Communications in Mathematical Analysis Volume 4, Number 1, pp. 45-57 (2008) ISSN 1938-9787 www.commun-math-anal.org MORE G ENERAL C ONSTRUCTIONS OF W AVELETS ON THE I NTERVAL H. BIBI , A. J OUINI , AND M. KRATOU ∗ Faculty of Sciences of Tunis Department of Mathematics Campus, 1060 Tunis, Tunisia (Communicated by Yong Ding) Abstract In this paper we present general constructions of orthogonal and biorthogonal mul- tiresolution analysis on the interval. In the first one, we describe a direct method to define an orthonormal multiresolution analysis. In the second one, we use the integra- tion and derivation method for constructing a biorthogonal multiresolution analysis. As applications, we prove that these analyses are adapted to study regular functions on the interval. AMS Subject Classifications : 42C15 ; 44A15. Keywords : Multiresolution analysis, wavelet, Sobolev space 1 Introduction The search for wavelet bases on a bounded domain has been an active field for many years, since the beginning of the 1990’s. All these constructions use either the basis of I. Daubechies or the spline basis. In his fundamental paper on wavelets on the interval [14], Y. Meyer proved that one can take restrictions of the orthonormal multiresolution analysis of I. Daubechies to the interval [0, 1] and then we can study functions known only on the interval. More precisely, he proves that the restrictions of Daubechies scaling functions on the interval are linearly independent but the restrictions of associated wavelets on the interval are not linearly independent. In 1992, we have constructed multiresolution analysis on the interval by using Daubechies wavelets [9]. The associated bases have compact support and allow also the study of divergence-free vector functions on [0,1] n . There are related constructions as well by A. Canuto and coworkers [1] and by A. Jouini and P. G. Lemari´ e ([8] and [10]). ∗ E-mail address:abdellatif.jouini@fst.rnu.tn, hatem.bibi@univ-paris1.fr, Kratoumouna@yahoo.fr