Translates of exponential box splines and their related spaces
HTML articles powered by AMS MathViewer
- by Asher Ben-Artzi and Amos Ron PDF
- Trans. Amer. Math. Soc. 309 (1988), 683-710 Request permission
Abstract:
Exponential box splines ($EB$-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space $\mathcal {H}$ spanned by exponential polynomials. This space can be defined as the intersection of the kernels of certain partial differential operators with constant coefficients. The main part of this paper is devoted to algebraic analysis of the space ${\mathbf {H}}$ of all entire functions spanned by the integer translates of an $EB$-spline. This investigation relies on a detailed description of $\mathcal {H}$ and its discrete analog $\mathcal {S}$. The approach taken here is based on the observation that the structure of $\mathcal {H}$ is relatively simple when $\mathcal {H}$ is spanned by pure exponentials while all other cases can be analyzed with the aid of a suitable limiting process. Also, we find it more efficient to apply directly the relevant differential and difference operators rather than the alternative techniques of Fourier analysis. Thus, while generalizing the known theory of polynomial box splines, the results here offer a simpler approach and a new insight towards this important special case. We also identify and study in detail several types of singularities which occur only for complex $EB$-splines. The first is when the Fourier transform of the $EB$-spline vanishes at some critical points, the second is when $\mathcal {H}$ cannot be embedded in $\mathcal {S}$ and the third is when ${\mathbf {H}}$ is a proper subspace of $\mathcal {H}$. We show, among others, that each of these three cases is strictly included in its former and they all can be avoided by a refinement of the mesh.References
- C. de Boor and R. DeVore, Approximation by smooth multivariate splines, Trans. Amer. Math. Soc. 276 (1983), no. 2, 775–788. MR 688977, DOI 10.1090/S0002-9947-1983-0688977-5
- C. de Boor and K. Höllig, $B$-splines from parallelepipeds, J. Analyse Math. 42 (1982/83), 99–115. MR 729403, DOI 10.1007/BF02786872
- Wolfgang Dahmen and Charles A. Micchelli, Translates of multivariate splines, Linear Algebra Appl. 52/53 (1983), 217–234. MR 709352, DOI 10.1016/0024-3795(83)80015-9
- Wolfgang Dahmen and Charles A. Micchelli, On the local linear independence of translates of a box spline, Studia Math. 82 (1985), no. 3, 243–263. MR 825481, DOI 10.4064/sm-82-3-243-263
- Wolfgang Dahmen and Charles A. Micchelli, On the solution of certain systems of partial difference equations and linear dependence of translates of box splines, Trans. Amer. Math. Soc. 292 (1985), no. 1, 305–320. MR 805964, DOI 10.1090/S0002-9947-1985-0805964-6 N. Dyn and A. Ron, Local approximation by certain spaces of exponential polynomials, approximation order for exponential box splines and related interpolation problems, CAT Rep. 160, Texas A&M Univ., College Station, Tex. (January 1988).
- D. I. Gurevič, Counterexamples to a problem of L. Schwartz, Funkcional. Anal. i Priložen. 9 (1975), no. 2, 29–35 (Russian). MR 0390759
- Rong Qing Jia, Linear independence of translates of a box spline, J. Approx. Theory 40 (1984), no. 2, 158–160. MR 732698, DOI 10.1016/0021-9045(84)90026-1
- Rong Qing Jia, Local linear independence of the translates of a box spline, Constr. Approx. 1 (1985), no. 2, 175–182. MR 891538, DOI 10.1007/BF01890029
- Serge Lang, Algebra, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR 0197234
- Amos Ron, Exponential box splines, Constr. Approx. 4 (1988), no. 4, 357–378. MR 956173, DOI 10.1007/BF02075467 —, Linear independence for the translates of an exponential box, Rocky Mountain J. Math. (to appear).
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Laurent Schwartz, Théorie générale des fonctions moyenne-périodiques, Ann. of Math. (2) 48 (1947), 857–929 (French). MR 23948, DOI 10.2307/1969386
- Laurent Schwartz, Analyse et synthèse harmoniques dans les espaces de distributions, Canad. J. Math. 3 (1951), 503–512 (French). MR 44754, DOI 10.4153/cjm-1951-051-5
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 309 (1988), 683-710
- MSC: Primary 41A15; Secondary 33A10, 41A63
- DOI: https://doi.org/10.1090/S0002-9947-1988-0961608-7
- MathSciNet review: 961608