Ten lectures on wavelets
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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: the general theorem which declares necessary and sufficient conditions for the possibility of multiresolution analysis in the case of arbitrary scaling function; the reformulation of this theorem for the case of B-spline scaling function from W$_{\rm 2}^{m}$; the complete description of the family of wavelet bases generated by B-spline scaling function; 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.