B–Splines and Nonorthogonal Wavelets

Abstract

The necessary and sufficient conditions for the (non orthogonal) 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 W $_{\rm 2}^{m}$;

• 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 W $_{\rm 2}^{m}$.

These wavelet basesare simple and convenient for applications. In spite of their nonorthogonality, these bases possess the following advantages: 1) compactness of set $\mbox{supp\,}\psi$ and minimality of its measure; 2) simple explicit formulas for the change of level. These advantages compensate the nonorthogonality of described bases.