# zbMATH — the first resource for mathematics

Multivariate nonhomogeneous refinement equations. (English) Zbl 0945.42017
Summary: We give necessary and sufficient conditions for the existence and uniqueness of compactly supported distribution solutions $$f= (f_1,\dots, f_r)^T$$ of nonhomogeneous refinement equations of the form $f(x)= h(x)+ \sum_{\alpha\in A} c_\alpha f(2x- \alpha)\quad (x\in \mathbb{R}^s),$ where $$h= (h_1,\dots, h_r)^T$$ is a compactly supported vector-valued multivariate distribution, $$A\subset \mathbb{Z}^s_+$$ has compact support, and the coefficients $$c_\alpha$$ are real-valued $$r\times r$$ matrices. In particular, we find a finite-dimensional matrix $$B$$, constructed from the coefficients $$c_\alpha$$, such that there are precisely as many linearly independent compactly supported distribution solutions to the refinement equation as there are linearly independent vector solutions $${\mathbf q}$$ of the equation $$(I- B){\mathbf q}={\mathbf p}$$, where the vector $${\mathbf p}$$ depends on $$h$$. Our proofs proceed in the time domain and allow us to represent each solution regardless of the spectral radius of $$P(0):= 2^{-s} \sum c_\alpha$$, which has been a difficulty in previous investigations of this nature.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
Full Text:
##### References:
  Cohen, A., Daubechies, I., and Plonka, G. (1997). Regularity of refinable function vectors,J. Fourier Anal. Appl.,3, 295-324. · Zbl 0914.42025 · doi:10.1007/BF02649113  Dinsenbacher, T. and Hardin, D. (1997). Nonhomogeneous refinement equations,Wavelets, Multiwavelets and Their Applications, Aldroubi and Lin, Eds., Contemporary Mathematics Series. · Zbl 0945.42017  Daubechies, I. and Lagarias, J. (1991). Two-scale difference equations; I. Existence and global regularity of solutions,Siam. J. Math. Anal,22(5), 1388-1410. · Zbl 0763.42018 · doi:10.1137/0522089  Geronimo, J., Hardin, D., and Massopust, P. (1994). Fractal functions and wavelet expansions based on several scaling functions,J. Approx. Th.,78(3), 373-401. · Zbl 0806.41016 · doi:10.1006/jath.1994.1085  Halperin, I. (1952).Introduction to the Theory of Distributions, (based on lectures by L. Schwartz), University of Toronto Press, Toronto. · Zbl 0046.12603  Heil, C. and Colella, D. (1996). Matrix refinement equations: existence and uniqueness,J. Fourier Anal. Appl.,2, 363-377. · Zbl 0904.39017  Hardin, D. and Marasovich, J. (1999). Biorthogonal multiwavelets on [?1, 1],Appl. and Comp. Harmonic Anal.,7, 34-53 · Zbl 0952.42019 · doi:10.1006/acha.1999.0261  Jia, R. (1995). Refinable shift-invariant spaces: from splines to wavelets,Approx. Th.,VIII(2), 179-208. · Zbl 0927.42021  Jia, R., Jiang, Q., and Shen, Z. (1998). Distributional solutions of nonhomogeneous discrete and continuous refinement equations, preprint. · Zbl 0980.42035  Jia, R., Riemenschneider, S., and Zhou, D. (1997). Approximation by multiple refinable functions,Can. J. Math.,49, 944-962. · Zbl 0904.41010 · doi:10.4153/CJM-1997-049-8  Jiang, Q. and Shen, Z. On existence and weak stability of matrix refinable functions, to appear,Constr. Approx. · Zbl 0932.42028  Rudin, W. (1991).Functional Analysis, 2nd ed., McGraw-Hill, New York. · Zbl 0867.46001  Shen, Z. (1998). Refinable function vectors,SIAM J. Math. Anal.,29, 235-250. · Zbl 0913.42028 · doi:10.1137/S0036141096302688  Strang, G. and Zhou, D. (1998). Inhomogeneous refinement equations,J. Fourier Anal. Appl.,4, 733-747. · Zbl 0932.42026 · doi:10.1007/BF02479677  Sun, Q. (1998). Nonhomogeneous refinement equation: existence, regularity and biorthogonality, preprint.  Zhou, D. (1997). Existence of multiple refinable distributions,Michigan Math. J.,44, 317-329. · Zbl 0901.42022 · doi:10.1307/mmj/1029005707
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.