Non-trivial solutions to the Cauchy problem for the heat equation on the infinite rod \( \partial_tu=\partial_{xx}u, \ u(x,0)=0 \) have been known since Tychonoff's classical example. We connect this problem to the existence of Laplace transforms whose determining function changes across adjacent vertical strips without loss of analyticity.
Building on a method pioneered by Mittag-Leffler and later developed by Kaneko in the context of the theory of hyperfunctions, we construct explicit examples of such Laplace transforms. This approach generalizes a method of Chung, yielding new non-trivial solutions of the Cauchy problem for the heat equation which are uniformly bounded in the space variable.