Springer-verlag Electronic Production Hk 23 Xii 1997 0:45 A.m. Construction of Compactly Supported Aane Frames in L 2 (ir D ) 1. Wavelet Frames: What and Why?
نویسنده
چکیده
Since the publication, less than ten years ago, of Mallat's paper on Mul-tiresoltion Analysis Ma], and Daubechies' paper on the construction of smooth compactly supported wavelets D], wavelets had gained enormous popularity in mathematics and in the application domains. It is suucient to note that there are currently more than 10,000 subscribers to the monthly Wavelet Digest. At the same time, tailoring concrete wavelet systems to speciic applications is still a challenge, especially in more than one dimension (although a few constructions are available, such as tensor products, or the methods developed in RiS] and JRS]). The main search is for simple and feasible constructions of orthonormal and bi-orthogonal systems of wavelets with small (and of desirable shape) support, high smoothness and many symmetries. In a series of recent articles (RS1-7] and GR]), a theory that changes the previous state-of-the-art had been developed. That theory makes wavelet constructions simple and feasible, and it is the intent of the present article to provide a brief glance into it, with an emphasis on particular examples of univariate and multivariate constructs. We want to start with somewhat philosophical discussion: anyone who is familiar with wavelets knows that the simplest wavelet system is the Haar family. The Haar function is piecewise-constant, has a very small support, and the algorithms based on it are fast and simple. Had the Haar
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