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Excerpt from Zero Crossings and the Heat Equation
Thus a representation of the v(x,t) data provides an implicit representation of the f(x) and u(x,t) data. Since v(x,t) is a solution to the heat equation, we are henceforth concerned with representations of the data in a solution to the heat equation such that some form of the initial data can be reconstructed. In what follows, u(x,t) will stand for any generic solution to the heat equation, with f(x) as initial data, although in terms of the above notation, v(x,t) and g(x) are likely instances of the representation problem.
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