Constrained smoothing of histograms by quadratic splines
Let \(x_ 0<x_ 1<\dots<x_ n\) be fixed points and \(\ell > 0\), \(p_ i > 0\), \(i=1,\dots,n\), be given parameters. The author studies the problem of constructing the quadratic spline \(s(x)\) with knots \(x_ 1,\dots,x_{n-1}\), which minimizes the functional \(K_ q(s):=\ell\| s'\|_{L_ q[x_ 0,x_ n]}+\| p_ i(f_ i-{1\over h_ i}\int^{x_ i}_{x_{i-1}}s(x)dx)\|_{\ell_ q},\) where \(h_ i:=x_ i-x_{i-1}\), \((q=2,\infty)\) under certain constraints like monotonicity, convexity or positivity of \(s\).
scientific article; zbMATH DE number 63397