On the decay of infinite products of trigonometric polynomials

We consider infinite products of the form (see article). We show that (see article) can decrease at infinity not faster than (see article) and present conditions under which this maximal decay attains. This result proves the impossibility of the construction of infinitely differentiable nonstationary wavelets with compact support and resricts the smoothness of nonstationary wavelets by the length of their support. Also this generalizes well-known similar results obtained for stable sequences of polynomials (when all mk coincide). In several examples we show that by weakening the boundedness conditions one can achieve an exponential decay.