Analytic Version of Weyl Bases for the Heat Kernel on the Half Axis

In this chapter, following faithfully an ingenious idea due to Iwasaki (Osaka J. Math. 31:663–728, 1994, Definition 7) we introduce a family of symbol functions that are supposed to correspond to an analytic version of Weyl bases for the heat equation associated with the Dirichlet, Neumann and Robin conditions. By virtue of these symbol functions, we can express the transport equation associated with the initial boundary value problem (IBP) in matrix form in Chap. 19 . This algebraic aspect is a great advantage of Iwasaki’s approach to the initial boundary value problem (IBP). We give three examples of symbol functions on the half axis \({\mathbf {R}}_{+}\) for the Dirichlet, Neumann and regular Robin cases as in Tables 17.1, 17.2 and 17.3, respectively.