AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL

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1 ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 33, Number 3, Fall 2003 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL ROBERTO ANDREANI AND DIMITAR K. DIMITROV ABSTRACT. For any positive integer n, the sine polynomials that are nonnegative in [0,π] which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials K n θ) are constructed in such awaythat{k n θ)} is a summability kernel. Thus, for each p, 1 p for any 2π-periodic function f L p [ π, π], the sequence of convolutions K n f is proved to converge to f in L p [ π, π]. The pointwise almost everywhere convergences are also consequences of our construction. 1. Introduction statement of results. There are various reasons for the interest in the problem of constructing nonnegative trigonometric polynomials. Among them are the Gibbs phenomenon [16, Section 9], univalent functions polynomials [7], positive Jacobi polynomial sums [1] orthogonal polynomials on the unit circle [15]. Our study is motivated by a basic fact from the theory of Fourier series by an intuitive observation which comes from an overview of the variety of known nonnegative trigonometric polynomials. The sequence {k n θ)} of even, nonnegative continuous 2π-periodic functions is called an even positive kernel if k n θ) are normalized by 1/2π) π π k nθ) dθ = 1 they converge locally uniformly in 0, 2π) uniformly on every compact subset of 0, 2π)) to zero. It is a slight modification of the definition in Katznelson s book [8]. In what follows we denote by k n f the convolution 1/2π) π π k nt)fθ t) dt. Itiswell known that, for every 2π-periodic function f L p [ π, π], 1 p, the sequence of convolutions k n f converges to f in the L p [ π, π]- norm provided k n is a sequence of even positive summability kernels. The convolutions converge also pointwise almost everywhere. We refer to the first chapter of [8] for the details AMS Mathematics Subject Classification. Primary 42A05, 26D05. Key words phrases. Nonnegative sine polynomial, positive summability kernel, extremal polynomial, ultraspherical polynomials, convergence. Research supported by the Brazilian Science Foundations CNPq under grants / /96-6, FAPESP under grants 97/ / Received by the editors on January 21, 2000, in revised formon June 4, Copyright c 2003 Rocky Mountain Mathematics Consortium 759

2 760 R. ANDREANI AND D.K. DIMITROV On the other h, most of the classical positive summability kernels are sequences of nonnegative cosine polynomials which obey certain extremal properties. Fejér [3] proved that the cosine polynomials 1.1) F n θ) =1+2 1 k ) cos kθ, n +1 are nonnegative established the uniform convergence of the sequence F n f to f for any continuous 2π-periodic function f. It is easily seen that Fejér s cosine polynomial 1.1) is the only solution of the extremal problem { max a a n :1+ } a k cos kθ 0. A basic tool for constructing positive kernels is the Fejér-Riesz representation of nonnegative trigonometric polynomials see [4]). It states that for every nonnegative trigonometric polynomial T θ), 1.2) T θ) =a 0 + a k cos kθ + b k sin kθ), there exists an algebraic polynomial Rz) = n c kz k of degree n such that T θ) = Re iθ ) 2, conversely, for every algebraic polynomial Rz) ofdegreen, the polynomial Re iθ ) 2 is a nonnegative trigonometric polynomial of order n. F ejér [4] showed that 1.3) a b2 1 2cos π/n +2) ) for any nonnegative trigonometric polynomial 1.2) with a 0 =1 that this bound is sharp. As a consequence he obtained the estimate 1.4) a 1 2cos π/n +2) ) for the first coefficient of any nonnegative polynomial of the form 1+ a k cos kθ.

3 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 761 Moreover, Fejér determined the nonnegative trigonometric cosine polynomials for which inequalities in 1.3) 1.4) are attained. A nice simple proof of Jackson s approximation theorem in Rivlin [10, Chapter 1] makes an essential use of the extremal property of this cosine polynomial. These observations already suggest that many sequences of nonnegative trigonometric polynomials whose coefficients obey certain extremal properties are positive summability kernels. While there are many results concerning extremal nonnegative cosine polynomials [2, 12], only a few results of the same nature about nonnegative sine polynomials are known. Since sine polynomials are odd functions, in what follows we shall call s n θ) = b k sin kθ a nonnegative sine polynomial if s n θ) 0 for every θ [0,π]. It is clear that, if s n θ) is nonnegative, then b 1 0b 1 =0ifonly if s n is identically zero. Rogosinski Szegő [11] considered some extremal problems for nonnegative sine polynomials. Among the other results, they proved that 1.5) s n0) = 1+2b 2 + +nb n { nn +2)n +4)/24 n even n +1)n +2)n +3)/24 n odd, provided b k are the coefficients of a sine polynomial s n θ) inthespace S + n = { s n θ) =sinθ + b k sin kθ : s n θ) 0 k=2 } for θ [0,π]. However, the sine polynomials for which the above limits are attained were not determined explicitly. The first objective of this paper is to fill this gap. Theorem 1. The inequality 1.5) holds for every s n θ) S + n. Moreover, if n = 2m +2 is even, then the equality S 2m+2 0) = m +1)m +2)m +3)/3 is attained only for the nonnegative sine

4 762 R. ANDREANI AND D.K. DIMITROV polynomial 1.6) S 2m+2 θ) = m { 1 k ) 1 k ) 2k+1 + kk 1) m+1 m+2 m+3 sin2k+1)θ +k+1) 1 k ) m+1 2 k+3 m+2 kk+1) m+2)m+3), if n =2m +1 is odd, the equality S 2m+10) = m +1)m + 2)2m +3)/6 ) ) sin2k +2)θ is attained only for the nonnegative sine polynomial 1.7) m { S 2m+1 θ) = 1 k ) 1+2k kk+2) m+1 m+2 kk+1)2k+1) ) m+2)2m+3) sin2k+1)θ +2k+1) 1 k ) 1 k+2 ) m+1 m+2 1+ k+1 ) } sin2k +2)θ. 2m+3 }, We shall obtain a close form representation of the extremal polynomials 1.6) 1.7) in terms of the ultraspherical polynomials P n 2) x). Recall that, for any λ > 1/2, {P n λ) x)} are orthogonal in [ 1, 1] with respect to the weight function 1 x 2 ) λ 1/2 are normalized by P n λ) 1) = 2λ) n /n! wherea) n is the Pochhammer symbol. Section 4.7 of Szegő s book [13] provides comprehensive information on the ultraspherical polynomials. Theorem 2. For any positive integer m the polynomials 1.6) 1.7) are given by 1.8) S 2m+2 θ) = 12 m+1)m+2)m+3) sin θ[ cosθ/2)p 2) m cos θ) ] 2

5 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL ) S 2m+1 θ) = 6 m+1)m+2)2m+3) sin θ[ P 2) m cos θ)+p 2) m 1 cos θ)] 2. Since 2n +2)P 2) n x) = T n+2x), where T n x) denotes the nth Chebyshev polynomial of the first kind, then we can represent S 2m+2 θ) S 2m+1 θ) intheform 1.10) 1.11) S 2m+2 θ) = S 2m+1 θ) = 3 m+1)m+2) 3 m+3) sin θ[ cosθ/2)t m+2cos θ) ] 2 3 2m+1) 3 m+2) 3 2m+3) sin θ [ m+1)t m+2cos θ)+m+2)t m+1cos θ) ] 2. Then the well-known representation of the second derivative of the Chebyshev polynomial T n cos θ) = n sin 3 {cos θ sin nθ + n sin θ cos nθ} θ yields the following equivalent closed-form representations of the above extremal sine polynomials: S 2m+2 θ) = 3cos2 θ 2 )[ m+2)sinθ cos m+2)θ ) cos θ sin m+2)θ )] 2 m+1)m+2)m+3)sin 5 θ S 2m+1 θ) = 3 1 2m +1)m + 2)2m +3) sin 3 θ [ m +2)cos m +2)θ ) +m +1)cos m +1)θ ) +cotθ sin m +2)θ ) +sin m +1)θ ))] 2.

6 764 R. ANDREANI AND D.K. DIMITROV Set K n θ) =1/2π)sinθS n θ), for any function fx) whichis2πperiodic integrable in [ π, π], define the trigonometric polynomial K n f; x) = π π K n θ)fx θ) dθ. Observe that K n θ) is a cosine polynomial of order n+1. In Section 4 we shall prove that {K n θ)} is a positive summability kernel then the following results on L p, pointwise almost everywhere convergence of K n f; x) will immediately hold. Theorem 3. For any p, 1 p, for every 2π-periodic function f L p [ π, π], the sequence K n f; x) converges to f in L p [ π, π]. Theorem 4. Let f be a 2π-periodic function which is integrable in [ π, π]. If, for x [ π, π], thelimit lim fx + h)+fx h)) exists, h 0 then ) K n f; x) 1/2) lim fx + h)+fx h) as n diverges. h 0 Theorem 5. Let f be a 2π-periodic function which is integrable in [ π, π]. ThenK n f; x) converges to f almost everywhere in [ π, π]. It is worth mentioning that, while the sequences {k n θ)} of classical summability kernels, namely, Fejér s, de la Vallée Poussin s Jackson s one, converge to infinity at the origin, in our case K n 0) vanishes for any positive integer n. 2. Preliminary results. The above-mentioned Fejér-Riesz s theorem a result of Szegő [13, p. 4] imply a representation of nonnegative cosine polynomials. Lemma 1. Let c n θ) =a 0 +2 a k cos kθ

7 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 765 be a cosine polynomial of order n which is nonnegative for every real θ. Then an algebraic polynomial Rz) = n c kz k exists of degree n with real coefficients, such that c n θ) = Re iθ ) 2. Thus, the cosine polynomial c n θ) of order n is nonnegative if only if there exist real numbers c k, k =0, 1,...,n, such that 2.12) a 0 = n k c 2 k a k = c k+ν c ν ν=0 for k =1,...,n. The following relation between nonnegative sine polynomials s n θ) nonnegative cosine polynomials c n 1 θ) is an immediate consequence of the relation s n θ) =sinθc n 1 θ) see[11]). Lemma 2. Thesinepolynomialofordern s n θ) = b k sin kθ is nonnegative in [0,π] if only if the cosine polynomial of order n 1 n 1 c n 1 θ) =a 0 +2 a k cos kθ, where 2.13) is nonnegative. b k = a k 1 a k+1 for k =1,...,n 2, b n 1 = a n 2, b n = a n 1, These two lemmas imply a parametric representation for the coefficients of the nonnegative sine polynomials. Lemma 3. Thesinepolynomialofordern s n θ) = b k sin kθ

8 766 R. ANDREANI AND D.K. DIMITROV is nonnegative if only if there exist real numbers c 0,...,c n 1 such that n 1 n 3 b 1 = c 2 ν c ν c ν+2, ν=0 ν=0 ν=0 n k n k ) b k = c k+ν 1 c ν c k+ν+1 c ν, for k =2,...,n 2, b n 1 = c 0 c n 2 + c 1 c n 1, b n = c 0 c n 1. ν=0 Set q k =k+1)/2k). It can be verified that, if n is even, n =2m+2, then b 1 is given by 2.15) m 1 b 1 = {q k+1 c 2k c 2k+2 /2q k+1 )) 2 +q k+1 c 2k+1 c 2k+3 /2q k+1 )) 2} + q m+1 c 2 2m + q m+1 c 2 2m+1,, if n is odd, n =2m +1,then 2.16) m 1 b 1 = {q k+1 c 2k c 2k+2 /2q k+1 )) 2 +q k+1 c 2k+1 c 2k+3 /2q k+1 )) 2} + q m c 2m 2 c 2m /2q m )) 2 + q m c 2 2m 1 + q m+1 c 2 2m. 3. Proof of Theorem 1. We need to maximize s n0) subject to the conditions s n θ) 0in[0,π]b 1 = 1. Apply Lemmas 1 2 to represent the derivative of the nonnegative sine polynomial s n θ) at the origin in terms of the parameters c k.weobtain s n 0) = n n 2 kb k = ka k 1 a k+1 )+n 1)a n 2 + na n 1 n 1 = a 0 +2 a k = n 1 c 2 j +2 j=0 0 j<k n 1 c j c k = n 1 ) 2 c k.

9 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 767 Set Gc) =Gc 0,...,c n 1 )= Then the extremal problem to be solved is n ) max{gc) :b 1 c) =1}, ) 2 c k. where b 1 = b 1 c) = b 1 c 0,...,c n 1 ) is defined by 2.15) or 2.16) depending on the parity of n. We employ the Lagrange multipliers approach see [6, Section 9.1]) to solve this problem. The necessary conditions for c =c 0,...,c n 1 ) to be a point of extremum for 3.17) are 3.18) Gc) =λ b 1 c) b 1 c) =1, where denotes the gradient operator λ is the Lagrange multiplier. It is obvious that every solution c of 3.18) which corresponds to λ = 0 minimizes Gc). Indeed, in this case we have n 1 i=0 c i =0 these are the only points where the nonnegative function Gc) vanishes. It is worth mentioning that this observation means that a nonnegative sine polynomial has a derivative which vanishes at the origin if only if its coefficients are given by 2.14) the parameters satisfy n 1 i=1 c i =0. Thus, for the points of maximum the Lagrange multiplier λ is nonzero. Define n 1 r e r o )=λ 1 j=1 c j if n even odd), ξ 2k = c 2k 2q k+1 ) 1 c 2k+2, ξ 2k+1 = c 2k+1 2q k+1 ) 1 c 2k+3, k =0,...,m, where we set c 2m+2 = c 2m+3 =0.

10 768 R. ANDREANI AND D.K. DIMITROV Consider first the case n =2m + 2. The conditions Gc) =λ b 1 c) reduce to the parametric system of linear equations q 1 ξ 0 = q 1 ξ 1 = r e ξ 2k +2q k+2 ξ 2k+2 = ξ 2k+1 +2q k+2 ξ 2k+3 =2r e, k =0,...,m 1 for the unknowns ξ k, k =0,...,2m + 1 the parameter r e. The explicit form of q k yields ))) = k +1. 2q k 2q 2 2q 1 q k+1 Then we obtain the solution ξ k, k =0,...,2m + 1, of the above linear system explicitly in terms of r e : ξ 2k = ξ 2k+1 = ))) r e q k+1 2q k 2q 2 2q 1 =k +1)r e, k =0,...,m. Hence, for the parameters c k, k =0,...,2m, weobtain c 2k = c 2k+1 =k +1)m k +1)r e, k =0,...,m. Now the condition b 1 c) =1gives 3.19) r 2 e =3/ m +1)m +2)m +3) ). In the case when n =2m + 1 similar observations yield c 2k =k +1)m k +1)r o, k =0,...,m, c 2k+1 =k +1)m k)r o, k =0,...,m 1, 3.20) r 2 o =6/ m +1)m + 2)2m +3) ). Now the relations 2.14) straightforward calculations imply the explicit form of the coefficients b k.

11 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL Proofs of Theorems 2, 3, 4 5. Proof of Theorem 2. Observe that the statement of Lemma 2 is equivalent to the equality S n θ) =sinθc n 1 θ), where n 1 C n 1 θ) =a 0 +2 a k cos kθ, the coefficients of C n 1 θ) are given by 2.12) for the parameters c 0,...,c n 1 determined in the proof of Theorem 1. Then Lemma 1 the explicit form of c k yield 4.21) S n θ) =r 2 sin θ Rn 1 e iθ ) 2, where r = r e or r = r o depending on the parity of n R 2m+1 z) =1+z) R 2m z) = m k +1)m k +1)z 2k m m 1 k +1)m k +1)z 2k + z k +1)m k)z 2k. Following Fejér Szegő [5], consider the Cesàro sums S j mζ) of order j of the geometric series, S j mζ) = m m + j k j ) ζ k. Then R 2m+1 z) =1+z) ds1 m+1ζ) dζ R 2m z) = ds1 m+1ζ) dζ ζ=z 2 + z ds1 mζ) ζ=z 2 dζ. ζ=z 2

12 770 R. ANDREANI AND D.K. DIMITROV On the other h, Turán [14] proved that, for any pair n, j of positive integers, d/dζ) j S j n+j ζ) =j!ζn/2 P n j+1) ζ 1/2 + ζ 1/2 )/2 ), where P n λ) denotes the ultraspherical polynomial. Applying this result for j =1,weobtain R 2m+1 z) =1+z)z m P 2) m z + z 1 )/2 ) R 2m z) =z m{ P m 2) z + z 1 )/2 ) + P m 1 2) z + z 1 )/2 ) }. Substitute z = expiθ) in these representations use 4.21) to complete the proof. The next is a basic technical result. Lemma 4. inequality For every positive integer n for any real θ, the 4.22) sin 2 θ/2)k n θ) c1/n), holds with an absolute constant c. Proof. We provide three different proofs of the lemma depending on the representation of the extremal sine polynomials. It suffices to prove the above inequality for θ [0,π]. The first proof uses representations 1.8) 1.9) of S n θ). First we establish 4.22) in the case n =2m + 2. It follows from the definition of the kernel K n θ) from 1.8) that 4.23) sin 2 θ/2)k 2m+2 θ) = 1 2π sin2 θ/2) sin θs 2m+2 θ) = r2 e 2π sin2 θ/2) sin 2 θ [ 2cosθ/2)P m 2) cos θ) ] 2 = r2 [ e sin 2 θp m 2) cos θ) ] 2. 2π

13 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 771 Kogbetliantz [9] provedthat,foranyλ 0, 4.24) sin θ) λ P λ) n cos θ) 2Γn + λ) Γλ)Γn +1) An application of this result for λ =2yields sin 2 θ/2)k 2m+2 θ) 6 m +1 π m +2)m +3). When n =2m +1,weneedalittlemoreeffort.Wehave for 0 θ π. sin 2 θ/2)k 2m+1 θ) = r2 o 2π sin2 θ)sin 2 θ/2) [ P m 2) cos θ)+p 2) m 1 cos θ)] 2. The recurrence relation for the ultraspherical polynomials see ) in [13]) implies P m 2) x)+p 2) 2) m 1 x) =1+x)P m 1 x)+m+2 2) 2) xp m 1 x) P m 2 m x)) formulae ) ) in [13] yield xp 2) 2) m 1 x) P m 2 x) =1 x d 2 dx P m 1) x) d ) dx P 1) m 1 x) = m 2 P m 1) x). Therefore, sin 2 θ/2)k 2m+1 θ) [ = r2 o 2π sin2 θ)sin 2 θ/2) 2 2cos 2 θ/2)p 2) m 1 cos θ)+m+2 2 P m 1) cos θ)]. Then we use 4.24) for λ = 1 for λ = 2 to obtain sin 2 θ/2)k 2m+1 θ) { r2 o cos 2 θ/2) [ sin 2 θp 2) m 1 2π cos θ)] 2 + m +2 sin θ [ sin 2 θ 2) P m 1 2 cos θ) ][ sin θ P 1) m cos θ) ] m +2)2 + sin 2 θ/2) [ sin θp m 1) cos θ) ] } 2 4 { r2 o 4m 2 cos 2 θ 2π 2 +2mm +2)sinθ +m +2)2 sin 2 θ } 2 = r2 o 2π 1 2π 2m cosθ/2) + m +2)sinθ/2) 2 6 5m 2 +4m +4) 2 m +1)m + 2)2m +3) 4m 2.

14 772 R. ANDREANI AND D.K. DIMITROV Here the latter inequality follows from the fact that the maximal value in [0,π] of the function 2m cosθ/2) + m +2)sinθ/2), which is positive in [0,π], is attained at the point θ 0 for which 2m sinθ 0 /2) = m +2)cosθ 0 /2). This completes the first proof of the lemma. The second proof is based on the representations 1.10) 1.11) of S n θ) in terms of the Chebyshev polynomials. It is essentially equivalent to the first one. We only sketch the proof for n =2m +2 because, for n = 2m + 1, it is similar. The representation 4.23) the relation between P 2) n x) T m+2x) yield sin 2 θ/2)k 2m+2 θ) = r 2 e 8πm +2) 2 [ sin 2 θt m+2cos θ) ] 2. Then the second order differential equation for the Chebyshev polynomials 1 x 2 )T m+2x) =xt m+2x) m +2) 2 T m+2 x) the inequalities T m+2 x) 1 T m+2x) m +2) 2 for x [ 1, 1] imply sin 2 θ/2)k 2m+2 θ) r 2 e 8πm +2) 2 4m +2)4. The third proof is straightforward, it was the one we obtained before discovering the nice relation between the extremal polynomials S n θ) ultraspherical Chebyshev polynomials. The coefficients β k in the representation are given by sin 2 θ/2)k n θ) = 1 n+2 β k cos kθ 8π β 0 =3/m +2), β 2k = 12k/ m +1)m +2)m +3) ), β 2k+1 =0, k =0,...,m+1, β 2m+4 =3m +1)/ m +2)m +3) ), k =1,...,m+1,

15 AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 773 when n =2m +2,by β 0 =6/2m +3), β 2k = 12k/ m +1)m + 2)2m +3) ), k =1,...,m+1, β 2k+1 = 62k +1)/ m +1)m + 2)2m +3) ), k =0,...,m, β 2m+3 =6m +1)/ m + 2)2m +3) ), when n =2m + 1. It is easy to see that the sum of the modulus of these coefficients is less than c/n where c is an absolute constant. More precisely, we have sin 2 θ/2)k 2m+2 θ) 3 2πm+3), sin 2 θ/2)k 2m+1 θ) 3 2πm+2). As it has already been mentioned, Lemma 4 implies the truth of Theorems 3, 4 5. REFERENCES 1. R. Askey G. Gasper, Inequalities for polynomials, in The Bieberbach conjecture: Proceedings of the symposium on the occasion of proof A. Baernstein II, et al., eds.), Amer. Math. Soc., Providence, 1986, pp E. Egerváry Szász, Einige Extremalprobleme im Bereiche der trigonometrischen Polynome, Math.Z ), L. Fejér, Sur les functions bornées et integrables, C.R.Acad.Sci.Paris ), , Über trigonometriche polynome, J.ReineAngew.Math ), L. Fejér G. Szegő, Special conformal maps, Duke Math. J ), ; reprinted in Gabor Szegő: Collected papers R. Askey, ed.), Vol. 3, Birkhäuser, Boston, 1982, pp Fletcher, Practical methods of optimization, 2nd ed., John Wiley & Sons, Chichester, A. Gluchoff F. Hartmann, Univalent polynomials non-negative trigonometric sums, Amer. Math. Monthly ), Y. Katznelson, An introduction to harmonic analysis, John Wiley, New York, E. Kogbetliantz, Recherches sur la sommabilité desséries ultrasphériques par la méthode des moyennes arithmétiques, J. Math. III ), T.J. Rivlin, An introduction to the approximation of functions, Dover Publications, New York, 1981.

16 774 R. ANDREANI AND D.K. DIMITROV 11. W.W. Rogosinski G. Szegő, Extremum problems for non-nonegative sine polynomials, Acta Sci. Math. Szeged ), ; reprinted in Gabor Szegő: Collected papers R. Askey, ed.), Vol. 3, Birkhäuser, Boston, 1982, pp G. Szegő, Koeffizientenabschätzungen bei ebenen und räumlichen harmonischen Entwicklungen, Math. Ann /27), ; reprinted in Gabor Szegő: Collected papers R. Askey, ed.), Vol. 2, Birkhäuser, Boston, 1982, pp , Orhtogonal polynomials, Amer. Math. Soc. Colloq. Pub., vol. 23, 4th ed., AMS, Providence, Turán, Remark on a theorem of Fejér, Publ. Math. Debrecen ), W. Van Assche, Orthogonal polynomials in the complex plane on the real line, Fields Inst. Comm ), A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, Departamento de Ciências de Computação e Estatística, Ibilce, Universidade Estadual Paulista, São José doriopreto,sp,brazil address: reani@dcce.ibilce.unesp.br Departamento de Ciências de Computação e Estatística, Ibilce, Universidade Estadual Paulista, São José doriopreto,sp,brazil address: dimitrov@dcce.ibilce.unesp.br