ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR Technical Report No. 1 SOME PROPERTIES OF THE FEJER POLYNOMIALS FRITZ; -ZOGGEORGET PIRANIAN Project 2316 DETROIT ORDNANCE DISTRICT, U. S. ARMY CONTRACT NO. DA-20-018-ORD-13585, ARMY PROJECT NO. 5899-01-004 ORDNANCE R AND D NO. TB2-0001, OOR PROJECT NO. 1299 DETROIT, MICHIGAN April 1955

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN OBJECT To investigate the distribution of the zeros of the polynomials Pn(z) =1 + z + znl zn rz2n+l 2 n n-l 1 1 2 n and to determine the maximum modulus of Pn(z) on the unit circle C. ABSTRACT The polynomial Pn has a zero at z 1, and it has no other positive zero and no other zero on the unit circle. It has no negative zeros if n is odd; if n is even, it has two negative zeros which are given asymptotically by the formula log n log logl +00 Each of the n - 1 sectors (2k - 1) it/n < arg z < (2k + l)it/n (k = 1, 2,..., n - 1) contains exactly two zeros of Pn- For e > 0 and n sufficiently large, each of the zeros of Pn(z)/(l z) lies in one of the annuli 1_ (Q4 + c ) log n < I z < 1 _e - c n n 1+(1. + c) log n +> Z >1 + n n As n-*o, the maximum modulus of Pn on C approaches 2 o t41 sin t dt ii

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN I. INTRODUCTION The polynomials zn-1 zn zn+1,2n-1 Pn(z) 1+ + +... + zn- _ zn _ z+ln n n-l 1 1 2 n were introduced, by way of their real and imaginary parts on the unit circle C, by Fejer [6, 7]. Fejer showed that the number 2 + ir is an upper bound for the modulus both of the real and of the imaginary part of Pn on C (n = 1, 2,...); a very simple proof that the polynomials Pn are uniformly bounded on C is given in [3, page 43]. In view of recent applications of the Fejer polynomials in the study of Taylor series (see, for example [3, 4, 8]), we have undertaken an investigation of their least upper bound on C (see Section III) and of the distribution of their zeros (see Section II). Elementary considerations show that limn,, n Pn(z) = 1/(1 - z) for Izi < 1, and that the convergence is uniform in every disc Iz | r < 1. From this and the fact that the reciprocal of every zero of Pn is also a zero of Pn, it follows that the zeros of Pn lie on or near the circle C. The theorem of Jentzsch and Szego" [9, 11] implies further that the arguments of the zeros are uniformly distributed in the interval [0, 2X]. Somewhat stronger results on the distribution of the arguments could be obtained by applying a theorem of Erdo5s and Turan [5, page 106]; but by using a method which involves nothing deeper than Rouche's theorem, we prove that each of the n-' sectors Z = reig (2k - 1): < < (2k + 1)r 1 2,..., n-) n n contains precisely two zeros of Pn. II. THE ZEROS OF THE FEJER POLYNOMIALS We will state and prove various lemmas concerning the zeros of the polynomials Pn. At the end of the section, the contents of the lemmas will be gathered into a theorem. LEMMA 1. The point z =1 is the only positive zero of Pn. The reciprocal and the complex conjugate of every zero of Pn are also zeros of Pn

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN This lemma follows at once from Descartes' rule of signs, from the fact that the coefficients of the polynomial Pn are real, and from the relation Pn(z'l) = -z-2n+l Pn(z). LEMMA 2. Except for z = 1, Pn has no zeros on the unit circle. It is easily verified that Pn(eiQ) = -2i ei(n-1/2) C(n, Q), (1) where C(n, 9) sin(k;-. 2) k=l k Using the identity in [10, Section VI, Problem 17, page 78] and Abel's summation,we obtain the formula C(n, Q) = i sin2(n//2) + n sin2(k. /2)3 sin(/2).n k(k + 1) The lemma follows from the fact that the right member is positive for O < Q < 2it. In the sequel, it will be useful to deal with the function Wn(z) = z-n(l - z) P(z) zn Z+ zn 2 + zk + (2) n k=l k(k + 1) LEMMA 3. Each of the n sectors (2k - 1)t < arg z < (2k + 1). (k = 1,. n-l) n n contains exactly two zeros of Wn(z). Let zn n -,+ z+fn(z) _Z + Z - 2 B g(z) zk + z k n If z lies on one of the raysarg z = (2k + 1)x (k = 0, 1,, n -l), (3) n~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN then, with izl = r, lfn(z) = (rn + r-n)/n + 2, and therefore Ifn(z)! - Ign(Z) I _ r(r), where fr(r)= r n + rrk n k- k(k+ 1) To show that lr(r) > 0 for r > O0 it is sufficient to consider the values r > 1. Now 4r(l) = 4/n > 0; and 1' (r) > 0 when r > 1, since the function n-l k -k r *'(r) = rn _ rn r k=l k + 1 vanishes at r = 1 while its derivative lr nn -n nK7 k(rk + 1r-k) r_-1 n~rn+ r-) _ k(rk + r+r ) ka- k + 1 is obviously positive. It follows that Ifn(z)I > Ign(z)I (4) on each of the rays (3). The inequality (4) is also satisfied on the circle Izl = rn, provided rn is sufficiently small or sufficiently large. By applying Rouche's theorem to the functions fn and fn + gn = Wn, with reference to a region which is bounded by two neighboring rays (3) and by arcs of two circles Iz = rn and Izi = rn-l, we conclude that fn and Wn have the same number of zeros in this region. The lemma now follows from the fact that the zeros of fn are the 2n numbers (n +n2 _ 1)/n e2k-i/n (k = 0O 1,..., n-l). LEMMA 4. Pn has two or no negative zeros, depending on whether n is even or odd. The function Wn has 2n zeros of which two lie at z = 1. If n is odd, Lemma 3 disposes of the remaining 2n - 2 zeros. If n is even, Lemma 3 implies that at most two zeros are negative, and Lemmas 1 and 2 imply that at least two zeros are negative. LEMMA 5. If E > 0 and n is sufficiently large, the zeros of Pn lie in the annulus exp[-(4 + e)n-1 log n] < jzl < exp [(4 + E)n1 log n]. 3

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN It suffices to show that Wn(z) f 0 for Izi < exp[-(4 + E)n-l log n] and n > no(E). By (2), zn Wn(z) = 1/n + Vn(z), where Vn(z) + z+ + 2z n(n - 1) (n - l)(n - 2) 2 1 zn+l zn+2 z2n-1 z2n 1P2 2.3 (n - 1)n n We will now show that IVn(z)l < 1/n for the values of z indicated above. Let f = 2 + e/2. The sum of the moduli of the first [n/i] terms on the right-hand side of (5) is less than 1 1< 1 1 1 n - [n/i] n- n - n/ n n( - 1) Since the remaining terms have coefficients of modulus at most 2, the modulus of their sum is at most 21zln/i 2n-2 1 + o(1) 1 - iz- 1 - exp.[-2i n-l log n] in log n It follows that, for large n, IVn(z)I < 1 + 1 < 1 n(~ 1 l) log n n and the lemma is proved. LEMMA 6. If n is even, the two negative zeros of Pn are given asymptotically by the formula z = - [n log 16 + o(n)]+- 1/n To prove this lemma, suppose that Wn(-rn) 0, with rn > 1, and we write rn = nPn/n exp(pnn7l log n). (For the sake of typographical simplicity, the subscript n is dropped, in the remainder of the proof.) By Lemma 5, p < 5 when n is sufficiently large. From Equation (2) we obtain the relation nPi1 + n-P1 = h(r), (6) 1 4 He~~~~~~~~~~~~~~~~~~~~~L

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN where h( r) + r 2 + r2 r +r- r-l + r2 r3 + h(r)=2' 23 324 (n + (7) The proof of the lemma hinges on an effective computation of h(r). For the purpose of this computation, we divide the right-hand member of (7) into two sections. The first section contains the number 2 and the first [3n/(p log n)] terms that follow. The kth of these terms has the modulus 2/k(k + 1) + ek, where ECk= 2 cosh(kpnl1 log n) - 2 k(k + 1) Since cosh u < 1 + 2u2 when O < u <_ 3, 0 < Ek < 4(k log n) < 4(pnLk log n)2, k(k + 1) and Ek < 3n 4(pn-1 log n)2 = 12 pn-1 log n = o( l). k _ 3n/(plogn) log n) o It follows that the sum of the terms in the first section on the right of (7) is log 16 + o(l). To show that the sum of the terms in the second section is small, we will first prove that each term is numerically larger than its predecessor. It will then follow that the sum of the terms in the second section is positive and smaller than the last term. We write rk+l + r-k-l k(k + 1) rk + rk (k + l)(k + 2) cosh[(k + l)pn-1 log n] 2 4 cosh(kpn-1 log n) k k2' The second factor on the right is greater than 1 - 2/k, that is, greater than 1 - (2/3)pn-1 log n. To obtain a lower bound on the first factor, we use the fact that the derivative of cosh x is an increasing function, and we conclude that the factor is greater than cosh(kpn-1 log n) + pn-1 log n sinh(kpn- log n) cosh(kpn' log n) 1+ (pn 1 log n) tanh(kpn-l log n) > 1 + (tanh 3) pn-l log n 5

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN Since tanh 3 > 2/3, the ratio between the numerical value of the (k + l)st term and that of the kth term on the right of (7) is greater than 1 when k > 3n/(p log n). Since the last term in (7) is less than (nP + l)/(n l 1)n < 3nP 2 it follows that log 16 + o(1) < h(r) < log 16 + 3nP'2 + o(l). (8) We conclude from equation (6) and the first inequality in (8) that p > 1, and from (6) and the second inequality in (8) that p - 1 as n + oo. It follows, again from (8), that h(r) = log 16 + o(1). From (6) we now deduce that rn nP = n[h(r) - n-P-l], that is, r = In[log 16 + o(l)]1)/n, as was to be proved. LEMMA 7. If E > 0 and n is sufficiently large, the annulus 1- X _ E< jzI < l+_e - e n n contains no zeros of Pn(z)/(l - z). We write Pn(z)/(l - z) = ao + alz + a2z2 + + a2n.2z2n-2 where an-k = an-2+k = + + + (k = 1, 2., n). n n-l k We now apply a theorem of Egervary [2, page 81] which, slightly specialized for our purpose, reads as follows: If am > 0 (O < m < 2n - 2) and if, for some p > 1 and for m = O, 1,..., n - 2, n, n + 1,..., 2n - 2, the condition am_l - (p + p-l) am + am+l > 0 (9) (with the notation a_1 = a2nl = O) is satisfied, then the polynomial >'m am zm has no zeros in the annulus p-l < z I K P. To establish a value of p for which our polynomial satisfies the condition (9), we write p = 1 + b. Applied to the case mn = n k, condition (9) becomes, after some elementary computations, b2 1 1 + b k(k - l)[l/n + l/(n-l) +.:o + 1/k] (= 2' "~ n 6

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN In order to find an upper bound for the denominator on the right side of (10), we note that this denominator has the value k(k 1)I + n 1 +. k 1 1 + k - 1 < k2 log(n/k) + n. The maximum of this, for k > 1, is n + n2/2e. It follows that condition (10) is satisfied provided b2 < 1j(n + n2/2e); that is, condition (9) is satisfied, for large n, if b = (2 e - e)/n. The following proposition summarizes the results of this section: THEOREM 1. The polynomial Pn has a zero at z = 1, and it has no other positive zero and no other zero on the unit circle. It has no negative zeros if n is odd; if n is even, it has two negative zeros which are given asymptotically by the formula z = -1 + (log n + log log 16 + 0 Each of the n - 1 sectors (2k - l)r/n < arg z < (2k + l)ic/n (k = 1, 2,..., n - 1) contains exactly two zeros of Pn. For c > 0 and n sufficiently large, each of the zeros of Pn(z)/(l - z) lies in one of the annuli 1 - (4 + E) log n < zl < 1 - - E c n n 1 + (4 + c) log n > jz > I + 2e c n n III. THE MAXIMUM MODULUS OF THE FEJER POLYNOMIALS We denote by Mn the maximum modulus of Pn on the unit circle C. THEOREM 2. As n + co, t Mn + 2 sin t dt = 35704.. From (1) we note that Mn = 2 max sin(k l/2)Q O < O < t k=l k 7

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The sum on the right is a partial sum of the Fourier series of the function g(@) = sin(k - 1/2) k=l k which satisfies the relations g(G + 4t) = g(G), g(-@) = -g(G) and g(2t - Q) = g(9). We write () 2 sin(k - 1/2)1 0 sin(k - 1/2) 2k - 1 k(2k 1) k=l k=l and we observe that the first sum on the right is the well-known Fourier series of the function which takes the values c/2 and -T/2 in alternate intervals of length 2t, while the second series on the right converges uniformly for all real @. Our result now follows from the theory of the Gibbs phenomenon. For a concise statement, we refer the reader to Zygmund [12, Sections 8.5, 8.51]; interesting graphs and an excellent historical account will be found in [1, Chapter IX]. By a more detailed investigation, we have been able to show that the constant of Theorem 2 is actually an upper bound, and hence the least upper bound, for the maxima Mn (n = 1, 2,.. ). This investigation is based on the representation C(n, Q) = A(n, o) cos (0/2) - B(n, 0) sin(0/2), where n n A(n, 0) = k, B(n, ) o k k=l k=l The principal properties of these trigonometric polynomials are treated in [10, Section VI, Problems 23-28]. The computations involved are so tedious that their publication is not justified. REFERENCES 1. H. S. Carslaw, Introduction to the Theory of Fourier s Series and Integrals, 3rd edition, The Macmillanf Company, London, 1930. 2. E. Egervary, "On a Generalisation of a Theorem of Kakeya", Acta Szeged, 5, 78-82 (1930-1932). 3. P. Erdoss, F. Herzog, and G. Piranian, T'On Taylor Series of Functions Regular in Gaier Regions", Arch. Math., 5, 39-52 (1954). 4., "Sets of Divergence of Taylor Series and of Trigonometric Series, Math. Scand., 2, 262-266 (1954). 8

ENGINEERING RESEARCH INSTITUTE ~ UNIVERSITY OF MICHIGAN 5. P. Erdos and P. Turan, "On the Distribution of Roots of Polynomials, Ann. of Math. (2), 51, 105-119 (1950). 6. L. Fejer, "Lebesguesche Konstanten und divergente Fourierreihen", J. Reine Angew. Math., 138, 22-53 (1910). 7., "Sur les singularites de la serie de Fourier des fonctions continues", Ann. Sci. Ecole Norm. Sup. (3), 28, 63-103 (1911). 8. D. Gaier, "'Uber die Summierbarkeit beschrankter und stetiger Potenzreihen an der Konvergenzgrenze", Math. Z., 56, 326-334 (1952). 9. R. Jentzsch, "Untersuchungen zur Theorie der Folgen analytischer Funktionen"!, Inaugural Dissertation, Berlin, 1914. 10. G. Polya and G. Szego, Aufgaben und Lehrs.tze aus der Analysis, Vol II, Springer, Berlin, 1925. 11. G. Szego, "Uber die Nullstellen von Polynomen, die in einem Kreise gleichmassig konvergieren"', Sitzungsberichte Berlin Math. Ges., 59-64 (1922). 12. A. Zyygund, Trigonometrical Series, Warsaw, 1935. Michigan State College University of Michigan