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A smoothing spline that approximates Laplace transform functions only known on measurements on the real axis. (English) Zbl 1257.65074

This paper focuses on the construction of a generalized polynomial smoothing spline for approximating Laplace transform functions only known at a finite set of measurements along the real axis. Starting from the data set, the authors construct a generalized polynomial spline defined on the whole real line, which is a complete polynomial smoothing spline inside the data interval while it enjoys the Laplace transform properties outside the data interval. Both rational decay end behaviour and the exponential decay end behaviour are discussed.
The authors show results concerning existence and uniqueness and also give approximation error bounds. The experimental results are also obtained by running two real inversion algorithms. The results obtained by employing the smoothing spline as an input function are compared with the inversion algorithm with the numerical values of the original function obtained by using the Laplace transform function. The authors also compare the approximation provided by the generalized polynomial smoothing spline with the same obtained by using exponential splines inside the knot intervals. The numerical results confirm that approximation errors are quite the same.

MSC:

65R10 Numerical methods for integral transforms
44A10 Laplace transform
44A20 Integral transforms of special functions
65D07 Numerical computation using splines
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