Chapter 30 - Generalized Derivatives

Abstract

This chapter elaborates the different types of generalized derivatives. The types of generalized derivatives include Sobolev type, distributional one, and the one appearing in the Mikusinski operational calculus. Some elements from the theory of Sobolev and distribution spaces, as well as the construction of the Mikusinski operator field are discusses in the chapter. It is found that if all first-order generalized derivatives of a function exist and are zero, then this function is equal to a constant almost everywhere. It is observed that things are different with the generalized derivative giving an essential difference between the classical and generalized derivatives. A necessary and sufficient condition for a function to have a generalized derivative is that it is absolutely continuous. The last supposition of the existence of the derivative does not imply the absolute continuity of the function. In the upper analysis, one can take the space whose elements are square integrable on every compact set. The statements giving relations between Sobolev spaces and spaces of continuously differentiable functions are usually called “imbedding theorems.” The Mikusinski operational calculus is also elaborated in the chapter.