A scaling equation with only non-measurable orthogonal solutions

Abstract:

In this paper we construct a non-measurable scaling function for the scaling equation \[ \phi (x)=\phi (3x)+\phi (3x-2)+\phi (3x-4), \] using the Axiom of Choice. We prove that –apart from being non-measurable– it satisfies the classical conditions for a scaling function to lead to orthonormal wavelets. While non-measurable functions are not directly useful for numerical calculations, the example given here explains the possible anomalous behaviour of numerical methods. Indeed the origin of this paper lies in the solution of the scaling equation above, calculated on a finite grid consisting of the points $i/3^k$, $0\leq i<2.3^k$. The result is \[ \phi \left (\frac i{3^k}\right ) = \begin {cases} 0 & \text {$i$ odd},\\ 1&\text {$i$ even}. \end {cases} \] This finite approximation seemingly satisfies the conditions for a scaling equation but, as we show here, any extension is either non-measurable or not orthogonal. While in this case, and for the chosen grid, the strange behaviour is apparent from the graph of the approximating function which looks like a comb, there is of course no guarantee that this will be clear in different situations.