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Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames

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Abstract

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L 2(ℝ). In particular, we show that a Sobolev space of an arbitrary fixed order of smoothness can be characterized in terms of the weighted ℓ2-norm of the analysis wavelet coefficient sequences using a fixed compactly supported nonstationary tight wavelet frame in L 2(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame of L 2(ℝ) in [15] can be properly normalized into a pair of dual frames in the corresponding pair of dual Sobolev spaces of an arbitrary fixed order of smoothness.

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Authors and Affiliations

  1. Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1

    Bin Han

  2. Department of Mathematics, National University of Singapore, Singapore, Singapore

    Zuowei Shen

Authors
  1. Bin Han
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  2. Zuowei Shen
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Correspondence to Bin Han.

Additional information

Research supported in part by NSERC Canada under Grant RGP 228051.

Research supported in part by Grant R-146-000-060-112 at the National University of Singapore.

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Han, B., Shen, Z. Characterization of Sobolev spaces of arbitrary smoothness using nonstationary tight wavelet frames. Isr. J. Math. 172, 371–398 (2009). https://doi.org/10.1007/s11856-009-0079-9

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  • DOI: https://doi.org/10.1007/s11856-009-0079-9

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