C.1 Introduction

A counterexample to Mallat's conjecture about zero-crossings (section 8.4) was found by Yves Meyer in the early 1990s. It appeared in conference notes, but it has never been published. Since there is continuing interest in analyzing signals using zero-crossings, we have elected to present a complete discussion rather than the outline given in the first edition of this book. The following counterexample is based on the one announced by Meyer. The development given here is more "constructive" than that presented by Meyer, but the price paid is that the proof requires considerable computation.

The counterexample for two dimensions follows rather easily from the one-dimensional case, where the real work must be done. The construction given here is reminiscent of the one given for the counterexample to Marr's conjecture in section 8.3. However, in the case of Mallat's conjecture, there are other conditions to be satisfied, since both the zero-crossings and the first derivatives of the functions must agree. Fortunately, these conditions must be met only for dyadic values of the scaling parameter ?. This makes it possible to construct a smooth, compactly supported counterexample. The other difference between the two conjectures is that in Marr's case the kernel is the Gaussian and in Mallat's case the kernel is the basic cubic spline.

We begin with the function f ₀ defined by

We will show that there are infinitely many functions of the form

such that ( f ₀ * ? _?) ? and...