โ Scribed by B.I. Kvasov
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PARABOLIC ~-SPLINES IN INTERPOLATION PROBLEMS* B. I. KVASOV The use of local basis functions (B-splines) in interpolation by parabolic splines is studied.
Constraints on the mesh of interpolation nodes whereby diagonal dominance is ensured are obtained, and the limits of good stipulationof the matrix of the system of defining relations of the interpolation problem, are indicated. An expression is obtained for passing from one representation of a spline to another. Explicit expressions are obtained for constructing local approximation splines. It is shown that the latter have approximation properties similar to those of interpolation splines.
Parabolic splines are an alternative to the cubic splines most widely used in practice. Their use is specialiy justified when interpolating functions with poor smoothness properties. Algorithms for constructing interpolation parabolic splines when they are written in piecewise polynomial form were considered in detail in /I/. Order estimates of approximation by such splines were also obtained in /i/. In many applications, however, it proves more effective to represent the parabolic splines in terms of B-splines, i.e., non-negative local splines with supports of minimal length. For instance, when solving interpolation problems, the use of this representation demands minimal computer memory for storing the spline. This advantage is specially important when solving multi-dimensional problems. Efficient recursive algorithms /2/ have been developed for computing B-splines and their derivatives. Here, however, the matrix of the system of defining relations of the interpolation problem does not always have diagonal dominance and may even be poorly stipulated.
Below, we consider a method of constructing basis splines, applicable not only for polynomial splines, but also for various generalizations of them. We find correctness conditions for the construction of a spline by ordinary and by non-monotonic pivotal condensation. Since situations can arise when interpolation by B-splines is ineffective, we propose a method of constructing a local approximation spline. The determination of the latter does not require pivotal condensation while the accuracy of the approximation is orderwise the same as for an interpolation spline.We show, moreover that, in the case of a uniform mesh, the principal terms of the error are the same for interpolation and for local approximation splines. A method is given for extending our results to the case of two variables.
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