On the Decay of Infinite Products of Trigonometric Polynomials

We consider infinite products of the form $f(\xi )=\prod _{k=1}^\infty m_k(2^{-k}\xi )$, where $\{m_k\}$ is an arbitrary sequence of trigonometric polynomials of degree at most $n$ with uniformly bounded norms such that $m_k(0)=1$ for all $k$. We show that $f(\xi )$ can decrease at infinity not faster than $O(\xi ^{-n})$ and present conditions under which this maximal decay is attained. This result can be applied to the theory of nonstationary wavelets and nonstationary subdivision schemes. In particular, it restricts the smoothness of nonstationary wavelets by the length of their support. This also generalizes well-known similar results obtained for stable sequences of polynomials (when all ${m_k}$ coincide). By means of several examples, we show that by weakening the boundedness conditions one can achieve exponential decay.