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The L_4 norm of Littlewood polynomials derived from the Jacobi symbol
- Source :
- Pac. J. Math., vol. 257, no. 2, pp. 395-418, 2012
- Publication Year :
- 2011
-
Abstract
- Littlewood raised the question of how slowly the L_4 norm ||f||_4 of a Littlewood polynomial f (having all coefficients in {-1,+1}) of degree n-1 can grow with n. We consider such polynomials for odd square-free n, where \phi(n) coefficients are determined by the Jacobi symbol, but the remaining coefficients can be freely chosen. When n is prime, these polynomials have the smallest known asymptotic value of the normalised L_4 norm ||f||_4/||f||_2 among all Littlewood polynomials, namely (7/6)^{1/4}. When n is not prime, our results show that the normalised L_4 norm varies considerably according to the free choices of the coefficients and can even grow without bound. However, by suitably choosing these coefficients, the limit of the normalised L_4 norm can be made as small as the best known value (7/6)^{1/4}.<br />Comment: minor revisions
Details
- Database :
- arXiv
- Journal :
- Pac. J. Math., vol. 257, no. 2, pp. 395-418, 2012
- Publication Type :
- Report
- Accession number :
- edsarx.1105.5180
- Document Type :
- Working Paper