Polynomial approximation of CR singular functions

Abstract

We begin by recalling some basic facts about continuity and differentiability in the one real variable setting. Following a quick discussion on what classical mathematicians had in mind when dealing with continuous functions, we present Takagi's everywhere continuous but nowhere differentiable function. After recalling some basic facts about holomorphic functions, we present the theorems of Runge and Mergelyan. We mention why no direct generalization of Mergelyan's result is possible in the context of several complex variables and then move on to the theory of CR functions on CR submanifolds of n-dimensional complex space in order to state the polynomial approximation theorem of Baouendi—Treves. We then begin discussion of how much of the Baouendi—Treves theorem we can recover in the CR singular setting, that is, when our submanifold is no longer CR. For certain functions on a particular CR singular submanifold, we ask the question: Can we find approximating polynomials that are holomorphic in some variables, but perhaps not holomorphic in all variables? We end by answering the question in the affirmative.