These functions are both window-like functions, most interesting, not only in spatial domain, they are also window-like functions in spectral domain! This accounts for their unique localization property. The Gaussian function is another function with such property.
Another striking feature of wavelet is that the wavelet functions are designed to have large number of moments (zeros crossings), the expansion of functions on such wavelet bases needs much fewer terms than the Taylor expansion, the property leads to very sparse decompositions of differential operators, functions, which further leads to fabulous applications, including data compression.
Maintained by Zhaobo Meng