Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems

Dyn, N.; Ron, A.

doi:10.1090/S0002-9947-1990-0956032-6

Local approximation by certain spaces of exponential polynomials, approximation order of exponential box splines, and related interpolation problems
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by N. Dyn and A. Ron

Trans. Amer. Math. Soc. 319 (1990), 381-403

DOI: https://doi.org/10.1090/S0002-9947-1990-0956032-6

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Abstract:

Local approximation order to smooth complex valued functions by a finite dimensional space $\mathcal {H}$, spanned by certain products of exponentials by polynomials, is investigated. The results obtained, together with a suitable quasi-interpolation scheme, are used for the derivation of the approximation order attained by the linear span of translates of an exponential box spline. The analysis of a typical space $\mathcal {H}$ is based here on the identification of its dual with a certain space $\mathcal {P}$ of multivariate polynomials. This point of view allows us to solve a class of multivariate interpolation problems by the polynomials from $\mathcal {P}$, with interpolation data characterized by the structure of $\mathcal {H}$, and to construct bases of $\mathcal {P}$ corresponding to the interpolation problem.

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