B-splines and nonorthogonal wavelets

Abstract

The necessary and sufficient conditions for the (nonorthogonal) wavelet multiresolution analysis with arbitrary (for example B-spline) scaling function are established. The following results are obtained: 1) the general theorem which declares necessary and sufficient conditions for the possibility of multiresolution analysis in the case of arbitrary scaling function; 2) the reformulation of this theorem for the case of B-spline scaling function from W2m; 3) the complete description of the family of wavelet bases generated by B-spline scaling function; 4) the concrete construction of the unconditional wavelet bases (with minimal supports of wavelets) generated by B-spline scaling functions which belongs to W2m. These wavelet bases are simple and convenient for applications. In spite of their nonorthogonality, these bases possess the following advantages: 1) compactness of set supp ψ and minimality of its measure; 2) simple explicit formulas for the change of level. These advantages compensate the nonorthogonality of described bases. © Springer-Verlag Berlin Heidelberg 2005.