The existence, characterization and essential uniqueness of solutions of lextremal problems

Let/= (a, b) be an interval in R and let Hn'°° consist of those real-valued functions f such that is absolutely continuous on I and /^ e L°°(I). Let L be a linear differential operator of order n with leading coefficient 1, a =< • • • < x - b be a partition of/and let the linear functionals L „ on Hn>°° be given by V = £0aii^v)(xh/= i.* • •. «■ = i.' * •. where 1 <k^<n and the k^ n-tuples • • i °^y are linearly inde pendent. Let r. be prescribed real numbers and let U = (/€ Hn*°°x j = 1, • • •, &., i — 1, • • •, mi. In this paper we consider the extremal problem (*) llislL»=a = infi£/1lLoo: feU\. We show that there are, in general, many solutions to (*) but that there is, under certain consistency assumptions on L and the L^., a fundamental (or core) interval of the form (x^, x^) on which all solutions to (*) agree; n^ is determined by the and satisfies > 1. Further, if s is any solution to (*) then on (x^9Ls = a a.e. Further, we show that there is a uniquely determined solution s* to (*), found by minimizing HL/H^oo over all subintervals (Xj, /= 1» • • • * m — 1, with the property that Ls* is constant on each subinterval (x^, ^y+1) and Ls* is a step function with at most n — 1 discontinuities on (x*y+j)* When L = Dnt s* is a piecewise perfect spline. Examples show that the results are essentially best possible.