SUMS OF MONOMIALS WITH LARGE MAHLER MEASURE. K.-K. Stephen Choi and Tamás Erdélyi
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1 SUMS OF MONOMIALS WITH LARGE MAHLER MEASURE K.-K. Stephen Choi and Tamás Erdélyi Dedicated to Richard Askey on the occasion of his 80th birthday Abstract. For n 1 let A n := { P : P(z) = n j=1 } z k j : 0 k 1 < k < < k n,k j Z, that is, A n is the collection of all sums of n distinct monomials. If α < β are real numbers then the Mahler measure M 0 (Q,[α,β]) is defined for bounded measurable functions Q on [α,β] as ( 1 β ) M 0 (Q,[α,β]) := exp log Q(e it ) dt. β α α Let I := [α,β]. In this paper we examine the quantities with 0 < I := β α π. L 0 M 0 (P,I) n (I) := sup and L 0 (I) := liminf P A n n n L0 n (I) 1. Introduction The large sieve of number theory [M-84] asserts that if n P(z) = a k z k k= n is a trigonometric polynomial of degree at most n, 0 t 1 < t < < t m π, and δ := min{t 1 t 0,t t 1,...,t m t m 1 }, t 0 := t m /pi, Key words and phrases. large sieve inequalities, Mahler measure, L 1 norm, constrained coefficients, Fekete polynomials, Littlewood polynomials, sums of monomials. 000 Mathematics Subject Classifications. 11C08, 41A17 1 Typeset by AMS-TEX
2 then m j=1 P ( e it j ) ( n π +δ 1) π 0 P ( e it ) dt. There are numerous extensions of this to L p norm (or involving ψ ( P ( e it ) p ), where ψ is a convex function), p > 0, and even to subarcs. See [LMN-87] and [GLN-01]. There are versions of this that estimate Riemann sums, for example, with t 0 := t m π, m ( ) P e it j (tj t j 1 ) C j=1 π 0 P ( e it ) dt, with a constant C independent of n, P, and {t 1,t,...,t m }. These are often called forward Marcinkiewicz-Zygmund inequalities. Converse Marcinkiewicz-Zygmund inequalities provide estimates for the integrals above in terms of the sums on the left-hand side, see [L-98], [MR-99], [ZZ-95], [KL-04]. A particularly interesting case is that of the L 0 norm. A result in [EL-07] asserts that if {z 1,z,...,z n } are the n-th roots of unity, and P is a polynomial of degree at most n, then (1.1) n P (z j ) 1/n M 0 (P), j=1 where ( 1 π ) M 0 (P) := exp log P(e it ) dt π 0 is the Mahler measure of P. In [EL-07] we were focusing on showing that methods of subharmonic function theory provide a simple and direct way to generalize previous results. We also extended (1.1) to points other than the roots of unity and exponentials of logarithmic potentials of the form ( ) P(z) = c exp log z t dν(t), where c 0 and ν is a positive Borel measure of compact support with ν(c) 0. Inequalities for exponentials of logarithmic potentials and generalized polynomials were studied by several authors, see [E-91], [E-9], [EN-9], ELS-94], [EMN-9], [BE-95], and [EL-07], for instance. Let α < β be real numbers. The Mahler measure M 0 (Q,[α,β]) is defined for bounded measurable functions Q defined on [α,β] as ( ) 1 β M 0 (Q,[α,β]) := exp log Q(e it ) dt. β α It is well known (see [HLP-5], for instance) that M 0 (Q,[α,β]) = lim p 0+ M p(q,[α,β]), α
3 where ( 1 M p (Q,[α,β]) := β α β α Q(e it ) p dt) 1/p, p > 0. It is a simple consequence of the Jensen formula that for every polynomial of the form M 0 (Q) := M 0 (Q,[0,π])= c Q(z) = c n max{1, z k } k=1 n (z z k ), c,z k C. k=1 Finding polynomials with suitably restricted coefficients and maximal Mahler measure has interested many authors. The classes L n := { p : p(z) = of Littlewood polynomials and the classes K n := { p : p(z) = } n a k z k, a k { 1,1} } n a k z k, a k C, a k = 1 of unimodular polynomials are two of the most important classes considered. Beller and Newman [BN-73] constructed unimodular polynomials of degree n whose Mahler measure is at least n c/logn with an absolute constant c > 0. For a prime number p the p-th Fekete polynomial is defined as p 1 f p (z) := k=1 ( ) k z k, p where ( ) k 1, if x k (modp) has a nonzero solution, = 0, if p divides k, p 1, otherwise is the usual Legendre symbol. Since f p has constant coefficient 0, it is not a Littlewood polynomial, but g p defined by g p (z) := f p (z)/z is a Littlewood polynomial, and has the same Mahler measure as f p. Fekete polynomials are examined in detail in [B-0]. In [EL-07] we proved the following result. 3
4 Theorem 1.1. For every ε > 0 there is a constant c ε such that ( ) 1 p M 0 (f p,[0,π]) ε for all primes p c ε. One of the key lemmas in the proof the above theorem formulates a remarkable property of the Fekete polynomials. A simple proof is given in [B-0, pp ]. Lemma 1. (Gauss). We have and f p (1) = 0, where f p (z j p ) = ε p ( ) j p 1/, p ( ) πi z p := exp p is the first p-th root of unity, and ε p { 1,1, i,i}. j = 1,,...,p 1, The choice of ε p is more subtle. This is also a result of Gauss, see [H-8]. Lemma 1.* (Gauss). In Lemma 1. we have { 1, if p 1 (mod4) ε p = i, if p 3 (mod4). The distribution of the zeros of Littlewood polynomials plays a key role in the study of the Mahler measure of Littlewood polynomials. There are many papers on the distribution of zeros of polynomials with constraints on their coefficients, see [ET-50], [BE-95], [BE-97], [B-97], [B-0], [BEK-99], [E-08], and P-11], for example. Results of this variety have been exploited in [EL-07] to obtain Theorem 1.1. From Jensen s inequality, M 0 (f p,[0,π]) M (f p,[0,π])= p 1. However, as it is observed in [EL-07], 1/ ε in Theorem 1.1 cannot be replaced by 1 ε. Indeed if p is prime of the form p = 4m+1, then the polynomial f p is self-reciprocal, that is, z p f p (1/z) = f p (z), and hence (p 3)/ f p (e it ) = e ipt a k cos((k +1)t), a k {,}. A result of Littlewood [L-66] implies that M 0 (f p,[0,π]) 1 π f p (e it ) dt = 1 π 0 π 4 π 0 f p (e it ) dt (1 ε) p 1,
5 for some absolute constant ε > 0. A similar argument shows that the same estimate holds when p is a prime of the form p = 4m+3. It is an interesting open question whether or not there is a sequence of Littlewood polynomials (f n ) such that for all ε > 0 and sufficiently large n N ε. M 0 (f n,[0,π]) (1 ε) n In [E-11] we proved the following sieve type lower bound for the Mahler measure of polynomials of degree at most n on subarcs of the unit circle. Theorem 1.3. Let ω 1 < ω ω 1 +π, ω 1 t 0 < t 1 < < t m ω, t 1 := ω 1 (t 0 ω 1 ), t m+1 := ω +(ω t m ), δ := max{t 0 t 1,t 1 t 0,...,t m+1 t m } 1 sin ω ω 1 There is an absolute constant c 1 > 0 such that. m j=0 t j+1 t j 1 log P(e it j ) ω ω 1 log P(e it ) dt+c 1 E(N,δ,ω 1,ω ) for every polynomial P of the form where P(z) = N b j z j, b j C, b 0 b N 0, j=0 E(N,δ,ω 1,ω ) := (ω ω 1 )Nδ +Nδ log(1/δ)+ ( N logr δlog(1/δ)+ δ ω ω 1 ) and R := b 0 b N 1/ max t R P(eit ). Observe that R appearing in the above theorem can be easily estimated by R b 0 b N 1/ ( b 0 + b b N ). As a reasonably straightforward consequence of our sieve-type inequality above, the lower bound for the Mahler measure of Fekete polynomials below follows. 5
6 Theorem 1.4. There is an absolute constant c > 0 such that M 0 (f p,[α,β]) c p for all prime numbers p and for all α,β R such that 4π p (logp)3/ p 1/ β α π. It looks plausible that Theorem. holds whenever 4π/p β α π, but we do not seem to be able to handle the case 4π/p β α (logp) 3/ p 1/ in this paper. In [E-13] we proved the following. Theorem 1.5. There are constants c 3 (q,ε) depending only on q > 0 and ε > 0 such that whenever p 1/+ε β α π. For n 1 let M 0 (f p,[α,β]) M q (f p,[α,β]) c 3 (q,ε) p, A n := {P : P(z) = n z k j : 0 k 1 < k < < k n,k j Z}, j=1 that is, A n is the collection of all sums of n distinct monomials. Let I := [α,β]. We define L 1 M 1 (P,[0,π]) n(i) := sup and L 1 (I) := liminf P A n n n L1 n(i) Σ(I) := sup n N L 1 n(i) In the case of I = [0,π] the problem of calculating Σ(I) appears in a paper of Bourgain [B-93]. Deciding whether Σ([0,π]) < 1 or Σ([0,π]) = 1 would be a major step toward confirming or disproving other important conjectures. Karatsuba [K-98] observed that Σ([0,π]) 1/ Indeed, taking, for instance, it is easy to see that n 1 P n (z) = z k, n = 1,,..., (1.) M 4 (P n,[0,π]) 4 = n(n 1)+n, and as n = M (P n,[0,π]) M 1 (P n,[0,π]) /3 M 4 (P n,[0,π]) 4/3, 6
7 the inequality (1.3) M 1 (P n,[0,π]) n n 1 follows. Similarly, if S n := {a 1 < a < < a n } is a Sidon set (that is, S n is a subset of integers such that no integer has two essentially distinct representations as the sum of two elements of S n ), then the polynomials P n (z) = a S n z a, n = 1,,..., satisfy (1.) and (1.3). Improving Karatsuba s result, by using a probabilistic method Aistleitner [A-13] proved that Σ([0,π]) π/ We note that P. Borwein and Lockhart [BL-01] investigated the asymptotic behavior of the mean value of normalized L p norms of Littlewood polynomials for arbitrary p > 0. Using the Lindeberg Central Limit Theorem and dominated convergence, they proved that lim n 1 n+1 (M p (f,[0,π]))p n p/ f L n n ( = Γ 1+ p ). An analogue of this result does not seem to be known for p = 0 (the Mahler measure). However, it follows simply from the p = 1 case of the the result in [BL-01] quoted above that Σ([0,π]) π/ Morover, this can be achieved by taking the sum of approximately half of the monomials of {x 0,x 1,...,x n } and letting n tend to. Littlewood asked how small the ratio M 4 (Q,[0,π]) M (Q,[0,π]) = M 4(Q,[0,π]) n+1 can be for polynomials Q L n as the degree tends to infinity. As it is remarked in [JKS- 13], since 1988, the least known asymptotic value of this ratio has been 4 7/6 which was conjectured to be minimum. In [JKS-13] this conjecture was disproved this by showing that there is a sequence (Q nk ) of Littlewood polynomials Q nk L nk, derived from the Fekete polynomials, for which M 4 (Q nk,[0,π]) lim < 4 /19. k nk +1 Hence, it follows by a simple application of Hölder s inequality that lim inf k M 1 (Q nk,[0,π]) nk +1 In this paper we examine the size of 19/ > L 0 M 0 (P,[α,β]) n (I) := sup and L 0 (I) := liminf P A n n n L0 n ([α,β]) with 0 < I := β α π. 7
8 New Results Theorem.1. There are polynomials S n A n P N with N := n+o(n) such that M 0 (S n,[0,π]) ( ) 1 n, +o(1) n N. That is, L 0 ([0,π]) 1. Theorem.. There are polynomials S n A n P N with N := n + o(n) an absolute constant c 4 > 0, and a constant c 5 (ε) > 0 depending only on ε > 0 such that for all n N and α,β R such that M 0 (S n,[α,β]) c 4 n 4π n (logn)3/ n 1/ β α π, while M 1 (S n,[α,β]) c 5 (ε) n for all n N and α,β R such that n 1/+ε β α π. Theorem.3. There are polynomials P n L n such that M 0 (P n,[0,π]) ( ) 1 n, +o(1) n N. Theorem.4. There are polynomials P n L n, and absolute constant c 4 > 0, and a constant c 5 (ε) > 0 depending only on ε > 0 such that for all n N and α,β R such that M 0 (P n,[α,β]) c 4 n 4π n (logn)3/ n 1/ β α π, while M 1 (P n,[α,β]) c 5 (ε) n for all n N and α,β R such that n 1/+ε β α π. 8
9 3. Lemmas Theorem 1.3 will be used as a key lemma in the proof of Theorem.. Theorem 1.5 will play a key role in the proof of Theorem.3. In addition to these we need the following lemmas, We present the short proof of each lemma right after the lemma. The new results stated in Section will be proved in Section 4. Lemma 3.1. We have ( p 1 Q(z k p ) ) 1/p N/p M 0 (Q) for all polynomials Q of degree N with complex coefficients. Proof. Let Note that while Q(z) = c N (z z k ), c,z k C. k=1 a p k 1 1/p ( a k p ) 1/p = 1/p a k, a k 1, a p k 1 1/p 1/p, a k < 1. Multiplying these inequalities for k = 1,,...,N, we obtain ( p 1 ) 1/p ( N ) 1/p Q(zp k ) = a p k 1 N N/p max{ a k,1} N/p M 0 (Q). For r {1,3} Let π(x,4,r) denote the number of prime numbers of the form p = 4k+r in [1,x]. A well known result states that π((x,4,1)) π(x,4,3) lim = lim x x/logx x x/logx = 1. An immediate consequence of this is the following. Lemma 3.. For every n 5 there is a prime of the form p = 4k +3 and there integers m > 0 and r {0,1} such that n = m+q +1+r, m = p 1, q = o(n). Associated with a prime number p let m := (p 1)/ and let 0 < q < m be an integer. 0 < a m < a m 1 < < a 1 m denote the quadratic residues and 0 < b m < b m 1 < < b 1 m denote the quadratic non-residues modulo p. Let G q (z) := q z b j +z b j. j=1 9
10 Note that my the Polya Vinogradov inequality b q q +o(q). For an integer µ N we also define µ 1 D µ (z) := z j. Using a possible decomposition given by Lemma 3.3, for any integer n 5 we define S A n P N, N := n+o(n) by (3.1) S n (z) := 1 (f p(z)+d p 1 (z)+1)+z p (G q (z)+rz p+b q+1 Combining (3.1), Lemma 3. we can easily deduce the following. j=0 Lemma 3.3. Let S n A n be defined by (3.1). We have S n (z j p ) p 1, j = 1,,...,p 1, where z p := exp(πi/p) is the first p-th root of unity. We also have S n (1) = n. Proof. When j = 0 we have S n (zp 0 ) = n, wjich is better than what is stated in the lemma. Now let j {1,,...,p 1}. Lemma 1.* implies that Im(f p (zp)) j = p. Also D p (zp) j = 0, 1+zp jp (D q (zp) j R, and Im(rzp j(p+q+1) ) = 1. Hence S n (z j p ) Im(S n(z j p )) p 1, j = 1,,...,p 1, k p. Lemma 3.4. There is always a prime number in the interval [n,n n 3/4 ] for all sufficiently large n N. Lemma 3.4 has the following straightforward consequence. Lemma 3.5. For every n 5 there are primes p and q and r N such that n = p+q +r q = O(n /3 ), and r = O(n 4/9 ). µ 1 D µ (z) := z j. Using a possible decomposition given by Lemma.3, for any integer n 5 we define (3.) P n := 1+f p (z)+z p f q (z)+z p+q D r L n The following property of the Fekete polynomials f p is due to Montgomery [M-80]. Lemma 3.6. There are absolute constants c 1 > 0 and c > 0 such that j=0 c 1 ploglogp max t R f p(e it ) c plogp. Combining (3.), Lemma 3.5, and the upper bound of Lemma 3.6 we can easily deduce the following. Lemma 3.7. Let P n L n be defined by (3.). We have P n (z j p) = (1+o(1)) p, j = 1,,...,p 1, and P n (1) = 1+r 1, where z p := exp(πi/p) is the first p-th root of unity. 10
11 4. Proof of the Theorems Proof of Theorem.1. The theorem follows from Lemmas 3.1 and 3.3 in a straightforward fashion. Let S n A n P N be defined by (3.1). 0 t 0 < t 1 < t < < t p 1 π be chosen so that z k p = e it k, k = 0,1,...,p 1. Using Lemma 3.3 and then applying Lemmas 3.1 with Q := S n A n P N, and recalling that N = p+o(p), we obtain and hence ((1+o(1)) n/) (p 1)/p and the theorem follows. ( p 1 M 0 (S n ) 1+o(1) n, S n (e it j ) ) 1/p N/p M 0 (S n ), Proof of Theorem.. Let S n A n P N be defined by (3.1). Note that (M 0 (f,[α,β])) β α = (M 0 (f,[α,γ])) γ α (M 0 (f,[γ,β])) β γ, for all α < γ < β α+π and for all functions f continuous on [α,β]. Hence, to prove the lower bound of the theorem, without loss of generality we may assume that β α π. The lower bound of the theorem now follows from Theorem.1 and Lemmas 3.3 and 3.5 in a straightforward fashion. Let (4.1) ω 1 := α t 0 < t 1 < t < < t m β =: ω be chosen so that e it j, j = 0,1,...,m, are exactly the primitive p-th roots of unity lying on the arc connecting e iα and e iβ on the unit circle counterclockwise. The assumption on n together with p = n+o(n) guarantees that the value of δ defined in Theorem.1 is at most 4π/p. Observe also that R n. Applying Theorem.1 with P := S n A n P N and (4.1), we obtain m j=0 t j+1 t j 1 where the assumption log S n (e it j ) β (logn) 3/ n 1/ α log S n (e it ) dt+c 1 E(N,4π/p,α,β), β α π together with p = n+o(n), N = n+o(n), and R n implies that ( (β α)n E(N,4π/p,α,β) c 6 + N logp p p + ( logp N logn + p c 7 (β α) 11 )) 1 p (β α)
12 with absolute constants c 6 > 0 and c 7 > 0. Therefore M 0 (S n,[α,β]) exp( c 1 c 7 )(1+o(1)) n/, and the lower bound of the theorem follows. Now we prove the upper bound of the theorem. Using the well known estimate M 1 (D µ,[α,β]) 1 β α M 1(D µ,[0,π]) c 8logµ β α with an absolute constant c 8 > 0 and applying Theorem 1.5, we obtain that there is a constant c 9 (ε) > 0 depending only on ε > 0 such that M 1 (S n,[α,β]) =M 1 ( 1 (f p +D p )+ 1 zp (f q +D q )+z p+q D r : [α,β]) 1 (M 1(f p,[α,β])+m 1 (f q,[α,β])) + 1 M 1(D p,[α,β])+ 1 M 1(D q,[α,β])+m 1 (D q,[α,β]) c 3 (ε) p+c 3 ε) q + 3c 8log(n) β α c 9 (ε) n whenever (n+o(n)) 1/+ε β α π. Proof of Theorem.3. The theorem follows from Lemmas 3.4 and 3.7 in a straightforward fashion. Let SP n L n be defined by (3.). let 0 t 0 < t 1 < t < < t p 1 π be chosen so that z k p eit k, k = 0,1,...,p 1. Using Lemma 3.7 and then applying Lemmas 3.4 with Q := P n L n, and recalling that n = p+o(p), we obtain ((1+o(1)) n) (p 1)/p and the theorem follows. M 0 (S n ) ( p 1 P n (e it j ) ) 1/p n/p M 0 (P n ), ( ) 1 +o(1) ) n, Proof of Theorem.4. The proof is quite similar to that of Theorem.4 by replacing Lemma 3.3 with Lemma 3.7. We leave the straightforward modifications to the reader. (?) 1
13 References A-13. Ch. Aistleitner, On a problem of Bourgain concerning the L 1 -norm of exponential sums, Math. Z. (to appear). BN-73. E. Beller and D.J. Newman, An extremal problem for the geometric mean of polynomials, Proc. Amer. Math. Soc. 39 (1973), B-0. P. Borwein, Computational Excursions in Analysis and Number Theory, Springer, New York, 00. BE-95. P. Borwein and T. Erdélyi, Polynomials and Polynomial Inequalities, Springer, New York, BE-97. P. Borwein and T. Erdélyi, On the zeros of polynomials with restricted coefficients, Illinois J. Math. 41 (1997), BEK-99. P. Borwein, T. Erdélyi, and G. Kós, Littlewood-type problems on [0, 1], Proc. London Math. Soc. (3) 79 (1999), 46. BL-01. P. Borwein and R. Lockhart, The expected L p norm of random polynomials, Proc. Amer. Math. Soc. (001), B-93. J. Bourgain, On the sprectal type of Ornstein s class one transformations, Israel J. Math. 84 (1-) (1993), B-97. D. Boyd, On a problem of Byrnes concerning polynomials with Restricted Coefficients, Math. Comput. 66 (1997), E-91. T. Erdélyi, Bernstein and Markov type theorems for generalized nonnegative polynomials, Canad. J. Math. 43 (1991), E-9. T. Erdélyi, Remez-type inequalities on the size of generalized polynomials /a,, J. London Math. Soc. 45 (199), 55i 64. E-08. T. Erdélyi, An improvement of the Erdős-Turán theorem on the zero distribution of polynomials, C.R. Acad. Sci. Paris Sér. I Math. 346 (008), E-11. T. Erdélyi, Sieve-type lower bounds for the Mahler measure of polynomials on subarcs, Computational Methods and Function Theory 11 (011 (1)), E-13. T. Erdélyi, Upper bounds for the L q norm of Fekete polynomials on subarcs, Acta Arith. 153 (1), ELS-94. T. Erdélyi, X. Li, and E.B. Saff, Remez and Nikolskii type inequalities for logarithmic potentials, SIAM J. Math. Anal. 5 (1994), EL-07. T. Erdélyi and D. Lubinsky, Large sieve inequalities via subharmonic methods and the Mahler measure of Fekete polynomials, Canad. J. Math. 59 (007), EMN-9. T. Erdélyi, A. Máté, and P. Nevai, Inequalities for generalized nonnegative polynomials, Constr. Approx. 8 (199), EN-9. T. Erdélyi and P. Nevai, Generalized Jacobi weights, Christoffel functions, and zeros of orthogonal polynomials, J. Approx. Theory 69 (199 no. ), ET-50. P. Erdős and P. Turán, On the distribution of roots of polynomials, Ann. Math. (1950), JKSch. J. Jedwab, D.J. Katz, and K.-U. Schmidt, Littlewood polynomials with small L 4 norm, Adv. Math. 41 (013), GLN-01. L. Golinskii, D.S. Lubinsky, and P. Nevai, Large sieve estimates on arcs of a circle, J. Number Theory 91 (001),
14 HLP-5. G.H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, Cambridge Univ. Press, London, 195. H-8. L.K. Hua, Introduction to Number Theory, Springer-Verlag, New York-Berlin, yr 198. IP-84. H. Iwaniec and J. Pintz, Primes in Short Intervals., Monatsh. Math. 98, (1984), K-98. A.A. Karatsuba, An e stimate for the L 1 -norm of an exponential sum, Mat. Zametki 64 (3) (1998), KL-04. C.K. Kobindarajah and D.S. Lubinsky, Marcinkiewicz-Zygmund inequalities for all arcs of the circle, in Advances in Constructive Approximation, Vanderbilt 003, 55 64, Mod. Methods Math., Nashboro Press, Brentwood, TN, 004, L-66. J.E. Littlewood, The real zeros and value distributions of real trigonometric polynomials, J. London Math. Soc. 41 (1966), L-98. D.S. Lubinsky, Marcinkiewicz-Zygmund inequalities, in Methods and Results, Recent Progress in Inequalities (ed. G.V. Milovanovic et al.), Kluwer Academic Publishers, Dordrecht, 1998, LMN-87. D.S. Lubinsky, A. Máté, and P. Nevai, Quadrature sums involving pth powers of polynomials, SIAM J. Math. Anal. 18 (1987), MR-99. G. Mastroianni and M.G. Russo, Weighted Marcinkiewicz inequalities and boundedness of the Lagrange operator, in Mathematical Analysis and Applications (ed. T.M. Rassias), Hadronic Press, Palm Harbor, 1999, M-84. H.L. Montgomery, The analytic principle of the large sieve, Bull. Amer. Math. Soc. 84 (1978), P- 11. I.E. Pritsker, Distribution of algebraic numbers, J. Reine Angew. Math. 657 (011), ZZ-95. L. Zhong and L.Y. Zhu, The Marcinkiewicz-Zygmund inequality on a simple arc, J. Approx. Theory 83 (1995), Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6 (K.-K. S. Choi) Department of Mathematics, Texas A&M University, College Station, Texas (T. Erdélyi) address: terdelyi@math.tamu.edu 14
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