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Mixed interpolating-smoothing splines and the \(\nu\)-spline. (English) Zbl 1100.41005

J. Math. Anal. Appl. 322, No. 1, 28-40 (2006); corrigendum ibid. 452, No. 1, 726-728 (2017).
Let \(X\), \(Y\), \(Z_{1}\) and \(Z_{2}\) be Hilbert spaces with inner products \(\langle\;,\;\rangle_{X}\), \(\langle\;,\;\rangle_{Y}\), \(\langle\;,\;\rangle_{Z_{1}}\) and \(\langle\;,\;\rangle_{Z_{2}}\) respectively, and let \(T:X\rightarrow Y\), \(\Lambda _{1}:X\rightarrow Z_{1}\), \(\Lambda _{2}:X\rightarrow Z_{2}\) and \(\pi :Z_{2}\rightarrow Z_{2}\) be bounded linear maps. For \(z_{1}\in Z_{1}\), \(z_{2}\in Z_{2}\) and \(\rho >0\), the mixed interpolating-smoothing spline is a solution to the following problem: \(\underset{f\in \Lambda _{1}^{-1}\{z_{1}\}}{\text{ minimize }}\left\| Tf\right\| _{Y}^{2}+\rho \left\| \pi (\Lambda _{2}f-z_{2})\right\| _{Z_{2}}^{2}\).
In the present paper the solutions for this problem are characterized by the theory of the best approximation in a Hilbert space. This abstract characterization is refined by specializing the above mentioned problem to the case that \(Z_{1}\) and \(Z_{2}\) are finite dimensional, and further refined when the splines are represented in a certain dual basis. The result is applied to a few well-known problems, including the \(\nu \)-spline (which is actually a mixed spline). The author also shows that the \(\nu \)-spline is a limiting case of smoothing splines as certain weights increase to infinity, and a limiting case of near-interpolants as certain tolerances decrease to zero. The paper concludes with the construction of a \(C^{1}\) curvature-continuous parametric curve with curvature values bounded from below by prescribed values at the knots.

MSC:

41A15 Spline approximation
65D05 Numerical interpolation
65D07 Numerical computation using splines
65D10 Numerical smoothing, curve fitting
41A50 Best approximation, Chebyshev systems

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