Exponential box splines

Abstract

In a recent paper by Nira Dyn and the author, univariate cardinal exponential B-splines are shown to have a representation similar to the wellknown box spline representation of the univariate cardinal polynomialB-splines. Motivated by this, we construct, for a set ofn directions inZs and a vector of constants λ Rn, an “exponential box spline” which has the same smoothness and support as the polynomial box spline, and is a positive piecewise exponential in its support. We derive recurrence relations for the exponential box splines which are simpler than those for the polynomial case. A relatively simple structure of the space spanned by the translates of an exponential box spline is obtained for λ in a certain open dense set ofRn—the “simple” λ. In this case, the characterization of the local independence of the translates and related topics, as well as the proofs involved, are quite simple when compared with the polynomial case (corresponding toλ = 0).