A general theorem on generation of moments-preserving cosine families by Laplace operators in C[0,1]

We use Kelvin’s method of images (Bobrowski in J. Evol. Equ. 10(3):663–675, 2010; Semigroup Forum 81(3):435–445, 2010) to show that given two non-negative integers $i\not= j$ there exists a unique cosine family generated by a restriction of the Laplace operator in C[0,1], that preserves the moments of order i and j about 0, if and only if precisely one of these integers is zero.