In the paper we investigate the properties of the Abel-Poisson transformation of the Hermite formal series (differentiability property, boundary properties). Such series are identified with linear continuous functionals defined on the space $S_{1/2}^{1/2}$, which belongs to spaces of type $S$. The space $S_{1/2}^{1/2}$ coincides with the class of analytic vectors of the harmonic oscillator - the operator $d^2/dx^2+x^2$, which is integral and self-adjoint in the Hilbert space $L_2(\mathbb{R})$. An explicit form of the function, which is the core of the Abel-Poisson transformation, was found, and the properties of this function were investigated. The application of such transformation is given when studying the well-posedness of the Cauchy problem for a degenerate partial differential equation.

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