A new type of algebraic spline is used to derive a filter for smoothing or interpolating discrete data points. The spline is dependent on control parameters that specify the relative importance of data fitting and the derivatives of the spline. A general spline of arbitrary order is first formulated using matrix equations. We then focus on eighth-order splines because of the continuity of their first three derivatives (desirable for motor and robotics applications). The spline's matrix equations are rewritten to give a recursive filter that can be implemented in real time for lengthy data sequences. The filter is lowpass with a bandwidth that is dependent on the spline's control parameters. Numerical results, including a simple image processing application, show the tradeoffs that can be achieved using the algebraic splines.