Optimal nonlinear approximation with exponential splines

The present paper is concerned with the L 2 -approximation properties of exponential splines with n knots for functions defined on an interval Ω = ( a , b ) . We first specify the Banach native space for these splines and show that it continuously embeds in L 2 ( Ω ) . We derive sharp bounds on the approximation error rate for functions in this native space by exponential splines with n knots. To prove the optimality of these rates, we show that these native spaces are equivalent (as Banach spaces) to spaces of higher-order bounded variation on Ω .