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

In this paper we shall characterize Sobolev spaces of an arbitrary order of smoothness using nonstationary tight wavelet frames for L2(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 L2(ℝ) derived from masks of pseudosplines in [15]. This implies that any compactly supported nonstationary tight wavelet frame of L2(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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