On the Determination of a Density Function by Its Autoconvolution Coefficient

ISSN： 0163-0563

Source： Numerical Functional Analysis and Optimization, Vol.27, Iss.3-4, 2006-04, pp. : 357-375

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Abstract

We deal with a modification of the well-known ill-posed autoconvolution equation x * x = y on a finite interval, e.g., analyzed in [8]. In this paper, we focus on solutions that are probability density functions and assume to have data of the autoconvolution coefficient k of the density function x , which we define as the quotient of the autoconvolution function x * x and x itself. The corresponding inverse problem leads to the nonlinear integral equation kx − x * x = 0 of the third kind. For this equation, we give results on existence and make notes on uniqueness and stability. We show the ill-posedness of the equation by an example and make assertions on its regularization by Tikhonov's method. In this context, we prove the weak closedness of the forward operator for some appropriate domain.