Box splines are multivariate spline functions defined on uniform triangulations. See a monograph on box splines by Carl de Boor, Klaus Hoelig, and Sherman Riememschneider, published by Spring Verlag, 1993. Also see Chapter 12 in book "Spline Functions over Triangulations" by Ming-Jun Lai and Larry Schumaker, published by Cambridge University Press, 2007.
A wavelet is a square integrable function whose translates and dilates form an orthonormal basis for Hilbert space L_2(R). To be more precise, letting
f_{m,n}(x)=2^(m/2)f(2^m x-n)
for all integer m,n, if { f_{m,n}(x), m, n in Z} is an orthonormal basis for L_2(R), f is called a wavelet function. In the multivariagte setting, one may need more than one function to form an orthonormal basis for L_2(R^N) with N>1. Together with multiresolution analysis, the study of wavelets was a hot subject in mathematics and many other sciences and engineering in last twenty years. One typical reference is the monograph "Ten Lectures on Wavelets" by Ingrid Daubechies, published by SIAM Publication in 1992. One of the important applications of wavelest is image compression, image segmentation, and image denoising in signal and image processing. For example, FBI uses wavelets for its fingerprint compression.To know more about wavelets, there are several webpages available. Check them out: wavelet digest and wavelet idr
We can use box splines to construct various wavelets. For example, one can use uniform B-splines (univariate box spline functions) to construct compactly supported orthonormal wavelets , multivariate box splines to construct orthonormal multi-wavelets , biorthogonal wavelets , tight wavelet frames , and pre-wavelets, Riesz basis, and pre-Riesz basis in the multivariate setting.