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Research Institute for Advanced Computer Science 

NASA Ames Research Center 


//v -&y 

New Bernstein Type Inequalities OS / 

for Polynomials on Ellipses 

Roland Freund and Bernd Fischer 


March 1990 


Research Institute for Advanced Computer Science 
NASA Ames Research Center 


RIACS Technical Report 90.12 


NASA Cooperative Agreement NCC 2-387 




(NASA-CR-186626) NEW BERNSTEIN TYPE N92-11724 

INEQUALITIES FOR POLYNOMIALS ON ELLIPSES 
(Research Inst, for Advanced Computer 

Science) 23 p CSCL 12A Unclas 

G 3/64 0043051 




New Bernstein Type Inequalities 
for Polynomials on Ellipses 


Roland Freund and Bernd Fischer 


March 1990 


Research Institute for Advanced Computer Science 
NASA Ames Research Center 


RIACS Technical Report 90.12 


NASA Cooperative Agreement NCC 2-387 


NEW BERNSTEIN TYPE INEQUALITIES 
FOR POLYNOMIALS ON ELLIPSES 


ROLAND FREUND 


BERND FISCHER 


Institut fur 

Angewandte Mathematik 
Universitat Hamburg 
D - 2000 Hamburg 13, F.R.G. 

and 

RIACS, Mail Stop 230-5 
NASA Ames Research Center 
Moffett Field, CA 94035, U.S.A. 


Institut fur Angewandte 
Mathematik und Statistik 
Universitat Wurzburg 
D - 8700 Wurzburg, F.R.G. 


Abstract. We derive new and sharp estimates for the growth in the com- 
plex plane of polynomials known to have a curved majorant on a given ellipse. 
These so-called Bernstein type inequalities are closely connected with certain con- 
strained Chebyshev approximation problems on ellipses. We also present some 
new results for approximation problems of this type. 


1980 Mathematics Subject Classification (1985 Revision). 41A17, 41A50, 26D05, 
33A65. 

Key words and phrases. Bernstein type inequalities, curved majorant, complex Cheby- 
shev approximation, Chebyshev polynomials of the second kind. 

The first author was partially supported by Cooperative Agreement NCC 2-387 be- 
tween the National Aeronautics and Space Administration (NASA) and the Universities 
Space Research Association (USRA). The second author was partially supported by the 
German Research Association (DFG). 


2 


Roland Freund and Bernd Fischer 


1. INTRODUCTION AND STATEMENT OF THE MAIN RESULTS 


A classical result due to Bernstein [1] is 
Theorem A. Let n £ W. Then 

(L1) c = ± l( fi + s)’ *»• 

for any p £ II n which satisfies 

(1.2) \p(z)\ < -j= 1 — _ for all - 1 < z < 1 . 

The estimate (1.1) is best possible with equality holding only for p = e xa U n , a £ 1R. 


Here and in the sequel, 


(1.3) 


U„ 


v n+1 - l/v n+1 1/ 1\ 

( * )s v-i/v ■ ‘Mr*;)- 


is the nth Chebyshev polynomial of the second kind. Note that U n = T( l+1 /(n + 1) where 


(1.4) 



is the usual nth Chebyshev polynomial. Furthermore, throughout this paper, H n denotes 
the set of all complex polynomials of degree at most n. Bernstein and Markov type 
inequalities for polynomials with a curved majorant of the form (1.2) were studied by 
several authors (see [10, p. 90], [11,12,9,8] and the references in the recent paper by 
Rahman and Schmeisser [13]). Note that often (1.2) is written in the form 


\q(z)\ < y/l — z 2 for all — 1 < z < 1, 


which, obviously, is equivalent to ( 1 . 2 ) with q{z) = (1 - z 2 )p(z). 

Interestingly, for the case of complex c, sharp estimates ( 1 . 1 ) for polynomials satisfying 
(1.2) are known only for special cases. For polynomials p £ H n with recil coefficients, 
Rahman [ 11 , Theorem 4], [12] has shown that 

lp(c)l - v/|i - c»| I (*" +I + jF«) foran C = <~I’ R )- 

Here c £ C \ [ — 1 , 1 ] is arbitrary and parametrized in the form 

(1.5) c = c( 7 ,R) := ^)cos 7 + ^) sin 7> 0 < 7 < 2tt, R > 1 . 


Bernstein Type Inequalities 


3 


For complex polynomials with (1.2) it follows as a special case of Corollary 1 in [5] that 

< L6) W«>'*7|IT3J 5 (*” +,+ “*■’)• r>1 ■ 

Moreover, (1.6) is best possible with equality holding if, and only if, 

(1.7) p(z) = ^—-(R2 Un (z) ± 2 iU n -i(z) - U n - 2 (z)), a € R. 

It seems that, for complex polynomials satisfying (1.2), the cases (1.1) and (1.6) are the 
only ones for which sharp bounds are known. In particular, it can be shown that, for 
c € C \ IR, the polynomials U n are never extremal for best possible inequalities of the type 
( 1 . 1 ). 

Since the interval [—1,1] can be viewed as the degenerated case r = 1 of the family of 
ellipses 


( 1 . 8 ) 


S r = { Z e C 


1| + |* + 1| <r + - }, r > 1, 


with foci at ±1 and semi-axes (r ± l/r)/2, it is natural to ask for estimates of the form 
(1.1) for polynomials 


(1.9) 


p 6 II n (r) := { p 6 II n I |\/l - z 2 p(z) | < 1 for all z E S T } 


which satisfy (1.2) on £ r . In this note, we present several new Bernstein inequalities of 
this type. In particular, it turns out that, somewhat surprisingly and in contrast to the 
case S\ = [—1,1], the polynomials (1.3) still lead to optimal estimates for the true ellipse 
case r > 1, as long as c = c( 7, R) is not “too close” to S T . Note that, for fixed R, (1.5) is a 
parametrization of the boundary of the ellipse Sr, and R — r is a measure of the distance 
of c(j,R) to E r . More precisely, we will prove the following result. 


Theorem 1.1. Let n £ IV and r > 1. There exists a number R*(n,r ) (> r) such that, 
for all p E II n (r), 


( 1 . 10 ) 


\p(c)\ < 


l yj ( + l/IE n+1 ) 2 — 4cos 2 ((n + 1) 7 ) 

V^ll - c 2 1 r n+1 + l/r n+1 


for all c — c(j, R) with R > R*(n, r). The estimate (MO) is best possible with equality 
holding if, and only if, 


( 1 . 11 ) 


P( z ) 


2e ta 

r ^+l _j_ lj r n+l 


Un(z), 


a € JR. 


4 


Roland Freund and Bernd Fischer 


Furthermore, 


( 1 . 12 ) 


R*(n,r ) < 


r 


65 r 4 - 1 
r 4 — 1 


Remark 1.2. The upper bound (1.12) for R*(n,r ) is very pessimistic. In particular, 
numerical tests indicate that R*(n,r) r for r or n large. We were not able to prove this. 

For special values of 7 , the estimate (1.10) is true for c(j, R) with arbitrary R > r, as 
long as r is sufficiently large. 

Theorem 1.3. Let n E IN and m £ {0,1,..., 2 n + 1 }. Then there exists a number 
r*(n) > 1 such that, for all p £ II n (r), 


(1.13) 


\p(c) | < 


1 

\/|l-c 2 | 


fl n+1 + 1/JT+ 1 

pTl-fl _|_ JyfpTl-fl * 


c = 


/m -f 1/2 
V n + 1 





R> r > r*(n). 


The estimate (1.13) is best possible with equality holding only for the polynomials (1.11). 

Actually, it turns out that the inequalities (1.10) and (1.13) also hold true for poly- 
nomials p which, instead of p £ II n (r), satisfy the weaker condition 

(1.14) p £ n i?\r) := { p £ n n | I sj\ - zf p(z t )\ <1, l - 0, 1, . . . , 2n + 1 }. 


Here and in the sequel, 

(1.15) zi = zi(r) := i (r + ^ cos <p t + (r - i) sin <pi, <p t := 

Theorem 1.4. Theorems 1.1 and 1.3 remain true ifp £ II n (r) is replaced by p £ II (r). 

By means of this last theorem, we will deduce the following corollary to Theorem 1 . 1 . 

Corollary 1.5. Let m £ 2 IN be even and r > 1. There exists a number R*(m,r) (> r) 
such that, for all “ self-inverse ” polynomials 


s £ E m := {s £ n m | s(v) = — i? m s(-)}, 


(1.16) 

' v 

the inequality 

W«0I < R > R‘(m, r). 


Bernstein Type Inequalities 


5 


holds. The estimate (1.17) is best possible with equality only for s(v) = <r(v m — 1), <t € C. 
Furthermore, 


R*(m,r ) < r 


65r 4 - 1 


r 4 — 1 

Remark 1.8. The inequality (1.17) was motivated by a recent result of Frappier, Rahman, 
and Ruscheweyh. In [4, Theorem 9], they showed that to each given polynomial p £ II m 
there exists a number p(p) > 0 depending on p such that 


R m 

max: |p(u)| < — — max |p(re’- J ’ r / m )| for all R > rp(p), r > 0. 

|v[<H 7* 771 j=1,2,...,2t7i 


For real c and the true ellipse case r > 1, we obtain the following extension of Bern- 
stein’s result (1.1). 

Theorem 1.7. Let n £ IN. 

a) Let r > 1 and c £ 1R with |c| > (r 4 + r 2 + l)/(r(r 2 + 1)). Then, for all p £ n n (r), 


(1.18) 


\p(c) | < 



R n+1 - 1/R n+1 

r n+l _j_ 1 j T n+l ’ 



The estimate (1.18) is best possible with equality holding only for the polynomials (1.11). 
b) There exists a number r(n) > 1 such that to any r > £(n) one can find numbers R > r 
and polynomials p £ D n (r) for which (1.18) is not fulfilled. 

Remark 1.8. For r = 1, the estimate (1.18) reduces to (1.1). Moreover, note that 
Theorem 1.7 leaves open the problem of finding sharp Bernstein type inequalities for c £ 1R 
with 1 < (r + l/r)/2 < |c| < (r 4 + r 2 + l)/(r(r 2 + 1)). 

The rest of the paper is organized as follows. In Section 2, we collect some auxiliary 
results. The problem of obtaining sharp Bernstein type inequalities of the type (1.1) can 
be reformulated via weighted complex Chebyshev approximation. In Section 3, we derive 
some new results for such approximation problems. Finally, the proofs of the results stated 
in the introduction and in Section 3 are given in Section 4. 


2. PRELIMINARIES 

In this section, we introduce some further notation and list some auxiliary results. 
In the sequel, it is always assumed that k = 0,1,... and r > 1. Moreover, let n £ IN 


6 


Roland Freund and Bernd Fischer 


be fixed and set tpi := (/ + l/2)7r/(n + 1). Finally, let the branch of the square root in 
u;(z) := y/l — z 2 be chosen such that 



In view of (1.3), this choice guarantees 


( 2 . 1 ) 





Next, let c = 0(7, R) be as in (1.5) and set 


( 2 . 2 ) 


d k := w(c)Uk(c). 


With (1.5) and (2.1), one readily verifies that 

dk = -Ak+i sin ((k + 1)7) + iBk+i cos((fc + 1)7), where 

Ak+1 '' = 2( Rk+1 + R *+*) and Bk+1 := 2( Rk+1 ~ 

In particular, (2.3) yields 


(2.3) 


(2.4) 


= A 2 k+I - co s ! ((fc + 1)7). 


Let us introduce the Chebyshev norm ||/||£ T := max ie £ r |/(z)| on £ r . Using (1.5) (with R 
replaced by r) and (2.1), a straightforward computation shows that 

(2-5) Wl«. = j('" +1 + ^l) 

and the maximum is attained precisely for the points zi = zj(r), / = 0, 1 , . . . , 2n + 1, defined 
in (1.15). Moreover, 


( 2 . 6 ) 


u(zi)Uk(zi) = — aj. + i sin((fc + l)y?j) + ibk+i cos((k + 1 )<pi), where 

5( r ‘ + ' + ^r) “ d 


flfc+i : = 


and, in particular, 


(2.7) u)(z t )U n (zi) = -(-l) z a n+1 . 

Next, we state a criterion due to Rogosinski and Szego [15] for the nonnegativity of 
cosine polynomials. 


Bernstein Type Inequalities 


7 


Lemma 2.1. Let A 0 , Aj,.. . , A m be real numbers which satisfy A m > 0, A m _i — 2A m > 0, 
and A„_i — 2A„ + A„+i > 0 for v = 1, 2, ... ,m — 1. Then 


( 2 . 8 ) 




i(<p) := ^ A„ cos (u<p) > 0 for all <p € M. 


i/=i 


As a first application of Lemma 2.1, one readily obtains the following 
Proposition 2.2. Let n € W and j G {l,2,...,n}. Then 


2 (n + 1— ;)! ^ (n + 1 _ j - v)\ v Y) ~ Y 


| n+l-j 


(n + 1 — i/)! 


Finally, we collect some discrete orthogonality relations which will be used in the next 
section. 


Proposition 2.3. 

*) 

2n+l 


(~l) l e ijv> = | 2l ( n 
1=0 ' 0 


+ 1)(-1) TO if j = (n + l)(2m + 1), m G Z, 

otherwise. 


b) 


c ) 


= 0 fa,, u = 1,2, ... ,n. 


k = o 


1 re n + 1/2 if fc = 0 (mod 2(n + 1)), 

2 + ( — 1)* COS ( I/ n ^ 7r ) = ^ — 1/2 ifkj/z 0 (mod 2(n + 1)) is even, 

. 1/2 if k is odd. 


V=1 


d) 


n + 1 

2 


+ ^(n + i 

i/=i 



n -J- 1 




0 

1 

2 sin 2 (fcir/(2(n + 1))) 


if k is even, 
ifk is odd. 


= (-l)'(-i + (-l)‘ 2(1 ! n c ^ y|) ) 


8 


Roland Freund and Bemd Fischer 


Proof. The proofs of a) - c) and e) are straightforward. For example, in view of 

2n+l 2n+l 


‘"ti .. i 

£ (-1)V»' = £ (-««&) , 

1=0 1=0 V 


part a) is an immediate consequence of 

2n+l 


y f_ e ^rry = / 2 ( n + 1 ) if ie(" + 

,_ 0 ' ' \ 0 otherwise. 


+ \){2Z + 1 ), 


Parts b), c), and e) follow similarly. For part d), apply the well-known identity (see e.g. 
(15, p. 75]) 

» + l . A, . - x ,x 1 /sin((n + l)yj/2)\2 


+ £(» + 1 - ”) cos(u<p) = - ( 


i/=i 


2 V sin(y?/2) 


for <p = fc7r/(n + 1). 


3. A WEIGHTED COMPLEX CHEBYSHEV APPROXIMATION PROBLEM 

The problem of determining sharp estimates (1.1) for complex c and polynomials 
p € H n (r) is intimately related to the family of constrained approximation problems 


w(z) 


'1-Z 2 


(3.1) (£ n (r,c) :=) min max |w c (z)p(z)|, u> c (z) := , , - v . 

Here and in the sequel, it is always assumed that c = c(7,I2) (cf. (1.5)) and R > r > 1. 
Standard results from approximation theory (see e.g. [7]) then guarantee that there always 
exists a unique optimal polynomial for (3.1). Clearly, the minimal deviation E n (r, c) of 
the approximation problem (3.1) yields the best possible constant in the Bernstein type 
inequality 

1 1 


(3.2) 


\p(c)\ < 


max | v 1 - z 2 p(z)|, p 6 II n . 


v/iT^J E n (r,c)?e7: 

Furthermore, equality in (3.2) holds if, and only if, p is a scalar multiple of the optimal 
polynomial for (3.1). 

The solution of (3.1) is classical for the case r = 1, £ r = [—1, 1], and c E 1R. \ [—1, 1]. 
Here Bernstein [1] (cf. Theorem A) proved that the scaled Chebyshev polynomial of the 
second kind 

EM*) 


»M*;c) = 


EM<0 


(3.3) 


Bernstein Type Inequalities 


9 


is optimal for (3.1). For purely imaginary c and, again, r = 1, Freund [5] showed that 
the extremal polynomial for (3.1) is a suitable combination of u n (z; c), u n _i(z; c), and 
«n-2 ( z ; c ) (compare (1.7) and (1.6)). We are not aware of any other cases for which the 
solution of (3.1) is explicitly known. 

In this section, we will derive conditions for the polynomials (3.3) to be optimal for 
(3.1) in the general case r > 1 and c E C \ £ r . Our main tool is Rivlin and Shapiro’s 
characterization [14] of the best approximation for general linear approximation problems. 
Recall (see (2.5) and (2.6)) that ||o7 c u n ||f, is attained just for the points zj , 1=0,... ,2n+l, 
stated in (1.15). Using (2.7), we then deduce from [14] the following 

Criterion 3 . 1 . u n (z;c) is the unique optimal polynomial for (3.1) if, and only if, there 
exist real numbers cr 0 , <J\ , . . . , <T 2n +i (not all zero) such that 

2n+l 

(3.4) cri{— l) , u;(z/)g(zj) = 0 for all q E II n with 9(c) = 0. 

1=0 

We remark that (3.4) is a system of linear equations for the unknowns <To , . . . , <72 n +i- 
It turns out that one can derive explicit formulas for all real solutions of this linear system. 
To this end, note that it suffices to check (3.4) for 

(3.5) ?(*) = Ufc(z) - U k (c), fc = 1,2, ... ,n, 
only. Furthermore, we will use the ansatz 

n+1 

(3.6) <r\ = YXXj cos (jipi) + Pj sm(jipi)), l = 0,1,..., 2n + 1, 

j = 0 

where A j, pj E 1R, j = 0, 1, . . . ,n + 1, and the <pi are defined in (1.15), for the unknowns 
of (3.4). Clearly, every collection of real <r 0 , . . . , <72n+i admits a representation of the form 

(3.6) . Next, we insert (3.5) and (3.6) into (3.4) and, furthermore, rewrite w(zi)Uk(zi) by 
means of (2.6). Using part a) of Proposition 2.3, one readily verifies that the resulting 
linear system (3.4) decouples as follows: 

An-fcflfc+i - ipn-kbk+i - (A n ai - ip n h \ ) U*(c) = 0, k = 1,2 , . . . ,n — 1, 
2^o®n+i (A n ai ip n b 1) U n {c) — 0. 

Here aj, bj, j = 1,2, ...,n + 1, are defined in (2.6). By determining all real solutions 
A j, pj of (3.7) and inserting them into (3.6), we finally obtain all E 1R which satisfy the 


(3.7) 



10 


Roland Freund and Bernd Fischer 


linear system (3.4). A straightforward computation shows that this solution space of (3.4) 
is two dimensional and given by 

(3.8) = (-l) l p + rp t , l = 0,1,..., 2n + 1, /i, t € H, 


where 

(3.9) 


1 MnP / - (dndv— i) , , v Im (c£ n <£„_i) , v\ 

<■' = 2^7 +( - 1) S( — z — sm( ‘ / '" ) + — hi — 


with d„ defined in (2.2). 

Note that the sign of cri still depends on the choice of the free parameters p and r in 
(3.8). However, it is possible to restate Criterion 3.1 in terms of the pi only. 

Theorem 3.2. Let n € IN, R > r > 1, and c = c(j,R). Then, u n (z;c) is the unique 
optimal polynomial for (3.1 ) if, and only if, 


(3.10) i j k := P2j + P2k+i > 0 for all j, k = 0,1, . . . ,n. 


Moreover, if u n is optimal, then 


(3.11) 


r. ( \ r n+1 + 1 / rn+1 

’ y / (Ji n+1 + 1/J2 n+1 ) 2 — 4cos 2 ((n + 1)7) 


Proof. In view of Criterion 3.1 and (3.8), it remains to show that there exist p, r € 1R, p 
and r not both 0, such that 


(3.12) <n = (-l)V + rpi > 0, / = 0,l,...,2n + l, 

iff (3.10) is fulfilled. Clearly, we may assume that r E {— 1,+1}. First let r = 1. Then 
(3.12) holds iff there exists a p such that 


min Pik+i > P > - min p 2 j. 

0<fc<n 0 <]<n 

Obviously this condition is equivalent to (3.10). Analogously, for r = —1 we arrive at 


or, equivalently, 


— max p2k+\ > P > max p 2 i 

0<fc<n 0 < j<n 


(3.13) 


tjk < 0, for all j,k = 0,1 ,..., 


n. 


Bernstein Type Inequalities 

However, the case (3.13) can always be excluded since 


11 


(3.14) 


£ i4l = („+l)M>0 

m °”+> 


holds. It remains to verify (3.14). Using trigonometric formulas, one readily deduces from 
(3.9) that, for all j, k = 0, 1, . . . , n, 




\dr 


(3.15) 


®n+l 


+ 2 


v /'Re (d n d^_i) ( j + k + 1 ^ 
\ a v v n + 1 1 


u/=l 


Im (d n d v - 1) . , j + k + 1 


j - k - 1/2 


6, 


. ( J - r T x \ \ 1 . , J — K — i. A s 

smli/ — 7 r) sin i/ 7r) 

v n + 1 VI v n + 1 ' 


By considering (3.15) for j = k = 0 ,... ,n and applying Proposition 2.3 b), we arrive at 
(3.14). 

Finally, note that the formula (3.11) for E n (r,c) follows from (2.4) and (2.5). ■ 

Since every optimal polynomial is in particular optimal with respect to the set of its 
extremal points, we have 

Corollary 3.3. Theorem 3.2 remains true if S r in (3.1 ) is replaced by the set {z[ | l = 
0, 1, . . . , 2n + 1} of extremal points of u n (z; c). 

Every nontrivial solution of (3.4) always leads to a lower bound for the minimal 
deviation E n (r,c), which is sharp in a certain sense. 

Corollary 3.4. Let <tj, l = 0, 1, . . . ,2n + 1, be any nontrivial real solution of (3.4), 
normalized such that |<t/| = 1. Then 


(3.16) 


(l n (r,c) :=) 


2n+l 


E (-1 ) , <rjw c (zj) 


1=0 


< E n (r, c). 


Proof. Let q £ H n with q(c) = 0. We deduce from (3.4) 


2n+l 


XI (“l)Wc(*l) 


i=0 


2n+l 


E (-l) f *i«c(*i)(l - q{z{)) 


1=0 


2n+l 


< M*)(i - $(*))| e 


1=0 


and the result follows. ■ 


12 


Roland Freund and Bernd Fischer 


The following example illustrates the lower bound (3.16). 

Example 3.5. We computed the relative deviation 

D (r c) - hn(z;c)\\ £r -L n (r,c) 

of the lower bound (3.16) from the weighted Chebyshev norm of u n for various cases. In 
Figure 3.6 the result for n = 2, r — 3, and c G [-2.5, 2.5] x i[- 2.5, 2.5] \£ r is displayed. For 
<rj in (3.16), the numbers (3.8) with p = 0 and r = 1 were used. Note that Di(r, c) = 0 if U 2 
is optimal for (3.1). Moreover, for points c € S r inside the ellipse, we have set D n (r , c) = 0. 


Figure 3.6. Relative deviation of «2 


Figure 3.6 as well as our other numerical experiments suggest that the polynomials 
tt n (z; c) are optimal for (3.1) as long as c is not “too close” to £ r . Furthermore, for certain 
fixed values of 7 and r sufficiently large, it seems that u n (z ; c) is optimal for all R > r. In 
accordance with these observations, we obtained the following results. 

Theorem 3.7. Let n € W, r > 1, and c = c(j,R). If R > r(65r 4 — l)/(r 4 - 1), then 
u n (z;c) is the unique optimal polynomial for (3.1). 

Theorem 3.8. Let n G IV, and c = 0(7, R). There exists a number r*(n) > 1 such that, 
for all 7 = 7 m = (m + l/2)7r/(n + 1), m = 0, 1, . . . ,2n + 1, and R > r > r*(n), u n (z ; c) is 
the unique optimal polynomial for (3.1). 

Note that for fixed 7 m the points 

C = c(7 m ,R) = R + cos 7m + sin 7m, R > 1, 

describe a hyperbola which intersects 6 r just at the extremal point z m (r) of u n . 

Finally, for the special case of real c, we will prove in the next section the following 


Bernstein Type Inequalities 


13 


Theorem 3.9. Let n £ IN, R > r > 1, and c = ±(R + l/R)/2. 

a) If |c| > (r 4 + r 2 + l)/(r(r 2 + 1)), then u n (z;c) is the unique optimal polynomial for 

(3.1). 

b) There exists a number f(n) > 1, such that to any r > f(n) one can find numbers R > r 
for which u n (z;c) is not optimal for (3.1). 

Remark 3.10. An analogue to Theorem 3.9 for the case of unweighted problems (3.1), 
i.e. u> = 1, was derived by the authors in [3, Theorem lb), 2b)]. Futhermore, in [2] resp. 
[6], we obtained a result similar to Theorem 3.7 for approximation problems of type (3.1) 
with complex c and weight functions w(z) = 1 resp. w(z) = y/l -f z. 


4. PROOFS OF THE MAIN RESULTS 


In this section, we give the remaining proofs of the results stated in the introduc- 
tion and in Section 3. First, recall the connection between the constrained Chebyshev 
approximation problem (3.1) and the inequality (3.2). Moreover, note that, by (3.11), 


E n (r, c(j,R)) 


^/(R n + 1 -f l/i2 n+1 ) 2 — 4cos 2 ((n + 1 ) 7 ) 

fTi-hl _|_ 1 / r n+ 1 

g_ n+i ± v Rn+ ' if ..- (-+■/»- ra6iZ 

r n + l + l/ r n+l ' n- hi ? 771 t 


R n+1 - l/i? n+1 . 


r n - hi _|_ 


if 7 = 0, 7T, 


and, by (2.5), the polynomials (1.11) are in H n (r) (see (1.9)). Thus, in view of (3.2), 
Theorem A is an immediate consequence of Bernstein’s results [1], while Theorem 1.1, 1.3, 
and 1.4 follow directly from Theorem 3.7, 3.8, and Corollary 3.3, respectively. 

Corollary 1.5 follows from Theorem 1.1 by rewriting the discrete (cf. Theorem 1.4) 
version of (3.2) by means of the Joukowsky map 


»sK) 

for the disks |u| < R and |u| < r. Let m £ 21N be even and set n := m/2 — 1. Then, using 
(2.1), one readily verifies that 


m ? 




pe n n , 




14 


Roland Freund and Bemd Fischer 


defines a one-to-one mapping between II n and the class of polynomials (1.16). Therefore, 
we deduce from (3.2) and (3.11) 


max ls(v) I < 
M<H ' ~ 


J2 n+1 


1 


max 


r n +i C ££ t E n (r,c) |t>|<r 
ft™ + 1 i / „*(2/-l)»/m 


max 

r-^ + l 1 = 1,2 m 


max |*(»)l 

\v\<r 

s(re ii2l ~ 1)ir/m )\, 


where the last equality holds if u n (z;c) is optimal for (3.1) for all c € S r . 

It remains to prove Theorems 3.7 - 3.9. We start with the 
Proof of Theorem 3.7. Let j y k € {0,1,..., n} and tjk be given by (3.15). In view of 
Theorem 3.2, we need to show that 


(4.1) 


65r 4 - 1 

R>r— — , r> 1, 

r 4 — 1 


implies tjk > 0. To this end, note that, by (2.4) and (2.3), 
(4.2) 
and 


Kl 2 > A 2 n+ , - 1 > i(R 2n+2 - 2) 


(4.3) | Re (d n d v ~,)\ < i2 n+1+I ', | Im (d n dv-i)\ < R n+1+I/ , v= 

Using (3.15), (4.2), (4.3), and a n+J < r n+1 , one obtains 

Kl* " 


tjk > 


(4.4) 


fln+l 




Re (d n d„_i)| Im (d n d^_i) 


+ 


V=1 


> 


R 2n+2 _ 2 


4 7* 71-}- 1 

By means of the estimates 

R 2n+2 _ 2 > ±fl2n+2 &nd 


v— 1 V ' 


1 r 4i/ 


< 


-, ^ = 2,3, ... , 


2 r 4u — 1 r 4 — 1 

which are guaranteed by (4.1), we further deduce from (4.4) the inequality 


(4.5) 


tjk > 


R 2n+2 


h 


64r 5 


— rO 


8r n +i V (r 4 — l)(i2 — r). 

However, by the first condition in (4.1), the lower bound in (4.5) is nonnegative, and this 
concludes the proof. ■ 



Bernstein Type Inequalities 


15 


Proof of Theorem 3.8. Let r > 1, m E {0,1,..., 2n + 1} be arbitrary, but fixed and let 
7 = (m + l/2)7r/(n + 1). For l = 0,1,..., 2n + 1 and R > r, we consider the numbers 
pi defined in (3.9). A standard calculation, using (2.3), (2.6), and simple trigonometric 
identities, yields 


Pi 


1 R n+1 +l/R n+1 


(4.6) 


2 +l/r n + 1 

(<!)■-( *r)-( >^.)] 


I/=I 


= : MR)- 

By Proposition 2.3 c), we have 


n Jn + 1/2 ifm-laO, 

(4.7) fi(r) = — + ( — l) m_i £cos(, -7r^ = < —1/2 if m — l ^ 0 is even, 

v=l 1 1/2 if m — l is odd. 


^From (4.6) one easily deduces that for the derivatives of fi 
(4.8) 

holds. Furthermore, = 0 if j > n and 


fP M = i e 


Jj) 1 (n + 1)1 , (n + l-i/)! f m-l 

m ~ l 2(n + l-j)! + (n + l-i-^lTn + l 

Remark that, in view of Proposition 2.2, 

(4-9) > 0 for all j 6 IN, 



j = I,-- 


n. 


and, by Proposition 2.3 d), 

c m-i = \( n + 1) + J^(rc + 1 - I/ ) cos ( I/ ^T 7r ) 

l/=l 

(4.10) , o 

if m — / is even 


2 sin 2 ((m — /)7r/(2(n + 1))) 


if m — l is odd. 


16 


Roland Freund and Bernd Fischer 


Next define 
(4.11) 


R n+1 

MR) ■■= -xrrMR)- 


By (4.6), pi is a polynomial in R of degree not exceeding 2 n -f 2. By means of Leibniz’s 
rule, we obtain from (4.11) and (4.8) that 

[ — (fi(r) ~ — — — ‘ - -f e^_i) + Q( " ~ V;" ) ifl<i'<n + l, 

(4.12) p { "\r) = i ^ V + 1” *)'• ' rV+ 

Lr4:l, + 0(==r) 


if n + 2 < v < 2n + 2, 


where 


= E (■) j - = -"-!}• 

Note that (4.9) implies 
(4.13) 


, > 0 for all v € IN. 

m — i — - 


Next, let M > 1 be any fixed constant. Then, by inserting (4.12) into the Taylor series of 
pi , we deduce that, for all 1 < r < R < Mr , 

(4.14) 

2n-f2 (v) / \ 

MR)='E iL jr(K-'r 


v=0 


= c 


(1) 
m — / 


R — T 


+ E (" + >W(^)' + E %(^)' + <*;?>■ 

i/=0 ' ' v=2 


Now, let j, k £ {0, 1, . . . , n} and tjk be defined by (3.10). From (4.7) resp. (4.10), it follows 
that 


( 4 - 15 ) h j(r) + fik+i > 0 resp. c l m } _ 2j + c^ l j _ 2t _ 1 > 0. 

Finally, using (4.6), (4.11), (4.13)-(4.15), we conclude that to any fixed M > 1 there is a 
number r(M) > 1 such that, for all r(M) <r<R< Mr, 

r n+l 

(4.16) tjk = P 2 j+P 2 k +1 = A n+ i-R^(p 2 j{R) +p 2 k+i(R)) >0 for all ;, k = 0,1,..., n, 

and hence, in view of Theorem 3.2, u n is optimal for (3.1). Furthermore, recall that, by 
Theorem 3.7, u n is the extremal polynomial for (3.1) if R satisfies (4.1). With (4.1) and 


Bernstein Type Inequalities 


17 


(4.16), it follows that e.g. r*(n) := max{2 1 / 4 ,7*(129)} fulfills the requirements of Theorem 
3.8. ■ 

Proof of Theorem 3.9. Let r > 1 be fixed and set a := oj. Since, by (3.3) and (1.3), 
u n (z;c) = (— l) n u n (z; — c), it suffices to consider only the case c > 0, i.e. 7 = 0. Then, the 
representation (3.9) reduces to 


(4.17) 


Pi = Bn+l 


f 1 -Bn+1 
V2 a n -n 


+ E 


1 — v 
a n + \-v 


COS 



{ = 0, 1 , . . . , 2 n + !• 


First, we turn to the proof of part a) and assume that 


(4.18) 


c > c* := 2a — — 
2a 


r 4 + r 2 + 1 
r(r 2 + 1) ' 


Note that (4.17) can be rewritten in the form pi = B n +it(<pi) where t is a trigonometric 
polynomial of type (2.8) (with m — n) and coefficients 


Ao 


Bn + 1 
°n+l 


Bn+l—v 
&n+l-v ’ 


17 = 1 , 2 , 


n. 


Therefore, Theorem 3.2 in combination with Lemma 2.1 ensures that u n (z\ c ) is the optimal 
polynomial for (3.1) if 


(4.19) 

and 



a>2 


(4.20) ^±i_2^±l + ^l>0, 1/ = 1,2,. . . ,n — 1. 

It is readily checked that the condition (4.19) is equivalent to c > 2a — 1/a and thus 
satisfied by (4.18). Furthermore, a lengthy, but routine, calculation shows that (4.20) is 
fulfilled if 


F„(c ) := 4c 2 a^a„ + i — 4 ca v a u +i + a„ + i(a„+2 — a„) >0, v = 1,2, . . . ,n — 1. 

One easily verifies that c* is larger than the zeros of F v , and this completes the proof of 
part a). 

Finally, we turn to the proof of part b). Let a > 1 be arbitrary, but fixed. Using (1.3), 
(1.4), (2.3), and (2.6), we rewrite (4.17) in the form 


(4.21) 


Pi — Z? n +ipj(c), l — 0,1,..., 2n + 1, 



18 


Roland Freund and Bernd Fischer 


where 


(4.22) 


/ ^ _ 1 ^n+l( c ) , / Tl(c) ,_ f x 

(C) 2 (n + l)r„+i(a) + (_1) S 

71 ( m )/ \ 


m=l 


is a polynomial in c of degree n. Since T' v /T v is an odd function, it follows that 
(4.23) 


4 ^ m+1> W 1 ' 

vT„(a) a a 3 T„(a) a m+I * 


With (4.23) and Proposition 2.3 e), we deduce from (4.22) that 

W(«) = + (-l) i X^sin(i/yj 1 )) + 0(^-) 


(4.24) 


Now, let tjk, j,k £ {0,1,... ,n}, be given by (3.10). Using (4.21)-(4.24), we obtian 

tjk 


BiB n +i 


= P2j(c) +p 2 fe+l(c) 


= ;( 


sm <p 2 j 


2(1 — cos j) 2(1 


sin ^ 1 , + 0 ( 4 ) + 0 (— )). 

cos <p 2 k+i ) a 2 a V 


Thus, tjk < 0 if j > (n + l/2)/2 (e.g. j = n), k < (n — l/2)/2 (e.g. k = 0), a sufficiently 
large, and e.g. c — a < 1. This concludes the proof. ■ 


ACKNOWLED GEMENT 

Part of this work was done while the authors were visiting the Computer Science 
Department of Stanford University. We would like to thank Gene Golub for his warm 
hospitality. 


REFERENCES 

[1] S. Bernstein, Sur une classe de polynomes d’ecart minimum, C. R. Acad. Sci. Paris 
190 (1930), 237-240. 


Bernstein Type Inequalities 


19 


[2] B. Fischer and R. Freund, On the constrained Chebyshev approximation problem on 
ellipses , J. Approx. Theory (1990) (to appear). 

[3] , Chebyshev polynomials are not always optimal, J. Approx. Theory 

(to appear). 

[4] C. Frappier, Q. I. Rahman and St. Ruscheweyh, New inequalities for polynomials, 
Trans. Amer. Math. Soc. 288 (1985), 69-99. 

[5] R. Freund, On some approximation problems for complex polynomials, Constr. Ap- 
prox. 4 (1988), 111-121. 

[6] , On Bernstein type inequalities and a weighted Chebyshev approx- 

imation problem on ellipses, Proceedings of Computational Methods and Function 
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[7] G. Meinardus, “Approximation of functions: Theory and numerical methods ”, Sprin- 
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[8] D. J. Newman and T. J. Rivlin, On polynomials with curved majorants, Canad. J. 
Math. 34 (1982), 961-968. 

[9] R. Pierre and Q. I. Rahman, On a problem of Turin about polynomials, Proc. Amer. 
Math. Soc. 50 (1976), 231-238. 

[10] G. Polya and G. Szego, “Aufgaben und Lehrsatze aus der Analysis, Vol. II ”, 4th ed., 
Springer, Berlin, Heidelberg, New York, (1970). 

[11] Q. I. Rahman, On a problem of Turin about polynomials with curved majorants, 
Trans. Amer. Math. Soc. 163 (1972), 447-455. 

[12] , Addendum to “On a problem of Turin about polynomials with 

curved majorants”, Trans. Amer. Math. Soc. 108 (1972), 517-518. 

[13] Q. I. Rahman and G. Schmeisser , Maskov-DufBn-Schaeffer inequality for polynomials 
with a circular majorant, Trans. Amer. Math. Soc. 310 (1988), 693-702. 

[14] T. J. Rivlin and H. S. Shapiro, A unified approach to certain problems of approxima- 
tion and minimization, J. Soc. Indust. Appl. Math. 9 (1961), 670-699. 

[15] W. Rogosinski and G. Szego, Uber die Abschnitte von Potenzreihen, die in einem 
Kreis beschrankt bleiben, Math. Z. 28 (1928), 73-94. 



FIGURE CAPTION 


Figure 3.6. Relative deviation D n {r , c) of u n for c 6 C and fixed n = 2, r 



PRECEDING page blank not filmed 



Figure 3.6