THE L p VERSION OF NEWMAN S INEQUALITY FOR LACUNARY POLYNOMIALS. Peter Borwein and Tamás Erdélyi. 1. Introduction and Notation

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1 THE L p VERSION OF NEWMAN S INEQUALITY FOR LACUNARY POLYNOMIALS Peter Borwein and Tamás Erdélyi Abstract. The principal result of this paper is the establishment of the essentially sharp Markov-type inequality xp x) Lp[,1] 1/p+12 λ j +1/p) P Lp[,1] for every P span{x λ,x λ 1,...,x λn } with distinct real exponents λ j greater than 1/p and for every p [1, ]. A remarkable corollary of the above is the Nikolskii-type inequality 1/p y 1/p Py) L [,1] 13 λ j +1/p) P Lp[,1] for every P span{x λ,x λ 1,...,x λn } with distinct real exponents λ j greater than 1/p and for every p [1, ]. Some related results are also discussed. 1. Introduction and Notation Let Λ := {λ i } i= be a sequence of distinct real numbers. The span of over R will be denoted by {x λ,x λ1,...,x λn } M n Λ) := span{x λ,x λ1,...,x λn }. Elements of M n Λ) are called Müntz or lacunary) polynomials. We first present a simplified version of Newman s beautiful proof of a Markov-type inequality for 1991 Mathematics Subject Classification. Primary: 41A17, Secondary: 3B1, 26D15. Key words and phrases. Müntz polynomials, lacunary polynomials, Dirichlet sums, Markovtype inequality, L p norm. Research of the first author supported, in part, by NSERC of Canada. Research of the second author supported, in part, by NSF under Grant No. DMS and conducted while an NSERC International Fellow at Simon Fraser University. 1 Typeset by AMS-TEX

2 2 PETER BORWEIN AND TAMÁS ERDÉLYI Müntz polynomials. This modification gives a better constant, 9, than the constant 11 appearing in Newman s paper [5]. But more importantly, this modification allows us to prove the L p analogues of Newman s Inequality. Some related results are stated in Section New Results be a sequence of dis- Theorem 2.1 Newman s Inequality). Let Λ := {λ i } i= tinct nonnegative real numbers. Then 2 3 xp x) L [,1] λ j sup P M nλ) P L [,1] 9 λ j. Frappier [4] shows that the constant 11 in Newman s Inequality can be replaced by We believe on the basis of considerable computation that the best possible constant in Newman s Inequality is 4. We remark that an incorrect argument exists in the literature claiming that the best possible constant in Newman s Inequality is at least = ) Conjecture Newman s Inequality with Best Constant). Let Λ := {λ i } i= be a sequence of distinct nonnegative real numbers. Then n xp x) L [,1] 4 λ j ) P L [,1] for every P M n Λ) := span{x λ,x λ1,...,x λn }. Theorem 2.2 Newman s Inequality in L p ). Let p [1, ). Let Λ := {λ i } i= be a sequence of distinct real numbers greater than 1/p. Then n )) xp x) Lp[,1] 1/p+12 λ j +1/p) P Lp[,1] for every P M n Λ) := span{x λ,x λ1,...,x λn }. The following Nikolskii-type inequality follows from Theorem 2.2 quite simply. Theorem 2.3 Nikolskii-Type Inequality). Let p [1, ). Let Λ := {λ i } i= be a sequence of distinct real numbers greater than 1/p. Then n 1/p y 1/p Py) L [,1] 13 λ j +1/p)) P Lp[,1] for every P M n Λ) := span{x λ,x λ1,...,x λn }. Theorem 2.3 immediately implies the following result.

3 THE L p VERSION OF NEWMAN S INEQUALITY 3 Theorem 2.4 Müntz-Type Theorem in L p ). Let p [1, ). Let Λ := {λ i } i= be a sequence of distinct real numbers greater than 1/p satisfying λ j +1/p) <. Then is not dense in L p [,1]. span{x λ,x λ1,...} Much more about Müntz-type theorems in L p [,1] may be found in [2] and the references therin. 3. Proofs Proof of Theorem 2.1 Newman s proof modified). It is equivalent to prove that 3.1) 2 3 P [, ) λ j sup 9 P E nλ) P [, ) λ j where E n Λ) is the linear span of {e λt,e λ1t,...,e λnt } over R. Without loss of generality we may assume that λ =. By a change of scale we may also assume that n λ j = 1. We may also assume that n 2, otherwise the theorem is trivial. We begin with the first inequality. We define the Blaschke product and the function 3.2) Tt) := 1 By the Residue Theorem so T E n Λ). We claim that Bz) := n z λ j z +λ j e zt dz, where := {z C : z 1 = 1}. Bz) Tt) := B λ j )) 1 e λjt 3.3) Bz) 1 3, z. Indeed, it is easy to see that < λ j 1 implies z λ j z +λ j 2 λ j = 1 λ j/2 2+λ j 1+λ j /2, z

4 4 PETER BORWEIN AND TAMÁS ERDÉLYI so, for z, Here the inequality Bz) n λ j /2 1+λ j /2 λ j = λ 1 j 2 = x 1 y 1+x 1+y = 1 x+y) 1+x+y) + 2xyx+y) 1+x)1+y)1+x+y)) 1 x+y) 1+x+y, x,y is used. From 3.2) and 3.3) we deduce that 3.4) Tt) 1 2π and 3.5) T ) = 1 e zt Bz) dz 1 32π) = 3, t. 2π T t) = 1 ze zt dz Bz) z Bz) dz = 1 z z =1 Bz) dz. Also, for z > max 1 j n λ j we have the Laurent series expansion 3.6) ) z n Bz) =z 1+λ j /z n 1 λ j /z = z 1+2 λ j /z) k k=1 n n ) 2 =z 1+2 λ j )z 1 +2 λ j z 2 + =z +2+2z 1 + which, together with 3.5), yields that T ) = 2. Hence, by 3.4) T ) T [, ) 2 3 = 2 3 so the lower bound of the theorem is proved. To prove the upper bound in 2.1), first we show that if Ut) := 1 e zt dz, := {z C : z 1 = 1} 1 z)bz) λ j

5 THE L p VERSION OF NEWMAN S INEQUALITY 5 then 3.7) U t) dt 6. Indeed, observe that if z = 1 + e θ then z 2 = 2 + 2cosθ, so 3.3) and Fubini s Theorem yield that Now we show that U t) dt = 1 2π 3 2π = 3 2π 1 2π 2π 2π 2π z 2 e zt 1 z)bz) dz dt z 2 e zt Bz) dθdt 2+2cosθ)e 1+cosθ)t dθdt 1 2+2cosθ) dθ = 6. 1+cosθ 3.8) e λjt U t)dt = 3 λ j. To see this we write the left-hand side as = 1 = 1 e λjt U t)dt = z =2 e λjt 1 z 2 e z+λj)t 1 z)bz) dzdt = 1 z z 1 z +λ j 1 z Bz) dz z 2 e zt 1 z)bz) dzdt z 2 z +λ j )1 z)bz) dz where in the third equality Fubini s Theorem is used. Here, for z > 1, we have the Laurent series expansions and, as in 3.6), z z +λ j =1 λ j z 1 +λ 2 j z 2 + z 1 z =1+z 1 +z Bz) = 1+2z 1 +2z 2 +. Now 3.8) follows from the Residue Theorem. Let P E n Λ) be of the form Pt) = c j e λjt, c j R.

6 6 PETER BORWEIN AND TAMÁS ERDÉLYI Then therefore = Pt+a)U t)dt = c j e λja e λjt U t)dt c j e λja e λjt U t)dt = =3Pa) P a) 3.9) P a) 3 Pa) + Combining this with 3.7), we obtain c j 3 λ j )e λja Pt+a)U t) dt. and the theorem is proved. P [, ) 3 P [, ) +6 P [, ) = 9 P [, ) Proof of Theorem 2.2. Firstweshowthatitissufficienttoprovethatif := {γ i } i= is a sequence of distinct positive real numbers then n 3.1) P Lp[, ) 12 γ j ) P Lp[, ) for every P E n ) and p [1, ), where E n ) is, as before, the linear span of {e γt,e γ1t,...,e γnt } over R. Indeed, if {λ i } i= is a sequence of distinct real numbers greater than 1/p and γ i := λ i +1/p for each i then {γ i } i=1 is a sequence of distinct positive real numbers. Let Q M n Λ). Applying 3.1) with Pt) := Qe t )e t/p E n ) and using the substitution x = e t, we obtain that 1 ) p x x 1/p Qx) x 1 dx )1/p n ) 12 λ j +1/p) Q Lp[,1]. Now the product rule of differentiation and Minkowski s Inequality yield n )) xq x) Lp[,1] 1/p+12 λ j +1/p) which is the inequality of the theorem. Q Lp[,1]

7 THE L p VERSION OF NEWMAN S INEQUALITY 7 Let P E n ) and p [1, ) be fixed. As in the proof of Theorem 3.1, by a change of scale, without loss of generality we may assume that n γ j = 1. It follows from 3.9) and Hölder s Inequality that ) p ) P a) p 2 3 p 1 p Pa) p + Pt+a)U t) dt) ) 1/p 6 p Pa) p +2 p 1 Pt+a) p U t) dt ) ) 1/q p U t) dt for every a [, ) where q 1, ] is the conjugate exponent to p defined by 1/p+1/q = 1. Combining the above inequality with 3.7), we obtain that P a) p 6 p Pa) p +2 p 1 6 p/q Pt+a) p U t) dt for every a [, ). Integrating with respect to a, then using Fubini s Theorem and 3.7), we conclude P p L p[, ] 6p P p L p[, ] +2p 1 6 p/q and the proof is finished. 6 p P p L p[, ] +2p 1 6 p/q 6 p P p L p[, ] +2p 1 6 p/q P p L p[, ] 6 p P p L p[, ] +2p 1 6 p/q+1 P p L p[, ] Pt+a) p U t) dtda Pt+a) p U t) dadt =6 p +2 p 1 6 p ) P p L p[, ] 12p P p L p[, ] U t) dt Proof of Theorem 2.3. After the scaling x yx and the substitution x = e t, it is sufficient to prove that if := {γ i } i= is a sequence of distinct positive real numbers then n ) 3.11) P) 13 γ j 1/p P Lp[, ] for every P E n ) and p [1, ), where E n ) is, as before, the linear span of {e γt,e γ1t,...,e γnt } over R. Indeed, if {λ i } i= is a sequence of distinct real numbers greater than 1/p and γ i := λ i +1/p for each i then {γ i } i=1 is a sequence of distinct positive real numbers. Let Q M n Λ) and y [,1]. Applying 3.11) with Pt) := Qye t )e t/p E n ) and using the substitution x = e t, we obtain that n 1/p y 1/p Qy) 13 λ j +1/p)) Q Lp[,y] n 1/p 13 λ j +1/p)) Q Lp[,1].

8 8 PETER BORWEIN AND TAMÁS ERDÉLYI which is the inequality of the theorem. NowletP E n ). AsintheproofofTheorem3.1, byachangeofscale,without loss of generality we may assume that n γ j = 1. Using Hölder s Inequality, we obtain that P) Pt)e t L [, ) Pt)e t ) L1[, ) P t)e t L1[, ) + Pt)e t L1[, ) P Lp[, ) + P Lp[, ). Combining this with 3.1) and n γ j = 1, we conclude 3.11). Remarks. Theorem 2.1 is due to Newman [5] with 11 instead of 9. We presented a modified version of Newman s original proof of Theorem 2.1. He worked with T instead of U, and instead of3.9) he established a more complicated identity involving the second derivative of P. Therefore, he needed an application of Kolmogorov s Inequality to finish his proof. 4. Related Results In this section we state some related results without proof. Proofs will be presented in [1]. Theorem 4.1 Sharpness of Theorem 2.2). Let Λ := {λ i } i= be a sequence of distinct real numbers with λ = and λ k k for each k. Then there exists an absolute constant c > independent of Λ or p) so that for every p [2, ). xp x) Lp[,1] sup P M nλ) P Lp[,1] c λ j. It can be proved that under a growth condition xp x) L [,1] in Newman s Inequality can be replaced by p L [,1]. More precisely, the following result holds. Theorem 4.2 Newman s Inequality Without the Factor x). Let Λ := {λ i } i= be a sequence of distinct real numbers with λ = and λ k k for each k. Then n P [,1] 18 λ j ) P [,1] for every P M n Λ). The next result shows that the growth condition in Theorem 4.2 cannot be dropped in general.

9 THE L p VERSION OF NEWMAN S INEQUALITY 9 Theorem 4.3. For every δ,1) there exists a sequence Λ := {λ i } i= with λ =, λ 1 1, and λ i+1 λ i δ, i =,1,2,... so that lim sup n P M nλ) n λ j P ) =. ) P [,1] The following L 2 version of Newman s Inequality is proved in Borwein, Erdélyi, and J. Zhang [3]. This theorem offers an L 2 analogue of Theorem 3.1 even for complex exponents. It also improves the multiplicative constant 12 in the L 2 inequality of Theorem 3.2. Theorem 4.4 Newman s Inequality in L 2 [,1]). If Λ := {λ i } i= is a set of distinct complex numbers with Reλ i ) > 1/2, and M n Λ) denotes the linear span of {x λ,x λ1,...,x λn } over C then for every n N. xp x) L2[,1] sup P M nλ) P L2[,1] n λ j Reλ j )) k=j+1 1+2Reλ k )) ) 1/2 If λ < λ 1 < are real, and M n Λ) denotes the linear span of over R then for every n N. {x λ,x λ1,...,x λn } xp x) L2[,1] λ j sup P M nλ) P L2[,1] 1 n 1+2λ j ) 2 Note that the interval [,1] plays a special role in the study of Müntz polynomials. Analogues of the results on [a,b], a >, cannot be obtained by a linear transformation. We can however prove the following result. Theorem 4.5 Newman s Inequality on [a,b] [, )). Let Λ := {λ i } i=1 be a sequence of nonnegative real numbers. Assume that there exists an α > so that λ i λ i 1 α for each i. Suppose that [a,b] [, ). Then there exists a constant ca,b,α) depending only on a, b, and α so that n P [a,b] ca,b,α) λ j ) P [a,b]

10 1 PETER BORWEIN AND TAMÁS ERDÉLYI for every p M n Λ) where M n Λ) denotes the linear span of {x λ,x λ1,...,x λn } over R. It is also shown that the above result does not necessarily hold without the gap condition λ i λ i 1 α >. When p = 2 the best possible constant in the Nikolskili-type inequality of Corollary 2.3 is found in [3]. We have the following result. Theorem 4.6. Let Λ := {λ i } i= than 1/2. Then be a sequence of distinct real numbers greater max P M nλ) P1) P L2[,1] n 1/2 = 1+2λ j )). References 1. P. B. Borwein and T. Erdélyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, N.Y. to appear). 2. P. B. Borwein and T. Erdélyi, The full Müntz theorem in C[,1], L 2 [,1], and L 1 [,1] to appear). 3. P. B. Borwein, T. Erdélyi, and J. Zhang, Müntz systems and orthogonal Müntz Legendre polynomials, Trans. Amer. Math. Soc. 1994). 4. C. Frappier, Quelques problemes extremaux pour les polynomes at les functions entieres de type exponentiel, Ph. D. Dissertation, Universite de Montreal, D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory ), Department of Mathematics, Simon Fraser University, Burnaby B.C., Canada V5A 1S6

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS

EXTREMAL PROPERTIES OF THE DERIVATIVES OF THE NEWMAN POLYNOMIALS Tamás Erdélyi Abstract. Let Λ n := {λ,λ,...,λ n } be a set of n distinct positive numbers. The span of {e λ t,e λ t,...,e λ nt } over R