Let I = ( a , b ) I = (a,b) be an interval in R and let H n , ∞ {H^{n,\infty }} consist of those real-valued functions f such that f ( n − 1 ) {f^{(n - 1)}} is absolutely continuous on I and f ( n ) ∈ L ∞ ( I ) {f^{(n)}} \in {L^\infty }(I) . Let L be a linear differential operator of order n with leading coefficient 1 , a = x 1 > ⋯ > x m = b 1,a = {x_1} > \cdots > {x_m} = b be a partition of I and let the linear functionals L i j {L_{ij}} on H n , ∞ {H^{n,\infty }} be given by \[ L i j f = ∑ v = 0 n − 1 a i j ( v ) f ( v ) ( x i ) , j = 1 , ⋯ , k i , i = 1 , ⋯ , m , {L_{ij}}f = \sum \limits _{v = 0}^{n - 1} {a_{ij}^{(v)}{f^{(v)}}({x_i}),\quad j = 1, \cdots ,{k_i},i = 1, \cdots ,m,} \] where 1 ≤ k i ≤ n 1 \leq {k_i} \leq n and the k i {k_i} n-tuples ( a i j ( 0 ) , ⋯ , a i j ( n − 1 ) ) (a_{ij}^{(0)}, \cdots ,a_{ij}^{(n - 1)}) are linearly inde pendent. Let r i j {r_{ij}} be prescribed real numbers and let U = { f ∈ H n , ∞ : L i j f = r i j , j = 1 , ⋯ , k i , i = 1 , ⋯ , m } U = \{ f \in {H^{n,\infty }}:{L_{ij}}f = {r_{ij}},j = 1, \cdots ,{k_i},i = 1, \cdots ,m\} . In this paper we consider the extremal problem ( ∗ ) ‖ L s ‖ L ∞ = α = inf { ‖ L f ‖ L ∞ : f ∈ U } . \begin{equation}\tag {$\ast $}{\left \| {Ls} \right \|_{{L^\infty }}} = \alpha = \inf \{ {\left \| {Lf} \right \|_{{L^\infty }}}:f \in U\} .\end{equation} We show that there are, in general, many solutions to ( ∗ ) ( \ast ) but that there is, under certain consistency assumptions on L and the L i j {L_{ij}} , a fundamental (or core) interval of the form ( x i , x i + n 0 ) ({x_i},{x_{i + {n_0}}}) on which all solutions to ( ∗ ) ( \ast ) agree; n 0 {n_0} is determined by the k i {k_i} and satisfies n 0 ≥ 1 {n_0} \geq 1 . Further, if s is any solution to ( ∗ ) ( \ast ) then on ( x i , x i + n 0 ) , | L s | = α ({x_i},{x_{i + {n_0}}}),|Ls| = \alpha a.e. Further, we show that there is a uniquely determined solution s ∗ {s_ \ast } to ( ∗ ) ( \ast ) , found by minimizing ‖ L f ‖ L ∞ {\left \| {Lf} \right \|_{{L^\infty }}} over all subintervals ( x j , x j + 1 ) , j = 1 , ⋯ , m − 1 ({x_j},{x_{j + 1}}),j = 1, \cdots ,m - 1 , with the property that | L s ∗ | |L{s_ \ast }| is constant on each subinterval ( x j , x j + 1 ) ({x_j},{x_{j + 1}}) and L s ∗ L{s_ \ast } is a step function with at most n − 1 n - 1 discontinuities on ( x j , x j + 1 ) ({x_j},{x_{j + 1}}) . When L = D n , s ∗ L = {D^n},{s_ \ast } is a piecewise perfect spline. Examples show that the results are essentially best possible.