In the paper, we study an inverse problem for the heat equation. We introduce a class of bilinear forms on the space of harmonic polynomials ( called harmonic moments), which are represented by the Dirichlet-to-Neumann map. We investigate the uniqueness, stability, and reconstruction of the inverse problem.
The inverse data are given in the terms of the bilinear forms and can be exchanged into the data of the Dirichlet-to-Neumann map. The reconstruction ( of the density) is accomplished in two different ways: one is due to the idea of the mollifier and the other to the representation by the Carleman kernel in the complex analysis. The error terms are estimated depending on the degree of the harmonic polynomials. We estimates norms of the data on an arbitrary time interval by the norms on some fixed interval (e.g., 0 < t < 2).