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 polynomi- als that are nonnegative in [0, π] and which have the maximal derivative at the origin are determined in an explicit form. Associated cosine polynomials Kn(θ) are constructed in such a way that {Kn(θ)} is a summability kernel. Thus, for each p, 1 ≤ p ≤ ∞ and for any 2π-periodic function f ∈ Lp[−π, π], the sequence of convolutions Kn ∗ f is proved to converge to f in Lp[−π, π]. The pointwise and almost everywhere conver- gences are also consequences of our construction. 1. Introduction and 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 and polynomials [7], positive Jacobi polynomial sums [1] and orthogonal polynomials on the unit circle [15]. Our study is motivated by a basic fact from the theory of Fourier series and by an intuitive observation which comes from an overview of the variety of known nonnegative trigonometric polynomials. The sequence {kn(θ)} of even, nonnegative continuous 2π-periodic func- tions is called an even positive kernel if kn(θ) are normalized by (1/2π) ∫ π −π kn(θ) dθ = 1 and 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 kn∗f the convolution (1/2π) ∫ π −π kn(t)f(θ−t) dt. It is well known that, for every 2π-periodic function f ∈ Lp[−π, π], 1 ≤ p ≤ ∞, the sequence of convolutions kn ∗ f converges to f in the Lp[−π, π]- norm provided kn is a sequence of even positive summability kernels. The convolutions converge also pointwise and almost everywhere. We refer to the first chapter of [8] for the details. 1991 AMS Mathematics Subject Classification. Primary 42A05, 26D05. Key words and phrases. Nonnegative sine polynomial, positive summability kernel, extremal polynomial, ultraspherical polynomials, convergence. Research supported by the Brazilian Science Foundations CNPq under grants 300645/95-3 and 301115/96-6, and FAPESP under grants 97/6280-0 and 99/08381-4. Received by the editors on January 21, 2000, and in revised form on June 4, 2001. Copyright c©2003 Rocky Mountain Mathematics Consortium 759 760 R. ANDREANI AND D.K. DIMITROV On the other hand, 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) Fn(θ) = 1 + 2 n∑ k=1 ( 1− k n+ 1 ) cos kθ, are nonnegative and established the uniform convergence of the se- quence Fn ∗ 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 { a1 + · · ·+ an : 1 + n∑ k=1 ak cos kθ ≥ 0 } . A basic tool for constructing positive kernels is the Fejér-Riesz repre- sentation of nonnegative trigonometric polynomials (see [4]). It states that for every nonnegative trigonometric polynomial T (θ), (1.2) T (θ) = a0 + n∑ k=1 (ak cos kθ + bk sin kθ), there exists an algebraic polynomial R(z) = ∑n k=0 ckz k of degree n such that T (θ) = |R(eiθ)|2, and conversely, for every algebraic polynomial R(z) of degree n, the polynomial |R(eiθ)|2 is a nonnegative trigonometric polynomial of order n. Fejér [4] showed that (1.3) √ a2 1 + b21 ≤ 2 cos ( π/(n+ 2) ) for any nonnegative trigonometric polynomial (1.2) with a0 = 1 and that this bound is sharp. As a consequence he obtained the estimate (1.4) a1 ≤ 2 cos ( π/(n+ 2) ) for the first coefficient of any nonnegative polynomial of the form 1 + n∑ k=1 ak cos kθ. AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 761 Moreover, Fejér determined the nonnegative trigonometric and cosine polynomials for which inequalities in (1.3) and (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 nonnega- tive 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 sn(θ) = n∑ k=1 bk sin kθ a nonnegative sine polynomial if sn(θ) ≥ 0 for every θ ∈ [0, π]. It is clear that, if sn(θ) is nonnegative, then b1 ≥ 0 and b1 = 0 if and only if sn is identically zero. Rogosinski and Szegő [11] considered some extremal problems for nonnegative sine polynomials. Among the other results, they proved that (1.5) s′n(0) = 1+2b2+· · ·+nbn ≤ { n(n+ 2)(n+ 4)/24 n even (n+ 1)(n+ 2)(n+ 3)/24 n odd, provided bk are the coefficients of a sine polynomial sn(θ) in the space S+ n = { sn(θ) = sin θ + n∑ k=2 bk sin kθ : sn(θ) ≥ 0 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 sn(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 762 R. ANDREANI AND D.K. DIMITROV polynomial (1.6) S2m+2(θ) = m∑ k=0 {( 1− k m+1 )( 1− k m+2 )( 2k+1 + k(k−1) m+3 ) × sin(2k+1)θ + (k+1) ( 1− k m+1 ) × ( 2− k+3 m+2 − k(k+1) (m+2)(m+3) ) sin(2k + 2)θ } , and, if n = 2m+ 1 is odd, the equality S′ 2m+1(0) = (m+ 1)(m+ 2)(2m+ 3)/6 is attained only for the nonnegative sine polynomial (1.7) S2m+1(θ) = m∑ k=0 {( 1− k m+1 )( 1+2k − k(k+2) m+2 − k(k+1)(2k+1) (m+2)(2m+3) ) × sin(2k+1)θ + 2(k+1) ( 1− k m+1 )( 1− k+2 m+2 ) × ( 1 + k+1 2m+3 ) sin(2k + 2)θ } . We shall obtain a close form representation of the extremal polynomi- als (1.6) and (1.7) in terms of the ultraspherical polynomials P (2) n (x). Recall that, for any λ > −1/2, {P (λ) n (x)} are orthogonal in [−1, 1] with respect to the weight function (1 − x2)λ−1/2 and are normalized by P (λ) n (1) = (2λ)n/n! where (a)n is the Pochhammer symbol. Sec- tion 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) and (1.7) are given by S2m+2(θ) = 12 (m+1)(m+2)(m+3) sin θ [ cos(θ/2)P (2) m (cos θ) ]2(1.8) AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 763 and S2m+1(θ) = 6 (m+1)(m+2)(2m+3) sin θ [ P (2) m (cos θ)+P (2) m−1(cos θ) ]2 . (1.9) Since 2(n + 2)P (2) n (x) = T ′′ n+2(x), where Tn(x) denotes the nth Chebyshev polynomial of the first kind, then we can represent S2m+2(θ) and S2m+1(θ) in the form S2m+2(θ) = 3 (m+1)(m+2)3(m+3) sin θ [ cos(θ/2)T ′′ m+2(cos θ) ]2(1.10) and S2m+1(θ) = 3 2(m+1)3(m+2)3(2m+3) (1.11) × sin θ [ (m+1)T ′′ m+2(cos θ) + (m+2)T ′′ m+1(cos θ) ]2 . Then the well-known representation of the second derivative of the Chebyshev polynomial T ′′ n (cos θ) = n sin3 θ {cos θ sinnθ + n sin θ cosnθ} yields the following equivalent closed-form representations of the above extremal sine polynomials: S2m+2(θ) = 3 cos2( θ 2 ) [ (m+2) sin θ cos ( (m+2)θ )−cos θ sin ( (m+2)θ )]2 (m+1)(m+2)(m+3) sin5 θ and S2m+1(θ) = 3 2(m+ 1)(m+ 2)(2m+ 3) 1 sin3 θ × [ (m+ 2) cos ( (m+ 2)θ ) + (m+ 1) cos ( (m+ 1)θ ) + cot θ ( sin ( (m+ 2)θ ) + sin ( (m+ 1)θ ))]2 . 764 R. ANDREANI AND D.K. DIMITROV Set Kn(θ) = (1/2π) sin θSn(θ) and, for any function f(x) which is 2π- periodic and integrable in [−π, π], define the trigonometric polynomial Kn(f ;x) = ∫ π −π Kn(θ)f(x− θ) dθ. Observe thatKn(θ) is a cosine polynomial of order n+1. In Section 4 we shall prove that {Kn(θ)} is a positive summability kernel and then the following results on Lp, pointwise and almost everywhere convergence of Kn(f ;x) will immediately hold. Theorem 3. For any p, 1 ≤ p ≤ ∞, and for every 2π-periodic function f ∈ Lp[−π, π], the sequence Kn(f ;x) converges to f in Lp[−π, π]. Theorem 4. Let f be a 2π-periodic function which is integrable in [−π, π]. If, for x ∈ [−π, π], the limit lim h→0 (f(x+ h) + f(x− h)) exists, then Kn(f ;x)→ (1/2) lim h→0 ( f(x+ h) + f(x− h) ) as n diverges. Theorem 5. Let f be a 2π-periodic function which is integrable in [−π, π]. Then Kn(f ;x) converges to f almost everywhere in [−π, π]. It is worth mentioning that, while the sequences {kn(θ)} of classical summability kernels, namely, Fejér’s, de la Vallée Poussin’s and Jack- son’s one, converge to infinity at the origin, in our case Kn(0) vanishes for any positive integer n. 2. Preliminary results. The above-mentioned Fejér-Riesz’s theo- rem and a result of Szegő [13, p. 4] imply a representation of nonneg- ative cosine polynomials. Lemma 1. Let cn(θ) = a0 + 2 n∑ k=1 ak cos kθ AN EXTREMAL NONNEGATIVE SINE POLYNOMIAL 765 be a cosine polynomial of order n which is nonnegative for every real θ. Then an algebraic polynomial R(z) = ∑n k=0 ckz k exists of degree n with real coefficients, such that cn(θ) = |R(eiθ)|2. Thus, the cosine polynomial cn(θ) of order n is nonnegative if and only if there exist real numbers ck, k = 0, 1, . . . , n, such that (2.12) a0 = n∑ k=0 c2k and ak = n−k∑ ν=0 ck+νcν for k = 1, . . . , n. The following relation between nonnegative sine polynomials sn(θ) and nonnegative cosine polynomials cn−1(θ) is an immediate conse- quence of the relation sn(θ) = sin θcn−1(θ) (see [11]). Lemma 2. The sine polynomial of order n sn(θ) = n∑ k=1 bk sin kθ is nonnegative in [0, π] if and only if the cosine polynomial of order n− 1 cn−1(θ) = a0 + 2 n−1∑ k=1 ak cos kθ, where bk = ak−1 − ak+1 for k = 1, . . . , n− 2, bn−1 = an−2,(2.13) bn = an−1, is nonnegative. These two lemmas imply a parametric representation for the coeffi- cients of the nonnegative sine polynomials. Lemma 3. The sine polynomial of order n sn(θ) = n∑ k=1 bk sin kθ 766 R. ANDREANI AND D.K. DIMITROV is nonnegative if and only if there exist real numbers c0, . . . , cn−1 such that b1 = n−1∑ ν=0 c2ν − n−3∑ ν=0 cνcν+2, (2.14) bk = n−k∑ ν=0 ck+ν−1cν − n−k−2∑ ν=0 ck+ν+1cν , for k = 2, . . . , n− 2, bn−1 = c0cn−2 + c1cn−1, bn = c0cn−1. Set qk = (k+1)/(2k). It can be verified that, if n is even, n = 2m+2, then b1 is given by (2.15) b1 = m−1∑ k=0 { qk+1(c2k−c2k+2/(2qk+1))2+qk+1(c2k+1−c2k+3/(2qk+1))2 } + qm+1c 2 2m + qm+1c 2 2m+1, and, if n is odd, n = 2m+ 1, then (2.16) b1 = m−1∑ k=0 { qk+1(c2k−c2k+2/(2qk+1))2+qk+1(c2k+1−c2k+3/(2qk+1))2 } + qm(c2m−2 − c2m/(2qm))2 + qmc22m−1 + qm+1c 2 2m. 3. Proof of Theorem 1. We need to maximize s′n(0) subject to the conditions sn(θ) ≥ 0 in [0, π] and b1 = 1. Apply Lemmas 1 and 2 to represent the derivative of the nonnegative sine polynomial sn(θ) at the origin in terms of the parameters ck. We obtain s′n(0) = n∑ k=1 kbk = n−2∑ k=1 k(ak−1 − ak+1) + (n− 1)an−2 + nan−1 = a0 + 2 n−1∑ k=1 ak = n−1∑ j=0 c2j + 2 ∑ 0≤j