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Characterizations of orthonormal scale functions: A probabilistic approach

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Abstract

The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L 2(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al. [4]. The method also applies to more general dilation schemes that commute with translations by Z d.

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Authors and Affiliations

  1. Lehigh University, USA

    V. Dobrić

  2. Rutgers University, 00903-201, New Brunswick, NJ

    R. Gundy

  3. North Carolina State University, USA

    P. Hitczenko

Authors
  1. V. Dobrić
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  2. R. Gundy
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  3. P. Hitczenko
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Corresponding author

Correspondence to V. Dobrić.

Additional information

Communicated by Guido Weiss

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Dobrić, V., Gundy, R. & Hitczenko, P. Characterizations of orthonormal scale functions: A probabilistic approach. J Geom Anal 10, 417–434 (2000). https://doi.org/10.1007/BF02921943

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  • DOI: https://doi.org/10.1007/BF02921943

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