On the Density of Compactly Supported Functions in a Space with Multiweighted Derivatives

We define a space with multiweighted derivatives on the half-axis. A multiweighted derivative of a function is an operation under which each subsequent derivative is taken of the function multiplied by some weight function. All weight functions involved in the definition of a multiweighted derivative are assumed to be sufficiently smooth; therefore, the set of compactly supported infinitely smooth functions belongs to the space with multiweighted derivatives, and the closure of this set in the norm of the space is a subspace of the latter. We study the mutual relation between these spaces depending on the integral behavior of the weight functions in the neighborhood of zero and infinity.