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

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

2010-06-02

arxiv.org/abs/1006.0409v3

Mathematics

Classical Analysis and ODEs

19 pages,1 figure

Scientific paper

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 $0 \int_0^{\frac12}|1+e^{2\pi ix}+e^{2\pi i([\frac p2]+2)x}|^p$. The proof combines delicate calculus with numerical integration and precise error estimates.

Affiliated with

Also associated with

No associations

Rating

Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Three-term idempotent counterexamples in the Hardy-Littlewood majorant problem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-512974