Beyond B-splines: Exponential pseudo-splines and subdivision schemes reproducing exponential polynomials

The main goal of this paper is to present some generalizations of polynomial B-splines, which include exponential B-splines and the larger family of exponential pseudo-splines. We especially focus on their connections to subdivision schemes. In addition, we generalize a well-known result on the approximation order of exponential pseudo-splines, providing conditions to establish the approximation order of nonstationary subdivision schemes reproducing spaces of exponential polynomial function

Scalar multivariate subdivision schemes and box splines, Ergebnisberichte Angewandte Mathematik No. 397, Fakultät für Mathematik, Technische Universität

Abstract We study convergent scalar d-variate subdivision schemes satisfying sum rules of order k ∈ N, with dilation matrix 2I. Using the results of Möller and Sauer in [18], stated for general expanding dilation matrices, we characterize the structure of the mask symbols of such schemes by showing that they must be linear combinations of shifted box spline generators of a quotient polynomial ideal J k . The directions of the corresponding box splines are θ ∈ {0, 1} d \ {(0, . . . , 0)}. The quotient ideal J k , as shown in Our results open a way to a systematic study of subdivision schemes. For example, in the bivariate case, if the mask symbol of any convergent subdivision scheme is in J k , then the mask is an affine combination of smoothed versions of three-directional box splines. Many special cases, including affine combinations of convergent schemes, can be looked at this way; see, e.g., As in the univariate case, this characterization seems to be the proper way of matching the smoothness, as determined in [1], of the box spline building blocks with the order of polynomial reproduction of the corresponding scheme. Due to the interaction of the building blocks, the convergence and smoothness, however, are usually destroyed, if several convergent schemes are combined in this way. We illustrate our results with several examples