Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem

The Hardy-Littlewood majorant problem was raised in the 30's and it can be formulated as the question whether $\int |f|^p\ge \int|g|^p$ whenever $\hat{f}\ge|\hat g|$. 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 $\pm$ signs with which the signed exponential sum has larger $p^{\rm th}$ norm than the idempotent obtained with all the signs chosen + in the exponential sum. That conjecture was proved recently by Mockenhaupt and Schlag. \comment{Their construction was used by Bonami and R\'ev\'esz to find analogous examples among bivariate idempotents, which were in turn used to show integral concentration properties of univariate idempotents.}However, a natural question is if even the classical $1+e^{2\pi i x} \pm e^{2\pi i (k+2)x}$ 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 $\int_0^{\frac12}|1+e^{2\pi ix}-e^{2\pi i([\frac p2]+2)x}|^p > \int_0^{\frac12}|1+e^{2\pi ix}+e^{2\pi i([\frac p2]+2)x}|^p$. The proof combines delicate calculus with numerical integration and precise error estimates.Comment: 19 pages,1 figur

Carathéodory–Fejér type extremal problems on locally compact Abelian groups

We consider the extremal problem of maximizing a point value ∣f(z)∣∣f(z)∣ at a given point z∈Gz∈G by some positive definite and continuous function ff on a locally compact Abelian group (LCA group) GG, where for a given symmetric open set Ω∋zΩ∋z, ff vanishes outside ΩΩ and is normalized by f(0)=1f(0)=1. This extremal problem was investigated in RR and RdRd and for ΩΩ a 0-symmetric convex body in a paper of Boas and Kac in 1945. Arestov, Berdysheva and Berens extended the investigation to TdTd, where T:=R/ZT:=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 ZZ or Zm:=Z/mZZm:=Z/mZ. We extend their results to arbitrary LCA groups

The point value maximization problem for positive definite functions supported in a given subset of a locally compact group

AbstractThe century-old extremal problem, solved by Carathéodory and Fejér, concerns a non-negative trigonometric polynomial $T(t) = a_0 + \sum\nolimits_{k = 1}^n {a_k} \cos (2\pi kt) + b_k\sin (2\pi kt){\ge}0$, normalized by a0=1, where the quantity to be maximized is the coefficient a1 of cos (2π t). Carathéodory and Fejér found that for any given degree n, the maximum is 2 cos(π/n+2). In the complex exponential form, the coefficient sequence (ck) ⊂ ℂ will be supported in [−n, n] and normalized by c0=1. Reformulating, non-negativity of T translates to positive definiteness of the sequence (ck), and the extremal problem becomes a maximization problem for the value at 1 of a normalized positive definite function c: ℤ → ℂ, supported in [−n, n]. Boas and Kac, 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.</jats:p