Borwein, P., and T. Erd'elyi. [1995] "Polynomials and Polynomial Inequalities", SpringerVerlag, New York.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to note this, otherwise the work of this paper would be without object. In this paper we answer Sa#ari s Problem a#rmatively, namely we show that (1.7) or equivalently (1.8) is true for every ultraflat sequence (Pn ) n . An interesting related result to Kahane s breakthrough is given in [Be]. For an account of some of the work done till the mid 1960 s, see Littlewood s book [Li2] and [QS2] in general, not necessarily those produced by Kahane in his paper [Ka] With trivial modifications our results remain valid even if we study ultraflat sequences (Pnk ) of unimodular polynomials ....

.... 1. As a corol lary (consider e int Tn (t) if Tn (t) c k e c k n (t) #M for all t R, then it satisfies n (t) #Me nIm(t) for all t C. The main tool to prove Theorem 2.4 is the following well known Jensen s Formula. For its proof, see, for example, E.10 c] of Section 4. 2 in [BE]. Lemma 3.2 (Jensen s Formula) Suppose h is a nonnegative integer and f (z) k=h c k z ,c h #=0, is analytic on a disk D(0,R # ) with some R # R. Suppose that the zeros of f in D(0,R) 0 are a 1 ,a 2 , a m , where each zero is listed as many times as its multiplicity. Then c h ....

P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

....# # # # # 1,1] # # # # # # # # # # 1 ,a 2 , a n #C R ,and # # # # # # # # # # 1 ,a 2 , a n #C R . The spaces # # # # # # # j=1 a j ) 2) # # # # # # # 1 ,a 2 , a n #C Rand # # # # # # # # # # have been studied in [2] and [3], and the sharp Bernstein Szego type inequalities f # (# 0 ) Bn (# 0 ) f(# 0 ) B(# 0 ) K ,# 0 #K n (a 1 ,a 2 , a 2n ;K)with (a 1 ,a 2 , a 2n )#C R, Im(a j ) 0,j=1,2, 2n 0 )f # (x 0 ) Bn (x 0 ) 1,1] x 0 #( 1,1) n (a 1 ,a 2 , a n ; 1,1] with have ....

....1 ,a 2 , a n ;R) where B n (x) x ,x#R. We remark that equality holds in Theorem 5 if and only if x 0 is a maximum point of f (i.e. f(x 0 ) #f#R)orf is a Chebyshev polynomial for the space n (a 1 ,a 2 , a n ;R) which can be explicitly expressed by using the results of [2] and [3]. Note that Bernstein s classical inequalities are contained in Theorem 1, 2, and 3 as limiting cases, by taking in Theorems 1 and 3 so that lim for each j =1,2, n, and by taking in Theorem 2 so that a for each j =1,2, n. Further results can be obtained as limiting ....

. Borwein, P. & Erdelyi, T., Polynomials and Polynomial Inequalities, Springer-Verlag (to appear).

....conditions. For example, D. Boyd [Bo] shows that real polynomials with 1 coe cients have at most c log n= log log n zeros at 1. I. Schur [Sc] G. Szeg [Sz] and P. Erds and P. Turn [ET] establish that n degree polynomials with coe cients 0; 1 have at most c n log n real zeros. See also [BE]. P. Borwein, T. Erdlyi and G. Ks have since shown in 1999 that the optimal bound is c n [BEK] Many other problems come to mind concerning real zeros of real polynomials, some of which may be di cult. For example, nd all k for which there is a polynomial of degree n having exactly k ....

P. Borwein, T. Erdlyi, Polynomials and polynomial inequalities, Springer-Verlag 1995.

....the basis of considerable computation, in [3] we speculate that the best possible constant in Newman s inequality is 4. We remark that an incorrect argument exists in the literature claiming that the best possible constant in Newman s inequality is at least 4 # 15 = 7.87 . It is proved in [2] that under a growth condition, which is essential, in Newman s inequality can be replaced by [0,1] More precisely, the following result holds. Theorem 1.5 (Newman s Inequality Without the Factor x) Let #: # j ) # j=0 be a sequence of distinct real numbers with # 0 =0and # j j for ....

....to Newman [13] with the constant 11 rather than 8.29. The L# [0, 1] version of the above inequalities is proved in [9] A slightly simplified version of Newman s proof in the L# [0, 1] case as well as the above L p [0, 1] inequalities with the constant 12 rather than 8. 29 are given in both [1] and [2]. Here we will reduce the proof of the above L p [0, 1] inequalities to Newman s inequality given by Theorem 1.3, by recalling, as we have already remarked, that the constant 11 in Theorem 1.3 can be replaced by 8.29. 3. New Results: Newman s Inequality in L p [a, b] for [a, b] #) We ....

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, N.Y., 1996.

....present note. In fact, this will be accomplished in the context of generalized polynomials, and other norms on the unit circle will be discussed as well. It should be noted that, for m = o( # n) the constant in (1. 1) was substantially improved by Boyd [3] see also Borwein [1] and Borwein Erdelyi [2] for some recent developments in this area. # 1991 Mathematics Subject Classification: 30C10, 11C08. Written during the author s visit at Kent State University, Department of Mathematics and Computer Science, Kent, OH 44242 0001, U.S.A. 2 Results For # = # 1 , # n ) denote # n ....

....# z # # # # n (#) # # # # # # # # . 2.3) Moreover, the equality in (2.3) holds if and only if n is a solution of the minimization problem (2. 2) Functions p(z) j=1 z a are usually called generalized polynomials of degree # j = d n (#) # ) see [2]) Using Theorem 2.1, we can easily derive the following Corollary 2.2 Let p 1 (z) p m (z) be generalized complex polynomials such that p(z) p j (z) where # = # 1 , # n ) and n are arbitrary. Then . p # n (#) #p# # , 2.4) with ....

P. B. Borwein and T. Erd elyi, Polynomials and Polynomial Inequalities (SpringerVerlag, New York, 1995).

....1.1. We need two lemmas. The rst is the classical Descartes rule of signs. Lemma 5.1. The number of zeros on (0; 1) of any polynomial n X i=0 a i t i is at most the number of sign changes in the sequence of coecients (a i ) n i=0 after zero terms are discarded. Proof. See for example [1] or [10] The next lemma is of a computational nature. Lemma 5.2. Let m 1. a) For every a 2 [ 1; 1) we have 1 Z 1 a s 2m s t p (1 s) s a)ds = 2m X j=0 B j (a)t j t 2m 1 ; a t 1; 5.1) with coecients B j (a) satisfying B j (a) 0; j = 0; 1; 2m; a 2 [ 1; 1) ....

P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995. MR 97e:41001

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, N.Y. (to appear).

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P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, SpringerVerlag, New York (to appear).

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities (to appear).

....the more restricted class the analogous Erdos conjecture is unsettled to this date. It is a common belief that the analogous Erdos conjecture for is true, and consequently there is no ultraflat sequence of polynomials Pn n . An interesting result related to Kahane s breakthrough is given in [Be]. For an account of some of the work done till the mid 1960 s, see Littlewood s book [Li2] and [QS2] Let (# n ) be a sequence of positive numbers tending to 0. Let the sequence (Pn ) of unimodular polynomials Pn n be (# n ) ultraflat. We write (1.3) Pn (e ) R n (t)e i#n (t) R n (t) P n( ....

P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

....so that . Then, with c : 1 sin t# L2 (R) A [#, 1] of Lebesque measure at least # 0. 3. Lemmas Our first lemma shows that C(#,#) in the Remez type inequality is related to a much simpler (Chebyshev type) extremal problem. This is proved in both [1] and [2]. Lemma 3.1. Suppose 0=# 0 # 1 # 2 ,##(0, 1) and##(0, 1 #) p(x) #1 ## =sup . Our key lemma is the following. Lemma 3.2. Suppose 0=# 0 # 1 # 2 and 1 # k #. Given (0, 1) letN#Nbe chosen so that . 1 sin t# L2 (R) ....

....as the Fourier transform of some function f L 2 [ #, #] We will need it in the proof of Lemma 3.3. Theorem (Paley Wiener) Let # (0, #) Then f L 2 (R) if and only if there exists an f L 2 [ #, #] so that dt . The following comparison theorem for Muntz polynomials is proved in [2]. We will need it in the proof of Theorem 2.3. Lemma 3.4. Let #: # k ) # k=0 and #: # k ) # k=0 be increasing sequences of nonnegative real numbers with # 0 =0,# 0 =0,and# k## k for each k.Let0 a b. max #max . 4. Proofs Proof of Lemma 3.3. By the Paley Wiener ....

P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, SpringerVerlag, New York, N.Y., 1995.

....measure, due to Clarkson and Erdos [12] and Schwartz [24] whose works include the result that if then every function belonging to the uniform closure of M(#) on [a, b] can be extended analytically throughout the b . There are many generalizations and variations of Muntz s Theorem [1, 4, 5, 6, 7, 8, 9, 17, 19, 24, 26, 28, 29]. There are also still many open problems. The proper generalizations to many variables are still open. In Section 6 of this paper we show that the interval [0, 1] in Muntz s Theorem can be replaced by an arbitrary compact set A of positive Lebesgue measure. That is, if A is a compact set ....

....containing at least n 1 points. That is, M(#) C(A) and every p Mn (#) having at least n 1 (distinct) zeros in A is identically 0. In fact, Muntz spaces are the canonical examples for Chebyshev spaces and the following properties of Muntz spaces Mn (#) are well known (see, for example, [9, 11, 21]) Theorem 2.1 (Unique Interpolation Property) For every x 0 x 1 x n and y 0 ,y 1 , y n there exists a unique p Mn(#) so that p(x j ) y j,j =0,1, n. Theorem 2.2 (Existence of Chebyshev Polynomials) Let A be a compact subset of [0, containing at least n 1points. Then there ....

P. B. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, SpringerVerlag, New York (to appear).

....[a, b] #) Let #: # j ) # j=0 be an increasing sequence of nonnegative real numbers. Suppose #0 =0and there exists a # 0so that # j #j for each j.Suppose0 a b. Then there exists a constant c(a, b, #) depending only on a, b,and#so that X A # 0 over R. When [a, b] [0,1] and with [a,b] replaced with # (x)# [a,b] this was proved by Newman. Note that the interval [0, 1] plays a special role in the study of Muntz spaces Mn (#) A linear transformation y = #x # does not preserve membership in Mn(#) in general (unless # = 0) So the analogue of Newman s ....

....there exists a # 0so that # j #j for each j.Suppose0 a b. Then there exists a constant c(a, b, #) depending only on a, b,and#so that X A # 0 over R. When [a, b] 0,1] and with [a,b] replaced with # (x)# [a,b] this was proved by Newman. Note that the interval [0, 1] plays a special role in the study of Muntz spaces Mn (#) A linear transformation y = #x # does not preserve membership in Mn(#) in general (unless # = 0) So the analogue of Newman s Inequality on [a, b] for a 0 does not seem to be obtainable in any straightforward fashion from the [0,b]case. ....

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, N.Y. (to appear).

....is now the monic polynomial of degree n of smallest supremum norm on the interval [a, b] and it satisfies (1.3) a,b] 2 . The Chebyshev polynomials have a central role to play in minimization problems in the supremum norm as well as many other extremal problems. See, for example, [6]. The analogous problem where the polynomials are restricted to have integer coe#cients is very much harder. For a very nice discussion of this problem see [15] We define (1.4) n [a, b] and let (1.5) # [a, b] inf # n [a, b] n=0,1, lim n### n [a, b] We call any polynomial p n ....

....To this end, we need some results on orthogonal Muntz Legendre polynomials. Let # = i # i=0 be a fixed sequence of distinct, nonnegative real numbers. Let Ln (x) L n 0 ,# 1 , # n ( x) c j,n x # j ,x#(0,#) c j,n = i=0 (# j # i 1) i=0,i#=j (# j # i ) It is shown in [6] that Ln (x)Lm (x)dx = # n,m ,n,m=0,1, where # n,m is the Kronecker symbol. From this orthogonality it follows easily that c =min # # # : a 0 ,a 1 , a n 1 #R,a n =1 and hence that # c # for every a 0 , a 1 , a n #R. Let 0 n be fixed integers. ....

P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York (to appear).

....of non zero coe#cients in f and c 0 is an absolute constant. This was proved by Konjagin [45] and independently by McGehee, Pigno, and Smith [56] When the coe#cients are required to be integers, the questions have a Diophantine nature and have been studied from a variety of points of view. See [2,3,8,18,39,61]. One key to the analysis is a study of the related problem of how large an order zero these restricted polynomials can have at 1. In [17] we answer this latter question precisely for the class of polynomials of the form with fixed #=0. Variants of these questions have attracted ....

....Schmidt in the early thirties. His complicated proof was not published the first published proof is due to Schur [69] Later new and simpler proofs and generalizations were published by Szego [71] and Erdos and Turan [38] and others. A version of the approach of Erdos and Turan is presented in [8]. Theorem 7.1. Suppose has m positive real roots. Then 2n log a 0 a n . Our Theorem 7.2 below (see [17] for a proof) improves the above bound of c # n log n in the cases we are interested in where the coe#cients are of similar size. Up to the constant c it is the ....

[Article contains additional citation context not shown here]

P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

....in large measure, due to Clarkson and Erdos [12] and Schwartz [24] whose works include the result that if then every function belonging to the uniform closure of span x , on [a, b] can be extended analytically b . There are many generalizations and variations of Muntz s Theorem [1, 4, 5, 6, 7, 8, 9, 16, 17, 19, 24, 26, 28, 29]. There are also still many open problems. For example, the proper generalizations to many variables are still open. Schwartz [24] extended the results of Clarkson and Erdos to sequences (# i ) # i=0 of arbitrary real numbers. His main results in this direction are formulated by the next two ....

P. B. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, SpringerVerlag, New York, N.Y. (to appear).

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Borwein, P., and T. Erd'elyi. [1995] "Polynomials and Polynomial Inequalities", SpringerVerlag, New York.

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P. Borwein and T. Erdelyi [1995], Polynomials and Polynomial Inequalities, Springer-Verlag, New York.

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P. Borwein and T. Erdelyi, "Polynomials and Polynomial Inequalities," SpringerVerlag, New York, 1995.

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, 1995.

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Borwein, P.B., and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, N.Y., 1995.

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P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer-Verlag, New York, 1995.

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P. B. Borwein and T. Erdelyi, Polynomials and Polynomials Inequalities, Springer-Verlag, New York, 1995.

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Borwein P., Erdelyi T.: Polynomials and polynomial inequalities. Springer-Verlag, GTM 161.

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