TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM. Peter Borwein and Tamás Erdélyi. 1. Introduction

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1 TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS AND A LITTLEWOOD-TYPE PROBLEM Peter Borwei ad Tamás Erdélyi Abstract. We examie the size of a real trigoometric polyomial of degree at most havig at least k zeros i K := R (mod π) (coutig multiplicities). This result is the used to give a ew proof of a theorem of Littlewood cocerig flatess of uimodular trigoometric polyomials. Our proof is shorter ad simpler tha Littlewood s. Moreover our costat is explicit i cotrast to Littlewood s approach, which is idirect. 1. Itroductio The set of all polyomials of degree with coefficiets ±1 will be deoted by L. Specifically L := p : p(z) = a j z j, a j { 1,1}. j=0 Let D deote the closed uit disk of the complex plae. Let D deote the uit circle of the complex plae. Littlewood made the followig cojecture about L i the fifties. Cojecture 1.1 (Littlewood). There are at least ifiitely may values of N for which there are polyomials p L so that C 1 ( +1) 1/ p (z) C (+1) 1/ for all z D. Here the costats C 1 ad C are idepedet of. Sice the L ( D) orm of a polyomial from L is exactly (π) 1/ ( +1) 1/, the costats must satisfy C 1 1adC 1. See Problem 19 of [Li-68]. While there is much literature o this problem ad its variats this is still ope. See [Saf-90] ad [Bor-98]. I fact, fidig polyomials that satisfy just the lower boud i Cojecture 1.1 is still ope. The Rudi Shapiro polyomials satisfy the upper boud. There is a related cojecture of Erdős [Er-6] Mathematics Subject Classificatio. Primary: 41A17. Key words ad phrases. real trigoometric polyomials, real zeros, uimodular trigoometric polyomials, flatess. Research of the first author supported, i part, by NSERC of Caada. Research of the secod author is supported, i part, by NSF uder Grat No. DMS Typeset by AMS-TEX

2 PETER BORWEIN AND TAMÁS ERDÉLYI Cojecture 1. (Erdős). There is a costat ε>0(idepedet of ) so that max p (z) (1 + ε)( +1) 1/ z D for every p L ad N. That is, the costat C i Cojecture 1.1 must be bouded away from 1 (idepedetly of ). This cojecture is also ope. Kahae [Kah-85], however, shows that if the polyomials are allowed to have complex coefficiets of modulus 1 the Cojecture 1.1 holds ad Cojecture 1. fails. That is, for every ε>0 there are ifiitely may values of N for which there are polyomials p of degree with complex coefficiets of modulus 1 that satisfy (1 ε)( +1) 1/ p (z) (1 + ε)( +1) 1/ for all z D. Beck [Bec-91] exteds Kahae s result (with two costats C 1 > 0 ad C > 0 istead of 1 ε ad 1 + ε) for the class of polyomials of degree whose coefficiets are 400th roots of uity. Our mai result is a reprovig of Cojecture 1. for real trigoometric polyomials. This is Corollary.4 of the ext sectio. Littlewood gives a proof of this i [Li-61] ad explores related issues i [Li-6], [Li-66a], ad [Li-66b]. Our approach is via Theorem.1 which estimates the measure of the set where a real trigoometric polyomial of degree at most with at least k zeros i K := R (mod π) is small. There are two reasos for doig this. First the approach is, we believe, easier ad secodly it leads to explicit costats.. New Results Let K := R (mod π). For the sake of brevity the uiform orm of a cotiuous fuctio p o K will be deoted by p K := p L (K). Let T deote the set of all real trigoometric polyomials of degree at most, ad let T,k deote the subset of those elemets of T that have at least k zeros i K (coutig multiplicities). Theorem.1. Suppose p T has at least k zeros i K (coutig multiplicities). Let α (0, 1). The m{t K : p(t) α p K } α e k, where m(a) deotes the oe-dimesioal Lebesgue measure of A K. Theorem.. We have ( π 1 c ) k sup p T,k p L1(K) p L (K) ( π 1 c ) 1k for some absolute costats 0 <c 1 <c.

3 TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS 3 Theorem.3. Assume that p T satisfies (.1) p L(K) A 1/ ad (.) p L(K) B 3/. The there is a costat ε>0depedig oly o A ad B such that (.3) p K (π ε) 1 p L. (K) Here works. ε = π3 B 6 A 6 104e Corollary.4. Let p T be of the form p(t) = a k cos(kt γ k ), a k = ±1, γ k R, k =1,,...,. k=1 The there is a costat ε>0such that p K (π ε) 1 p L. (K) Here works. ε := π e 7 3. Proofs To prove Theorem.1 we eed the lemma below that is proved i [BE-95, E.11 of Sectio 5.1 o pages 36 37]. Lemma 3.1. Let p T,t 0 K,adr>0. The p has at most er p(t 0 ) 1 p K zeros i the iterval [t 0 r, t 0 + r]. Proof of Theorem.1. Suppose p T has at least k zeros i K, ad let α (0, 1). The {t K : p(t) α p K } ca be writte as the uio of pairwise disjoit itervals I j, j =1,,...,m. Each of the itervals I j cotais a poit y j I j such that p(y j ) = α p K.

4 4 PETER BORWEIN AND TAMÁS ERDÉLYI Also, each zero of p from K is cotaied i oe of the itervals I j. Let µ j deote the umber of zeros of p i I j. Sice p T has at least k zeros i K, wehave m j=1 µ j k. Note also that Lemma 3.1 implies that Therefore µ j e I j (α p K ) 1 p K = e α I j. k m µ j e α j=1 m I j e α m({t K : p(t) α p K}), j=1 ad the result follows. Proof of Theorem.. The upper boud of the theorem follows from Theorem.1 applied with α = 1/. The lower boud follows by cosiderig p(t) :=D m (0) D m (kt) T,k with m =, (k +1) where is the Dirichlet kerel of degree m. D m (t) = 1 m + cos jt Proof of Theorem.3. First ote that by Berstei s iequality for real trigoometric polyomials i L (K), we have B A. Assume that p T satisfies (.1) ad (.) but (.3) does ot hold with ε = π. The (3.1) M := p K (π π) 1/ p L(K) π 1/ A 1/. j=1 Combiig this with Berstei s iequality we obtai (3.) p K p K π 1/ A 3/. Usig (.), we obtai that is 3 p L (K) = p K K K p (t) dt p (t) dt π 1/ A 3/ p L1(K), (3.3) p 1/ B L1(K) π A 3/. Associated with p T,M= p K,adγ [0, 1], let (3.4) A γ = A γ (p) ={t K: p(t) (1 γ)m}

5 TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS 5 ad (3.5) B γ = B γ (p) ={t K: p(t) >(1 γ)m}. Sice every horizotal lie y = c itersects the graph of p T i at most poits with x coordiates i K, wehave (3.6) Bγ p (t) dt 4γM 4γπ 1/ A 1/ π1/ A 3/ if 4γπ 1/ A π1/ Now (3.3) (3.6) give A that is, if γ π 8 A. Aγ p (t) dt π1/ A 3/ with γ = π 8 A. From this, with the help of (3.1) we ca deduce that there is a such that p δ has at least π 1/ A 3/ (1 γ)m π1/ 4 δ ( (1 γ)m,(1 γ)m) A 3/ M π1/ 3/ 4 A π 1/ A = π 1/ 4 A zeros i K. Therefore Theorem.1 yields that { ( m t K : p(t) 1 γ } { ) p K m t K: p(t) δ γ } p K { m t K: p(t) δ γ } 4 p δ K Therefore π p K p L (K) = We ow coclude that K = π3 1 e γπ 44A π B 4 18e A 4. B 4 ( p K p(t) )dt π γ 18e A 4 p K B 6 104e A 6 p K. p L (K) ( π π3 104e B 6 A 6 ) p K, ad the result follows.

6 6 PETER BORWEIN AND TAMÁS ERDÉLYI Proof of Corollary.4. Let p T be of the give form. We have p L = π (K) a k = π, k=1 that is Also that is p L = π (K) k a k p L(K) = π 1/ 1/. k=1 = π ( + 1)( +1) 6 ( π ) 1/ p L(K) 3/. 3 π 3 3, Now the result follows from Theorem.3 with A := π 1/ ad B := (π/3) 1/. Refereces Bec-95. J. Beck, Flat polyomials o the uit circle ote o a problem of Littlewood, Bull. Lodo Math. Soc. 3 (1991), Bor-98. P. Borwei, Some Old Problems o Polyomials with Iteger Coefficiets, i Approximatio Theory IX, ed. C. Chui ad L. Schumaker, Vaderbilt Uiversity Press (1998), BE-95. P. Borwei ad T. Erdélyi, Polyomials ad Polyomial Iequalities, Spriger-Verlag, New York, Er-6. P. Erdős, A iequality for the maximum of trigoometric polyomials, Aales Poloica Math. 1 (196), Kah-85. J-P. Kahae, Sur les polyômes á coefficiets uimodulaires, Bull. Lodo Math. Soc 1 (1980), Li-61. J.E. Littlewood, O the mea value of certai trigoometric polyomials, Jour. Lodo Math. Soc. 36 (1961), Li-6. J.E. Littlewood, O the mea value of certai trigoometric polyomials (II), Jour. Lodo Math. Soc. 39 (1964), Li-66a. J.E. Littlewood, The real zeros ad value distributios of real trigoometrical polyomials, Jour. Lodo Math. Soc. 41 (1966), Li-66b. J.E. Littlewood, O polyomials P ±z m ad P e αmi z m, z = e θi, Jour. Lodo Math. Soc. 41 (1966), Li-68. J.E. Littlewood, Some Problems i Real ad Complex Aalysis, Heath Mathematical Moographs, Lexigto, Massachusetts, Saf-90. B. Saffari, Barker sequeces ad Littlewood s two sided cojectures o polyomials with ±1 coefficiets, Sémiaire d Aalyse Harmoique, 1989/90, Uiv. Paris XI, Orsay, 1990, Departmet of Mathematics ad Statistics, Simo Fraser Uiversity, Buraby, B.C., Caada V5A 1S6 (P. Borwei)

7 TRIGONOMETRIC POLYNOMIALS WITH MANY REAL ZEROS 7 address: pborwei@cecm.sfu.ca (Peter Borwei) Departmet of Mathematics, Texas A&M Uiversity, College Statio, Texas (T. Erdélyi) address: terdelyi@math.tamu.edu (Tamás Erdélyi)