A Gibbs phenomenon for spline functions (Q1178504)

Put \(F(x)=-1\) for \(-1\geq x<0\), \(F(x)=1\) for \(0\leq x\leq 1\). Suppose that \[ \tau_ n(x)={a_ 0 \over 2} +\sum_{k=1}^ n(a_ k\cos \pi kx+b_ k\sin \pi kx) \] is the polynomial of the best approximation of \(F\) in the norm \(L^ 2[-1,1]\). Then \[ \lim_{n\to\infty}\tau_ n \left( {x\over n}\right)={2\over\pi}\int_ 0^{\pi x} {\sin t \over t}dt \tag{1} \] locally uniformly in \(x\). I. J. Schoenberg conjectured that an analogue of (1) holds for spline functions. In the present paper this conjecture is proved for periodic splines with equal knot spacing.