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 ℓ
2-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.
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.