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Inf-convolution spline pour l’approximation de données discontinues. (Spline inf-convolution for approximating discontinuous data). (French) Zbl 0696.41011

The author introduces a new kind of spline function. In the classical theory, one approximates functions f whose regularity properties bear in general on the continuity of f and some of its derivatives. Here the author studies the approximation of continuous functions f for which some of the derivatives can have discontinuities at certain known points. Thus he takes these discontinuities into account. Concretely, a classical interpolation spline is often pictured as a single flexible rod. Here, however, the splines considered appear as a set of rods interlinked by articulations taking into account the discontinuities of the first derivative of the function to be approximated.
Mathematically, let \(H_ m(I)\) be the space of functions with a.e. a derivative \(f^{(m)}\in L^ 2(I)\), where \(I=[a,b]\subset {\mathbb{R}}\). In I, the author gives the interpolation points \(t_ i\), \(i=1,...,n\), and the abscissae \(\alpha_ 1,...,\alpha_ m\) contained in \((t_ 1,t_ n)\), and seeks a generalized spline function \(s(t)=g(t)+\sum^{m}_{j=1}d_ jp_ j(t)\), where \(p_ j(t)=(t- \alpha_ j)_+\), such that s minimizes \(E_ I^{(2)}(f)=\int_{I}(g''(t))^ 2dt\) as g varies over \(H_ 2(I)\) and \(d_ j\) varies over \({\mathbb{R}}\), under the constraints \(g(t_ i)+\sum^{m}_{j=1}d_ jp_ j(t_ i)=f(t_ i)\). He then proves that s exists, is unique and is characterized by the following conditions: (1) s is a polynomial function on every interval defined by the \(t_ i\) and \(\alpha_ j\) (the polynomials are of degree \(\leq 3\), except on \([a,t_ 1]\) and \([t_ n,b]\), where they are of degree \(\leq 1)\); (2) s, \(s'\), \(s''\) are continuous at \(t_ i\) \((i=1,...,n)\); (3) s, \(s''\), \(s'''\) are continuous at \(\alpha_ j\) and \(s''(\alpha_ j)=0\) \((j=1,...,m)\); (4) \(s(t_ i)=f(t_ i)\) \((i=1,...,n)\). By introducing \(q_ j(t)=(t- \alpha_ j)^ 2_+/2\) and minimizing the quantity \(\int_{I}(g'''(t))^ 2dt\) in \(H^ 3(I)\), the author obtains piecewise quintic splines. He also considers the case in which the \(\alpha_ j\) are unknown but can be localized. In this interesting article the author shows first how the minimization problems under study can be rewritten using the inf-convolution of two functions. He then recalls the notions of semi-Hilbert space and semikernel introduced by J. Duchon in his thesis to characterize spline functions. He shows that in a general way these spline functions are associated with a semi-Hilbert space the semikernel of which can be evaluated, making it possible to characterize the corresponding spline functions. The article concludes with some examples and some algorithms leading to the numerical results.

MSC:

41A15 Spline approximation
65D07 Numerical computation using splines
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References:

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