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On the Decay of Infinite Products of Trigonometric Polynomials

Author

Listed:
  • Vladimir Protassov

    (Erasmus University Rotterdam)

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.

Suggested Citation

  • Vladimir Protassov, 2001. "On the Decay of Infinite Products of Trigonometric Polynomials," Tinbergen Institute Discussion Papers 01-046/4, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20010046
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