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Constructing wavelets by welding segments of smooth functions

par Jansen, Maarten
Référence Journal of Computational and Applied Mathematics, 477, 117160
Publication Publié, 2025-10-20

Article révisé par les pairs

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Résumé :

The construction of B-spline wavelet bases on nonequispaced knots is extended to wavelets thatare piecewise segments from any combination of smooth functions. The extended wavelet familythus provides multiresolution basis functions with support as compact as possible and belongingto a user controlled smoothness class. The construction proceeds in two phases. In the firstphase of segmentation, a set of smooth functions is used in the welding of compact supported,piecewise smooth basis functions. These piecewise smooth basis functions are refinable, meaningthat they can be written as linear combinations of similar basis functions constructed on a finergrid of knots. The expression of the linear combination between the bases at two scales isknown as a refinement or two-scale equation. In the second phase, the refinability enablesthe construction of a wavelet transform. To this end, the refinement equation of the piecewisesmooth scaling functions is factored into a lifting scheme, to which the desired properties ofthe subsequent wavelet basis can then be added. Special attention is paid to proper boundarytreatment. Next to the details of the construction, the paper discusses the conditions for it tofit into the classical framework of multiresolution analyses.