As one of the major directions in applied and computational harmonic analysis, the
classical theory of wavelets and framelets has been extensively investigated in the
function setting, in particular, in the function space
L2(ℝd). A discrete wavelet
transform is often regarded as a byproduct in wavelet analysis by decomposing and
reconstructing functions in L2(ℝd)
via nested subspaces of L2(ℝd) in
a multiresolution analysis. However, since the input/output data and all filters in a
discrete wavelet transform are of discrete nature, to understand better the performance of
wavelets and framelets in applications, it is more natural and fundamental to directly
study a discrete framelet/wavelet transform and its key properties. The main topic of this
paper is to study various properties of a discrete framelet transform purely in the
discrete/digital setting without involving the function space
L2(ℝd). We shall develop a
comprehensive theory of discrete framelets and wavelets using an algorithmic approach by
directly studying a discrete framelet transform. The connections between our algorithmic
approach and the classical theory of wavelets and framelets in the function setting will
be addressed. Using tensor product of univariate complex-valued tight framelets, we shall
also present an example of directional tight framelets in this paper.