Theory of generalized Bessel potential space and functional completion

The article's objective is to present norms based on weighted Dirichlet integrals in the space of generalized Bessel potentials. The weighted Dirichlet integral is first defined and then that this integral may be written using the multidimensional generalized translation of the module's degree demonstrated. We then show that a defined previously norm cannot be specified in function space of arbitrary fractional order of smoothness. We present a new norm associated with the generalized Bessel potential kernel. We demonstrate the existence of a complete function space with a perfect functional completion for the class of all indefinitely differentiable finite even functions with the norm based on the generalized Bessel potential.