NEWMAN’S INEQUALITY FOR M¨UNTZPOLYNOMIALS ON POSITIVE INTERVALSPeter Borwein and Tam´as Erd´elyiAbstract. The principal result of this paper is the following Markov-type inequalityfor M¨untz polynomials.Theorem (Newman’s Inequality on [ a, b] ⊂ (0, ∞)). Let Λ := (λj)∞j=0be anincreasing sequence of nonnegative real numbe rs. Suppose λ0= 0 and there exists aδ > 0 so that λj≥ δj for each j. Suppose 0 < a < b. Then there exists a constantc(a, b, δ) depending only on a, b, and δ so thatkP′k[a,b]≤ c(a, b, δ)nXj=0λjkP k[a,b]for every P ∈ Mn(Λ), where Mn(Λ) denotes the linear span of {xλ0, xλ1, . . . , xλn}over R.When [a, b] = [0, 1] and with kP′k[a,b]replaced with kxP′(x)k[a,b]this was provedby Newman. Note that the i nterval [0, 1] plays a special role in the study of M¨untzspaces Mn(Λ). A linear transformation y = αx + β does not preserve membership inMn(Λ) in general (unless β = 0). So the analogue of Newman’s Inequality on [a, b]for a > 0 does not seem to be obtainable in any strai ghtforward fashion from the[0, b] case.1. Introduction and NotationLet Λ := (λj)∞j=0be a sequence of distinct real numbers. The span of{xλ0, xλ1, . . . , xλn}over R will be denoted byMn(Λ) := spa n{xλ0, xλ1, . . . , xλn}.Elements of Mn(Λ) are called M¨untz polynomials. Newman’s beautiful ineq uality[6] is an essentially sharp Markov-type inequality for Mn(Λ), whe re Λ := (λj)∞j=0is a seque nc e of distinct nonnegative real numbers. For notational convenience, letk · k[a,b]:= k · kL∞[a,b].1991 Mathematics Subject Classification. Primary: 41A17, Secondary: 30B10, 26D15.Key words and phrases. M¨untz polynomials, lacunary polynomials, Dirichlet sums, Markov-type inequality.Research of the first author supported, in part, by NSERC of Canada. Research of the secondauthor supported, in part, by NSF under Grant No. DMS-9024901 and conducted while anNSERC International Fellow at Si mon Fraser University.Typeset by AMS-TEX12 PETER BORWEIN AND TAM´AS ERD´ELYITheorem 1.1 (Newman’s Inequality). Let Λ := (λj)∞j=0be a sequence of dis-tinct nonnegative real numbers. Then23nXj=0λj≤ sup06=P ∈Mn(Λ)kxP′(x)k[0,1]kP k[0,1]≤ 11nXj=0λj.Frappier [4] shows that the constant 11 in Newman’s Ine quality can be replacedby 8.29. In [2], by modifying (and simplifying) Newman’s arguments, we showedthat the constant 1 1 in the above inequality can b e replaced by 9. But more impor-tantly, this modification allowed us to prove the following Lpversion of Newman’sInequality [2] (an L2version of which was proved earlier in [3]).Theorem 1.2 (Newman’s Inequality in Lp). Let p ∈ [1, ∞). Let Λ := (λj)∞j=0be a sequence of distinct real numbers greater than −1/p. ThenkxP′(x)kLp[0,1]≤ 1/p + 12 nXj=0(λj+ 1/p)!!kP kLp[0,1]for every P ∈ Mn(Λ) := spa n{xλ0, xλ1, . . . , xλn}.We believe on the basis of consider able computation that the best possible con-stant in Newman’s Ineq uality is 4. (We remark that an incorrect argument existsin the literature claiming that the best possible constant in Newman’s Inequalityis at least 4 +√15 = 7.87 . . . .)Conjecture (Newman’s Inequality with Best Constant). Let Λ := (λj)∞j=0be a sequence of distinct nonnegative real numbers. ThenkxP′(x)k[0,1]≤ 4 nXj=0λj!kP k[0,1]for every P ∈ Mn(Λ) := spa n{xλ0, xλ1, . . . , xλn}.It is proved in [1] that under a g rowth condition, which is essential, kxP′(x)k[0,1]in Newman’s Inequality ca n b e re placed by kP′k[0,1]. More precisely, the followingresult holds .Theorem 1.3 (Newman’s Inequality Without the Factor x). Let Λ :=(λj)∞j=0be a sequence of distinct real numbers with λ0= 0 and λj≥ j for eachj. ThenkP′k[0,1]≤ 18 nXj=1λj!kP k[0,1]for every P ∈ Mn(Λ).Note that the interval [0, 1] plays a special role in the study of M¨untz polynomials.A linear transformation y = αx + β does not preserve membership in Mn(Λ) ingeneral (unless β = 0), that is P ∈ Mn(Λ) does not necessarily imply that Q(x) :=P (αx + β) ∈ Mn(Λ). Analogues of the above results on [a, b], a > 0, cannot beobtained by a simple transformation. We can, however, prove the following result.NEWMAN’S INEQUALITY ON POSITIVE INTERVALS 32. New ResultsTheorem 2.1 (Newman’s Inequality on [a, b] ⊂ (0, ∞)). Let Λ := (λj)∞j=0bean increasing sequence of nonnegative real numbers. Suppose λ0= 0 and thereexists a δ > 0 so that λj≥ δj for each j. Suppose 0 < a < b . Then there exists aconstant c(a, b, δ) depending only on a, b, and δ so thatkP′k[a,b]≤ c(a, b, δ) nXj=0λj!kP k[a,b]for every P ∈ Mn(Λ), where Mn(Λ) denotes the linear span of {xλ0, xλ1, . . . , xλn}over R.Theorem 2.1 is sharp up to the constant c (a, b, δ). This follows from the lowerbound in Theorem 1.1 by the substitution y = b−1x. Indeed, take a P ∈ Mn(Λ) sothat|P′(1)| ≥23nXj=0λjkP k[0,1].Then Q(x) := P (x/b) satis fie skQ′k[a,b]≥ |Q′(b)| = b−1|P′(1)| ≥23bnXj=0λjkP k[0,1]≥23bnXj=0λjkQk[a,b].The following example shows that the growth condition λj≥ δj with a δ > 0 inthe above theorem cannot be dropped. It will also be used in the proof of Theorem2.1.Theorem 2.2. Let Λ := (λj)∞j=0, where λj= δj. Let 0 < a < b. Thenmax06=P ∈Mn(Λ)|P′(a)|kP k[a,b]= |Q′n(a)| =2δaδ−1bδ− aδn2where, with Tn(x) = cos(n arcco s x),Qn(x) := Tn2xδbδ− aδ−bδ+ aδbδ− aδis the Chebyshev “polynomial” for Mn(Λ) on [a, b]. In particularlimδ→0max06=P ∈Mn(Λ)|P′(a)| nXj=0λj!kP k[a,b]= ∞.Theorem 2.2 is a well-known property of differentiable Chebyshev spaces. See,for example, [5] or [1].4 PETER BORWEIN AND TAM´AS ERD´ELYI3. LemmasThe following comparison theorem for M¨untz polynomials is proved in [1, E.4 f]of Section 3 .3]. For the sake of completeness, in the next section we outline a shortproof suggested by Pinkus. This proof assumes familarity with the basic prope rtiesof Che byshev and Descartes systems. All of thes e may be found in [5].Lemma 3.1 (A Comparison Theorem). Let Λ := (λj)∞j=0and Γ := (γj)∞j=0beincreasing sequences of nonnegative real numbers with λ0= γ0= 0, and γj≤ λjfor each j. Let 0 < a < b. ThenmaxP ∈Mn(Γ)|P′(a)|kP k[a,b]≥ maxP ∈Mn(Λ)|P′(a)|kP k[a,b].The following result is essentially proved by Saff and Varga [7]. They assumethat Λ := (λj)∞j=0is an inc reasing seq uenc e of nonnegative integers and δ = 1 in