http://worldcat.org/entity/work/id/1909300226

Some Qualitative Properties of Bivariate Euler-Frobenius Polynomials

This is a further report in a series devoted to the study of box splines. Box splines provide a natural generalization of univariate cardinal splines, i.e., splines with a uniform knot sequence. The process of univariate spline interpolation becomes particularly simple in the cardinal case, and this report considers the corresponding bivariate process of interpolation at the integer points in the plane to a given function by a linear combination of integer translates of a box spline. In particular, the report shows that this process is well posed, i.e., any bounded continuous function has exactly one such bounded interpolant If. The argument uses the Fourier transform to identify a certain trigonometric polynomial (in two variables) whose nonvanishing is equivalent to the asserted well-posedness. The minimum value of this polynomial yields a bound on the norm of the resulting interpolation projector I.

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