Approximation by the solutions of the heat equation (Q1331076)

Solutions of the heat equations \[ u_{xx}(x,t)= u_ t(x,t)\text{ on }(- \infty,+\infty)\times \{t>0\},\;u(x,0)= F(x)\text{ on }(- \infty,+\infty), \] where \(F\) belongs to the weighted \(L^ 2[\mathbb{R},\exp(- ax^ 2)]\) space, are used in order to study best approximation problems. Solution \(h(x)\) is extensible analytically onto \(C\) except for a set of measure zero and this extension belongs to \(H_{K(a,t)}\). Most initial value problems for the heat equations in which one is interested deal with initial values with compact supports. The solution of them can be extended as entire functions for any fixed positive time point. Hence, the analyticity of such solutions may have a clue for representing the passed time in terms of the current heat distribution.