On approximation of the rational functions, whose integral is single-valued on c, by differences of simplest fractions

Komarov M.A.

Probl. Anal. Issues Anal. Vol. 7 (25), Special Issue, 2018, pp. 63-71

63

DOI: 10.15393/j3.art.2018.5510

The paper is presented at the conference "Complex analysis and its applications" (COMAN 2018), Gelendzhik - Krasnodar, Russia, June 2-9, 2018.

ON APPROXIMATION OF THE RATIONAL FUNCTIONS, WHOSE INTEGRAL IS SINGLE-VALUED ON C, BY DIFFERENCES OF SIMPLEST FRACTIONS

Abstract. We study a uniform approximation by differences ©i — ©2 of simplest fractions (s.f.'s), i. e., by logarithmic derivatives of rational functions on continua K of the class Qr, r > 0 (i. e., any points z0,z1 € K can be joined by a rectifiable curve in K of length < r). We prove that for any proper one-pole fraction T of degree m with a zero residue there are such s.f.'s ©1,©2 of order < (m — 1 )n that ||T + ©i — ©2||k < 2r-1 A2n+1n!2/(2n)!2, where the constant A depends on r, T and K. Hence, the rate of approximation of any fixed individual rational function R, whose integral is single-valued on C, has the same order. This result improves the famous estimate ||R + ©1 — ©2||C(K) ^ 2errn/n!, given by Danchenko for the case ||R||C(K) ^ 1. Key words: difference of simplest fractions, rate of uniform approximation, logarithmic derivative of rational function 2010 Mathematical Subject Classification: 41A25, 41A20

1. Introduction. By a simplest fraction (s.f.) of order n, n G N, we mean a logarithmic derivative of polynomial of degree n:

The function ©(z) = 0 is the s.f. of order n = 0.

The approximation properties of s.f.'s have become an object of intensive study after the paper [5] was published. It turned out, for example, that the rate of the approximation by s.f.'s for a wide class of functions and sets has the same order as for the polynomial approximation [5], [9].

UDC 517.538.5

M. A. Komarov

FKT

The first result on approximation by differences of s.f.'s, i.e., by logarithmic derivatives of rational functions, was also proved in [5] (Theorem A below). Let R*n be the class of rational functions of degree < n, whose integral is single-valued on C. We say that a set K c C is of the class Qr, r > 0, if any points G K can be joined by a rectifiable curve in K of length < r. Let || • ||K be a sup-norm over K.

Theorem A. [5] Let K G Or, R G R*N, ||R||K < 1. There are s.f.'s ©1; ©2 of order < (N + 1)n such that

||R +©1 - ©2|| k < 2errn/n! (n > 5r). (1)

The author has proved the following much more strong estimate in the case where R = M is a polynomial [7], [8] (hereinafter n0(x) = 14+ex2/4):

Theorem B. [8] Let M ^ 0 be a polynomial of degree N ^ 0, K G Or, ||M ||K < c. There are s.f.'s ©1; ©2 of order (N + 1)n, such that

- n |2

||M + ©i - ©2|| k < -(cr)2n+1 --— (n > no(cr)). (2)

r (2n)!2

In this paper we prove that the approximation of any given function R G RN has the same order. The crucial point is the following theorem on approximation of a one-pole fraction.

Denote by Kthe unbounded component of the complement of continuum K, and let K0 = C\{K U K~}.

Theorem 1. Let K G Qr, a G C\K, 6 = dist(a, K) > 0,

m

T(z) = E j, m > (3)

and C = ||T(z)(z — a)2||K. There are s.f.'s ©1,©2 of order < (m — 1)n such that

2 / Cr \2n+1 n!2 A:= ||T + ©1 — ©2||k < ^j ^ (n > no(Cr/62)). (4)

If |cm| ^ 1 and a G K0, then

A < - (l6r3||T|K+2/m)2"+1 ^ (n > no(16r3|T|K+2/m)). (5)

In the case where |cm| ^ 1, S ^ (diam K)/6 and a G K, the estimate (5) is also true, but the factor 16 must be replaced by (28/3)2.

Theorem 1 is proved in Section 4. In Section 2 we consider the general case where R G R*N. The following example shows that the conditions R G RN and T G R*m are essential for Theorem A and Theorem 1, respectively.

Denote by dn = dn(f) the best approximation of the function

f (x) = , 1 ,, a G R, a > 1, 2(x + a)

over x G [-1,1] by all differences of s.f.'s of order at most n. Proposition 1. If a =: 1 (p + p-1) > § (p > ), then

d„(f) > Mn(1+ o(1)), Mn := 21-2n(p + TP2-! - Ap-1)-2"-1 as n ^ to for some A G [—2,1 ].

Proof. Set || • || = || • ||[-1,1]. There is a difference D(x) of s.f.'s of order < n, such that ||D - f || = d„ • (1 + o(1)) as n ^ to (|| • || := || • |[-1,1]). Let R(x) be the rational function of degree at most n such that R(0) = y/a and D = R'/R.

Set I(x) = JX (D(t) - f (t)) dt. Obviously, ||11| < d„ • (1 + o(1)), (e1(x) - 1)Vx + a = R(x) - Vx + a, -1 < x < 1. Since dn ^ 0 as n ^ to, we have M = m(R) := ||(R(x) - Vx + a)/Vx + a|| < el|J11 - 1 < dn • (1 + o(1)).

But infRm = M"(1 + o(1)) (over all rationals R(x) of degree < n) [1]. □

In Section 5 we study the approximation of arbitrary rational functions by the quotients between two differences of s.f. 's. This useful method for the calculation of values of rational functions and polynomials was introduced in [2]. Recall, that the Horner scheme is usually applied for this. However, if the values of arguments and coefficients of these functions are large, using this scheme may lead to loss of accuracy because of multiple multiplications (see examples in [4, §3]).

2. Corollaries of Theorem 1. Set

P TOfc

R(z) = £ Tk (z), , mk > 2, (6)

k=1 j=2 (Z Zk)

Zk = Zj (k = j), mi +-----+ mp = N, p > 1.

Corollary 1. Let R be a function (6), m = maxk mk, c = maxkj |ckj|, 5k = dist(zk,K). If K € , S := mink Sk > 0 and A := c^j=2 , then there are s.f.'s ©1,©2 of order < (N — p)n such that

Ai := ||R + ©i — ©21|k < 2p(Ar)2n+1 ^ (n > no(Ar)). (7)

This assertion follows from (4), because R is the sum of p functions Tk of the form (3), k=1(mk — 1)n = (N — p)n and

c mk m

C" < E j ^ E j = A (Ck := ||Tk(z)(z — zk)2||k, 1 < k < p).

k j = 2 -k j=2

The estimate (7) is better than (1) for any fixed individual function R of the form (6). On the other hand, (1) is a universal estimate (i.e., (1) only depends on ||R||K and r), whereas (7) depends on the norms ||Tk||K of all p components of the function R = Tk, and it is easy to construct such a fraction R = T1 + T2 that ||Tk||k > 1 while ||R||k < 1.

We now consider the case where the set K has special form and in this case we get new estimates of A1 of the same order as in (7) but with more universal constants. Let R be a function of the form (6). We write K € (R) if K € Qr and all poles zk € K0, and every bounded component Kj0 of the complement of the set K (U Kj0 = K0) contains at most one of the poles zk, i.e., "poles of R(z) are separated by K".

Corollary 2. If K € ^(R) and ||R||k < 1, then (see (7))

2p ( 50mr3 \ 2n+1 n!2 A1 < — —^ 777-^ (n > no(50mr3/S2)).

r V -2 J (2n)!2 If, in addition, |ck,mk | > 1 for all 1 < k < p (see (6)), then

A1 < 2p (16 • 104r3)2n+1 (n > no(16 • 104r3)).

r (2n)!2

Indeed, because of K G Q*(R), the singularities of the function R = = ETfc are separated [3]: ||Tfc||k < 50mfc||R||k, 1 < k < p. Thus, the assertion follows from the estimates (4), (5) and ||z - a||K < r. To prove the last estimate of A1 we also use the fact that the function (50x)1+2/x is decreasing for x > 2, and hence (see (5)),

max ||Tk ||K 1+2/mfc < max(50mk)1+2/mk < (50 • 2)2 = 104.

Remark 1. Let R(z) = M(z) + R(z), where M be a polynomial and R be a fraction of the form (6). Let c := ||M||K > 0. Under the assumptions of Corollary 1 we have the following assertion: there are s.f.'s ©1, ©2 of order at most (deg M + 1 + N - p)n such that

n!2

||R + ©1 - ©21|k < 2r2n(c2n+1 + pA2n+1)—- (n > no(max{A,c}r)).

(2n)!2

3. Auxiliary results. Our first lemma is trivial:

Lemma 1. Let B(z) ^ 0 be a polynomial of degree N1 > 0, H(v) ^ 0 be a polynomial of degree N2 > 0,

F(z> = H (^) H. (8)

Let q1(v), q2(v) be polynomials of degree (N2 + 1)n > 0. Then the functions Sj(z) := (B(z))(N2+1)nqj(1/B(z)), j = 1,2, are polynomials of degree at most N1(N2 + 1)n, and the following identity holds:

F (z)+- sm. (H (v) - +q2'(v)

S1(z) S2(z) B2(z) V V ' 91 (v) ?2(v^ B(z) '

Let K and a be an arbitrary fixed set and a point in C. Put

Ka = {v : v = (z - a)-1, z G K}.

Lemma 2. If K G Qr, a G C\K and S := dist(a,K) > 0, then Ka G QTa, where ra := rS-2.

Proof. For any fixed points v0, v1 G Ka we put zj = a + v-1, j = 0,1. Since K G Qr, there is a rectifiable curve z(s), 0 < s < s1 (z(0) = z0, z(s1) = z1) in K of the length /0S1 |z'(s)|ds < r (s is a natural parameter).

Then the curve v(s) = (z(s) — a) 1, 0 ^ s ^ si (v(0) = v0, v(si) = vi) belongs to Ka, and the length of this curve

si si ^ si

|v'(s)|ds = —^-pyds ^^ |z'(s)|ds < r0-2.

7 7 |z(s) — a|2 02 7

o o o

Thus, the lemma is proved. □

Lemma 3. Let K be a continuum in C, T(z) be a function of the form (3). If 0 := dist(a,K) > 0 and cm = 0, then

1 < 4v V«TIIK/|cm|; v := { J; a I KC and 0 < (diamK)/6. (9)

Proof. Put v = 1/(z — a), T(z)/cm = tm(v),

tm(v) = C2V2 +-----+ Cm_ivm-i + vm, Cj = Cj/cm.

Let t(Ka) be the transfinite diameter of the set Ka. We have the following estimate [6]: t(Ka) < VI|tm||Ka = VIIT||K/|cm|. But K is a continuum, therefore [6], diam Ka < 4t(Ka) < 4 VI|T||K/|cm |. We now need to prove that diam Ka > 1/(v0). In the case a I K0, the estimate diam Ka ^ 1/0 is trivial. Let a I Kc and 0 ^ (diam K)/6. Let z1 I K be a point such that |z1 — a| = 0. Then we have maxz£K |z — a| ^ maxzeK |z — z1| — 0 and

diamK = max |z — CI ^ max |z — zJ + max |z1 — C| = 2max |z — z11,

z,2eK ze^ 2ek zek

therefore maxzeK |z — a| ^ (diamK)/2 — 0 ^ 30 — 0 = 20. Thus,

1 1 111

diam Ka > —-.----.-- > - — — = —,

minK |z — a| maxK |z — a| 0 20 20

and the lemma follows. □

4. Proof of Theorem 1. Firstly, we prove the estimate (4). Assume that T(z) ^ 0 (the other case is trivial). The function (3) has the form (8) with B(z) = z — a, H(v) = ^m=2 Cjvj-2 (deg H(v) = = m1 — 2 ^ m — 2). By Lemma 2 we have Ka I Qra, where ra = r0-2.

Obviously, H(v) = T(z)(z - a)2, therefore ||H||Ka = C. By Theorem B, there are s.f.'s 6j(v) = qj'(v)/qj(v), j = 1, 2, of order (m1 - 1)n, such that

||H - 01 + 02||k, < 2C(Cra)2"n!2/(2n)!2 (n > no(Cra)). (10)

Estimate (4) follows by (10), Lemma 1 and the equality ||B'/B2||K = S-2.

We have C < ||T||k(diamK)2 for a G K0. Thus, the estimate (5) follows by the estimates (4), (9) and diamK < r. Similarly, in the case a G Kand S ^ (diam K)/6 we have

C < ||T||K(S + diamK)2 < ||T||K((7/6)diamK)2,

and the theorem follows.

5. On approximation by special rational functions. Consider the following special fractions, introduced in [2, §8.2]:

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Theorem C. [2] Let K be a compact set, R be a rational function of degree N ^ 1, and r := ||R||K < to. There is a fraction © of the form (11) with orders mj < Nn such that

We now get a stronger estimate for the case K G Qr:

Corollary 3. Let P, Q be polynomials of degree at most N, K G Or, ||P||k < 1, infk |Q(z)| =: c0 > 0. Put c2 = ||Q|k. There is a fraction © of the form (11) with orders mj ^ (N + 1)n such that

Proof. Let ©1 - ©2 (©3 - ©4) be the difference of s.f.'s of order at most (N + 1)n that approximates the polynomial -P (-Q, respectively), as in Theorem B. Let n2 be an integer such that n > n0(r), n > n0(c2r) and

Corollary 3 is proved. □

Acknowledgment. The author is grateful to the referees for their useful suggestions.

This work was supported by RFBR project 18-31-00312 mol_a.

References

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[2] Danchenko V. I. Estimates of derivatives of simplest fractions and other questions. Sb. Math., 2006, vol. 197, no. 4, pp. 505-524.

DOI: https://doi.org/10.1070/SM2006v197n04ABEH003768.

[3] Danchenko V. I. O ratsionalnykh sostavlyayuschikh meromorfnykh funktsii i ikh proizvodnykh. Analysis Mathematica, 1990, vol. 16, no. 4, pp. 241255. (In Russian) DOI: https://doi.org/10.1007/BF02630358.

[4] Danchenko V. I., Chunaev P. V. Approximation by simple partial fractions and their generalizations. J. Math. Sci. (N.Y.), 2011, vol. 176, no. 6, pp. 844-859. DOI: https://doi.org/10.1007/s10958-011-0440-5.

[5] Danchenko V. I., Danchenko D. Y. Approximation by simplest fractions. Math. Notes, 2001, vol. 70, no. 4, pp. 502-507.

DOI: https://doi.org/10.1023/A:1012328819487.

[6] Goluzin G. M. Geometric theory of functions of a complex variable. Translations of Mathematical Monographs, vol. 26 American Mathematical Society, Providence, R.I. 1969 vi+676 pp.

[7] Komarov M. A. On approximation by differences of simple partial fractions. Abstracts International Conference "Complex Analysis and its Applications" (Gelendzhik-Krasnodar, June 2-9, 2018), Krasnodar, Kuban St. Univ., 2018, p. 65.

[8] Komarov M. A. On approximation by special differences of simple partial fractions. Algebra i analiz, 2018, vol. 30, no. 4, pp. 47-60. (St. Petersburg Math. J.)

[9] Kosukhin O. N. Approximation properties of the most simple fractions. Moscow Univ. Math. Bull., 2001, vol. 56, no. 4, pp. 36-40.

Received May 16, 2018. In revised form, September 14, 2018. Accepted September 15, 2018. Published online September 27, 2018.

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