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Littlewood Polynomials with Small $L^4$ Norm
- Publication Year :
- 2012
-
Abstract
- Littlewood asked how small the ratio $||f||_4/||f||_2$ (where $||.||_\alpha$ denotes the $L^\alpha$ norm on the unit circle) can be for polynomials $f$ having all coefficients in $\{1,-1\}$, as the degree tends to infinity. Since 1988, the least known asymptotic value of this ratio has been $\sqrt[4]{7/6}$, which was conjectured to be minimum. We disprove this conjecture by showing that there is a sequence of such polynomials, derived from the Fekete polynomials, for which the limit of this ratio is less than $\sqrt[4]{22/19}$.<br />Comment: minor revisions
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.1205.0260
- Document Type :
- Working Paper