On the convolution of a box spline with a compactly supported distribution: The exponential-polynomials in the linear span

AbstractFor a given box spline B and a compactly supported distribution μ, we examine in this note the convolution B ∗ μ and the space H(B ∗ μ) of all exponential-polynomials spanned by its integer translates. The main result here provides a necessary and sufficient condition for the equality H(B ∗ μ) = H(B). This condition is given in terms of the distribution of the zeros of the Fourier-Laplace transform of B ∗ μ and allows us to reduce the above equality to much simpler settings. The importance of this result is for the determination of the approximation properties of the space spanned by the integer translates of B ∗ μ. Typical examples are discussed