normalized pole counting measures τ s and τ 1 of r * s , respectively, r * s 1 , onto −1, of 5.7 and 5.8 follows the same lines as the proof of Theorem 2the 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 lim sup ≥ M s − M 1 .5.13 ∈ Λ, ξ cannot be an isolated point of supp ν .Consequently, since z 0 ∈ supp ν there exists a sequence {ξ k } k∈N in U, ξ k ∈ supp ν , such that Λ 1 , n ≥ n 1 , 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 n 2 ∈ N, n 2 ≥ n 1 , such that } n∈Λ , ξ n ∈ −1, 1 , and | f − r * n ξ n | f − r * n .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 * n , r * n 1 .Hence, Theorem 2.2 is true for all a ∈ C.

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 ∈ N 0 × N 0 5.1 depend on parameters s ∈ N.For abbreviation, let * s have no common factor.As above, let