Stable Bases of Spline Wavelets on the Interval

Following the approach of Chui and Quak, we investigate semi-orthogonal spline wavelets on the unit interval [0, 1]. We give a slightly different construction of boundary wavelets. As a result, we are able to prove that the inner wavelets and the newly constructed boundary wavelets together constitute a Riesz basis for the wavelet space at each level with the Riesz bounds being level-independent. With the help of multiresolution analysis, the combination of the bases at multi-levels forms a Riesz basis for L2[0, 1]. Furthermore, we show that such a basis is also a stable basis in the Sobolev space H(0, 1) for a certain range of μ. §