Roulier, J.A

Taylor, G.D

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JOURNAL OF APPROXIMATION THEORY 11, 208-215 (1974) 
Rational Chebyshev Approximation on [0, +oo) 
J. A. ROULIER 
North Carolina State University, Raleigh, North Carolina 27607 
AND 
G. D. TAYLOR* 
Michigan State University, East Lansing, Michigan 48824 
Communicated by G. Meinardus 
1. INTRODUCTION 
For any nonnegative integer m, let flm denote the collection of all real 
polynomials of degree at most m. For given r > 0 and s > 1, let E(r, S) 
denote the unique ellipse in the complex plane with foci at x = 0 and x = r 
and semi-major and semi-minor axes a and b, respectively, such that 
b/a = (9 - l)/(? + 1). Iff(z) is any entire function, set 
tif(r, s) = max{ 1 f(z)1 : z E E(r, s)}. 
In a recent paper [2], a Bernstein type of theory has been developed for 
the problem of approximating real valued functions on the half line [0, co). 
Precisely, the following two results were proved. 
THEOREM 1. Let f(x) be a real continuous function ($0) on [0, oo), and 
assume that there exist a sequence of real polynomials {p,(x)}~~,, , with plz E 17, 
for each n > 0, and a real number q > 1 such that 
Then, there exists an entire function F(z) with F(x) = f (x) for all x > 0, and 
F is of finite order p. In addition, for every s > 1, there exist constants 
K = K(s, q) > 0, 8 = 6(s, q) > 0 and r0 = r&, q) > 0 such that 
n&, s> < Wf IIL~~o,Tl)e for all r > rO. (2) 
* Supported in part by NSF Grant GP-12088. 
208 
Copyright 0 1974 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
RATIONAL CHEBYSHEV APPROXIMATION ON [o, +C0) 209 
Also included in the theorem is a best possible upper estimate for p. It 
should also be noted that (1) implies that either f is identically a constant 
or limz+m If(x)1 = co. 
THEOREM 2. Let f(z) = C,“=, a,zn be an entire function with nonnegative 
coeficients and a,, > 0. If there exist real numbers s > 1, A > 0, 0 > 0 and 
r0 > 0 such that l@?(r, s) < A(jlfllLm[o.r~)e for all r b r,, then there exists a 
sequence of real polynomials {p,(~)}~=~ with pn E IIn for n > 0 such that 
Thus, Theorem 1 states that geometric convergence on the half line of the 
reciprocals of polynomials to the reciprocal of a real continuous function 
implies that the function is the restriction of an entire function satisfying a 
growth condition (2) on certain ellipses. Furthermore, this growth condition 
implies that F is an entire function of finite order. 
In the converse direction, if F is an entire function satisfying this growth 
condition and having nonnegative Taylor coejicients then there exists a 
sequence of real polynomials whose reciprocals converge geometrically to 
the reciprocal of F on the half line. 
It is this additional assumption of nonnegative Taylor coefficients that 
motivated the work of this paper. This condition is not necessary for the 
conclusion of Theorem 2. For example, the function F(z) = ez + 2e-” 
satisfies the other two hypotheses of Theorem 2 and the conclusion of 
Theorem 2 is valid for this function. 
In this paper we shall present a new sufficient condition for geometric 
convergence to occur. It is our conjecture that Theorem 2 is true if one 
replaces the requirement on the Taylor coefficients of F with the assumption 
that F is either identically a constant or lim,,, I F(X)] = + cc. However, 
we have not succeeded in proving this and we would be pleased if an answer 
to this question could be found. 
In the remainder of this paper we shall say that an entire function f has 
geometric convergence whenever there exist a sequence of polynomials 
{pn(~))~zo , pn E 17, for n 3 0, and a real number q > 1, for which (1) holds. 
Ah we shall write II . lhr~ and II - /Ic~,~) for II - IIL~,~I and II . lI.cm[o,~) , 
respectively. 
II. MAIN RESULT 
In this section we wish to prove a new theorem for geometric convergence 
to occur. This result will be a comparison type theorem. That is, we shall 
210 ROIJLIER AND TAYLOR 
show that if f does not differ too much from a function known to have 
geometric convergence then f has geometric convergence. The proof that 
we shall give is self-contained; however, many of results that we shall use 
are special cases of some recent work by J. A. Roulier [3] and [4] on w- 
approximation. This work studies the problem of approximating continuous 
function on [a, b] with polynomials in IIm with respect to a weight function 
of the form 
w(x) = 
( 
fi / x - xi p 
1 
-1 
) 
i-1 
where a < x1 < x2 < .*. < x, < b and ai is a nonnegative real number for 
each i = l,..., n. Since the functions that we are approximating are always 
entire functions and the powers oli are always integers here, we decided to 
simply develop the specific facts of w-approximation that we need within 
the proof without explicit reference to this more general study. We refer 
the reader to the papers referenced above for the details of w-approximation. 
THEOREM 3. Let f be an entire function having nonnegative zeros at 
precisely {Xi}:=, , 0 < x1 < x2 < *** < xk , with respective orders fil ,..., & 
and assume there exist real numbers K > 0, s, > 1, 0 > 0 and r0 > 0 such that 
@(r, so> G Wlf Iko.dB for all r > r, . (3) 
Further, assume there exist entire functions h and g such that 
(i) f(z) = h(z) + g(z) for all z E C, 
(ii) h(z) = Crzo a,z” with a, > Ofor n = 0, I,..., where h is not apoly- 
nomial and h has geometric convergence, 
(iii) there exists B > 0 such that g(x) > -B for all x > 0, 
(iv) there exist rl > 0, # > 0 and A > 0, such that g(x) < Ah”(x) for 
all x b rI , 
(v) there exists a sequence of positive integers {n3}~~,, for which 
0 < n,,, - n3 < p, p a fixed positive integer and 
g (nj+l)(~) < 0 for all x >, 0, j = 0, l,... . (4) 
Then there exist a sequence of real polynomials {s,(x)}E,, with s, E l7, for each 
n 3 0 such that 
Iii5 Ill n-am f;x) 1 1/n --- II I dx) [o&d < 1. 
RATIONAL CHEESYSHEV APPROXIMATION ON [o, +03) 211 
Proof. Set 
m=p1+*-+/L, r2 = max(r, , r, , xk + I), w(x) = i (x - xp 
i=l 
and select j,, > 0 such that njO 2 3m. Let p&x) and qam(x) be the Hermite 
interpolating polynomials from I&m to h(x) and g(x), respectively, at the 
points x1 ,..., xk with respective orders 3/3, ,..., 38, . Define h1 and g, by 
hl(z) = N4 - Pd4 
w”(z) 
and 
g (z) 
1 
= g(z) - %(4 
w"(z) * 
Note that both h1 and g, are entire functions. Select r, 3 r, such that 
1 z - xi 1 3 1 for i = I,..., k and z on the boundary of E(r3, s,,) and 
h(x) 2 max (E I bi 1 xg’, E 1 ci 1 xi) 
i=o i=o 
for x 2 r, , where J&X) = xi;“, &xi and q&c) = Crb CiX’. THUS, for 
r 2 r3, 
For each r 3 r, and j Z j, , let P,+~(x, r) be the best uniform approxima- 
tion to h1 from 17,,-, on [0, r], 
It is well known that pnlmanz is the Lagrange interpolating polynomial to h1 
on a certain set of points 0 < y1 < *a* < yn,-3m+1 < r. Set p,*,(x, r) = 
pn,&x, r) w”(x) + p,,(x). Then, it is easily seen that pz, is the Hermite 
212 ROULJJZR AND TAYLOR 
interpolating polynomial to h on a certain set of points in [0, r] including 
the set x1 ,..., xk with respective orders (at least) 38, ,..., 3pk and 
II w-3(x)(w - P,*,(X, ~))ll~&Tl = Gj-3&l). (8) 
Similarly, let qnj-3m E L7ni-3m be the best uniform approximation to g, on 
[0, r] with error E&,&) and set q,+(x, r) = q,+&, r) w”(x) + q3&). 
Then, q,*i(x, r) is the Hermite interpolatmg polynomial for g from II,) on a 
certain set of nodes in [0, r] and 
II w-“(x)(g(x) - d&G ~~)llb.rl = -c,-S&l). (9) 
Select r, > r, such that h(r,) >, max(2,2B) and w(x) < h(x) for all x > r4 
which is possible since h is not a polynomial. Write f(z) = j@(z) w(z) where J” 
is entire. Since g(x) 3 -B for x 2 0, we have thatf(x) 3 &h(x) for x >, r4 . 
This, in turn, implies thatf(x) 3 #z(x)/w(x)] for x > r, . Thus, there exists 
6 > 0 such that f*(x) > S for all x > 0. Now, fix r 3 r, and set s,*,(x, I) = 
p:Jx, r) + qZ&, r) + EFp3(x), where EZj = ~&&3 + Eij-3&l). As 
noted earlier, p,*,(x, r) and qz,(x, r) are Hermite interpolating polynomials 
to h and g, respectively, on certain sets of nodes in [0, r]. Thus, p$(x, r) 
has all nonnegative coefficients since P(x) >, 0 for j = 0, l,... and x 3 0. 
Also, the standard remainder formula for Hermite interpolation implies 
that g,*,(x, r) >, g(x) for x > r since gcnj+l)(x) < 0 for x > 0. Combining 
these facts and estimate (8), we have for x >, r and j 3 j, , 
S,*,(x, r) 3 P,*,<x, r) + -Gj--3&3 w”(x) + q,*,(x, r-1 
2 fW1 + g(x) 
>, !&h(r) 
and 
Since g(x) < Ah@(x) for x 3 r, we have that 
for r 3 r, and j 2 j, , where y = {max(l, I/J)}-’ and KI = 4(A + 1). 
Due to the special form of p,*,(x, r) and q:Jx, r) we may write .$(x, r) = 
RATIONAL CHEBYSHEV APPROXIMATION ON [o, +oO) 213 
fn,l-,(x, r) w(x), where +&, r) = Qnjsm(x, r) + E&w2(x). Now, for 
O,<x,<randjbj,,, 
where 
d &E;, , (11) 
, 
Using a result due to S. N. Bernstein [l, p. 911, we may estimate E& by 
since h, and g, are both entire functions. Using (5) and (6), we get 
where K3 = 2 s~/(so - 1). Since h has geometric convergence there exist 
by Theorem 1 real numbers K’ > 0, 0’ > 0 and r’ > 0 such that 
J%(r, soI < K’ II h Ifi.+. for all r > r’. Without loss of generality we shall 
assume r, 3 r’. Combining this with (3), gives 
E;, < 2 (4K' II h l/$.7, + ~llfllfo.d. 04) 
From the inequality f(x) 3 @(.x) for all x 3 r, and (i), we get that there 
exists a positive constant K4 , such that Ilfllro,rl > K4 II h Il[o,rl for all r > r, . 
Hence, using this, (ll), and (14) 
(15) 
214 ROULIER AND TAYLOR 
forr>r,,j>j,,andO<x<rwhere 
‘p = max(8, 0’) and K5 = K2 - K3(4K’KT1 + K). 
Now the fact that iim,,,f(r) = + co g ives a positive integer j, 3 j,, such 
that to each j > j, there corresponds an rj 3 r4 for which 
n,/b+d Ilfllto.~,l = .m = so * 
Consequently, if we set s,*,<x) = s~,(x, rj) for each j > j, we see from (10) 
and (15) that 
where s1 = s;‘(~+@ and K8 = max(K, , K6). Finally, using the sequence of 
polynomials s,(x) where s,(x) = S:,(X) for nj d n < nifl and j > j, , gives 
since nj, -rzn,<pforallj. 1 
We would like to remark that this Theorem remains true if one drops 
the requirement that h is not a polynomial. For in this case, it can be shown 
that the remaining hypotheses imply thatf is also a polynomial. 
Using this theorem, it readily follows thatf(x) = ez + cc-x has geometric 
convergence for each real constant c. We feel that an approach in this direc- 
tion may prove that an entire function satisfying the growth condition and 
tending to + co as x -+ co has geometric convergence. We had hoped to 
apply Theorem 3 to such a function by carefully separating its Taylor series 
into two parts. However, we have not succeeded and this remains open. 
We also feel that the hypotheses (iii)-(v) may be successfully weakened 
without affecting the truth of the theorem and this also is an open question. 
For example, Theorem 2 implies that f(x) = ex + cos x has geometric 
convergence. We conjecture that g(x) = ex + e4 cos x has geometric con- 
vergence. However, Theorem 3 is not readily applied here as the obvious 
decomposition of g does not satisfy the hypotheses of Theorem 3. 
REFERENCES 
1. G. MEINARDUS, “Approximation of Functions: Theory and Numerical Methods,” 
Springer-Verlag, New York, 1967. 
2. G. MEINARLXJS, A. R. REDDY, G. D. TA~OR, AND R. S. VARGA, Converse theorems and 
extensions in Chebyshev rational approximation to certain entire functions in [O, co), 
Bull. Am. Math. Sot. 77 (1971), 46C461; Trans. Am. Math. Sot. 170 (1972), 171-185. 
RATIONAL CHEBYSHEV APPROXIMATION ON [&+a) 21.5 
3. J. A. ROW, A note on weighted approximation on a closed interval, An. Acad. 
Brasileira Ciencias 43 (1971), 547-551. 
4. J. A. ROULDZR, A note on the rate of convergence of best weighted approximation, 
J. Approximation Theory, to appear. 


Rational Chebyshev approximation on [0, + ∞) 

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Rational Chebyshev approximation on [0, + ∞)

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