Abstract
Cubic spline histopolation with arbitrary placement of histogram knots and spline knots between them is studied. Classical boundary conditions are used. Histopolating spline is represented with the help of second moments and particular integrals. The systems determining these parameters are investigated in different cases where diagonal dominance in matrices takes place or may be absent.
Highlights
The histopolation problem is more practical than the interpolation problem as, e.g., the statistical information is rather given in form of histograms
We treat in this paper the histopolation problem with cubic splines
Instead of implicit theory via quartic spline interpolation we develop explicit theory of cubic spline histopolation
Summary
The histopolation problem is more practical than the interpolation problem as, e.g., the statistical information is rather given in form of histograms. For the boundary value problem with second order linear differential equation, on uniform mesh, the collocation with cubic splines has the rate O(h2) [10, 17, 18] but the subdomain method has O(h4) [15, 18]. The same idea works well in case of Volterra integral equations [5] These circumstances are a great motivation to give a special attention to histopolation problem with cubic splines. Rational interpolating or histopolating splines do not exist for any data [7, 16], cubic spline interpolants or histopolants exist always Another idea to preserve geometrical properties is to add some auxiliary spline knots (see, e.g., [14] to preserve monotonicity). The convexity preserving combined spline theory similar to [8] should use cubic spline histopolation which we develop in this paper
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