New bounds on the zeros of spline functions
For a finite real sequence \({\mathbf a} = (a_ 0,\dots,a_ k)\) the number of strong (weak) sign changes is defined as the minimum (maximum) number of sign changes in \({\mathbf a}\) gained by assigning signs to zero entries. In Theorem 1, the author shows that, under a certain condition, the number of zeros of a spline function is bounded by the number of strong sign changes in its sequence of \(B\)-spline coefficients. By writing a general spline function as a sum of functions which satisfy the given condition, he deduces known bounds on zeros and sign changes and he can show that the number of zeros of any spline function is bounded by the number of weak sign changes in its sequence of \(B\)-spline coefficients, where the zero count is stronger than that previously used.
scientific article; zbMATH DE number 503002