Regularity of the multidimensional scaling functions: estimation of the $L^p$-Sobolev exponent

 Access to full text

 Full (PDF)

1. [1] N. K. Bari, Trigonometric Series, Fizmatgiz, 1961 (in Russian). 
2. [2] A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana 12 (1996), 527-591. Zbl0879.65102
3. [3] A. Cohen and R. D. Ryan, Wavelets and Multiscale Signal Processing, Appl. Math. Math. Comput. 11, Chapman & Hall, 1995. Zbl0848.42021
4. [4] I. Daubechies and J. Lagarias, Two-scale difference equation I. Existence and global regularity of solutions, SIAM J. Math. Anal. 22 (1991), 1388-1410. Zbl0763.42018
5. [5] I. Daubechies and J. Lagarias, Two-scale difference equation II. Local regularity, infinite products of matrices, and fractals, ibid. 23 (1992), 1031-1079. Zbl0788.42013
6. [6] K. Deimling, Nonlinear Functional Analysis, Springer, 1985. 
7. [7] T. Eriola, Sobolev characterization of solution of dilation equations, SIAM J. Math. Anal. 23 (1992), 1015-1030. Zbl0761.42014
8. [8] C. Heil and D. Colella, Sobolev regularity for refinement equations via ergodic theory, in: C. K. Chui and L. L. Schumaker (eds.), Approximation Theory VIII, Vol. 2, World Sci., 1995, 151-158. 
9. [9] P. N. Heller and R. O. Wells Jr., The spectral theory of multiresolution operators and applications, in: Wavelets: Theory, Algorithms, and Applications, C. K. Chui, L. Montefusco and L. Puccio (eds.), Wavelets 5, Academic Press, 1994, 13-31. Zbl0845.42018
10. [10] L. Hervé, Construction et régularité des fonctions d'échelle, SIAM J. Math. Anal. 26 (1995), 1361-1385. Zbl0848.42023
11. [11] J. Kotowicz, On existence of a compactly supported $L^p$ solution for two-dimensional two-scale dilation equations, Appl. Math. (Warsaw) 24 (1997), 325-334. Zbl0946.39009
12. [12] K. S. Lau and J. Wang, Characterization of $L^p$-solutions for the two-scale dilation equations, SIAM J. Math. Anal. 26 (1995), 1018-1048. Zbl0828.42024
13. [13] C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870. Zbl0668.65011
14. [14] O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal. 23 (1992), 1544-1576. Zbl0761.42016
15. [15] N. A. Sadovnichiĭ, Theory of Operators, Moscow Univ. Press, 1979 (in Russian). 
16. [16] L. Villemoes, Energy moments in time and frequency for $2$-scale dilation equation solutions and wavelets, SIAM J. Math. Anal. 23 (1992), 1519-1543. Zbl0759.39005