The relationship between the spectral properties of the transfer operator corresponding to a wavelet refinement equation and the -Sobolev regularity of solution for the equation is established.
@article{bwmeta1.element.bwnjournal-article-zmv25i4p431bwm, author = {Jaros\l aw Kotowicz}, title = {Regularity of the multidimensional scaling functions: estimation of the $L^{p}$-Sobolev exponent}, journal = {Applicationes Mathematicae}, volume = {26}, year = {1999}, pages = {431-447}, zbl = {0995.42019}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-zmv25i4p431bwm} }
Kotowicz, Jarosław. Regularity of the multidimensional scaling functions: estimation of the $L^{p}$-Sobolev exponent. Applicationes Mathematicae, Tome 26 (1999) pp. 431-447. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-zmv25i4p431bwm/
[000] [1] N. K. Bari, Trigonometric Series, Fizmatgiz, 1961 (in Russian).
[001] [2] A. Cohen and I. Daubechies, A new technique to estimate the regularity of refinable functions, Rev. Mat. Iberoamericana 12 (1996), 527-591. | Zbl 0879.65102
[002] [3] A. Cohen and R. D. Ryan, Wavelets and Multiscale Signal Processing, Appl. Math. Math. Comput. 11, Chapman & Hall, 1995. | Zbl 0848.42021
[003] [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. | Zbl 0763.42018
[004] [5] I. Daubechies and J. Lagarias, Two-scale difference equation II. Local regularity, infinite products of matrices, and fractals, ibid. 23 (1992), 1031-1079. | Zbl 0788.42013
[005] [6] K. Deimling, Nonlinear Functional Analysis, Springer, 1985.
[006] [7] T. Eriola, Sobolev characterization of solution of dilation equations, SIAM J. Math. Anal. 23 (1992), 1015-1030. | Zbl 0761.42014
[007] [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.
[008] [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. | Zbl 0845.42018
[009] [10] L. Hervé, Construction et régularité des fonctions d'échelle, SIAM J. Math. Anal. 26 (1995), 1361-1385. | Zbl 0848.42023
[010] [11] J. Kotowicz, On existence of a compactly supported solution for two-dimensional two-scale dilation equations, Appl. Math. (Warsaw) 24 (1997), 325-334. | Zbl 0946.39009
[011] [12] K. S. Lau and J. Wang, Characterization of -solutions for the two-scale dilation equations, SIAM J. Math. Anal. 26 (1995), 1018-1048. | Zbl 0828.42024
[012] [13] C. A. Micchelli and H. Prautzsch, Uniform refinement of curves, Linear Algebra Appl. 114/115 (1989), 841-870. | Zbl 0668.65011
[013] [14] O. Rioul, Simple regularity criteria for subdivision schemes, SIAM J. Math. Anal. 23 (1992), 1544-1576. | Zbl 0761.42016
[014] [15] N. A. Sadovnichiĭ, Theory of Operators, Moscow Univ. Press, 1979 (in Russian).
[015] [16] L. Villemoes, Energy moments in time and frequency for -scale dilation equation solutions and wavelets, SIAM J. Math. Anal. 23 (1992), 1519-1543. | Zbl 0759.39005