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Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation. (English) Zbl 0855.41006
In this paper, the author constructs, on a bounded interval $$I= [a, b]$$, an unconditional semi-orthogonal cubic spline wavelet basis for the homogeneous Sobolev space $$H^2_0 (I)= \{f: f''\in L^2 (I)$$, $$f(a)= f'(a)= f(b)= f'(b) =0\}$$, equipped with the inner product $$\int_I f'' g''$$. In this setting, the orthogonal projections on scaling and wavelet spaces can be represented in terms of interpolatory operators. As these cubic spline wavelets also form a basis for the space of continuous functions with the homogeneous boundary conditions, the (global or local) decay of wavelet coefficients for such a function can then be described in terms of (global or local) Lipschitz smoothness of that function, and vice versa. Finally, a numerical scheme for the fast wavelet transform and adaptive wavelet approximation is presented, based on the efficient solution of linear systems of equations with banded coefficient matrices.
Reviewer: E.Quak (Oslo)

##### MSC:
 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 65D07 Numerical computation using splines 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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