The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether |f | p ≥ |g| p whenever f ≥ | g|. It has a positive answer only for exponents p which are even integers. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exists some finite set of exponentials and some ± signs with which the signed exponential sum has larger p th norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. However, a natural question is if even the classical 1+e 2πix ±e 2πi(k+2)x three-term exponential sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. We investigate the sharpened question and show that at least in certain cases there indeed exist three-term idempotent counterexamples in the Hardy-Littlewood majorant problem; that is we have for 0 < p < 6, p / ∈ 2NThe proof combines delicate calculus with numerical integration and precise error estimates. (2000): Primary 42A05. Mathematics Subject Classification
The Hardy-Littlewood majorant problem has a positive answer only for exponents p which are even integers, while there are counterexamples for all p / ∈ 2N. Montgomery conjectured that even among the idempotent polynomials there must exist some counterexamples, i.e. there exist some finite set of characters and some ± signs with which the signed character sum has larger p th norm than the idempotent obtained with all the signs chosen + in the character sum. That conjecture was proved recently by Mockenhaupt and Schlag.However, Mockenhaupt conjectured that even the classical 1 + e 2πix ± e 2πi(k+2)x threeterm character sums, used for p = 3 and k = 1 already by Hardy and Littlewood, should work in this respect. That remained unproved, as the construction of Mockenhaupt and Schlag works with four-term idempotents. In our previous work we proved this conjecture for k = 0, 1, 2, i.e. in the range 0 < p < 6, p / ∈ 2N. Continuing this work here we demonstrate that even the k = 3, 4 cases hold true. Several refinement in the technical features of our approach include improved fourth order quadrature formulae, finite estimation of G ′2 /G (with G being the absolute value square function of an idempotent), valid even at a zero of G, and detailed error estimates of approximations of various derivatives in subintervals, chosen to have accelerated convergence due to smaller radius of the Taylor approximation. (2000): Primary 42A05. Mathematics Subject Classification
Abstract. We consider the extremal problem of maximizing a point value |f (z)| at a given point z ∈ G by some positive definite and continuous function f on a locally compact Abelian group (LCA group) G, where for a given symmetric open set Ω ∋ z, f vanishes outside Ω and is normalized by f (0) = 1.This extremal problem was investigated in R and R d and for Ω a 0-symmetric convex body in a paper of Boas and Kac in 1943. Arestov and Berdysheva extended the investigation to T d , where T := R/Z. Kolountzakis and Révész gave a more general setting, considering arbitrary open sets, in all the classical groups above. Also they observed, that such extremal problems occurred in certain special cases and in a different, but equivalent formulation already a century ago in the work of Carathéodory and Fejér.Moreover, following observations of Boas and Kac, Kolountzakis and Révész showed how the general problem can be reduced to equivalent discrete problems of "Carathéodory-Fejér type" on Z or Z m := Z/mZ. We extend their results to arbitrary LCA groups.Mathematics Subject Classification (2000): Primary 43A35, 43A70. Secondary 42A05, 42A82.
The century old extremal problem, solved by Carathéodory and Fejér, concerns a nonnegative trigonometric polynomial T ptq " a 0`ř n k"1 a k cosp2πktq`b k sinp2πktq ě 0, normalized by a 0 " 1, and the quantity to be maximized is the coefficient a 1 of cosp2πtq. Carathéodory and Fejér found that for any given degree n the maximum is 2 cosp π n`2 q. In the complex exponential form, the coefficient sequence pc k q Ă C will be supported in r´n, ns and normalized by c 0 " 1. Reformulating, nonnegativity of T translates to positive definiteness of the sequence pc k q, and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c : Z Ñ C, supported in r´n, ns.Boas and Katz, Arestov, Berdysheva and Berens, Kolountzakis and Révész and recently Krenedits and Révész investigated the problem in increasing generality, reaching analogous results for all locally compact Abelian groups. We prove an extension to all the known results in not necessarily commutative locally compact groups. (2000): Primary 43A35, 43A70. Secondary 42A05, 42A82. Mathematics Subject Classification
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