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Spline is a continuous function piecewise-defined by polynomials and is widely used for interpolation and smoothing of observed data. In 1994, Heβ and Schmidt proposed a positive quartic C 2 -spline interpolation for estimation of a non-negative and twice continuously differentiable function. In this paper, first we generalize the positive quartic C 2 -spline interpolation to the positive quartic C 2 -spline smoothing. Then we propose two estimation methods of a probability density function from its histogram by extending the ideas of the positive quartic C 2 -spline interpolation and smoothing. Finally numerical experiments show the effectiveness of the proposed methods.
Kitahara et al. (Wed,) studied this question.