The refinability of step functions
Abstract: Refinable functions have been widely investigated because of their importance in wavelet theory and multiresolution analysis, as well as because of intrinsic interest. Problems involving refinability can be challenging and interesting problems in mathematics. Several papers have investigated refinability of splines and other classes of functions. The purpose of this paper is to develop necessary and sufficient conditions for the refinability of the class of step functions on the real line taking complex values.
- 1. Alfred S. Cavaretta, Wolfgang Dahmen, and Charles A. Micchelli, Stationary subdivision, Mem. Amer. Math. Soc. 93 (1991), no. 453, vi+186. MR 1079033, 10.1090/memo/0453
- 2. Xin-Rong Dai, De-Jun Fend, and Yang Wang, Classification of Refinable Splines, Applied and Computational Harmonic Analysis.
- 3. Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
- 4. Wayne Lawton, S. L. Lee, and Zuowei Shen, Characterization of compactly supported refinable splines, Adv. Comput. Math. 3 (1995), no. 1-2, 137–145. MR 1314906, 10.1007/BF03028364
- A.S. Cavaretta, W. Dahman and C.A. Micchelli, Stationary subdivision, Memoirs Amer. Math. Soc. # 453, 93 (1991) 1-182. MR 1079033 (92h:65017)
- Xin-Rong Dai, De-Jun Fend, and Yang Wang, Classification of Refinable Splines, Applied and Computational Harmonic Analysis.
- I. Daubechies, Ten Lectures on Wavelets, CBMS 61 (Society of Industrial and Applied Mathematics, Philadelphia, PA, 1992). MR 1162107 (93e:42045)
- Wayne Lawton, S.L. Lee and Zuowei Shen, Characterization of compactly supported refinable splines, Advances in Computational Mathematics 3(1995)137-145. MR 1314906 (95m:41020)
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 39B22
Retrieve articles in all journals with MSC (2000): 39B22
Matthew J. Hirn
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York
Address at time of publication: Department of Mathematics, University of Maryland, College Park, Maryland
Keywords: Refinable equations, step function
Received by editor(s): July 20, 2004
Received by editor(s) in revised form: August 20, 2006
Published electronically: November 30, 2007
Communicated by: David R. Larson
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.