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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 818907, 7 pages
http://dx.doi.org/10.1155/2013/818907
Research Article

Some Bivariate Smooth Compactly Supported Tight Framelets with Three Generators

1Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain
2Departamento de Análisis Matemático, Universidad de Alicante, 03080 Alicante, Spain
3Department of Mathematics and Statistics, Auburn University, Auburn, AL 36849-5310, USA

Received 17 December 2012; Accepted 11 April 2013

Academic Editor: Sung Guen Kim

Copyright © 2013 A. San Antolín and R. A. Zalik. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

For any dilation matrix with integer entries and , we construct a family of smooth compactly supported tight wavelet frames with three generators in . Our construction involves some compactly supported refinable functions, the oblique extension principle, and a slight generalization of a theorem of Lai and Stöckler. Estimates for the degrees of smoothness are given. With the exception of a polynomial whose coefficients must in general be computed by spectral factorization, the framelets are expressed in closed form in the frequency domain, in terms of elementary transcendental functions. By means of two examples we also show that for low degrees of smoothness the use of spectral factorization may be avoided.