This analysis explores a Reproducing Kernel Particle Methods which incorporates several inviting features. The emphasis is away from classical mesh generated elements in favor of a mesh free system which only requires a set of nodes or particles in space. Using a Gaussian distribution, flexible window functions are implemented to provide refinement in the solution process. It also creates the ability to analyze a specific frequency range in dynamic problems reducing the computer time required. This advantage is achieved through an increase in the critical time step when the frequency range is low and a large window is used. The stability of the window function is investigated to provide insight on Reproducing Kernel Particle Methods. Furthermore, there are no explicit elements in the formulation, allowing the derivatives to also be continuous, or C∞. The analytic theory is confirmed through numerical experiments by performing reconstructions and solving an elastic-dynamic one dimensional problem.