A Bernstein-Markov Theorem for Normed Spaces

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1 A Bernstein-Markov Theore for Nored Spaces Lawrence A. Harris Departent of Matheatics, University of Kentucky Lexington, Kentucky Abstract Let X and Y be real nored linear spaces and let φ : X R be a non-negative function satisfying φ(x + y) φ(x) + y for all x, y X. We show that there exist optial constants c,k such that if P : X Y is any polynoial satisfying P (x) φ(x) for all x X, then ˆD k P (x) c,k φ(x) k whenever x X and 0 k. We obtain estiates for these constants and present applications to polynoials and ultilinear appings in nored spaces. 1. Introduction. This note considers the growth of the Fréchet derivatives of a polynoial on a nored linear space when the polynoial has restricted growth on the space. Our ain concern is with real nored linear spaces. Here we obtain an estiate for the kth derivative of a polynoial bounded by an th power where the constant c,k in our estiate is best possible even when only the special case of the real line is considered. Moreover, we show that this inequality for the real line is equivalent to a Markov inequality for hoogeneous polynoials. Our estiates are applied iteratively to obtain a bound for the values of a syetric ultilinear apping where certain of its arguents are repeated. This bound is a constant ultiple of the nor of the associated hoogeneous polynoial. Although we are unable to obtain a general forula for the constants c,k, we do establish eleentary upper and lower bounds and provide a good estiate on their asyptotic growth. We deterine the value of the constants in soe low diensional cases and find associated extreal polynoials with the aid of an interpolation forula for hoogeneous polynoials. In the case of coplex nored linear spaces, we give a siple derivation of an estiate for the kth derivative which extends an inequality given in [2, Theore 2] by allowing ore general growth conditions. For coparison with the real case, 1

2 we deduce an extension of an inequality of Bernstein and derive a bound for syetric ultilinear appings where certain of its arguents are repeated. In both the real and coplex cases, we show that equality holds in ost of our estiates for soe scalar-valued hoogeneous polynoial defined on two diensional l 1 space. 2. Main results. Let be a positive integer and let 0 k. We define c,k to be the supreu of the values p (k) (0) where p varies through all polynoials satisfying p(t) (1 + t ) (1) for every t R. Clearly any such polynoial p has degree at ost. Note that if p satisfies (1) then so does the polynoial q(t) = t p(1/t). Hence c,k /k! = c, k /( k)! for 0 k. In particular, c,0 = 1 and c, =!. Proposition 1. k k ( k) c,k k k! ( k ) /2 (2) k k/2 ( k) ( k)/2 for 0 k. There exists an absolute constant M such that c,k (M log ) k (3) for 0 k and > 1. Estiates (2) and (3) follow fro work of Sarantopoulos [9] and Nevai and Totik [8], respectively. See Section 5 for a proof and for values of soe of the constants. For coparison with (3), note that (2) iplies only that c,k = O( 3k/2 ). Theore 2. Let X and Y be nored linear spaces over F, where F = R or F = C. Let φ : X R be a non-negative function satisfying φ(x + y) φ(x) + y (4) for all x, y X and let P : X Y be a polynoial satisfying P (x) φ(x) (5) for all x X. If F = R then ˆD k P (x) c,k φ(x) k (6) 2

3 and if F = C then ˆD k P (x) k! k k ( k) k φ(x) k (7) whenever x X and 0 k. Moreover, in the case where X = l 1 (F 2 ), Y = F and φ(x) = x, for each inequality (6) and (7), there exists a hoogeneous polynoial P as above and depending on and k such that equality holds in the given inequality for soe x with φ(x) any given non-negative nuber. Basic definitions and facts concerning polynoials on nored spaces can be found in [6] and [7]. Throughout, ˆDk P (x) denotes the hoogeneous polynoial associated with the kth order Fréchet derivative D k P (x) and is given by ˆD k P (x)y = dk P (x + ty). dtk t=0 In Theore 2 and in all our other estiates we take 0 0 = 1. The hypothesis (4) of Theore 2 is satisfied, for exaple, when φ(x) = f( x ), where f : [0, ) R is a continuous non-negative function satisfying f (t) 1 for all t > 0. In particular, hypothesis (4) is satisfied when φ(x) = x and when φ(x) = (1 + x p ) 1/p, where p 1. It is easy to verify that this hypothesis is also satisfied when φ(x) = ax{1, x }. The degree of any polynoial P satisfying (5) is at ost since P (y) (M + y ), M = φ(0), for all y X by (4). Proof of Theore 2. By coposing P with a given linear functional and applying the Hahn-Banach theore, we ay suppose that Y = F. Let x, y X with y 1. Given r > φ(x), define p(α) = P (x + αry) r. Then p(α) is a polynoial with p (k) (0) = ˆD k P (x)(ry)/r and p(α) φ(x + αry) r [ ] φ(x) + α r (1 + α ) r for all α F. If F = R then p (k) (0) c,k by the definition of c,k. Hence, ˆD k P (x) c,k r k 3

4 for all r > φ(x), and (6) follows. If F = C, the function p(α) is entire and by the Cauchy estiates, p (k) (0) k!(1 + R) R k, for a given R > 0. Hence, ˆD k P (x) k!(1 + R) r k R k for all r > φ(x). Taking R = k/( k) when k and letting R otherwise, we obtain (7). To prove the reainder of Theore 2, recall that by definition X is F 2 with the nor x = x 1 + x 2, where x = (x 1, x 2 ). Suppose 0 k. We first consider the case F = R. It follows fro Proposition 1 that there exists a polynoial p satisfying (1) for which p (k) (0) = c,k. Define a hoogeneous polynoial P : X R of degree by P (x 1, x 2 ) = x 1 p( x 2 x 1 ) for x 1 0. (8) Then P (x) x for all x X and ˆD k P (r, 0)(0, 1) = dk P (r, t) = r k c dtk,k. t=0 Thus equality holds in (6) with x = (r, 0) for r 0. We next consider the case F = C. Define P (x 1, x 2 ) = M k x k 1 x k 2, M k = Since the geoetric ean is less than the arithetic ean, ( ) x1 k ( ) 1 x2 k x 1 + x 2, k k. (9) k k ( k) k so P (x) x for all x X. Moreover, ˆDk P (r, 0)(0, 1) = M k k!r k. Thus equality holds in (7) with x = (r, 0) for r 0. In the applications which follow, we use the notation X 1 = {x X : x 1}, P = sup{ P (x) : x X 1 } 4

5 when P : X Y is a polynoial. Since by definition a polynoial is a su of continuous hoogeneous polynoials, P is finite and thus clearly a nor. 3. Applications to coplex spaces. The coplex case of Theore 2 can be applied to obtain an extension of the Bernstein theore given in [2, Corollary 2] to the case x > 1. Corollary 3. Let X and Y be coplex nored linear spaces and let P : X Y be a polynoial of degree at ost. Then ˆD k P (x) ˆD k P (x) k! P, k k ( k) k x 1, k! k k ( k) P k x k, x > 1, for 0 k and x X. Moreover, for each and k there exists a nontrivial hoogeneous polynoial P as above for which equality holds in the above inequalities (for soe x X with x any given nuber 1) when X = l 1 (C 2 ) and Y = C. Proof. As in the previous proof, we ay suppose that Y = C. Since the case where P 0 is obvious, we ay also suppose that P = 1. By Theore 2, all we need to establish is that (5) holds when φ(x) = ax{1, x }. This is clear when x 1. Let x X with x > 1 and define f(λ) = λ P (x/λ) for all coplex λ 0. Then f extends to a polynoial on C satisfying f(λ) x for all λ on the circle λ = x, so by the axiu principle, P (x) = f(1) x. Thus (5) holds in all cases, as required. One can apply Corollary 3 to obtain the case p = 1 of [2, Theore 1] given next. See [3] for a discussion of this and related inequalities. Corollary 4. Let X and Y be coplex nored linear spaces and let F : X X Y be a continuous syetric -linear apping with associated hoogeneous polynoial ˆF defined by ˆF (x) = F (x,..., x). If x1,..., x n are vectors in X 1, then F (x k 1 1 x kn n ) k 1! k n! k k 1 1 k kn n! ˆF (10) for all non-negative integers k 1,..., k n with k k n =. The constant in this inequality cannot be replaced by a saller one. Here and elsewhere, F (x k 1 1 x kn n ) = F (x 1,..., x }{{ 1,..., x } n,..., x n ). }{{} k 1 k n 5

6 Proof. Define f(k) = k k /k! for all non-negative integers k. By the binoial theore for hoogeneous polynoials [6, Th ], ( ) 1 k! ˆD k ˆF (x)y = F (x k y k ) k and hence by Corollary 3 (or Theore 2 with φ(x) = x ), F (y k x k ) f() f(k)f( k) ˆF (11) for x, y X 1 and 0 k. Now x F (x k 1 1 x k 1 ) is a hoogeneous polynoial of degree k 1 and hence (11) applies again to show that F (x k 1 1 x k 2 2 x k 1 k 2 ) f() f(k 1 )f(k 2 )f( k 1 k 2 ) ˆF for x X 1. Continuing in this way, we obtain (10). An exaple given in [2, p. 148] (which generalizes (9)) shows that the constant in (10) is best possible. 4. Applications to real spaces. The following is an iediate consequence of Theore 2 with φ(x) = 1 + x. Corollary 5. Let X and Y be real nored linear spaces. If P : X Y is a polynoial satisfying P (x) (1 + x ) for all x X, then ˆD k P (x) c,k (1 + x ) k whenever x X and 0 k. It is easy to see fro Corollary 5 with X = Y = R that c,n c,k c k,n k whenever 0 n and 0 k n. Theore 6. Let X and Y be real nored linear spaces. If P : X Y is a hoogeneous polynoial of degree, then ˆD k P c,k P for 0 k. Moreover, for each and k there exists a non-zero polynoial P as above for which equality holds when X = l 1 (R 2 ) and Y = R. 6

7 Corollary 7. Let X and Y be real nored linear spaces and let F : X X Y be a continuous syetric -linear apping with associated hoogeneous polynoial ˆF. If x1,..., x n are vectors in X 1, then F (x k 1 1 x kn n ) k k 1 1 k kn n ˆF (12) for all non-negative integers k 1,..., k n with k k n =. The case n = 2 of the above corollary is essentially part (a) of the Corollary in [9]. The constants we give are rather far fro being the best. For exaple, when k 1 = 1,..., k n = 1, the constant in (12) is /2 while the best constant in this case (deterined by R. S. Martin in 1932) is /!. The proble of deterining the best constant in (12) is open. (See [3].) Proof of Theore 6 and Corollary 7. Theore 6 follows fro the fact that if P 0, Theore 2 applies with P replaced by P/ P and φ(x) = x. Thus (12) follows fro the upper bound given in Proposition 1 and the proof of Corollary 4 with f(k) = k k/2. Note that in Corollary 5, a bound on the derivative which holds for X = Y = R continues to hold when X and Y are any real nored linear spaces. According to a theore of Sarantopoulos [9, Theore 2], this is also the case with Markov s inequality for the first derivative. It is an open question whether Markov s inequality for the higher derivatives continues to hold for arbitrary real Banach spaces. However, in general, one cannot expect Bernstein-type estiates for polynoials on the real line to hold for arbitrary real (or even coplex) Banach spaces. For exaple, it was shown by S. N. Bernstein [1, p. 56] that q (t) (1 + t 2 ) 1 2 (13) for any polynoial q satisfying q(t) (1 + t 2 ) /2 for all t R. Suppose X = l 1 (R 2 ) and let P : X R be the hoogeneous polynoial defined by (9) with k = 1. Then, P (x) (1 + x 2 ) /2 for all x X since P 1. If the result analogous to (13) held for X, we would have DP (x) (1 + x 2 ) 1 2 for all x X. Now x DP (x) is a hoogeneous polynoial of degree 1. Hence, replacing x by tu, where u X with u 1 and letting t, we obtain DP (u). Thus, M 1, which is ipossible when > 1. Inequality (13) and other classical polynoial inequalities are extended to Hilbert spaces in [2] and [4]. 7

8 5. Estiating c,k. Proof of Proposition 1. The lower bound for c,k in (2) follows with p(t) = at k, where a is the iniu of the function (1 + t) /t k for t > 0. This iniu occurs at t = k/( k). To obtain the upper bound in (2), suppose p satisfies (1) and put q(t) = p(st), where s > 0. Then q (k) (0) = s k p (k) (0) and by the Cauchy-Schwarz inequality, Hence, q(t) (1 + s t ) (1 + s 2 ) /2 (1 + t 2 ) /2. p (k) (0) k! ( k ) (1 + s 2 ) /2 s k by k applications of (13). The iniu of the right-hand side of the above occurs when s 2 = k/( k) and this gives the asserted estiate. (Copare [9, p. 311].) To prove (3), suppose p satisfies (1) and put q(t) = p(t/). Then q is a polynoial of degree at ost satisfying ( q(t) 1 + t ) e t for all t R. By k applications of a Markov-Bernstein theore given in [8, Theore 3] with weight exp( t ), we obtain sup e t q (k) (t) (M log ) k sup e t q(t), <t< <t< where M is an absolute constant. Thus p (k) (0) = k q (k) (0) (M log ) k, copleting the proof. (I a grateful to Prof. Taás Erdélyi for providing the arguent in the above paragraph.) In view of Theores 2 and 6, it is iportant to know the value of c,k as accurately as possible, especially for the case k = 1. Below is a table of a few values with a corresponding extreal polynoial which attains this value. Note that these extreal polynoials can be converted to extreal polynoials for Theore 6 using (8). 8

9 c,k Extreal polynoial p(t) 1 c 1,1 = 1 t + b, b 1 2 c 2,1 = 4 c 2,1 t 3 c 3,1 = c 3,1 t t 3 4 c 4,1 = 6 3 c 4,1 t(1 t 2 ) c 4,2 = c 4,2t 2 t c 6,1 = c 6,1 t(1 t 2 ) 2 64t 3 c 6,3 = c 6,3t(1 t 2 ) 2 32t(1 + t 4 ) The values of c,1 and the corresponding extreal polynoials given in the table can be deduced fro the estiate below. Lea 8. If t 1,..., t are any distinct real nubers, then c,1 i=1 (1 + t i ) j i t i t j (14) To obtain the extreal polynoials of the table, choose the interpolation points t 1,..., t syetric with respect to the origin (with the origin included for odd values of ) and select the positive points fro t, 1/t and 1, in that order, where t > 0. The value of c,1 is obtained by iniizing over t. For exaple, when t 1 = t, t 2 = 1/t, t 3 = t 2, t 4 = t 1, where 0 < t < 1, the right-hand side of (14) reduces to f(t) = t(1 + t)4 (1 + t)4 + 1 t 4 t(1 t 4 ) = (1 + t)4 t(1 t 2 ) = (1 + t)3 t(1 t). Let a = in{f(t) : 0 < t < 1} and define p(t) = at(1 t 2 ). Then c 4,1 a by (14) and clearly p(t) (1 + t) 4 for 0 < t < 1. It follows fro the identities p( t) = p(t) and t 4 p(1/t) = p(t), that p(t) (1 + t ) 4 for all t R. Since p (0) = a, we have c 4,1 = a. By calculus, a = f(2 3) = 6 3. The entries in the table for c 4,2 and c 6,3 can be obtained fro an interpolation forula for hoogeneous polynoials given in [3, (4)]. (The oversize Γ given there is a isprint for the sybol denoting a product.) For the case of c 4,2, take the interpolation sets to be r, 0, r and s, 0, s and set t = r/s. For the case of c 6,3, take both interpolation sets to be s, r, r, s and again set t = r/s. It is easy to deduce that c 6,3 = 12(c 6,1 + 32) by coparison of the expressions being iniized; hence the last two extreal polynoials in the table are the sae. 9

10 Proof of Lea 8. One can deduce the estiate (14) easily fro the interpolation forula given in [3, (4)], where t 1,..., t and 1, 1 are the sets of interpolation points and where W is the hoogeneous polynoial P defined by (8). To give a direct proof, suppose p satisfies (1) and define q(t) = t p 1 ( 1 t ), where p 1 (t) = p(t) p( t). 2 Then q extends to a polynoial on R of degree at ost 1 satisfying q(t) (1 + t ) for all t R. Moreover, the coefficient of t 1 in q is the coefficient of t in p, i.e., p (0). By equating the coefficients of t 1 in both sides of the Lagrange interpolation forula for q, we obtain p (0) = q(t i ) j i(t i t j ) i=1 and (14) follows. See [5] for further discussion of the deterination of the values c,k and related probles. References [1] S. N. Bernstein, Lecons sur les Propriétés Extréales et la Meilleure Approxiation des Fonctions Analytiques d une Variable Réele, Gauthier-Villars, Paris, [2] L. A. Harris, Bounds on the derivatives of holoorphic functions of vectors, Proc. Colloq. Analysis, Rio de Janeiro, 1972, , Act. Sci. et Ind., Herann, Paris, [3], Coentary on probles 73 and 74, The Scottish Book, R. D. Mauldin, Ed., Birkhäuser 1981, [4], Bernstein s polynoial inequalities and functional analysis, Irish Math. Soc. Bull. 36(1996), [5], Coefficients of polynoials of restricted growth on the real line, (to appear in J. Approx. Theory). 10

11 [6] E. Hille and R. S. Phillips, Functional Analysis and Sei-Groups, Aer. Math. Soc. Colloq. Publ., Vol. 31, AMS, Providence, [7] L. Nachbin, Topology on Spaces of Holoorphic Mappings, Ergebnisse 47, Springer-Verlag, New York, [8] P. Nevai and V. Totik, Weighted polynoial inequalities, Constr. Approx. 2(1986), [9] Y. Sarantopoulos, Bounds on the derivatives of polynoials on Banach spaces, Math. Proc. Cab. Phil. Soc. 110(1991),