6.4 Spline Wavelets

In this last section we construct the so-called BattJe-Lemari wavelets. The starting material are certain spline functions known from numerical analysis, and that's why these wavelets are occasionally called spline wavelets as well. The Battle-Lemari wavelets, in contradiction to the title of the current chapter, don't have compact support any more. Nevertheless it will be possible to use the formalism that we have erected in the foregoing sections for the treatment of these wavelets as well. But let's take everything in turn!

Another glance at the scaling equation in the form 5.3.(4) shows that, given two pairs ( , H ₁) and ( , H ₂), each of them satisfying such an equation, the pair ( , H ₁ H ₂) satisfies such an equation as well. To multiplication in the Fourier domain corresponds convolution in the time domain; in other words, if ? ₁ and ? ₂ are scaling functions, then ? ₁ * ? ₂ will satisfy a scaling equation as well. Therefore, beginning with ? ₀ : = ? _Haar and setting up the recursion scheme ? _{n +1} : = ? ₀ * ? _n ( n ? 0), we should obtain a sequence of ever more regular functions that a priori satisfy scaling equations and could maybe be adapted to be useful in the construction of wavelets.

We are going to change our notation to some extent,...