H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....strong constraint. We show that the acceptance probability is close to a bivariate polynomial in g and another parameter of degree at most 2T . We then obtain a lower bound by restricting to a univariate polynomial and generalizing a classical approximation theory result of Ehlich and Zeller [11] and Rivlin and Cheney [22] Much of the proof deals with the complication that g does not divide n in general. Shi [24] has recently improved our method to obtain a lower bound for the collision problem. In addition, he gives a tight lower bound when the x i range from 1 to 3n 2 rather ....

.... n 0.818 for all su#ciently large n. Thus, since 0 P (g, N) and we are done. 5. LOWER BOUND We are now ready to prove a lower bound for the collision problem. To do so, we generalize an approximation theory result due to Rivlin and Cheney [22] and (independently) Ehlich and Zeller [11]. That result was applied to query complexity by Nisan and Szegedy [20] and later by Beals et al. 3] Theorem 3. Q2 (Coln ) Proof. Let g have range 1 G. Then the quasilattice points (g, N) all lie in the rectangular region R = 1, G] n, n n (10T ) Recalling the polynomial q ....

H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964. Available at www.arxiv.org.

....theorem by A. A. Markov. Let for a polynomial P (x) kPk stand for the maximum absolute value of P on [ Gamma1; 1] Let P (x) be a polynomial of degree d. Markov s theorem (see, for example, 8,14,18] says that kP 0 k d 2 kPk. The following simplified version of Ehlich and Zeller lemma [10] is an easy corollary of the Markov s theorem. Let Y = f Gamma1; Gamma1 2=m; Gamma1 4=m; 1 Gamma 2=m; 1g and let kPY k stand for the maximum absolute value of P (x) on Y . Then (33) kPk 1 Gamma 2d 2 m kP Y k: 18 NIKOLAI K. VERESHCHAGIN Indeed, by Lagrange theorem ....

H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Mathematische Zeitschrift 86 (1964), 41--44.

....Chebyshev norms are replaced by p norms, then the corresponding estimates are named after Marcinkiewicz and Zygmund; see [13, Ch.X.7] or [9] for a more recent paper. For the case of oversampling, the only results known to us are the estimate c n;2m 1 cos n 2m due to Ehlich and Zeller [4], which is sharp if and only if njm, and the recent result by Wunder and Boche [12] c n;N q N 2n 1 N Gamma(2n 1) We embark on this important problem by elaborating on basic methods and results from classical Approximation Theory. Three different aspects will be considered. In Section 2, ....

H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Math. Zeitschr. 86 (1964), pp. 41--44.

....C(q) # cos( q 2 ) # ## for 0 # q # 1 (5) is valid for every P # P# and every interval J . Inequality (4) is remarkable because the norm #P# ###### on the righthandside of (4) depends on the values of P at the Chebychev points only. This result was given by Ehlich and Zeller in [2]. Using (4) the following inequalities P # ### # 1 2 # # C # n N # 1 # P ###### ### # # C # n N # # 1 # P ###### ### # ; 6) P # ### # 1 2 # # C # n N # 1 # P ###### ### # # C # n N # # 1 # P ###### ### # (7) which are valid for ....

H. Ehlich and K. Zeller, "Schwankung von Polynomen zwischen Gitterpunkten", Math. Z., vol. 86, pp. 41-44, 1964.

....with a slightly more complex polynomial, see [33, footnote 1 on p.560] Below we give Nisan and Szegedy s proof that deg(f) can be no more than quadratically smaller than bs(f) 32] This shows that the gap of the last example is close to optimal. The proof uses the following theorem from [12,36]: Theorem 3 (Ehlich Zeller; Rivlin Cheney) Let p : R R be a polynomial such that b 1 p(i) b 2 for every integer 0 i n, and its derivative has jp 0 (x)j c for some real 0 x n. Then deg(p) q cn= c b 2 Gamma b 1 ) Theorem 4 (Nisan Szegedy) bs(f) 2 deg(f) 2 . Proof Let ....

H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

....with a slightly more complex polynomial, see [NW95, footnote 1 on p.560] Below we give Nisan and Szegedy s proof that deg(f) can be no more than quadratically smaller than bs(f) NS94] This shows that the gap of the last example is close to optimal. The proof uses the following theorem from [EZ64, RC66] Theorem 3 (Ehlich Zeller; Rivlin Cheney) Let p : R R be a polynomial such that b 1 p(i) b 2 for every integer 0 i n, and its derivative has jp 0 (x)j c for some real 0 x n. Then deg(p) p cn= c b 2 Gamma b 1 ) Theorem 4 (Nisan Szegedy) bs(f) 2 deg(f) 2 . 3 It ....

H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

....# 2.5m 1 2 . Numerical examples that illustrate the behavior of the Gram polynomials are presented in Section 5. The relevance of the ratio n m 1 2 has previously been noted by Bjork [5] and Zaremba [22] in their investigation of Gram polynomials. Closely related problems are also considered in [9, 10, 13, 16, 18]. Our method of investigation also can be used to analyze classes of orthogonal polynomials other than Gram polynomials. 2. Gram Polynomials. The Gram polynomials introduced in Section 1 satisfy the three term recurrence relation, for 1 # n m, #n (x) 2#n 1x#n 1 (x) #n 1 #n 2 #n 2 (x) 17) ....

H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten,Math.Z.,86 (1964), 41-44.

....factor, to the time needed to compute f on a CREW PRAM [11] We show that the degree of f is also polynomially related to all these measures. 2 Theorem 2 For every boolean function we have deg(f) D(f) 16deg(f) 8 The proof of this result requires results from real approximation theory [4] [6] [13] We strongly suspect that the exponent 8 is not optimal. The strongest separation we can obtain is a function for which D(f) deg(f) 1:58: 1.3.3 Approximation in Lmax norm Our techniques are strong enough to allow us to give strong bounds on the degree needed even to approximate ....

....polynomial p : R R of degree at most n such that for all x 1 ; xn 2 f0; 1g n we have p sym (x 1 ; xn ) p(x 1 Delta Delta Delta xn ) Moreover, deg( p) deg(p) 3.2 A Theorem from Approximation Theory We will need the following result of H. Ehlich and K. Zeller [6] and T. J. Rivlin and E. W. Cheney [13] Theorem 2 (Echlich, Zeller; Rivlin, Cheney) Let p be a polynomial with the following properties: 1. For any integer 0 i n we have b 1 p(i) b 2 . 2. For some real 0 x n the derivative of p satifies jp 0 (x)j c. Then deg(p) p cn= c b 2 ....

H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Mathematische Zeitschrift, volume 86, pages 41-44, 1964.

....P (x) Henceforth let P denote a polynomial of degree d. Let P 0 denote the derivative of P . A. A. Markov s theorem, which can be found in Cheney (1966) Lorentz (1966) or P olya Szego (1972, Vol. II, Part VI, Prob. 83) says that jjP 0 jj d 2 jjP jj: The following lemma is due to Ehlich Zeller (1964) . See Rivlin Cheney (1966) for an improved version. Lemma 3.1 (Ehlich and Zeller) Let m be a positive integer, and let Y = f Gamma1; Gamma1 2 m ; Gamma1 4 m ; 1 Gamma 2 m ; 1g. Assume that d 2 (d 2 Gamma 1) m 2 6. Then jjP jj jjP jj Y Delta 1 1 Gamma 1 ....

H. Ehlich and K. Zeller, Schwankung von polynomen zwischen gitterpunkten. Math. Z. 86 (1964), 41--44.

....theorem by A. A. Markov. Let for a polynomial P (x) kPk stand for the maximum absolute value of P on [ Gamma1; 1] Let P (x) be a polynomial of degree d. Markov s theorem (see, for example, 6, 12, 14] says that kP 0 k d 2 kPk. The following simplified version of Ehlich and Zeller lemma [8] is an easy corollary of the Markov s theorem. Let Y = f Gamma1; Gamma1 2=m; Gamma1 4=m; 1 Gamma 2=m; 1g and let kP Y k stand for the maximum absolute value of P (x) on Y . Then kPk 1 Gamma 2d 2 m kPY k: 9) Indeed, by Lagrange theorem and by Markov s theorem we have ....

H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

....: 0) and B i = fig, then flipping B i in X flips the value of the OR function from 0 to 1. We can adapt the proof of [25, Lemma 3.8] on lower bounds of polynomials to get lower bounds on the number of queries in a quantum network in terms of block sensitivity. 2 The proof uses a theorem from [11, 28]: Theorem 4.12 (Ehlich, Zeller; Rivlin, Cheney) Let p : R R be a polynomial such that b 1 p(i) b 2 for every integer 0 i N , and jp 0 (x)j c for some real 0 x N . Then deg(p) p cN= c b 2 Gamma b 1 ) Theorem 4.13 If f is a Boolean function, then QE (f) p bs(f) 8 and Q 2 ....

H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

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H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

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EHLICH, H., AND ZELLER, K. 1964. Schwankung von Polynomen zwischen Gitterpunkten. Math. Z. 86, 41--44.

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H. Ehlich and K. Zeller. Schwankung von Polynomen zwischen Gitterpunkten. Mathematische Zeitschrift, 86:41--44, 1964.

No context found.

H. Ehlich and K. Zeller. Schwankung von polynomen zwischen gitterpunkten. Mathematische Zeitschrift, 86:41-44, 1964.

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H. Ehlich and K. Zeller, Schwankung von Polynomen zwischen Gitterpunkten, Math. Zeitschr. 86 (1964), pp. 41--44.

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