On the Decay of Infinite Products of Trigonometric Polynomials

Vladimir Protassov ()
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Vladimir Protassov: Erasmus University Rotterdam

No 01-046/4, Tinbergen Institute Discussion Papers from Tinbergen Institute

Abstract: We consider infinite products of the form ,where {mk} is an arbitrary sequence of trigonometric polynomials of degree at most n with uniformly bounded normssuch that mk(0)=1 for all k. We show that can decrease at infinity not faster than and present conditions underwhich this maximal decay attains. This result proves the impossibility of the construction of infinitely differentiablenonstationary wavelets with compact support and restricts the smoothness of nonstationary wavelets by thelength of their support. Also this generalizes well-known similar results obtained for stable sequences ofpolynomials (when all mk coincide). In several examples we show that by weakening the boundedness conditionsone can achieve an exponential decay.

Keywords: trigonometric polynomial; infinite product; wavelets; roots (search for similar items in EconPapers)
Date: 2001-04-27
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Persistent link: https://EconPapers.repec.org/RePEc:tin:wpaper:20010046

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