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We consider the problem of deconvolving the expression f(y, t) = ix g(x, y)h(x, t)dX, where it is required to estimate h(z, t) from the square-integrable functions g(z, y) and f(y, t), and the latter function is a noise corrupted version of f(y, t). This problem a.rises in the setting defined by a linear elliptic partial differential equation, where the unknown time-varying inhomogeneous term (the source) is to be computed from incomplete field data. Performance of such a deconvolution constitutes the Inverse Problem of Electrocardiography, for example. The standard approach to this problem constructs a global solution by collecting individually regularized solutions to the ill-posed problems defined by the above first kind Fredholm equations for different fixed values of para.meter t. That approach is shown to contain a. flaw. The corrected approach leads to more accurate deconvolution, particularly evident in the setting of Gaussian noise (a. theorem). In the presence of dominating geometric noise (noise generated by rotation, translation, or other systematic imprecision in the knowledge of g(x, y)), it may be expected that the accuracy advantage will be diminished, but there will be persisting benefits relating to superior time stability of the solutions. Keywords: inverse problem of electrocardiography, first kind Fredholm integral equations, deconvolution, compact opera.tors, cardiac electrical source imaging
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