What are unconditional bases and which wavelets have this property?

Question

What are unconditional bases and which wavelets have this property? Haar wavelet seems to be one but how common is it? Unconditional bases are related to Riesz basis condition (ordering of the terms in the sum) for convergence of series.

Answer 1

The answer can be found in "Ten lectures on wavelets" by Ingrid Daubechies (1992), chapter 9, theorem 9.1.6. Orthonormal bases can be conditional or unconditional.

My professor explained it like this: "Sufficient conditions for a mother wavelet to generate an unconditional basis are as follows: the wavelet must be continuously differentiable and both the wavelet and its derivative must be less than a constant times $(1+|x|)^{-1-\epsilon}$ for some positive $\epsilon$."

It is an unconditional basis not only for L2-space but also for higher orders of Lebesgue spaces and this is an improvement over Fourier basis functions.

Remark: "Another perhaps unfamiliar concept is that of an unconditional basis used by Donoho, Daubechies, and others [3], [10], [2] to explain why wavelets are good for signal compression, detection, and denoising [6], [5]." Source: http://cnx.org/content/m45090/latest/?collection=col11454/latest