The Toeplitz theorem and its applications to approximation theory and linear PDEs

Author:
Rong Qing Jia

Journal:
Trans. Amer. Math. Soc. **347** (1995), 2585-2594

MSC:
Primary 41A15; Secondary 35E99, 39A10, 41A63

DOI:
https://doi.org/10.1090/S0002-9947-1995-1277117-8

MathSciNet review:
1277117

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We take an algebraic approach to the problem of approximation by dilated shifts of basis functions. Given a finite collection of compactly supported functions in , we consider the shift-invariant space generated by and the family , where is the -dilate of . We prove that provides -approximation order only if contains all the polynomials of total degree less than . In particular, in the case where consists of a single function with its moment , we characterize the approximation order of by showing that the above condition on polynomial containment is also sufficient.

The above results on approximation order are obtained through a careful analysis of the structure of shift-invariant spaces. It is demonstrated that a shiftinvariant space can be described by a certain system of linear partial difference equations with constant coefficients. Such a system then can be reduced to an infinite system of linear equations, whose solvability is characterized by an old theorem of Toeplitz. Thus, the Toeplitz theorem sheds light into approximation theory. It is also used to give a very simple proof for the well-known Ehrenpreis principle about the solvability of a system of linear partial differential equations with constant coefficients.

**[1]**C. de Boor,*The polynomials in the linear span of integer translates of a compactly supported function*, Constr. Approx.**3**(1987), 199-208. MR**889555 (88e:41054)****[2]**-,*Quasiinterpolants and approximation power of multivariate splines*, Computation of Curves and Surfaces (W. Dahmen, M. Gasca, and C. A. Micchelli, eds.), Kluwer, Dordrecht, 1990, pp. 313-345. MR**1064965 (91i:41009)****[3]**C. de Boor, R. DeVore, and A. Ron,*Approximation from shift-invariant subspaces of*, Trans. Amer. Math. Soc.**341**(1994), 787-806. MR**1195508 (94d:41028)****[4]**C. de Boor and K. Höllig,*Bivariate box splines and smooth**functions on a three direction mesh*., J. Comput. Appl. Math.**9**(1983), 13-28. MR**702228 (85f:41004)****[5]**C. de Boor and R. Q. Jia,*Controlled approximation and a characterization of the local approximation order*, Proc. Amer. Math. Soc.**95**(1983), 547-553. MR**810161 (87d:41025)****[6]**C. K. Chui, K. Jetter, and J. D. Ward,*Cardinal interpolation by multivariate splines*, Math. Comp.**48**(1987), 711-724. MR**878701 (88f:41003)****[7]**L. Ehrenpreis,*Fourier analysis in several complex variables*, Wiley-Interscience, New York, 1970. MR**0285849 (44:3066)****[8]**R. Q. Jia,*A characterization of the approximation order of translation invariant spaces*, Proc. Amer. Math. Soc.**111**(1991), 61-70. MR**1010801 (91d:41018)****[9]**-,*A dual basis for the integer translates of an exponential box spline*, Rocky Mountain J. Math.**23**(1993), 223-242. MR**1212738 (94a:41022)****[10]**R. Q. Jia and J. J. Lei,*Approximation by multiinteger translates of functions having global support*, J. Approx. Theory**72**(1993), 2-23. MR**1198369 (94f:41024)****[11]**R. Q. Jia, S. Riemenschneider, and Z. W. Shen,*Solvability of systems of linear operator equations*, Proc. Amer. Math. Soc.**120**(1994), 815-824. MR**1169033 (94e:47020)****[12]**J. J. Lei and R. Q. Jia,*Approximation by piecewise exponentials*, SIAM J. Math. Anal.**22**(1991), 1776-1788. MR**1129411 (92i:41016)****[13]**Yu. I. Lyubich,*Functional Analysis*. I, Encyclopaedia of Math. Sci., vol. 19, Springer-Verlag, Berlin, 1992. MR**1300017****[14]**U. Oberst,*Multidimensional constant linear systems*, Acta Appl. Math.**20**(1990), 1-175. MR**1078671 (92f:93007)****[15]**A. Ron,*A characterization of the approximation order of multivariate spline spaces*, Studia Math.**98**(1991), 73-90. MR**1110099 (92g:41017)****[16]**G. Strang and G. Fix,*A Fourier analysis of the finite-element variational method*, Constructive Aspects of Functional Analysis (G. Geymonat, ed.), C.I.M.E., II Ciclo 1971, Edizione Cremonese, Rome, 1973, pp. 793-840.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
41A15,
35E99,
39A10,
41A63

Retrieve articles in all journals with MSC: 41A15, 35E99, 39A10, 41A63

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1995-1277117-8

Keywords:
Approximation order,
shift-invariant spaces,
infinite systems of linear equations,
partial differential equations,
partial difference equations

Article copyright:
© Copyright 1995
American Mathematical Society