We consider infinite products of the form , where {mk} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded norms such that mk(0)=1 for all k. We show that can decrease at infinity not faster than 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 restricts 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.

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hdl.handle.net/1765/6868
Tinbergen Institute Discussion Paper Series
Tinbergen Institute

Protassov, V. (2001). On the Decay of Infinite Products of Trigonometric Polynomials (No. TI 01-046/4). Tinbergen Institute Discussion Paper Series. Retrieved from http://hdl.handle.net/1765/6868