Spline approximations in $L_ p$ on an interval

We consider approximations in the space $L_p[0,a]$ to differentiable functions whose $l$th derivative belongs to $L_p[0,a]$. The function to be approximated is extended to the entire axis by Lagrange interpolation polynomials, and spline approximation with equally spaced nodes on the entire axis is then applied. This procedure results in a good approximation to the original function.