FACTORIZATION THEOREMS FOR UNIVARIATE SPLINES ON REGULAR GRIDS

For a given univariate compactly supported distribution φ, we investigate here the space S(φ) spanned by its integer translates, the subspace H(φ) of all exponentials in S(φ) and the kernel K φ{symbol} of the associated semi-discrete convolution φ*. The paper addresses a variety of results including a complete structure of H(φ) and K φ{symbol} and a characterization of splines of minimal support. The main result shows that each φ can be expressed as φ = p(∇)τ * M, where p(∇) is a finite difference operator, τ is a distribution of smaller support satisfying H(τ) =K τ = {0}, and M is a spline which depends on H(φ) but not on φ itself, and which in the generic case (termed here "regular") is the exponential B-spline associated with H(φ). The approach chosen is direct and avoids entirely the Fourier analysis arguments. The fact that a distribution is examined, rather than a function, is essential for the methods employed. © 1990 Hebrew University.