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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 Theory, Valparaiso, Chile, March 1989, to appear. [7] G. Meinardus, “Approximation of functions: Theory and numerical methods ”, Sprin- ger, Berlin, Heidelberg, New York, (1967). [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