Mixing product and approximation

Linear approximations of functions by scaling functions are well known thanks to a theorem by Strang and Fix. Things get more complicated if a product is performed on the functions.

As often, it is simpler to start with the Haar basis.
If f is a polynomial, u a signal , e its projection on a resolution space, i.e. the sequence of its averages over contiguous intervals, and let w=u-e and denote by E the (averaging) projection operator, we have

which is a simple formula with a strong statistical appeal.

However, the Haar approximation has a poor performance on smooth signals. It occurs that generalizing the above equation to other approximations is equivalent to a couple of conditions on the approximation space and its supplement when they are combined by product.

The first condition requires that the approximation space be invariant under product. This is not possible for general scaling functions, or, more generally, when the finite elements used in the approximation if they overlap.

We can use, however, approximate products. Approximate products are characterized by a Strang and Fix like condition. If we we restrict signals to piecewise polynomial functions, then approximate product are entirely (and constructively) defined by a Hermite interpolation scheme.

In this framework, it is possible to define nested low resolution approximations, and the corresponding details. In the process, we recover the formula above.

You will find the corresponding journal paper here, and related results here.