Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles

Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles

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Дата:2007
Автори: Peherstorfer, F., Yuditskii, P.
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Мова:English
Опубліковано: Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України 2007
Назва видання:Журнал математической физики, анализа, геометрии
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Цитувати:Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles / F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 95-108. — Бібліогр.: 6 назв. — англ.

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spelling irk-123456789-1064392016-09-29T03:02:08Z Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles Peherstorfer, F. Yuditskii, P. 2007 Article Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles / F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 95-108. — Бібліогр.: 6 назв. — англ. 1812-9471 http://dspace.nbuv.gov.ua/handle/123456789/106439 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
format Article
author Peherstorfer, F.
Yuditskii, P.
spellingShingle Peherstorfer, F.
Yuditskii, P.
Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
Журнал математической физики, анализа, геометрии
author_facet Peherstorfer, F.
Yuditskii, P.
author_sort Peherstorfer, F.
title Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
title_short Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
title_full Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
title_fullStr Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
title_full_unstemmed Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles
title_sort uniform approximation of sgn(x) by rational functions with prescribed poles
publisher Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
publishDate 2007
url http://dspace.nbuv.gov.ua/handle/123456789/106439
citation_txt Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles / F. Peherstorfer, P. Yuditskii // Журнал математической физики, анализа, геометрии. — 2007. — Т. 3, № 1. — С. 95-108. — Бібліогр.: 6 назв. — англ.
series Журнал математической физики, анализа, геометрии
work_keys_str_mv AT peherstorferf uniformapproximationofsgnxbyrationalfunctionswithprescribedpoles
AT yuditskiip uniformapproximationofsgnxbyrationalfunctionswithprescribedpoles
first_indexed 2025-07-07T18:29:34Z
last_indexed 2025-07-07T18:29:34Z
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fulltext Journal of Mathematical Physics, Analysis, Geometry 2007, v. 3, No. 1, pp. 95�108 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles F. Peherstorfer Abteilung f�ur Dynamische Systeme und Approximationstheorie, Universit�at Linz 4040 Linz, Austria E-mail:Franz.Peherstorfer@jku.at P. Yuditskii Abteilung f�ur Dynamische Systeme und Approximationstheorie Universit�at Linz 4040 Linz, Austria Department of Mathematics and Statistics Bar-Ilan University, Israel E-mail:Petro.Yudytskiy@jku.at Received July 28, 2006 For a 2 (0; 1) let Lkm(a) be an error of the best approximation of the func- tion sgn (x) on two symmetric intervals [�1;�a][ [a; 1] by rational functions with the only possible poles of degree 2k � 1 at the origin and of 2m� 1 at in�nity. Then the following limit exists lim m!1 L k m(a) � 1 + a 1� a �m� 1 2 (2m� 1)k+ 1 2 = 2 � � 1� a 2 2a �k+ 1 2 � � k + 1 2 � : (0.1) Key words: Bernstein constant, Chebyshev problems, approximation, conformal mappings, Gamma function. Mathematics Subject Classi�cation 2000: 41A44 (primary); 30E (se- condary). Partially supported by the Austrian Founds FWF, Project No. P16390�N04 and Marie Curie International Fellowship within the 6-th European Community Framework Programme, Contract MIF1-CT-2005-006966. c F. Peherstorfer and P. Yuditskii, 2007 F. Peherstorfer and P. Yuditskii Dedicated to the memory of B.Ya. Levin 1. Introduction This is the second step (for the �rst one see [5]) on the way to understand better the di�culties that up to now do not allow to �nd the Bernstein constant. Recall that Sergey Natanovich Bernstein found [3, 4] that for the error En(p) of the best uniform approximation of jxjp, p being not an even integer, on [�1; 1] by polynomials of degree n the following limit exists: lim m!1 npEn(p) = �(p) > 0: For p = 1 this result was obtained by Bernstein in 1914, and he posed the question, whether one could express �(1) in terms of the known transcendental functions. This question is still open. Actually, we solve here a problem on asymptotics of the best approximation of sgn (x) on the union of two intervals [�1;�a][[a; 1] by rational functions. In 1877, E.I. Zolotarev [6, 2] found an explicit expression, in terms of elliptic functions, of the rational function of the given degree which is uniformly closest to sgn (x) on this set. This result was a subject of a number of generalizations, and it has applications in electric engineering. In Zolotarev's case position of the poles of the rational function is free, the natural question is to �nd the best approximation when the poles and their multiplicities are �xed. In [5] A. Eremenko and the second co-author of the current paper solved the polynomial case. Here we allow the rational function to have one more pole in (�a; a), more precisely, admitted are two poles � one at in�nity and one in the origin. Thus the problem is: Problem 1.1. For k;m 2 N, �nd the best approximation of the function sgn (x), jxj 2 [a; 1], by functions of the form f(x) = a�(2k�1) x2k�1 + :::+ a2m�1x 2m�1 and the approximation error Lk m (a). One can be interested in many di�erent asymptotics for Lk m (a) when m or k, or both of them go to in�nity in a certain prescribed way. In this paper we concentrate on the case when k is �xed and m ! 1. Note, however, that due to the evident symmetry Lk m (a) = Lm k (a) and a bit less evident (6.2) we have simultaneously asymptotic for k ! 1, m is �xed and k ! 1, m ! 1 so that k = m. As it appears the tricks which are used in [5] to �nd precise asymptotic work in this general case (so we have a method in hands): 96 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles 1. For each certain k and m we reveal the structure of the extremal function by representing it with the help of an explicitly given conformal mapping. 2. The system of conformal mappings (k is �xed, m is a parameter) converges (in the Caratheodory sense) after appropriate renormalization. The limit map does not depend on a, thus we obtain asymptotics for Lk m (a) in terms of a-depending parameters, that we use for renormalization, (an explicit for- mula) and a k-depending constant, say Yk, which is a certain characteristic of this �nal conformal map (kind of capacity). Of course, it is very tempting to guess Yk directly from the given explicitly con- formal map. It might be that we have here special functions that are given in such a form that we are unable to recognize them. In any case, we would consider this way of �nding Yk as a very interesting open problem. However we are able to �nd Yk using the third step below our strategy. Problem 1.1 in an evident way is equivalent to Problem 1.2. For p = 2k � 1 and n = 2(k +m � 1), �nd the best weighted polynomial approximation and the minimal deviation E� n(p; a) = inf fP :degP�ng sup jxj2[a;1] ���� jxjp � P (x) xp ���� : (1.2) Thus we have E� n (p; a) = Lk m (a). Note that Bernstein himself solved the un- weighted problem. Problem 1.3. For a �xed non even p, �nd asymptotics for the minimal devi- ation En(p; a) = inf fP :degP�ng sup jxj2[a;1] jjxjp � P (x)j ; (1.3) when n goes to in�nity through the even integers. 3. Due to the evident relation lim a!1 lim n!1 E� n(p; a) En(p; a) = 1; we can recalculate the constant in Probl. 1.3 to the constant related to Probl. 1.2 and thus to get explicitly eYk = �(k+ 1 2 ) � . This interplay between Problems 1.2 and 1.3 indicates that most likely one can �nd our asymptotic formula (0.1) by using original Bernstein's method, though up to the last step our consideration is very direct and simple. However we can go Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 97 F. Peherstorfer and P. Yuditskii in the opposite direction. In particular, in this work we show that the extremal polynomials of Probl. 1.3, at least for p = 1, also have special representations in terms of conformal mappings. The boundaries of the corresponding domains are not so explicit as in Probl. 1.1, they are described in terms of certain functional equations with an unknown function being involved, its Hilbert transform and independent variable (7.2). Precise constants that characterize these equations (counterparts of the constants Yk), related to the conformal mappings and their asymptotics leave enough space for the hope that for a = 0 one also would be able to characterize very similar equations in terms of classical constants. Acknowledgment. We are thankful to Alex Eremenko for friendly conver- sations while writing the paper. 2. Special Functions In this section we introduce certain special conformal mappings that we need in what follows. They are marked by a natural parameter k, but in this section k can be just real, k > 1=2. For the given k, consider the domain �k = C + n fw : Rew = � log t; jImw � k�j � arccos t; t 2 (0; 1]g: (2.1) De�ne the conformal map Hk : C + ! �k; normalized by Hk(0) =11, Hk(1) =12 (on the boundary we have two in�nite points denoted 11;12 respectively), and moreover Hk(�) = � + : : : ; � !1 (that is the leading coe�cient is �xed). By Dk we denote the positive number such that Hk(�Dk) = 0. Note that for Hk we have the following integral representation Hk(�) = � +Dk + 1Z 0 � 1 t� � � 1 t+Dk � �k(t)dt; (2.2) where �k(t) = 1 � ImHk(t). Evidently �k(t)! k + 1 2 , t! +1. Lemma 2.1. The function Hk possesses the asymptotic lim �!�1 � Hk(�)� � + � k + 1 2 � log(��) � = Yk; (2.3) 98 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles where Yk := Dk + � k + 1 2 � logDk � 1Z 0 �k(t)� � k + 1 2 � t+Dk dt: (2.4) P r o o f. Since 1Z 0 � 1 t� � � 1 t+Dk �� �k(t)� � k + 1 2 �� dt! � 1Z 0 �k(t)� � k + 1 2 � t+Dk dt (2.5) and � k + 1 2 � 1Z 0 � 1 t� � � 1 t+Dk � dt = � � k + 1 2 � (log(��)� logDk); (2.6) we get (2.3). Finally, note that Yk, as it was de�ned here, has sense for all real k > 1 2 . As it is shown in Sect. 5, for the integer k we have Yk = log � � k + 1 2 � � log �: We do not know wether these values coincide for non integers k. 3. Extremal Problem Problems 1.1 and 1.2 are related in a trivial way. Recall, for p = 2k � 1 and n = 2(k +m� 1), we have E� n (p; a) = Lk m (a) = inf fP :degP�2(m+k�1)g sup jxj2[a;1] ���� jxj2k�1 � P (x) x2k�1 ���� ; (3.1) where a 2 (0; 1), k;m 2 N. Evidently, Lk m (a) can be rewritten in the terms of the best approximation of the function sgn (x) by functions of the form f(x) = a�(2k�1) x2k�1 + :::+ a2m�1x 2m�1: Also, it is trivial that in the �rst case the extremal polynomial is even and the ex- tremal function f = f(x; k;m; a) is odd. For a parameter B > 0, and k;m 2 N, we denote by k m(B) a subdomain of the half-strip fw = u+ iv : v > 0; 0 < u < (k +m)�g Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 99 F. Peherstorfer and P. Yuditskii that we obtain by deleting the subregion fw = u+ iv : ju� �kj � arccos � coshB cosh v � ; v � Bg: (3.2) Let �(z) = �(z; k;m;B) be a conformal map of the �rst quadrant onto k m (B) such that �(0) = 11, �(1) = (k +m)�, �(1) = 12. Let a = ��1(0). Then a is a continuous strictly increasing function of B, moreover limB!0 a(B) = 0 and limB!1 a(B) = 1. Thus we may consider the inverse function B(a) = Bk m (a), a 2 (0; 1). Theorem 3.1. The error of the best approximation is Lk m (a) = 1 coshBk m (a) (3.3) and the extremal function is of the form f(x; k;m; a) = 1� (�1)kLk m (a) cos �(x; k;m;B(a)); x > 0: P r o o f. Basically the proof is the same as in [5]. A comparably important di�erence is as follows. We have to note and prove that on the imaginary axis the extremal function has precisely one zero (there are no critical points and the behavior at i0 and at i1 is evident). At this point � = k�+ iB and we have (3.3). 4. Asymptotics Theorem 4.1. The following limit exists lim m!1 � Bk m(a)� � m� 1 2 � log 1 + a 1� a � � k + 1 2 � log(2m� 1) � = � k + 1 2 � log a 1� a2 � Yk: (4.1) P r o o f. As in [5], we use the symmetry principle and make convenient changes of variables to have a conformal map �m(Z) = �(Z; k;m;B) of the upper plane in the region i( k m (B) [ k m (B)) [ (0; i�(m + k)): This conformal map has the following boundary correspondence �m : (�Cm;�Am; 0; Am; Cm)! (�12;�11; 0;11;12); 100 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles here Am = aCm and Cm will be chosen later. For �m we have the following integral representation �m(Z) = � m� 1 2 � log 1 + Z Cm 1� Z Cm + 1Z Am � 1 X � Z � 1 X + Z � vm(X) dX; where vm(X) = ( 1 � Im�m(X); Am � X � Cm k + 1 2 ; X > Cm: (4.2) Put now Hk m (�) = �m(Z)�Bm; Z = Am + �; then Hk m (�) = � m� 1 2 � log 1 + a+ � Cm 1� a� � Cm + 1Z 0 � 1 t� � � 1 t+ 2Am + � � v̂m(t) dt�Bm; where v̂m(t) = vm(t + Am). Let us rewrite Hk m in the form that is close to the integral representation of Hk: Hk m (�) = � m� 1 2 � log 1 + � Cm(1+a) 1� � Cm(1�a) +Dk + 1Z 0 � 1 t� � � 1 t+Dk � v̂m(t) dt + � m� 1 2 � log 1 + a 1� a �Dk + 1Z 0 � 1 t+Dk � 1 t+ 2Am + � � v̂m(t) dt�Bm: (4.3) Now, we put Cm = 2m� 1 1� a2 : In this case the �rst line in (4.3) converges to Hk(�). Since lim m!1 1Z 0 � 1 t+Dk � 1 t+Am + � �� v̂m(t)� � k + 1 2 �� dt = 1Z 0 �k(t)� � k + 1 2 � t+Dk dt (4.4) and 1Z 0 � 1 t+Dk � 1 t+ 2Am + � � dt = log 2Am Dk + log � 1 + � 2Am � ; (4.5) Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 101 F. Peherstorfer and P. Yuditskii we have from the second line in (4.3) that lim m!1 � Bm � � m� 1 2 � log 1 + a 1� a � � k + 1 2 � log 2Am � = �Dk � � k + 1 2 � logDk + 1Z 0 �k(t)� � k + 1 2 � t+Dk dt = �Yk: (4.6) Thus we get (4.1). In order to prove (0.1) we have to �nd the constant 2eYk . 5. The Constant From the point of view of the best weighted polynomial approximation of the function jxjp (see Sect. 3) our current result has the form lim m!1 � 1 + a 1� a �n 2 +1 n p 2 +1E� n (p; a) = � (1 + a)2 2a � p 2 +1 c(p): (5.1) On the other hand for the uniform approximation of jxjp (see details in Appendix 1) lim m!1 � 1 + a 1� a �n 2 +1 n p 2 +1En(p; a) = 2 p 2 +1a p 2 �1 (1 + a)2 2 ��� ��p 2 ��� : (5.2) Since lim a!1 lim n!1 E� n(p; a) En(p; a) = 1; we obtain c(p) ������p 2 ���� = 2: Using ��� ��p 2 ���� �p2 + 1 � = �, we have c(p) = 2 � � �p 2 + 1 � : This �nishes the proof of (0.1). 6. Case m = k, m!1 It is quite evident that the �nal con�guration of the conformal mapping in this case should be just a symmetrization of the map that we had in the case k = 0, m ! 1. However it is even much simpler to make this reduction by a suitable change of variables. First, we put a = �2, then x 2 [a; 1] means y = x � 2 [�; ��1] 102 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles and we have one more symmetry y 7! 1=y. Therefore the extremal function is symmetric and possesses the representation ~f(y;m;m) := f(x;m;m; a) = P2m�1 � y + y�1 �+ ��1 � ; (6.1) where P2m�1(t) is the best polynomial approximation of sgn (t) on h �1;� 2� 1+�2 i [h 2� 1+�2 ; 1 i . Thus we have Lm m (a) = L0 m � 2 p a 1 + a � ; (6.2) and lim m!1 Lm m (a) � 1 + p a 1�pa �2m�1 (2m� 1) 1 2 = 1� ap � p a(1 + a) : (6.3) 7. Unweighted Extremal Polynomial via Conformal Mapping Let Pm(z; a) be the best uniform (unweighted) approximation of jxj by polyno- mials of degree not more than 2m on two intervals [�1;�a][ [a; 1] and L = Lm(a) be an approximation error. In this section we prove Theorem 7.1. There is a curve = m(a) inside the half�strip fw = u+ iv : u 2 (0; (m+ 1)�); v > 0g (7.1) such that the extremal polynomial possesses the representation Pm(z; a) = z + L cos�m(z; a); where �m(z; a) is the conformal map of the �rst quadrant onto the region in the half-strip (7.1) bounded on the left by m(a), which is normalized by �m(a; a) = 0, �m(1; a) = (m + 1)� and �m(1; a) = 1. Moreover, the curve is an image of the imaginary half�axis under this conformal map that satis�es the following functional equation m(a) = fu+ iv = �m(iy; a) : L sinu(y) sinh v(y) = y; y > 0g: (7.2) P r o o f. First we clarify the shape of the extremal polynomial. In particular, we prove that Pm(0; a) > L. On the way we show the fact that is probably interesting on its own: P 0 m(x; a) looks much similar to the polynomial of the best approximation of sgn (x) on two symmetric intervals [5], with the only di�erence Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 103 F. Peherstorfer and P. Yuditskii that the deviations in area should be equal, instead of the maximum modulus. However it can be shown that P 0 m (x; a) is not the best L1 approximation of sgn (x). Due to the symmetry of Pm(x; a), we can use the Chebyshev theorem with respect to the best approximation of p x on [a2; 1] by polynomials of degree m. It gives us that Pm(z; a) hasm+2 points fxjg on the interval [a; 1] where Pm(xj ; a) = xj � L (the right half of the Chebyshev set in this case). Moreover, x0 = a and xm+1 = 1. At all other points, in addition, we have P 0 m(xj ; a) = 1, 1 � j � m. Between each two of them we have a point yj, where P 00 m (yj) = 0. Therefore we obtain 2(m� 1) zeros of the second derivative in (�1;�a) [ (a; 1) and this is precisely its degree. Thus there is no other critical points of P 0 m (z; a), in particular, in (�a; a) and on the imaginary axis. From the �rst consequence, we conclude that on (�a; a) the P 0 m (z; a) increases. That is on (a; x1) the graph of Pm(z; a) is under the line x � L, depending on the value Pm(a; a), that, recall, should be a+ L or a� L. Therefore, it is under the line x + L and Pm(a; a) � a = L, Pm(x1; a) � x1 = �L. Continuing in this way we get values of Pm(xj ; a) at all other points xj by alternance principle. Note that as a byproduct we get xiZ xi�1 jP 0 m(x; a)� 1j dx = 2L for all 1 � i � m+ 1. From the second consequence we have that ImP 0 m (iy) � 0 on the imaginary axis, that is Pm(iy; a), being real, decreases with y, starting from Pm(0; a) > L to �1. From this remark and the argument principle we deduce that the equation Pm(z; a) � z = tL (7.3) has no solution in the open �rst quarter for all t 2 (�1; 1). Indeed, since Pm(z; a) � z alternates between �L in the interval [a,1], (7.3) has m + 1 solutions, which we denote by xj(t). Consider now the contour that runs on the positive real axis to xj(t)� �, then it goes around xj(t) on the half- circle of the radius � clockwise. After the last of xj 's we continue to go along the contour to the big positive R. The next piece of the contour is a quarter-circle up to imaginary axis. Finally, from iR we go back to the origin. On each half-circle of the radius � the argument of the function changes by ��. On the quarter- circle it changes by about degPm(z; a)� � 2 = m�. On the imaginary axis we have Re(Pm(iy; a) � iy) = Pm(iy; a) and Im(Pm(iy; a) � iy) = �y. Since Pm(iy; a) decreases much faster than �y (degree of Pm is at least two), the change of the argument on the last piece of the contour is about �. Thus the whole change is �(m + 1)� +m� + � = 0. Since the function has no poles, it has no zeros in the region. 104 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles Thus arccos Pm(z;a)�z L is well de�ned in the quarter-plane. We �nish the proof by inspection of the boundary correspondence. Note two facts: the curve (7.2) has the asymptote u! �, v ! +1 (y ! +1) and we have uniqueness of the solution of the functional equation (7.2) due to uniqueness of the extremal polynomial. 8. Appendix 1 As it is said in [1], problem 42: El � 1 (b+ x)s � � ls�1 j�(s)j (b�pb2 � 1)l (b2 � 1) s+1 2 ; b > 1; s 6= 0; (8.1) where El[f(x)] is an error of approximation of the function f(x) on the interval [�1; 1] by polynomials of degree not more than l. We change the variable y = b+ x b+ 1 and put a2 = b�1 b+1 . Then we have inf P :degP�l max y2[a2;1] jy�s � P (y)j = (1 + b)sEl � 1 (b+ x)s � : That is E2l(�2s; a) = (1 + b)sEl � 1 (b+ x)s � : (8.2) Note that b = 1 + a2 1� a2 ; b2 � 1 = 4a2 (1� a2)2 ; and therefore p b2 � 1 = 2a 1� a2 ; b� p b2 � 1 = 1� a 1 + a : Thus from (8.1) and (8.2) we get E2l(�2s; a) � � 2 1� a2 � s ls�1 j�(s)j � 1� a 1 + a � l � 1� a2 2a �s+1 =a�s ls�1 j�(s)j � 1� a 1 + a �l�1� a2 2a � =a�s�1 ls�1 j�(s)j � 1� a 1 + a �l+1 (1 + a)2 2 : Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 105 F. Peherstorfer and P. Yuditskii 9. Appendix 2 Here we present a "solvable model" for the problem under consideration: we replace the comparably complicated con�guration (3.2), that we remove from the strip, by just two slits. We used this model on the �rst step of rough un- derstanding of the form of asymptotic and it might be useful for the reader, in particular, it contains the hint that in a nonmodel case the asymptotic of L qm m (a) for m!1 can also be found for an arbitrary q 2 N �xed, see (9.12). For B > 0, consider the conformal mapping w = �(z) of the upper half-plane C + on the strip � = fw : 0 < Imw < (k +m)�g (9.1) with the cut B = fw : Imw = k�; jRewj � Bg (9.2) under the normalizations �(0) = 0; �(�1) = �12; (9.3) where12 denotes the point on the boundary of the domain when we go to in�nity on the level k� < Imw < (k+m)�. By11 we denote the point on the boundary that corresponds to the level 0 < Imw < k�. Put a = �(�1)(+11) (therefore �a = �(�1)(�11)). Let us �nd a precise formula for this map as well as the relation between a and B. We have �(z) =k 1Z a � 1 x� z � 1 x+ z � dx+m 1Z 1 � 1 x� z � 1 x+ z � dx +k log x� z x+ z ���� 1 a +m log x� z x+ z ���� 1 1 =k log a+ z a� z +m log 1 + z 1� z : (9.4) Further, for a < x < 1 we have Re�(x) = k log x+ a x� a +m log 1 + x 1� x (9.5) and B corresponds to the critical value of this function on the given interval. For the critical point c we have (Re�)0 (c) = � 2ka c2 � a2 + 2m 1� c2 = 0: (9.6) 106 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 Uniform Approximation of sgn(x) by Rational Functions with Prescribed Poles Therefore c = r ma2 + ka m+ ka (9.7) and B = k log c+ a c� a +m log 1 + c 1� c : (9.8) Let us mention that the relation between a and B is monotonic, and a runs from 0 to 1 as B runs from 0 to 1. As the next step, we calculate the asymptotic behavior of B for the �xed a as m!1. First, we write the asymptotic for c c = r ma2 + ka m+ ka = a+ k 2m (1� a2) + : : : : (9.9) Therefore B =k log � 2a+ k 2m (1� a2) + : : : � � k log � k 2m (1� a2) + : : : � +m log 1 + a+ k 2m (1� a2) + : : : 1� a� k 2m (1� a2) + : : : =k log 2a 1� a2 + k log 2m k + : : : +m log 1 + a 1� a +m log 1 + k 2m (1� a) + : : : 1� k 2m (1 + a) + : : : =m log 1 + a 1� a + k log 2m+ k log 2a 1� a2 + k � k log k + : : : : (9.10) Actually, it was important for us to note that in the second (logarithmic) term in asymptotic we have the factor k. To �nish this section let us discuss asymptotic for the case k = qm; m!1 for a �xed q. Note that now c is just a constant c = s a2 + qa 1 + qa (9.11) and we have B =m � q log c+ a c� a + log 1 + c 1� c � ; (9.12) and B = 2m log 1+ p a 1�pa for q = 1. Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1 107 F. Peherstorfer and P. Yuditskii References [1] N. Akhiezer, Theory of Approximation. Dover, NY, 1992. [2] N. Akhiezer, Elements of the Theory of Elliptic Functions. AMS, Providence, RI, 1990. [3] S. Bernstein, Sur la meilleure approximation de jxj par des polynomes des degr�es donn�es. � Acta Math. 27 (1914), 1�57. [4] S. Bernstein, On the Best Approximation of jxjp by Polynomials of Very High Degree. � Izv. Akad. Nauk SSSR. Ser. Mat. 2 (1938), 169�180. [5] A. Eremenko and P. Yuditskii, Uniform Approximation of sgn (x) by Polynomials and Entire Functions. � J. Anal. Math. (To appear) [6] E.I. Zolotarev, Anwendung der elliptischen Funktionen auf Probleme �uber Funktio- nen, die von Null am wenigsten oder am meisten abweichen. � Abh. St. Petersb. XXX (1877). 108 Journal of Mathematical Physics, Analysis, Geometry, 2007, v. 3, No. 1

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