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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 961209, 21 pages
On the Distribution of Zeros and Poles of Rational Approximants on Intervals
1Department of Mathematical Sciences, Kent State University, Kent, OH 44242-0001, USA
2Lehrstuhl für Mathematik-Angewandte Mathematik, Mathematisch-Geographische Fakulätt, Katholische Universität Eichstätt-Ingolstadt, 85071 Eichstätt, Germany
3Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
Received 29 February 2012; Accepted 8 May 2012
Academic Editor: Karl Joachim Wirths
Copyright © 2012 V. V. Andrievskii et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The distribution of zeros and poles of best rational approximants is well understood for the functions , . If is not holomorphic on , the distribution of the zeros of best rational approximants is governed by the equilibrium measure of 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, -values, and poles of best real rational approximants of degree at most to a function that is real-valued, but not holomorphic on . Generalizations to the lower half of the Walsh table are indicated.
Let be a subset of ; we denote by the measure of , where the infimum is taken over all coverings of by disks and is the radius of the disk .
Let be a region in and a function defined in with values in . A sequence of meromorphic functions in is said to converge to a function with respect to the measure inside if for every and any compact set we have (cf. Gončar ).
The sequence is said to converge to almost geometrically inside if for any there exists a set in with such that for any compact set . We note that is the supremum norm on a subset of .
For , we denote by the collection of all polynomials of degree at most , and let In , sequences , , on a region were investigated if the number of poles of in is bounded. It turns out that the geometric convergence of on a continuum implies that the sequence converges -almost geometrically inside to a meromorphic function in with at most a finite number of poles in .
To be precise, let and let denote the subset of meromorphic functions in with at most poles in , each pole counted with its multiplicity. The main result of  can be stated as follows.
Theorem A. Let be a continuum in and a region with . Let , , be a sequence of rational functions converging geometrically to a function on , that is, and assume that on . If there exists a fixed integer such that for all and for each compact set , then the sequence converges -almost geometrically inside to a meromorphic function .
Here, the number denotes the number of zeros of in , each zero counted with its multiplicity.
The above result was applied in  to Chebyshev approximation on . Let be the Green function of with pole at , and let be the Green domain to the parameter , that is, is the open Joukowski-ellipse with foci at and and major axis .
Let be real-valued on . For abbreviation, we will write for . Given , let denote the real rational function of best uniform approximation to with respect to , that is, Moreover, let be a sequence in with and let us consider a function that can be continued meromorphically into for some . Then the sequence converges -almost geometrically inside to . Using Theorem A, we obtain results about the distribution of the -values in the neighborhood of a point . For and , we denote by the number of -values of the rational function in and each -value is counted with its multiplicity. If cannot be continued meromorphically to , then for any neighborhood of and any , with at most one exception, Particulary, such a point is either an accumulation point of zeros or of poles of .
On the other hand, if is not holomorphic on , so far results about the distribution of the zeros of are only known in the case that for all (polynomial approximation) or in the case that is fixed (rational approximation with a bounded number of free poles). In the polynomial case, the normalized zero counting measures of converge in the weak*-sense to the equilibrium measure of , at least for a subsequence . This result was generalized to rational approximation with a bounded number of poles (cf. [5, Theorem 4.1]). Moreover, Stahl  and Saff and Stahl  have investigated for the function , , 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 , the behavior of the alternation points of for 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 be continuous on , possibly complex-valued. It is well known that the rate of approximation by rational functions does not guarantee the holomorphy of the function . Gončar (, p. 101) pointed out the example where the points are situated in such that any point of is a limit point of the sequence and the coefficients converge to zero sufficiently fast. Hence, it is possible that there exists a sequence , , such that and is continuous on , but nowhere holomorphic on .
But it turns out that in this case Theorem A immediately yields the following.
Theorem 2.1. Let be not holomorphic on , and let , , be a sequence such that Then for any non holomorphic point of any neighborhood of either or for all .
In the following we consider functions that are always real-valued on . Then the case that is not covered by Theorem 2.1. By Bernstein’s theorem, condition (2.6) implies that is not holomorphic on . Examples for (2.6) are functions which are piecewise analytic on (Newman , Gončar ).
In the following, we assume that is a sequence with For abbreviation, let where and have no common factor. We define as the defect of and . According to the alternation theorem of Chebyshev (cf. Meinardus , Theorem 98) there exist points , which satisfy where or is fixed. For each pair let denote an arbitrary, but fixed alternation set for the best approximation , and let denote the normalized counting measure of , that is, for any interval . Since is a probability measure on , there exists a subsequence such that in the weak*-topology and is again a probability measure on .
Theorem 2.2. Let be real-valued, and let (2.6) hold. Moreover, let be approximated with respect to , where the sequence satisfies (2.7). Then there exists a subsequence with the following properties: (i) (ii) let , , and let be a neighborhood of with on ; then
Applying to the approximation in the upper half of the Walsh table, we obtain the following.
Corollary 2.3. Let with (2.6) and let the subsequence satisfy Then there exists a subsequence with the following property: Let , , and let be a neighborhood of with on ; 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 denote the normalized counting measure of the poles of , counted with their multiplicities, and let us denote by the balayage measure of onto . Then the following distribution results hold for the interaction between the alternation points of and the poles of and .
We remark that in the proof of Theorem B in , the subsequence is defined by Inspecting the proof of (3.1) in , it turns out that we can modify the definition of by The existence of such sequences is based on the divergence of the infinite product to 0 if is not a rational function. This argument has already been used by Kadec  in his proof for the distribution of the alternation points in polynomial approximation.
Concerning the distribution of the zeros of best polynomial approximations to , the asymptotic behavior of the highest coefficient plays an essential role, namely, where and is the logarithmic capacity of .
If is not holomorphic on , then and we can choose a subsequence such that and moreover, If , then the polynomial is monic and satisfies for all which are sufficiently large, where can be chosen arbitrarily. Then the Erdős-Turán Theorem  (cf. ) implies a weak*-version of Kadec’s result, namely, the weak*-convergence of the normalized counting measures of alternation sets of to the equilibrium measure of , at least for a subsequence , .
The objective of this section is to show that there exists a subsequence 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 with (2.6). Then there exists a subsequence such that Moreover, let be a sequence in with ; then
Proof. Using the above arguments of the beginning of this section, there exists a subsequence such that
First, we show that there exists such that (3.13) holds.
For proving this, we define Since , , and is not finite, hence the complement of in has the property that If is a finite set, then there exists such that satisfies property (3.13).
If is an infinite set, then observing that is not a finite set, we can define subsequences and of such that Next, we consider a fixed integer . If then and we deduce Since the infinite product converges, there exists a constant , , such that all partial products of are bounded by from below, that is, .
By (3.22), and Let us define for
Next, we choose a subsequence of such that and If , then we are done. As for the general case, let us define then and (3.25)–(3.27) imply Hence, (3.13) is proved.
Moreover, for , Hence, and (3.14) is proved.
Proof of Theorem 2.2. First we will prove the theorem for .
According to the lemma in Section 3, there exists a subsequence such that (3.13)–(3.14) hold. Then Theorem B applies and (3.1) holds for . Because are probability measures on , we may assume that Let and a neighborhood of such that on .
Let us assume that (ii) of Theorem 2.2 does not hold. Hence, there exists Of course, we may assume that is a bounded symmetric region with respect to the real axis. Let be the number of poles of in counted with their multiplicities. Then we define Because , there exists a subsequence and such that Together with for , this implies that there exists an interval , and a constant such that Let be the number of zeros (with multiplicities) of in . If , let , , be the zeros of in and let Because of (4.3), as and we obtain for any compact set in . Now, let us define with Then is holomorphic in and harmonic in .
Consider and as before. Then by (4.5)–(4.7) there exists such that for , , and , . Then for where According to a Lemma of Gončar [20, Lemma 1, page 153], for any compact set there exists a constant such that for . For example, can be chosen as where is the Green function of with pole at .
Next, we choose a region , symmetric to the real axis, with , and , then for . Hence for , the sequences are uniformly bounded in from above as , , . By Harnack’s theorem, either or there exists a subsequence such that converges locally uniformly to as , , in the region and the function is harmonic in .
Next, let us show that the first situation cannot occur: if is such that for and sufficiently large, then Hence, by (4.5)–(4.7) there exists a constant such that Since is a monic polynomial and as , this is a contradiction to Next, we consider (4.17) for . Again by Harnack’s theorem, either or there exists a subsequence such that converges locally uniformly to a function in and is harmonic in .
As above for , the first situation cannot occur. Consequently, On the other hand, using (4.13) we deduce for that Summarized, we have for that By definition, the regions are symmetric to as well as the functions for . This symmetry, together with (4.26), implies that for . Hence, for all compact sets in , .
Combining (4.29) for , we obtain for all compact sets . Hence, the function is a harmonic majorant for the sequence of subharmonic functions in , where Next, we want to show that is an exact harmonic majorant for and also for any for any subsequence .
Let us assume that this assertion would be false: then there exists a subsequence ( as in the Corollary of Section 3) and a continuum such that Since is a harmonic majorant for in , then (4.32) implies that the inequality (4.32) holds for any continuum .
First, let us note that under the condition (4.2) a point cannot be an isolated point of .
To prove this, let us denote by the Dirac measure of the point , and let be the associated balayage measure of to the interval . For the density of the balayage measure at the point is given by where (resp., ) denotes the normal at the point to the upper half (resp., lower half) plane and is the Green function for with pole at , continuously extended by to .
Then for any interval Let , , and ; then Consider the exterior of the -neighborhood of ; that is, let Then we can obtain a sharpening of (4.35), namely, Since and (4.2) holds, (4.34)–(4.37) imply Because (3.1) and (4.1) hold for , cannot be an isolated point of .
Consequently, since there exists a sequence in , , such that and each is not an isolated point of . Hence, for any and any open interval with we have . Taking into account (4.39) and the fact that the zero set of the polynomials , in (4.5) is finite, there exists an interval , , with Using (4.5) we conclude that there exists and a constant such that where , , and .
Let us choose for in (4.32) the interval . Then there exists, by definition of in (4.31), a constant , , and , , such that for all , . By (4.42) we obtain contradicting the property (3.14) and .
Hence, is an exact harmonic majorant for and for any subsequence , , in the region .
This is now the situation that a distribution result of Walsh about the zeros of the sequence of holomorphic functions in can be applied (Walsh , Theorem 16, page 221): for every compact set in we have Choosing for the interval , then the number of alternations of in is a lower bound for the number of zeros of in . Because of (4.1) and , which contradicts (4.47).
Hence, the theorem is proved for . The case can be reduced to by defining If , we note that the inequality (4.30) is equivalent to and (3.14) is equivalent to where , , and . Therefore, all arguments for the sequence are invariant by replacing in definition (4.10) the functions by . Hence, Theorem 2.2 is true for all .
Proof of the Corollary. In the proof of Theorem 2.2, the subsequence was chosen such that where Since fulfills (2.17), we obtain Hence, by (3.1) for any interval . Therefore, property (i) of Theorem 2.2 implies that and Theorem 2.2 holds for all .
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 depend on parameters . For abbreviation, let where and have no common factor. As above, let be the defect of , and let be an alternation point set to , where We denote by the normalized counting measure of . Then Theorem 2.2 can be generalized in the following way.
Theorem 5.1. Let , , be a strictly increasing subsequence of with and let us approximate , with respect to , where If satisfies (2.6), then there exists a subset with the following properties: (i) as , . (ii)let ; then for any and any neighborhood of with on either or
For the proof, we use a generalization of Theorem B to the previous situation (see ): if (5.5) and (5.6) hold, then there exists a subsequence such that Again, we use in (5.9) the balayage measures of the normalized pole counting measures and of , respectively, , onto and Then the proof of (5.7) and (5.8) follows the same lines as the proof of Theorem 2.2 if Because of (5.5), the index runs from to . Moreover, let , ; then runs from to and since
For the function , , the distribution of alternation points of the optimal error curves, as well as the zeros and poles of is very well investigated .
Let , and let with Since all best approximants of are even functions, we can assume that are even. Moreover, the error function has always exactly points . By we denote the normalized alternation counting measure and denotes the normalized pole counting measure of and the normalized zero counting measure of . Then (cf. Theorems 1.6 and 1.7 in ).
For , we would obtain by (3.1) and by the corollary of Theorem 2.2 that any point of is either a limit point of poles or of -values of , , as . Since by (6.3) the normalized pole counting measures converge to the Dirac measure at 0, any point of , with 0 as only possible exception, is a limit point of -values.
For , . Hence Theorem 2.2 can only tell us that the point 0 is either a limit point of poles or of -values, . But (6.3) and (6.4) show that 0 is as well a limit point of zeros as of poles of . Hence, the investigations in  for the special functions lead to deeper results for the zeros and poles of the best approximants.
But the example of shows an interesting area for further investigations, namely, a weak*-type analogue of relation (3.1) for the distribution of zeros, respectively, -values, and poles of rational approximation would be desirable. Moreover, the approximation problem should be moved from the interval to more general compact sets in .
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