The authors extend Schoenbergâ€™s family of polynomial splines with uniform knots to all non-integral degrees

$\alpha >-1$. They study two approaches to the construction of the fractional B-splines and show that both approaches are equivalent. They show that the fractional splines share virtually all the properties of the conventional polynomial splines, except that the support of the B-splines for non-integral orders

$\alpha $ is no longer compact. They satisfy a two-scale relation and for

$\alpha >-1/2$ they satisfy all the requirements for a multi-resolution analysis of

${L}_{2}$. As for the usual splines the symmetric fractional splines are solutions of a variational interpolation problem.