  We study the copolynomials of $n$ variables, i.e. $K$-linear mappings from
the ring of polynomials $K[x_1,...,x_n]$ into the commutative ring $K$.
  We prove an existence and uniqueness theorem for a linear differential
equation of infinite order which can be considered as an algebraic version of
the classical Malgrange-Ehrenpreis theorem for the existence of the fundamental
solution of a linear differential operator with constant coefficients. We find
the fundamental solutions of linear differential operators of infinite order
and show that the unique solution of the corresponding inhomogeneous equation
can be represented as a convolution of the fundamental solution of this
operator and the right-hand side. We also prove the existence and uniqueness
theorem of the Cauchy problem for some linear differential equations in the
module of formal power series with copolynomial coefficients.
