Summary: The integrodifferential equations of mathematical physics are objects of research, and construction of interpolating polynomials to obtain approximate solutions of such equations is the subject of investigation. This paper lays out a technique for constructing approximate expressions for functionals on solutions of integrodifferential equations which are an analog of the Hermite polynomial used to interpolate functions. In the example of the diffusion equation, it is shown that the use of such basis solutions allows a substantial increase in the accuracy of the approximate representation of the functionals in comparison to the first approximation of perturbation theory with practically the same computational costs.