Results in Mathematics

, Volume 57, Issue 1, pp 121–162

New Stable Biorthogonal Spline-Wavelets on the Interval


DOI: 10.1007/s00025-009-0008-6

Cite this article as:
Primbs, M. Results. Math. (2010) 57: 121. doi:10.1007/s00025-009-0008-6


In this paper we present the construction of new stable biorthogonal spline-wavelet bases on the interval [0, 1] for arbitrary choice of spline-degree. As starting point, we choose the well-known family of compactly supported biorthogonal spline-wavelets presented by Cohen, Daubechies and Feauveau. Firstly, we construct biorthogonal MRAs (multiresolution analysis) on [0, 1]. The primal MRA consists of spline-spaces concerning equidistant, dyadic partitions of [0, 1], the so called Schoenberg-spline bases. Thus, the full degree of polynomial reproduction is preserved on the primal side. The construction, that we present for the boundary scaling functions on the dual side, guarantees the same for the dual side. In particular, the new boundary scaling functions on both, the primal and the dual side have staggered supports. Further, the MRA spaces satisfy certain Jackson- and Bernstein-inequalities, which lead by general principles to the result, that the associated wavelets are in fact L2([0, 1])-stable. The wavelets however are computed with aid of the method of stable completion. Due to the compact support of all occurring functions, the decomposition and reconstruction transforms can be implemented efficiently with sparse matrices. We also illustrate how bases with complementary or homogeneous boundary conditions can be easily derived from our construction.

Mathematics Subject Classification (2000)

Primary 42C40Secondary 65F30


Multiresolution on the intervalbiorthogonal spline waveletsRiesz basescharacterization of Sobolev spacesboundary conditions

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Geibelstr. 12DuisburgGermany