Cubic spline wavelet bases of Sobolev spaces and multilevel interpolation (Q1917558)

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.