### Symmetric Orthonormal Scaling Functions\\
and Wavelets With Dilation Factor $d=4$

### Bin Han

## Abstract

It is well known that in the univariate case, up to an integer shift and possible sign
change,
there is no dyadic compactly supported symmetric orthonormal scaling function
except for the Haar function. In this
paper we are concerned with the construction of symmetric orthonormal
scaling functions with dilation factor $d=4$. Several
examples of such scaling functions are provided in this paper. In particular, two
examples of $C^1$ orthonormal scaling functions, which are symmetric about $0$ and
$\frac{1}{6}$, respectively, are presented.
We will then discuss how to construct symmetric wavelets from these scaling functions.
We explicitly construct the corresponding orthonormal symmetric wavelets for
all the examples given in this paper.

Back to Preprints and Publication