Adaptive wavelet frame methods with B-spline bases

Dana Cerna


It is known that a function f that is smooth, except at some isolated
singularities, typically has a sparse representation in a wavelet
basis, i.e. only a small number of numerically significant wavelet
coefficients carry most of the information on f. This compression
property of wavelets led to design of adaptive wavelet methods for
solving operator equations. The effectiveness of these methods is
strongly influenced by the choice of a wavelet basis. In our
contribution, we propose a construction of B-spline wavelet basis on
the interval and its adaptation to complementary boundary conditions.
The resulting bases are well-conditioned and corresponding stiffness
matrices have small condition numbers. Furthermore, we show that this
construction combined with frame approach to adaptation to a fairly
general bounded domain leads to effective adaptive wavelet frame
methods.