Speaker: Vladimir Protassov

Title: One optimization problem on nonnegative trigonometric polynomials

Abstract:

We consider infinite products of the form $f(\xi ) = \prod\limits_{k=1}^{\infty}m_k(2^{-k }\xi)$, where $\{ m_k \}$ is an arbitrary sequence of nonnegative trigonometric polynomials of degree at most $n$ with uniformly bounded norms such that $m_k(0) = 1$ for all~$k$. The problem is to find for a given $n \in N$ a n optimal sequence of polynomials $\{ m_k \}$ such that the function $f(\xi )$ has the fastest possible decay as $\xi \to \infty$ along the real axis. This problem arises in the construction of nonstationary wavelets with compact support, it also plays an exceptional role in the theory nonstationary subdivision algorithms used in appproximation theory and computer design. The estimating of the rate of decay for a given sequence of polynomials is a very difficult problem even for stationary sequences, where all the polynomials coincide.For every $n$ we find the fastest decay and characterize optimal sequences of polynomials $\{ m_k \}$, for which this decay is attained. The answer is given in terms of the asymptotic behavior of zeros of the polynomials. Surprisingly, by weakening the norm boundedness condition one can achieve much faster decay. We also consider possible generalizations of this problem to multivariate polynomials and formulate several open problems.