6.3 Binary Interpolation

In the two foregoing sections we obtained scaling functions and corresponding wavelets by means of constructions in the Fourier domain, and also as limiting functions of an iteration procedure. In neither approach, however, did we discuss the convergence behaviour in the time domain. Now there is a third, called the direct method for constructing scaling functions ?. This method yields without a limiting process the exact values ?( x) at all "binary rational" points x ? ?, and it is with the help of this method that one obtains the best regularity results, e.g. for the Daubechies wavelets _N ?

In order to fix ideas, we assume that an N > 1 has been chosen once and for all and, furthermore, that

as agreed upon in connection with the Daubechies wavelets. The following abbreviations will prove useful:

For the description of the binary rational numbers we use the handy notation

therefore we have the inclusions

and is dense in ?.

The scaling equation now has the form

(1)

The "direct method" is founded on the following three simple facts:

• If t ? for some r ? 1, then the numbers 2 t - k ( k ? J) belong to .

• If t < 0, then the numbers 2 t - k ( k ? J) are < 0 as well.

• If t > 2 N -...