Tilings, Scaling Functions, and a Markov Process
نویسنده
چکیده
We discuss a class of Markov processes that occur, somewhat unexpectedly, in the construction of wavelet bases obtained from multiresolution analyses (MRA). The processes in question have been around for a long time. One of the first references that should be cited is a paper by Doeblin and Fortêt [10] (1937), entitled “Sur les chaînes à liaisons complètes”. In English, they are sometimes called “historical Markov processes” and have been used extensively to study the Ising model. However, their wavelet connection does not seem to have filtered into the standard texts on time-scale analysis. The material for this article is drawn from the publications [9], [12], [13], as well as the prior contributions by various people who are cited in the appropriate places. To keep the exposition self-contained and as elementary as possible, we discuss the special case of the “quincunx matrix” in which the proofs are somewhat simpler and, in some cases, radically different from those found in the above references. In the first section, we describe a remarkable coincidence: two discoveries, the first concerning a gambling strategy and the second concerning a wavelet basis, both leading to the same mathematics. It is remarkable that these discoveries, each of
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تاریخ انتشار 2010