In this talk we will present quantitative lower bounds of the L1 norm of a non-harmonic trigonometric polynomial of the following form:
– let T > 1;
– let (λj)j≥0 be a sequence of non-negative real numbers with λj+1 − λj ≥ 1
– let (aj)j=0,...,N be a finite sequence of complex numbers
then
T-1∫-T/2≤t≤T/2 |Σ0≤j≤N aj e2πiλj t| dt ≥ C(T) Σ0≤j≤N |aj|/(j+1)
where C(T) is an explicit constant that depends on T only. The L2 analogue is Inham’s Inequality and the harmonic case (λj integers) is McGehee, Pigno, Smith’s solution of the Littlewood conjecture while the version with non explicit constant is due to Nazarov.
This is joint work with K. Kellay and C. Saba (U. Bordeaux)
https://univienna.zoom.us/j/67922750549?pwd=Ulh5L1QxNFhBOC9PUjlVdG9hc0tmUT09