Proof of Saffari's near-orthogonality conjecture for ultraflat sequences of unimodular polynomials Academic Article uri icon

abstract

  • Let Pn(z)=∑k=0nak,nz k∈C[z] be a sequence of unimodular polynomials (ak,n=1 for all k, n) which is ultraflat in the sense of Kahane, i.e., limn→∞maxz=1(n+1)-1/2Pn(z)-1=0. We prove the following conjecture of Saffari (1991): ∑k=0nak,nan-k,n=o(n) as n→∞, that is, the polynomial Pn(z) and its "conjugate reciprocal" Pn*(z)=∑k=0na n-k,nzk become "nearly orthogonal" as n→∞. To this end we use results from [3] where (as well as in [5]) we studied the structure of ultraflat polynomials and proved several conjectures of Saffari. © 2001 Académie des sciences/Éditions scientifiques et médicales Elsevier SAS.

author list (cited authors)

  • Erdélyi, T.

citation count

  • 7

publication date

  • October 2001