Translates of exponential box splines and their related spaces

Authors:
Asher Ben-Artzi and Amos Ron

Journal:
Trans. Amer. Math. Soc. **309** (1988), 683-710

MSC:
Primary 41A15; Secondary 33A10, 41A63

MathSciNet review:
961608

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Abstract: Exponential box splines (-splines) are multivariate compactly supported functions on a regular mesh which are piecewise in a space 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 of all entire functions spanned by the integer translates of an -spline. This investigation relies on a detailed description of and its discrete analog . The approach taken here is based on the observation that the structure of is relatively simple when 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* -splines. The first is when the Fourier transform of the -spline vanishes at some critical points, the second is when cannot be embedded in and the third is when is a proper subspace of . 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.

**[BD]**C. de Boor and R. DeVore,*Approximation by smooth multivariate splines*, Trans. Amer. Math. Soc.**276**(1983), no. 2, 775–788. MR**688977**, 10.1090/S0002-9947-1983-0688977-5**[BH]**C. de Boor and K. Höllig,*𝐵-splines from parallelepipeds*, J. Analyse Math.**42**(1982/83), 99–115. MR**729403**, 10.1007/BF02786872**[DM]**Wolfgang Dahmen and Charles A. Micchelli,*Translates of multivariate splines*, Linear Algebra Appl.**52/53**(1983), 217–234. MR**709352**, 10.1016/0024-3795(83)80015-9**[DM]**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****[DM]**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**, 10.1090/S0002-9947-1985-0805964-6**[DR]**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).**[G]**D. I. Gurevič,*Counterexamples to a problem of L. Schwartz*, Funkcional. Anal. i Priložen.**9**(1975), no. 2, 29–35 (Russian). MR**0390759****[J]**Rong Qing Jia,*Linear independence of translates of a box spline*, J. Approx. Theory**40**(1984), no. 2, 158–160. MR**732698**, 10.1016/0021-9045(84)90026-1**[J]**Rong Qing Jia,*Local linear independence of the translates of a box spline*, Constr. Approx.**1**(1985), no. 2, 175–182. MR**891538**, 10.1007/BF01890029**[L]**Serge Lang,*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234****[R]**Amos Ron,*Exponential box splines*, Constr. Approx.**4**(1988), no. 4, 357–378. MR**956173**, 10.1007/BF02075467**[R]**-,*Linear independence for the translates of an exponential box*, Rocky Mountain J. Math. (to appear).**[Ru]**Walter Rudin,*Functional analysis*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR**0365062****[S]**Laurent Schwartz,*Théorie générale des fonctions moyenne-périodiques*, Ann. of Math. (2)**48**(1947), 857–929 (French). MR**0023948****[S]**Laurent Schwartz,*Analyse et synthèse harmoniques dans les espaces de distributions*, Canadian J. Math.**3**(1951), 503–512 (French). MR**0044754**

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DOI:
https://doi.org/10.1090/S0002-9947-1988-0961608-7

Article copyright:
© Copyright 1988
American Mathematical Society