NDLI: On a problem due to Littlewood concerning polynomials with unimodular coefficients

Content Provider	Springer Nature Link
Author	Schmidt, Kai Uwe
Copyright Year	2013
Abstract	Littlewood raised the question of how slowly $\lVert f_{n}\rVert_{4}^{4}-\lVert f_{n}\rVert_{2}^{4}$ (where $\lVert.\rVert _{r}$ denotes the L r norm on the unit circle) can grow for a sequence of polynomials f n with unimodular coefficients and increasing degree. The results of this paper are the following. For $$g_n(z)=\sum_{k=0}^{n-1}e^{\pi ik^2/n} z^k $$ the limit of $(\lVert g_{n}\rVert_{4}^{4}-\lVert g_{n}\rVert_{2}^{4})/\lVert g_{n}\rVert_{2}^{3}$ is 2/π, which resolves a mystery due to Littlewood. This is however not the best answer to Littlewood’s question: for the polynomials $$h_n(z)=\sum_{j=0}^{n-1}\sum _{k=0}^{n-1} e^{2\pi ijk/n} z^{nj+k} $$ the limit of $(\lVert h_{n}\rVert_{4}^{4}-\lVert h_{n}\rVert_{2}^{4})/\lVert h_{n}\rVert_{2}^{3}$ is shown to be 4/π 2. No sequence of polynomials with unimodular coefficients is known that gives a better answer to Littlewood’s question. It is an open question as to whether such a sequence of polynomials exists.
Ending Page	466
Page Count	10
Starting Page	457
File Format	PDF
ISSN	10695869
e-ISSN	15315851
Journal	Journal of Fourier Analysis and Applications
Issue Number	3
Volume Number	19
Language	English
Publisher	SP Birkhäuser Verlag Boston
Publisher Date	2013-03-16
Publisher Place	Boston
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Polynomial Special sequences and polynomials Signal, Image and Speech Processing Littlewood problem Restricted coefficients Abstract Harmonic Analysis Trigonometric polynomials, inequalities, extremal problems Mathematical Methods in Physics Fourier Analysis Shift register sequences and sequences over finite alphabets Merit factor Approximations and Expansions Norm Partial Differential Equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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